Microeconomic Theory
3rd Edition1
Susheng WANG
Department of Economics
Hong Kong University of Science and Technology
May 2016
© Hong Kong University of Science & Technology, 2016
1
The first edition was published by People’s University Publisher in Beijing in 2006. The second edition was
published by McGill-Hill in 2012. This edition is online for free.
Susheng Wang, HKUST
Contents in Brief
Part 1. Neoclassical Economics
Chapter 1. Producer Theory
Chapter 2. Consumer Theory
Chapter 3. Risk Theory
Chapter 4. Equilibrium Theory
Part 2. Applications of GE to Financial and Industrial Markets
Chapter 5. Financial Theory
Chapter 6. Industrial Markets
Part 3. Game Theory
Chapter 7. Imperfect Information Games
Chapter 8. Incomplete Information Games
Chapter 9. Cooperative Games
Part 4. Information Economics
Chapter 10. Signalling
Chapter 11. Mechanism Design
Chapter 12. Incentive Contracts
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Contents In Detail
Contents in Brief
2
Contents In Detail
3
Preface
11
Notation and Terminology
13
Chapter 1 Producer Theory
14
1.
Technology
14
2.
The Firm’s Problem
24
3.
Short-Run and Long-Run Cost Functions
33
3.1.
Definitions
33
3.2.
The relationship between the AC and MC curves
33
3.3.
The relationship between SR and LR cost functions
34
4.
Properties
35
5.
Aggregation
41
Notes
42
Chapter 2 Consumer Theory
43
1.
Existence of the Utility Function
43
2.
The Consumer’s Problem
48
3.
Properties
52
4.
Aggregation
60
5.
Integrability
61
6.
Revealed Preferences
62
7.
Intertemporal Analysis
65
Notes
71
Chapter 3 Risk Theory
72
1. Introduction
72
2. Expected Utility
73
3. Mean-Variance Utility
79
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4. Measurement of Risk Aversion
80
5. Demand for Insurance
83
6. Demand for Risky Assets
84
7. Portfolio Analysis
87
8. Stochastic Dominance
90
Notes
93
Chapter 4 Equilibrium Theory
94
1.
The Equilibrium Concept
94
2.
GE in a Pure Exchange Economy
95
3.
Pareto Optimality
102
4.
Welfare Theorems
107
5.
General Equilibrium with Production
112
6.
5.1.
General Equilibrium with Production
113
5.2.
Efficiency of General Equilibrium
116
General Equilibrium with Uncertainty
121
Notes
123
Chapter 5 Financial Markets
1.
124
Security Markets
124
1.1.
Contingent Markets
124
1.2.
Security Markets
125
2.
Static Asset Pricing
129
3.
Representative Agent Pricing
130
4.
The Capital Asset Pricing Model
132
5.
Dynamic Asset Pricing
134
6.
5.1.
The Euler Equation
134
5.2.
Dynamic CAPM
136
Continuous-Time Stochastic Programming
137
6.1.
Continuous-Time Random Variables
137
6.2.
Continuous-Time Stochastic Programming
138
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7.
The Black-Scholes Pricing Formula
140
Notes
143
Chapter 6 Industrial Markets
144
1.
A Competitive Output Market
144
2.
A Monopoly
151
2.1.
The Single-Price Monopoly
152
2.2.
The Price-Discriminating Monopoly
153
2.3.
Monopoly Pricing under Asymmetric Information
155
3.
Allocative Efficiency
156
4.
Monopolistic Competition
159
5.
Oligopoly
162
5.1.
Bertrand Equilibrium
163
5.2.
Cournot Equilibrium
164
5.3.
The Stackelberg Equilibrium
167
5.4.
Cooperative Equilibrium
167
5.5.
Competition vs Cooperation
169
5.6.
Cooperation in a Repeated Game
170
6.
Production Differentiation by Location
171
7.
Location Equilibrium
172
8.
Entry Barriers
175
9.
Strategic Deterrence Against Potential Entrants
177
10.
A Competitive Input Market
179
10.1.
Demand and Supply
180
10.2.
Equilibrium and Welfare
180
11.
A Monopsony
184
12.
Vertical Relationships
187
12.1.
Independent Firms
188
12.2.
An Integrated Firm
188
12.3.
Explanation
189
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Notes
191
Chapter 7 Imperfect-Information Games
1.
2.
3.
4.
192
Two Game Forms
192
1.1.
The Extensive Form
192
1.2.
The Normal Form
195
1.3.
Mixed Strategy
197
Equilibria in Normal-Form Games
199
2.1.
Nash Equilibrium
199
2.2.
Dominant-Strategy Equilibrium
202
2.3.
Trembling-Hand Nash Equilibrium
205
2.4.
Reactive Equilibrium
207
Equilibria in Extensive-Form Games
211
3.1.
Nash Equilibrium
211
3.2.
Subgame Perfect Nash Equilibrium
214
3.3.
Bayesian Equilibrium
218
Refinements of Bayesian Equilibrium
232
4.1.
BE under Subgame Consistency: Perfect Bayesian Equilibrium
232
4.2.
Sequential Equilibrium
237
4.3.
BE under Complete Dominance Criterion: CDBE
243
4.4.
BE under Equilibrium Dominance Criterion: EDBE
246
Notes
251
Chapter 8 Incomplete-Information Games
252
1.
Bayesian Nash Equilibrium
252
2.
Signalling Games
258
2.1.
Pure Strategies in Signalling
258
2.2.
Mixed Strategies in Signalling
260
2.3.
Cheap Talk
265
Notes
274
Chapter 9 Cooperative Games
275
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1.
The Nash Bargaining Solution
275
8.1. The Nash Solution
275
8.2. Implementation of the Nash Solution
277
9. The Alternating-Offer Bargaining Solution
279
9.1. The Alternating-Offer Solution
279
9.2. The Outside Option Principle
280
9.3. A Risk of Breakdown
281
10. The Core
282
11. The Shapley Value
286
11.1. The Balanced Contributions Property
286
11.2. The Shapley Value
287
Notes
291
Chapter 10 Market Information
292
12. Akerlof Model: the Used Car Market
2.
3.
4.
292
1.1.
The Used Car Market with Asymmetric Information
292
1.2.
The Used Car Market with Perfect Information
295
1.3.
The Used Car Market with Symmetric Information
296
1.4.
Discussions
296
The RS Model: Adverse Selection in the Insurance Market
297
2.1.
Insurance with Symmetric Information: A Contingent Market
297
2.2.
Insurance with Symmetric Information: Insurance Market
298
2.3.
Insurance with Asymmetric Information
301
2.4.
Extensions
305
Job Market without Signals
307
3.1.
The Model
307
3.2.
Bayesian Equilibrium
307
3.3.
Subgame Perfect Nash Equilibrium
309
3.4.
Constrained Pareto Optimum
310
Spence’s Model: Job Market Signalling
311
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5.
4.1.
The Perfect Information Solution
314
4.2.
The No Signalling Solution
315
4.3.
Separating Equilibria
315
4.4.
Pooling Equilibria
320
4.5.
Partial Pooling or Partial Separating Equilibrium
322
4.6.
Government Intervention
323
4.7.
Equilibrium Refinement
324
4.8.
Questions
326
Job Market Screening
327
5.1.
The Pooling Equilibrium
327
5.2.
The Separating Equilibrium
328
5.3.
Discussion
330
Notes
331
Chapter 11 Mechanism Design
1.
332
A Story of Mechanism Design
332
1.1.
Market Mechanism vs Direct Mechanism
332
1.2.
The Optimal Allocation: A Graphic Illustration
333
1.3.
The Optimal Allocation: Mathematical Presentation
335
1.4.
The Optimal Allocation: A General Case
337
1.5.
Market Mechanisms vs Direct Mechanism
338
2.
The Revelation Principle
338
3.
Examples of Allocation Schemes
343
4.
IC Conditions in Linear Environments
350
5.
IR Conditions and Efficiency Criteria
352
6.
5.1.
Individual Rationality Conditions
352
5.2.
Efficiency Criteria
353
Optimal Allocation Schemes
355
6.1.
Monopoly Pricing
355
6.2.
A Buyer-Seller Model with Linear Utility Functions
357
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6.3.
Labor Market
360
6.4.
Optimal Auction
362
6.5.
A Buyer-Seller Model with Quasi-Linear Utility
364
Notes
367
Chapter 12 Incentive Contracts
1.
368
The Standard Agency Model
368
12.1. Verifiable Effort: The First Best
369
12.2. Nonverifiable Effort: The Second Best
370
13. Two-State Agency Models
375
13.1. Verifiable Effort
375
13.2. Nonverifiable Effort
376
13.3. Example: Insurance
377
14. Linear Contracts under Risk Neutrality
379
14.1. Single Moral Hazard
380
14.2. Double Moral Hazard
382
15. A Conditional Fixed Contract
387
15.1. The Model
387
15.2. The Optimal Contract
388
16. A Suboptimal Linear Contract
390
Appendix A: General Optimization
392
1.
Gateaux Differentiation
392
2.
Positive Definite Matrix
393
3.
Concavity
394
4.
Quasi-Concavity
397
5.
Unconstrained Optimization
399
6.
Constrained Optimization
400
7.
The Envelope Theorem
402
8.
Homogeneous Functions
402
Notes
403
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Appendix B: Dynamic Optimization
1.
2.
404
Discrete-Time Stochastic Model
404
1.1.
The Markov Process
404
1.2.
The Bellman Equation
405
1.3.
The Lagrange Method
406
The Continuous-Time Deterministic Model
407
2.1.
The General Model
407
2.2.
Special Models
408
Notes
409
References
410
Index
415
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Preface
This book covers microeconomic theory at the Master’s and Ph.D levels for students in
business schools and economics departments. It concisely covers major mainstream microeconomic theories today, including neoclassical microeconomics, game theory, information
economics, and contract theory.
The coverage. Microeconomics is a required subject for all students in a business school,
especially for students in economics, accounting, finance, management, and marketing. It is a
standard practice nowadays that a Ph.D program in a business school requires a one-year
course in microeconomic theory. This one-year course is generally divided into two onesemester courses. Chapters 1–4 are the core subjects for the first semester. These chapters
cover neoclassical economics, which includes demand and supply theories, risk theory, and
general equilibrium. It is the foundation of modern microeconomics. Chapters 7–12 are the
core subjects for the second semester. Chapters 7–9 cover game theory. Chapters 7 and 8
cover noncooperative games, dealing respectively with imperfect and incomplete information
games. Chapter 9 covers cooperative games, which is useful for advanced topics in information
economics and organization theory. Chapters 10–12 cover information economics, including
asymmetric information and incentive contracts. Chapters 5 and 6 are optional; these chapters
are the foundations of some field courses, which may or may not be covered depending on the
emphasis of the instructor. Chapter 5 covers some basic theories in financial economics, especially asset pricing. Chapter 6 contains the basic theory of industrial organizations, which can
be covered in either semester.
Special features of this book. Mas-Colell et al. (1995) is today the standard textbook
for Ph.D microeconomics. In terms of its contents, our book covers all the important materials
in Mas-Colell et al. (1995). However, it does not include materials that I consider to be out of
the mainstream. Those materials that I think should belong to field courses (such as social
choice and public finance) are also left out. On the other hand, I strengthen many topics that I
think are important, particularly in Chapters 3, 5, 6, 10, 11 and 12. Roughly half of the materials in this book can be found in Mas-Colell et al.(1995) and the rest are from many different
sources. Some advanced materials exist only in papers. Our book is intended to be a more
focused and complete coverage on mainstream topics in microeconomic theory. In particular,
this book is intended for postgraduate students in business schools, rather than just for students in economics departments.
This book is intended as a handbook for graduate students and professors. It is concise
with many of detailed explanations deliberately left out. Such a book is good for students who
will learn most of the details and explanations from lecture. It is also good for instructors who
would want to fill in the details by themselves in their own words. Teachers generally do not
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like a textbook to have too many details. With a detailed textbook, some students may skip
class and the instructor may even feel like reading the book aloud in class. This book can also
serve as a reference book for those who have already learned all the materials. When such a
person wants to refresh some materials, he or she can quickly find the key points of the material in the book without going through all the details.
The main enhancement of this edition over the first edition published by People’s University Publisher in 2006 is the addition of one chapter, Chapter 8, on incomplete information
games. Such games are essential for information economics, but are not included in MasColell et al. (1995).
Supporting materials. Exercises and their solutions are available in PDF format at
www.bm.ust.hk/sswang/micro-book/. Errors are inevitable in such a book with so many
diverse materials. We will offer corrections at the site. We may also provide more materials
such as more exercises, new sections and chapters on that site.
Acknowledgement. I would like to thank Dr. Virginia A. Unkefer for professional English editing.
Susheng WANG
Hong Kong, CHINA
May 2016
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Notation and Terminology
is the set of all real numbers. Denote
Given
and
denote the inner product of two vec-
tors as
FOC stands for ‘first-order condition’. SOC stands for ‘second-order condition’. FOA
stands for ‘first-order approach’, defined by Chapter 12.
The word ‘iff’ means ‘if and only if’.
A function
is homogeneous of degree
particular, when
say that
we also say that
if
In
is linearly homogeneous; and when
we also
is zero homogeneous.
For
denote
or
gradient, i.e.,
as the vector of partial derivatives or the
Denote
or
as the Hessian
matrix.
If an
square symmetric matrix
negative definite, we denote
is negative semi-definite, we denote
if
is
We denote positive semi-definiteness and positive defi-
niteness by the same way.
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Chapter 1
Producer Theory
1. Technology
Efficient Production
Suppose that a firm has
units of good
goods to serve as inputs and/or outputs. If the firm uses
as an input and produces
is the net output of good
units of the good, where
,
The firm’s production plan
then
is a
list of net outputs of all the goods that it produces as outputs and/or uses as inputs. Practically,
we can treat positive numbers in a net output vector as outputs and negative numbers as
inputs.
For any two vectors
A set
denote
of production plans that are technologically feasible is called the firm’s produc-
tion possibility set. A production plan
that
2
is technologically efficient if there is no
A production plan is economically efficient if it maximizes profits
given price vector
over the production possibility set
such
for a
Technological efficiency is dependent
only on the firm’s technological characteristics; economic efficiency is further dependent on
the market.
We can easily see that economic efficiency implies technological efficiency. Technological
efficiency has nothing to do with the market. No matter what the prices of inputs and outputs
are, the firm needs to achieve technological efficiency in order to achieve profit maximization;
see Figure 1.1. We can think of the firm’s choice problem in two steps: the firm has to achieve
technological efficiency first and then achieve economic efficiency on the efficient production
set (the production frontier). The production frontier (PPF) is defined to be the set of all technologically efficient production plans.
2A
weak version of technological efficiency is as follows: a production plan
there is no
such that
is technologically efficient if
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y2
.

y1
PPF
Isoprofit line
p0 = p ⋅ y
Figure 1.1. Technological Efficiency vs. Economic Efficiency
Proposition 1.1. Economic efficiency implies technological efficiency.
We now use a function to define a firm’s technology. There is a twice differentiable function
that defines the production possibility set:
In order for
this definition to make sense, we have to impose two assumptions on
Assumption 1.1 (Strict Monotonicity).
By Assumption 1.1, since
any
such that
is technologically inefficient:
one can increase the production (or decrease the use) of one product
constant (some of the
’s are inputs). Therefore,
while keeping others
is the technologically
efficient production set or the production frontier (PPF).
Proposition 1.2. Set
is the production frontier.
We now define the marginal rate of transformation (MRT) as
Assumption 1.1 implies a positive MRT. MRT
there are only two goods
Assumption 1.2.
is the slope of the production frontier when
and
is quasi-convex.
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Proposition 1.3. Under Assumption 1.1, for
is quasi-convex iff3
Proof. By Theorem A.6, this inequality is necessary. Let us show that it is also sufficient. Let
the unique solution from
Function Theorem.
be
which is guaranteed by the Implicit
Then,4
This means that the production frontier is concave, implying a convex set
Since we can go through the same derivation with
by definition,
is quasi-convex.
Proposition 1.3 offers us a convenient tool to verify Assumption 1.2. It also relates the
quasi-convexity to the SOC of the firm’s problem. To see this, consider the profit-maximizing
problem:

∈ℝ
where
is given,
By Assumption 1.1, the production frontier is the set of efficient
production. Since technological efficiency is necessary for economic efficiency (profit maximization), the problem is equivalent to
∈ℝ
Let
Then, by the Lagrange Theorem, the first-order condition (FOC) is
∗
3
This is from Arrow–Enthoven (1961, p.796). For sufficient conditions of quasi-convexity when
(1.2)
, see
Wang (2008, Chapter 3).
4 Denote
(
,
)
(
,
)
Then,
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and the SOC is
∗
We have
∗
∗
∗
By the FOC, since
∗
and
we have
There-
fore, the SOC becomes
∗
By Theorem A.7, if
∗
(1.3)
is strictly quasi-convex, then the inequalities in Proposition 1.3 will be
strict and hence (1.3) will be satisfied. Therefore, under Assumptions 1.1 and 1.2, a solution
∗
from the FOC (1.2) must be the solution of the firm’s problem (1.1). This is illustrated in Figure
1.2.
y2
Profit line
.
G( y) = 0
y1
Figure 1.2. Profit Maximization with Implicit Production Technology
We will concentrate on the case in which the firm produces only one output; we may think
of this output as an index of the firm’s outputs. The case of multiple outputs will be dealt with
later. When there is only one output, we will use
to denote the firm’s output and
to denote the firm’s inputs. Then, a typical production plan is
set in
Given any vector
where
is a
of inputs, denote the maximum technologically feasible
output as
( ,
We call this function
)∈
the production function. We can easily verify that
if
is technologically efficient. In this sense, the production function fully characterizes
technologically efficient plans. An isoquant is a curve defined by
The isoquant for output
is the set of all inputs that produces
as the maximum technologi-
cally feasible output.
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Proposition 1.4. For the production function
Example 1.1. Cobb-Douglas Technology. For
if
is technologically efficient, then
let
x2
Q( y)
x1
Q(y) shifts to the right as y increases
Figure 1.3. Cobb-Douglas Technology
By definition, as shown in Figure 1.3, we have
Technological Characteristics
We often have special interest in several important characteristics of a production function, from which some economic insight can be obtained.
define
First, for production function
The MRT between two inputs is
Alternatively, we can define MRT directly on the production function. For production function
decrease
given output
with one unit increase in
to maintain the same level of output ? That is,
by how much do we need to
. measures the certain
substitutability of the two inputs. By differentiating both sides the following equation
and using the fact that
and
for
we have
→
Hence, the marginal rate of transformation (MRT) between
and
at
is defined as
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In Figure 1.4, we can see that the MRT is the slope of the isoquant.
x2
Marginal Rate of Transformation
Δx 2
direction = ( f x , f x )
Δx1
slope =
1
fx
1
fx
2
f ( x) = y
2
x1
Figure 1.4. The Marginal Rate of Transformation
Example 1.2. Suppose that
Then,
Therefore, the MRT is
Second, we say that a production function
(CRS) if
for all
and
has global constant returns to scale
With this production technology, the firm can
double its output by doubling its inputs. Similarly, we say that the production function has
global increasing returns to scale (IRS) or decreasing returns to scale (DRS) if, respectively,
or
for all
and
Note also that, if
for all
This also applies to global CRS and DRS.
Example 1.3. For the Cobb-Douglas production function
We immediately see that the production function has global IRS if
is globally IRS, then
we have
global CRS if
and global DRS if
Third, the elasticity of scale at
measures the percentage increase in output due to one
percentage increase in scale:
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In other words,
portionally by
is the percentage increase in output when all inputs have expanded proWe say that the technology exhibits local increasing, constant, or decreas-
ing returns to scale if
is greater, equal, or less than
respectively.
Proposition 1.5 (Returns to Scale).
1. We have
Conversely,
2. For
we have
3. For
we have
where
4. For
Hence, for
and
we have
we have
implying
(1.6)
Proof. 1. If
is globally IRS, then, for
and any
we have
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implying
Conversely, if, for all
and
then we have
implying
implying
implying that
is globally IRS. The proof for global DRS is symmetric.
2. By (1.4), global IRS immediately implies
immediately implies
3. For
at any point
we have
at any point
Similarly, global DRS
Hence,
By this, we can determine the regions for the
( )
three returns to scale on a diagram; see Figure 1.5.
4. We can also illustrate the returns to scale using cost curves. The cost function is
The Lagrange function is
The FOC is
∗
and the Envelope Theorem is
We then have
∗
∗
∗
∗
∗
hence,
∗
∗
∗
∗
∗
Hence, (1.6) holds and it can be illustrated in Figure 1.6.
The property in (1.5) is illustrated in the following figure.
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f
e( x) < 1
.
e( x) = 1
e( x) > 1
x
.
f ' ( x)
IRS
CRS
f ( x) / x
x
DRS
Figure 1.5. Local Returns to Scale
The property in (1.6) is illustrated in the following figure.
IRS
CRS
MC
AC
DRS
y
Figure 1.6. Local Returns to Scale
By the argument for the local returns to scale and Figure 1.5, we immediately have the diagrams for the global returns to scale in Figure 1.7.5
5The
equation
gives the general solution
where
is an arbitrary constant, implying
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f ¢( x)
f ( x) / x
f ¢( x) = f ( x) / x
f ( x) / x
f ¢( x)
x
Global IRS
x
Global DRS
x
Global CRS
Figure 1.7. Global Returns to Scale
By the argument for the local returns to scale and Figure 1.6, we immediately have the diagrams for the global returns to scale in Figure 1.8.
MC
AC
AC
MC
MC=AC
y
Global IRS
y
Global DRS
y
Global CRS
Figure 1.8. Global Returns to Scale
Finally, let
output
be the cost-minimizing input vector for given input price vector
The elasticity of substitution between the two inputs
to be the ratio of percentage change in
( , )
( , )
By the optimality of
is defined
to the percentage change in
Note that this variable is well defined only if
is zero homogenous 6
and
and
( , )
( , )
is a function of
is a function of
Therefore,
As shown later, since
( , )
( , )
is a function of
using (1.9), we can alternatively write this elasticity as
This is evaluated at the optimal point.
6 See
Appendix A for the definition of homogenous functions.
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Notice that MRT is based on technological efficiency and elasticity of substitution is based
on economic efficiency.
2. The Firm’s Problem
The firm is assumed to maximize its profits. Profits consist of two distinct parts:
Revenue is the money received from the sales of the firm’s products. Costs are the economic
cost or more popularly called the opportunity cost, which typically includes three components:
the cost of labor (and raw materials), the cost of capital (including depreciation), and the cost
of land (and natural resources).
actions accomplished by choosing a vector
In general, suppose that a firm takes
The actions may include output levels, labor inputs, capital inputs, and even prices. Suppose
that the firm gains revenue
and pays costs
from actions
The profit-
maximizing firm will choose its optimal actions from
By the FOC, the optimal action
∗
is the solution of
∗
∗
(1.7)
condition; see Figure
This is the well-known “marginal revenue = marginal cost”
1.9. The intuition is clear. By increasing a small amount
∗
∗
∗
of
at
∗
(
That is, the firm approximately gains
the firm should increase the level of
of
∗)
and if
∗
(
and loses
(
∗)
(
∗)
∗)
∗
we have
∗
Hence, if
(
∗)
(
∗)
the firm should decrease the level
The firm will be satisfied when (1.7) holds.
MC
p-cf
.
MR
a
Figure 1.9. MR=MC
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We will only deal with competitive firms until Chapter 6. A competitive firm is not able to
manipulate prices. It takes market prices as given (including the output and input prices).7
Under this assumption, we can now state the firm’s problem more specifically. The firm’s
profit function is
defined by
(1.8)
∈ℝ
where
is the output price and
the vector of input prices. Here, the action variables are the
quantity variables. In Chapter 6, when the firm is not competitive, both prices and quantities
can be action variables. Denote the optimal choice of factors
call it the demand function of the firm. Then,
∗
from (1.8) as
∗
and
is called the supply function.
Example 1.4. Given a production function
the production possibility set is then
defined by
Let
and
Then, the production possibility set is
and
and
Therefore, if
Since conditions
and
and
by Theorem A.6, Assumptions 1.1 and 1.2 are satisfied.
are usually imposed for single-output production functions,
this example shows that the two assumptions are not restrictive in the sense that they do not
impose unnecessary restrictions on production functions. In fact, single-output production
functions are just a special case of the more general production functions.
Let us now analyze the profit maximization problem (1.8). The FOC for the profit maximization problem (1.8) is
∗
where
7 In
Chapter 6, we will take a more sensible approach to firms and consider when a firm can be competitive in
output markets but monopolistic in input markets or vice versa.
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We can rewrite problem (1.8) into the following form:
∈ℝ ,
This problem can be illustrated graphically in Figure 1.10.
y
Isoprofit Line, slope=
y = f (x)
.
x
Figure 1.10. Profit Maximization
The second-order condition (SOC) for (1.8) is
∗
∗
×
As can be seen from Figure 1.10, this condition can be used to rule out the other possible solution of the FOC equation.
The cost function is
(1.11)
which is the minimum expenditure on producing
given price vector
If
is strictly
increasing, we can rewrite (1.11) as
∈ℝ
Denote the optimal choice of factors
∗
from (1.11) as
(1.12)
∗
and call it the conditional
demand function. Notice that this demand function is different from the previous demand
function. It is conditional because the output is given instead of being optimal. For problem
(1.11), consider the Lagrange function
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where
is some constant. By the Lagrange Theorem in Appendix A, the FOC is
∗
or
∗
(1.13)
∗
Condition (1.13) can also be derived graphically using Figure 1.11.
x2
Isocost line
slope= w1 / w2
.
x 2∗
f ( x) = y
x1
x1∗
Figure 1.11. Cost Minimization
As shown, the slope
of the isocost line is equal to the slope
of the isoquant
at the optimal solution. Therefore, (1.13) must be true.
Condition (1.13) can also be derived by economic intuition. We know the right side is the
MRT; the left side represents the economic rate of substitution — at that rate, factor can be
substituted for factor while maintaining a constant cost. When
∗
∗
factor is too costly relative to factor at
the firm should reduce the input of
∗
compared to the relative benefit
relative to the input of
In this case,
The same argument holds for
the opposite inequality. The best choice is therefore where (1.13) holds.
By the Lagrange Theorem, the SOC for (1.11) is
∗
That is, (since
∗
),
∗
∗
(1.14)
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which can be dealt with using Theorem A.8 in Appendix A. This SOC basically means that
∗
is quasi-concave at
In fact, the conditions used to verify quasi-concavity and this SOC are
almost the same.
at
What is the intuition for the SOC? Concavity of
any
Hence, condition (1.14) means that
∗
space orthogonal to
∗
is concave at
∗
∗
for
relative to the vector
In Figure 1.11, we can see that, when
concave relative to the vector space orthogonal to
∗
means that
is convex,
is
(it is actually the isocost line). Hence,
implies (1.14). At the same time, we also know that convexity of
convexity of
implies quasi-concavity of
quasi-concavity of
at
Therefore, condition (1.14) basically means the
∗
we can proceed to solve the following problem for
After we find the cost function
profit maximization:
(1.15)
The FOC for this problem is the well-known condition
for a competitive firm, i.e.,
∗
The SOC is the convexity of the cost function in
at
∗
∗
We will establish later that
is always concave in
Hence,
∗
is a so-called saddle
point.
In the short run, some inputs
(such as capital stock) are fixed. Let
be the price
vector for the fixed inputs. We then have the short-run total profit function:
the short-run (variable) profit function:
and the short-run total cost function:
and the short-run (variable) cost function:
Example 1.5. The Cost Function of the Cobb-Douglas Technology. Consider
,
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By solving
from the production function and substituting it into the objective function, the
problem becomes
By the FOC, we can solve for the conditional demand function:
Given
the other conditional demand function can be solved from the production
function or by symmetry:8
The cost function is then
Example 1.6. The Short-run Profit Function for the Cobb-Douglas Technology. Suppose that
in the short-run
Consider
,
This is equivalent to:
By the FOC, we can find the demand function:
the short-run supply function:
and the short-run profit function:
Example 1.7. The Profit Function for the Cobb-Douglas Technology. Let us write the cost
function from Example 1.5 as
The profit maximization problem is:
8 This
works since, after changes
and
the original problem is the same.
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If
the supply function is:
and the profit function is:
When
(CRS), the profit maximization problem is
which implies the following supply function:
where, when
can take any value in
In this case, since the supply curve
is horizontal, the only possible equilibrium price is
Hence, if the demand is down-
ward sloping, there is a market equilibrium, in which the price is
∗
and the quan-
tity is solely determined by demand. In this sense, we say that this market is demand driven.
When
has global CRS, the FOC (1.9) does not lead to a solution. For example, for the
case with two goods, since
is zero homogenous, the FOCs in (1.9) can be written as
That is, both equations give us
example, if
and these two equations may contradict each other. For
then the two equations imply
which implies
This is generally not true. The cause is that the solution is not an interior solution and thus the
FOC does not hold generally. To see this, consider a special case. With one input
tion function with CRS is
where
a produc-
is a constant. Then, the situation in Figure 1.10
becomes the following situation, which clearly indicates a corner solution. Thus, with a CRS
technology, we need to derive the cost function first, from which we can then determine the
supply function, just as we did in Example 1.7.
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y
Isoprofit line
y = f (x)
.
x
x*
Figure 1.12. A Homogenous Production Function
Example 1.8. The CES Production Function. Consider production function:
/
By (1.13),
Thus,
Using the fact that
we then have
Therefore, this function has constant elasticity of substitution (CES):
The CES function contains several well-known production functions as special cases.
When
or
When
or
as shown in Figure 1.13, we have a linear production function:
we further assume
→
Since, by L’Hopital’s rule,
→
we have
(
→
)
→
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which is the Cobb-Douglas function. When
and
or
we further assume that
Using the fact that
/
/
for
/
/
/
/
we have
/
/
/
/
/
→
/
/
/
which is the Leontief Production Function.
x2
.
σ =0
σ =∞
σ =1
x1
Figure 1.13. Isoquants for the CES Technology
For changing market conditions, or more precisely, changing price ratios, the two special
technologies with
and
have special reactions. As shown in Figure 1.14, if the
technology is Leontief, the firm typically does not react to changing market conditions at all; if
the technology is linear, the firm often reacts in a dramatic fashion — from demanding only
to demanding only
or vice versa.
x2
x2
Isoquant y
.
Isoquant y
.
w1
w2
Isocost c
slope =
slope =
Isocost c
x1
.
w1
w2
x1
Figure 1.14. The Two Special Technologies
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3. Short-Run and Long-Run Cost Functions
3.1. Definitions
In the short run, suppose that a vector
of factors is fixed. Let
be the vector
of prices for the variable and fixed factors, respectively. Then, the short-run total cost is:
Define
In the long run, when all factors are variable, the firm will choose an optimal
∗
Then, we will have the long-run cost (LC):
Similarly, define
3.2. The relationship between the AC and MC curves
Let us first find the relation between the AC and MC curves. Suppose that
short-run cost function with fixed input vector
and that
is a
is the short-run variable cost
function. We have
This means that
Furthermore, since
by L’Hopital’s rule, we have
→
→
Similarly,
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These are also true for long-run cost functions except in that case when the variable cost is the
same as the total cost. We thus have the general graphic situations for SMC, SAC and SAVC
and for LMC and LAC in Figure 1.15.
SMC
SAC
LMC
SAVC
y
0
LAC
y
0
Figure 1.15. AC, AVC, MC Curves
3.3. The relationship between SR and LR cost functions
Let us now find the relations between short-run and long-run cost functions. Suppose that,
in the short run, a vector of factors
is fixed and, in the long run, the minimum point of
Then,
is
implying
That is,
(1.17)
By the Envelope Theorem, we have, for any
( )
(1.18)
The above two equations (1.17) and (1.18) mean that, for any point
(a) the short-run average cost
cost
must be at least as great as the long-run average
for all output levels
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(b) the short-run average cost is equal to the long-run average cost at output
and
(c) the long- and short-run cost curves are tangent at
These mean that the long-run cost curve is simply the lower envelope of the short-run curves,
as shown in Figure 1.16.
SAC ( y , z1 )
LAC ( y )
SAC ( y , z 3 )
SAC ( y , z 2 )
.
.
.
zi = z ( y i )
y1
y
y3
y2
Figure 1.16. Long-Run and Short-Run Average Cost Curves
we have
Also, by the Envelope Theorem, from (1.16), for any output level
( )
This is illustrated in Figure 1.17.
SMC ( y , z1 )
LMC
SMC ( y , z3 )
LAC
SMC ( y , z 2 )
.
.
.
.
.
zi = z ( y i )
y1
y2
y3
y
Figure 1.17. Long-Run and Short-Run Marginal Cost Curves
4. Properties
This section discusses the properties of the various functions involved in production.
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Proposition 1.6. If the production function is homogeneous of degree
then
in output.
i.e., the cost function is homogenous of degree
Proof. We have
/
/
/
/
/
/
See Example 1.5 for a confirmation of Proposition 1.6.
has the following
Proposition 1.7 (Properties of Cost Functions). The cost function
properties:
(1) increasing in
(2) linearly homogeneous in
(3) concave in
Furthermore, if
is continuous, then the three conditions are sufficient for
to be
a cost function.9
Proof. We show the necessity of the three conditions only. First, given
satisfying
we must have
for any
Hence,
i.e.,
The second property is obvious.
For the third property, by the definition of
for any
we have
Then,
9A
function
is a cost function if there exists an increasing and concave function
( , ) ≡ min
⋅ | = ( ) ,
∀( , )∈ ℝ
such that
.
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What are the assumptions that we have used in the proof of the concavity? Only the costminimizing behavior! Hence, the concavity of the cost function comes solely from costminimizing behavior. The concavity of the cost function means that the cost rises less than
proportionally when the prices increase. The intuition is clear: when the price of a factor rises,
as one factor becomes relatively more expensive than other factors, the cost-minimizing firm
will shift away from that factor to other factors, which results in a less than proportional increase in cost. In other words, the cost increases at a decreasing rate. This is what concavity
means.
In Figure 1.17, at an input point
ways spend
∗ ∗
∗
∗
∗
for price vector
to maintain input
∗
optimal cost will generally be less than
when
changes, the firm can al-
However, this may not be optimal. The
∗ ∗
Since the optimal cost curve is al-
ways below the straight line, it must be concave.
cost
n
w1 x1* + å wi* xi*
i =2
c ( w1 , w2* , , wn* , y )
.
w1
w1*
Figure 1.18. Concavity of the Cost Function
Similarly, we have the following result for profit functions:
Proposition 1.8 (Properties of Profit Functions). The profit function
has the follow-
ing properties:
(1) increasing in
decreasing in
(2) linearly homogeneous in
(3) convex in
Proof. By comparing (1.10) with (1.12), we can see that
is defined just like
with
linear objective functions and the same constraint, except that one is minimization and the
other is maximization. Hence, the properties of
Alternatively, we can borrow the concavity of
definition of
for any
can be proven just like those of
to prove the convexity of
. By the
let
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We have
implying
The following Hotelling’s lemma establishes a relationship between demand and profit
functions.
Proposition 1.9 (Hotelling’s Lemma). Let
be the firm’s demand function for factor
Assume that it is an interior solution of the profit maximization problem. Let
and
be respectively the supply and profit functions. Then,
Proof. Let
be the production function. Then,
is the solution of
By the Envelope Theorem,
Similarly,
Shephard’s lemma establishes a relationship between cost and conditional demand functions.
Proposition 1.10 (Shephard’s Lemma). Let
tion for factor
be the firm’s conditional demand func-
Assume that it is an interior solution of the cost minimization problem. Let
be the cost function. Then,
Proof.
is the solution of the following problem:
By the Envelope Theorem, we immediately have
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We already know the properties of cost and profit functions. We can now find the properties of demand and supply functions through Hotelling’s and Shephard’s lemmas.
Proposition 1.11 (Properties of Conditional Demand). Suppose that
is twice contin-
uously differentiable. Then,
is zero homogeneous in
(1)
(2) Cross-price effects are symmetric:
(3) Substitution matrix
(4)
( , )
( , )
is negative semidefinite.
is decreasing in
Proof. (1) By Theorem A.9, since
neous in
is linearly homogeneous in
is zero homoge-
See Figure 1.11 for a graphic explanation.
(2) By Shephard’s lemma, we have
By continuous differentiability of
implying
That is,
is symmetric.
(3) By the concavity of
Therefore, the substitution matrix
is
symmetric and negative semidefinite.
(4) Since
is symmetric and negative semi-definite, its diagonal entries must be
negative, i.e.,
This immediately implies property (4).
Similarly, we have
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Proposition 1.12 (Properties of Demand and Supply). Let
and
be twice con-
tinuously differentiable demand and supply functions. Then,
(1)
and
are zero homogeneous in
(2)
is decreasing in
10
is increasing in
(3) cross-price effects are symmetric:
( , )
( , )
Notice that property (2) in the above proposition says that the firm’s demand function is
always downward sloping. This is very different from the Marshallian demand for consumers.
Proposition 1.13 (The LeChatelier Principle). Firms will respond more to price change in
the long run than in the short run. More precisely, suppose that
the short run, and
interior prices
is the profit maximization choice of
is a fixed vector of inputs in
in the long run for given
Then,
( , )
Proof. Let
and
( , )
be the long-run profit function,
the short-run profit function,
the long-run supply and demand functions, respectively,
the short-run supply and demand functions, respectively, and
tion of
Given any price vector
∗
and
the solu-
define
By definition, we have
Thus,
By
is the minimum point of
the diagonal elements of
This implies
and
must be positive, that is,
By Hotelling’s lemma, we then have
Example 1.9. Given a function
where
under what conditions, is
this function a cost function? Using Proposition 1.7, increasingness in
implies
Linear homogeneity implies that
10 See
Figure 1.10 for a graphic explanation.
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We have
Since
this implies that
(
)
(
)
Then, concavity requires that
These conditions are implied by (1.19) and (1.20). Hence, these two conditions are necessary
and sufficient conditions for
to be a cost function. The corresponding regression model
is
Using condition (1.20), the regression model becomes
By (1.19), we require
5. Aggregation
In applied economics, data are often from observations of aggregate variables, such as the
total output of an industry, the industry demand for a certain factor, and the total profit in a
sector. We have so far only discussed a firm’s production, input and profits. There is a need to
extend these discussions to aggregate variables. A classical approach to this problem is to
show that aggregate variables can be viewed as being generated by a single firm — a fictional
firm. If so, all the results that I have obtained so far become readily applicable to aggregate
variables.
Specifically, given
independent production units, can the aggregate supply and profit
functions from these units be viewed respectively as supply and profit functions of a firm? The
answer is yes under certain conditions.
Suppose that there are
duction possibility sets
production units (firms or plants) in an economy, having proin
respectively. Assume that each
and with the free disposal property.11 Let
let
be the supply function. Here,
11 The
free disposal property means that, if
is nonempty, closed
be the profit function for the unit with
and
can be a set-valued function, or called a corre-
then
for any
This means that output can be
freely thrown away at no cost.
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spondence, if, for each
is a subset of
instead of a single value. The aggregate supply
function is
where the sum is generally a sum of
Let
and
sets. The aggregate production set is
be respectively the profit and supply functions of
Proposition 1.14. For all
we have
In other words, the aggregate variables
and
can be viewed respectively as the
profit and supply functions of the fictional firm defined by production possibility set
Notes
Materials in this chapter are fairly standard. Many books cover them. Good references for
this chapter are Varian (1992), Jehle (2001), Dixit (1990), and Mas-Colell, Whinston and
Green (1995).
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Chapter 2
Consumer Theory
There are two distinct approaches to modeling individual behaviors. The first approach
starts with preferences and proceeds to rationalizing the preferences and then showing the
existence of a utility representation. The second approach is based on observations of individual choices and proceeds to reveal individual preferences with the aid of some consistency
assumptions on individual preferences. We will focus on the first approach. Although the
second approach has many attractive features, the existence of a utility representation gives us
a convenient tool for the development of demand theory. We briefly cover the second approach in Section 6.
1. Existence of the Utility Function
Consider a consumer faced with possible consumption bundles in some set
the consumption set. We will always assume that
we often use
called
is a closed and convex set. For convenience,
as the consumption set.
The consumer is assumed to have preferences for the consumption bundles of
the consumer thinks that bundle
preference relation on
is at least as good as bundle
we denote
When
We call
a
if it satisfies the following two axioms:
Axiom 1 (Reflexivity).
Axiom 2 (Transitivity).
if
then
These two axioms seem necessary for any consumer who is considered as rational. We expect that any rational individual has such preferences.12
Example 2.1. The relation
on
is a preference relation. The relation
is however not a
preference relation.
We introduce a bit more notation:
12 The
collective preferences of a group of rational consumers may not have such preferences at all.
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It will be convenient to have a function representing the preference relation. Given the function, we can solve the maximization problem using the Lagrange theorem and discuss the
solution using existing mathematical tools. In order to represent this preference relation by a
function, we need the following two axioms:
Axiom 3 (Completeness).
either
or
or
That is, any two consump-
tion bundles can be compared.
We say that a preference relation is rational if it is transitive and complete.
Axiom 4 (Continuity).
if
∗
and
and
as
It is obvious that if
then
are closed sets in
That is,
∗
is complete, then
Furthermore, by
Arrow and Intriligator (1981, p.386, Theorem 4.1),
iff
is
complete.
Further, the following concepts are often used in our discussion of consumer preferences.
Axiom 5 (Monotonicity).
Axiom 6 (Strict Monotonicity).
Axiom 7 (Local Nonsatiation).
such that
Axiom 8 (Convexity).
Axiom 9 (Strict Convexity).
Here
is the distance between
and
Local nonsatiation says that one can al-
ways find a better choice in any neighborhood of a consumption point. Thus, strong monotonicity implies local nonsatiation, but not vice versa.
Denote
A function
as the consumption space that has the preference relation
over the set
is said to represent the preferences if
•
if and only if
•
if and only if
and
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Such a function is called a utility function on
or a utility representation of preferences.
Convexity implies that the consumer prefers averages to extremes, which implies quasiconcavity of utility functions. The convexity is the neoclassical assumption of “diminishing
marginal rates of substitution.” By definition, we obviously have the following result.
Proposition 2.1. (Strict) convexity of preferences
(strict) quasi-concavity of the utility
function.
We now prepare to prove an existence theorem of the utility function. A set
is said
to be connected if it cannot be expressed as a union of two disjointed nonempty open sets in
That means that if
and
is connected and there are two open sets
then either
in
such that
or
is connected
Lemma 2.1. A subset
and
is an interval.
Lemma 2.2. Continuous functions map a connected set to a connected set.
Lemma 2.3.
is continuous
is open in
Lemmas 2.1 and 2.2 are well known and can be found in Armstrong (1983, Theorems 3.19
and 3.21). Lemma 2.3 can be easily shown from the definition of continuity.
Theorem 2.1 (Existence of the Utility Function). If the preferences
are reflexive, transitive,
complete, and continuous, then there exists a continuous utility function
on
Proof. For the simplicity of the proof, we assume that the preferences are strictly monotonic.
Let
For any
we want to find a unique number
As shown in Figure 2.1, the idea is to use one point on the
such that
to represent an indifference
curve.
45°
I ( x)
.
u ( x )e
.x
Indifference Curve
I ( x ) = { y | y ~ x}
Figure 2.1. Functional Representation of Preferences
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Define
By the continuity axiom,
Since
and
are closed sets in
By the completeness axiom,
and
This means that
nected. Thus,
or
and
are open sets in
Then,
monotonicity axiom,
then
By Lemma 2.1,
implying
Therefore,
is con-
By the strong
can contain only a single point. Denote this
Then,
represents the preferences, i.e.,
We now show that this function
only if
If
must be empty. This is a contradiction. Therefore,
Suppose
point as
By the monotonicity axiom,
and
if and only if
if and
By definition and strong monotonicity,
and
These relationships imply that
represents the preferences.
We now only need to show the continuity. For any
Since
we have13
we have
Thus, by the continuity axiom,
is open. Therefore, by Lemma 2.3,
is continuous.
What about the necessity of the conditions in Theorem 2.1? The first two are rationality
axioms, which are necessary for the rationality of preferences. Also, since any two real values
can be compared, completeness is obviously necessary. Let us now show that continuity is also
necessary in an example.
13 For
any sets
and
and any function we have
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Example 2.2. Nonexistence of Utility Functions for the Dictionary Order. The dictionary order
on
is defined by
Define
if
or
This order is obviously complete. By definition,
which is not closed. This means that the dictionary order violates the axiom of continuity.
x2
{x x  y}
y.
x1
Figure 2.2. Discontinuity of the Dictionary Order
We now show that the dictionary order cannot be represented by a continuous utility function.
Suppose that there exists a continuous utility function
and thus
Then, by Lemmas 2.1 and 2.2,
are connected sets. By Lemma 2.1 again,
must be an interval in
For
for some
Since
this dictionary order, for
we must have interval
by the definition of
is an interval and
Therefore, for any
is an interval
with a nonempty interior.
for
We also know that
Hence, each real number
can asso-
ciate a distinct interval with a nonempty interior. Thus, there are an uncountable number of
such intervals. Since each interval on
rational number, each interval
with an nonempty interior contains at least one
will contain at least one rational number. Since these inter-
vals are disjointed, an uncountable number of such intervals will then imply an uncountable
number of rational numbers. We know that there can only be a countable number of rational
numbers in
This is a contradiction. Therefore, such a function
must not exist for the
dictionary order.
Utility representation is not unique. In fact, if
increasing function
is a utility function, then, for any strictly
is also a utility function. However, we find that the
utility function is unique up to a strictly increasing transformation.
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Proposition 2.2. If
is a utility function, for any strictly increasing
is
also a utility function for the same preference relation. The utility function is unique up to a
strictly increasing transformation.
Proof. We need to prove the uniqueness only. Given two strictly increasing utility functions
and
if they represent the same preferences, we look for a strictly increasing function
such that
Let
Then, we identify
More rigorously, for each
Define
as
within the range
We now show that this
that
since
and
and
there is
such that
is well defined. If there is another
represent the same preferences, then
That is, although
Then, for any
of
is not unique for each
such
implying
the so-defined value
is unique.
we have
(2.1)
If
is not strictly increasing, then we can find two values
and
We can then find two arbitrary consumption bundles
We have
i.e.,
and
such that
such that
and
But, by (2.1), we have
which contradicts the fact that
and
represent the same preferences. Hence,
must be strictly increasing.
2. The Consumer’s Problem
The consumer’s problem is
∈ℝ
That is, given price vector
and income
the consumer chooses his consumption bundle
to maximize his utility subject to his budget constraint
problem will be denoted as
shallian demand function.
∗
∗
The solution from this
and called the ordinary demand function or the Mar-
is called the indirect utility function.
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Remark 2.1. If
and
By the Weierstrass Theorem, if
the consumption set
∗
is continuous,
is a compact set.
exists.
Remark 2.2. Strict convexity of preferences will imply a unique optimal consumption choice
∗
When the existence and uniqueness of the optimal choice
∗
are guaranteed, the consumer
demand function is well defined.
∗
Remark 2.3. The maximum choice
is independent of the particular choice of utility function
for given preferences. No matter what utility function is chosen to represent the preferences,
the optimal choice
∗
∗
has to satisfy
utility representations
for any
in the consumption set. Hence, for any two
and
∗
∗
∗
That is, any utility function for the preferences must have this
∗
as a solution.
By the local nonsatiation or strict monotonicity, the budget constraint must be binding
(i.e., it must be an equality). Hence, the consumer’s problem becomes
∈ℝ
(2.2)
Things become much easier to handle with an equality constraint. The Lagrange function for
the consumer’s problem is
The FOC is
∗
where
is the Lagrange multiplier. This condition can be rearranged to give:
∗
∗
As the explanation for (1.7), the left side is the relative gain in utility at
the relative cost for the consumption choice
∗
∗
and the right side is
By this interpretation, we understand that this
condition must hold in optimality. Denote the marginal rate of substitution (MRS) at a point
as
where
is the slope of an indifference curve at point
optimal solution is at the point where the slope
∗
Figure 2.3 shows that the
of the budget line is equal to the slope
of the indifference curve.
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x2
.
x∗
x1
Figure 2.3. Utility Maximization
By Theorem A.11 in Appendix A, the SOC is
∗
This condition can be checked using Theorem A.8. This SOC can be understood by the following argument. If
imum value at
∗
∗
is the maximum point, given any
The SOC for
at
is exactly the SOC for
to satisfy the budget constraint
∗
words, as long as
the SOC is the concavity of
satisfies
we must require
takes the max-
in the above. However,
to satisfy
In other
at
The dual problem of the utility maximization is the following problem of expenditure
minimization:
∈ℝ
is called the expenditure function, which is the minimum expenditure of achieving the
utility level
The solution from this problem is denoted as
and is called the com-
pensated demand function or Hicksian demand function. The expenditure function and the
compensated demand function are analogous respectively to the cost function and the conditional demand function in Chapter 1. Here,
is “compensated” simply because when
there is a change in prices, the income has to be changed or compensated accordingly to maintain the same level of utility. The Marshallian demand, on the other hand, keeps income constant but allows the level of utility to change. Compensated demand functions are not estimable since they depend on the unobservable utility value. Ordinary demand functions are, on
the other hand, estimable because they depend on observable prices and income.14
14 For
example, if we believe that ordinary demand has the form
we can try to collect data on
∗
∗
with positive constants
and
and use the following regression model to estimate the parameters
and
∗
Using the fact that
we can also identify
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Similar to the consumer’s problem (2.2), by the local nonsatiation or strict monotonicity,
the expenditure problem becomes
∈ℝ
The graphic presentation of this problem is also in Figure 2.3, just as for problem (2.2). The
FOC and SOC are therefore the same as the ones for (2.2).
The duality of utility maximization and expenditure minimization can be easily seen from
Figure 2.4.
x2
.
x∗
x1
Figure 2.4. Duality of Utility Maximization and Expenditure Minimization
Given a budget line, we can shift the indifference curve to get the optimal solution
versely, given an indifference curve going through
lelly to get
∗
∗
Given the two types of demand functions
∗
; con-
we can shift the expenditure line paral-
The two processes both yield the same solution
Duality theory talks about the duality of the following
∗
∗
This is the so-called duality.
pairs:15
and
we have some well-known
terminology for describing demand properties:
15 The
general duality is the equivalence between
for some arbitrary functions
and
Since for our case we have a linear function
we have stronger
properties.
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∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
Typically, if
∗
increases, we expect
to decrease; if goods and are substitutes, then gener-
increases. Hence, an increase in
ally
results in an increase in
if the two goods are
substitutes, i.e.,
If goods and are complements, then we expect
These relationships explain some of the above definitions.
3. Properties
This section discusses the properties of the various functions involved in consumer theory.
Proposition 2.3 (Continuity of Ordinary Demand Functions). Suppose that preferences are
strictly convex, strictly monotonic, and continuous. Then, the ordinary demand function
∗
is continuous.
Proof. We first show the continuity of
converging to
as
in
Let
be a sequence of price vectors
We know that when
∗
is large enough,
Since
sequence
∗
of
and a point
taking limit
We want to show that
∗
convexity,
that
∗
∗
and thus
is a compact set, there is a subsuch that
∗
as
Since
we have
∗
∗
If
Since
∗
and
∗
∗
then, by the strict
this contradicts the fact
is by definition the best choice among those ‘s satisfying
On the other
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∗
hand, if
∗
by the continuity of preferences, we have
Also by the continuity of preferences, since
∗
∗
∗
when
∗
and thus
such that
∗
is large enough
(here, strict monotonicity
of preferences is used to guarantee an equality budget constraint; it also used to imply that a
∗
thus
∗
must itself be convergent and the limit is
quence
∗
we have shown
We now show the continuity of
to
as
∗
∗
∗
∗
in
By zero homogeneity of
By the continuity in
∗
converges to the same point
∗
∗
sequence
Since for any convergent se∗
Let
∗
and
∗
This is again a contradiction. Therefore, we must have
Since any subsequence of
∗
∗
we then have
preferred consumption bundle costs more). Letting
is continuous in
be a sequence of incomes converging
in
we have
∗
∗
∗
which shows the continuity of
in
The result in Proposition 2.3 can be easily observed in Figure 2.5. With strictly quasi-
concave indifference curves, when the prices and income change slightly,
∗
changes slightly
too.
x2
I
p2
.
x∗
u
x1
I
p1
Figure 2.5. Utility Maximization
Proposition 2.4 (Properties of Indirect Utility Functions). The indirect utility function
has the following properties:
(1) decreasing in
increasing in
(2) zero homogeneous in
(3) quasi-convex in
Proof. Since
for
Similarly,
by the definition of
for
implies
that
These results can be easily observed in Figure 2.5.
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When the prices and income are both multiplied by a positive number, the budget constraint does not change at all. Therefore,
for
This can also be seen in
Figure 2.5.
and
Suppose
For any
∗
With
either
fore,
∗
or
consider
we must have either
∗
Thus,
∗
∗
or
∗
∗
Hence, we have
i.e.,
There-
is quasi-convex in
Expenditure function
of utility
is the minimum expenditure needed to obtain a given level
The expenditure function is completely analogous to the cost function
in
Chapter 1. They therefore have the same properties.
Proposition 2.5 (Properties of Expenditure Functions). The expenditure function
has
the following properties:
(1) increasing in
(2) linearly homogeneous in
(3) concave in
The following proposition provides some important identities that tie up the expenditure
function
the indirect utility function
the ordinary demand function
∗
and
the compensated demand function
Proposition 2.6 (Duality Equalities).
(1)
(2)
(3)
(4)
∗
∗
The proof of these identities is simple but tedious and is thus omitted. The proof can actually be easily seen in Figure 2.6.
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x2
I
p2
x * ( p, I )
u = v ( p, I )
x ( p, u )
I = e( p , u )
I
p1
x1
Figure 2.6. The Identities in Proposition 2.6
The duality equalities are very useful. For example, given an indirect utility function
if you let
mand function
be
and be
by letting
be
you can get
If you have the compensated de∗
you get the ordinary demand function
The following is an analog of Shephard’s lemma for the cost function.
and compensated
Proposition 2.7 (Shephard’s Lemma). For expenditure function
demand function
we have
Proposition 2.8 (Roy’s Identity). For ordinary demand function
function
∗
and indirect utility
we have
∗
provided that the right-hand side is well defined and that
and
Proof. We have
Then, by the Envelope Theorem,
(2.4)
which then implies Roy’s identity.
By
in (2.4),
is the utility value per unity of money. When price
one unity, since the consumer keeps
money, which means the loss of
goes up by
unit of consumption, the consumer loses
unit of
units of utility value. This explains why
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Example 2.3. The Constant Elasticity of Substitution (CES) Utility Function is given by
/
Consider the expenditure minimization problem:
which gives compensated demand functions:
(2.5)
where
We then have the expenditure function:
/
(2.6)
equation (2.6) implies the indirect utility function:
By letting
/
(2.7)
Using Roy’s identity, we then have
∗
We can also apply
in (2.5) to get this consumer demand function.
Since the compensated demand function
tional demand function
is mathematically the same as the condi-
in Chapter 1, they have the same properties.
Proposition 2.9 (Properties of Compensated Demand Functions). The compensated demand function
has the following properties:
is zero homogeneous in
(1)
is negative semi-definite;
(2) substitution matrix
(3) the cross-price effects are symmetric:
̅ ( , )
̅ ( , )
̅ ( , )
(4) the compensated own price effect is negative:
Proposition 2.10 (Properties of Ordinary Demand Functions). The ordinary demand function
∗
has the following properties:
(1) Homogeneity:
∗
is zero homogeneous in
∗
(2) Compensated Symmetry:
(3) Adding-up condition:
∗
∗
∗
∗
∗
∗
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Proof. (1) Since
homogeneous in
∗
and
∗
we find that
is zero homogeneous in
and
is zero
is zero homogeneous in
(2) The second result comes directly from the Slutsky equation in the next proposition
has symmetric cross-price effects.
and the fact that
(3) The third result is just the budget constraint.
Proposition 2.11 (The Slutsky Equation). For ordinary demand function
sated demand function
and indirect utility function
∗
∗
compen-
we have
∗
∗
Proof. By Proposition 2.6(4), we have
∗
By differentiating this equation on both sides w.r.t.
Let
∗
∗
∗
∗
which implies
and using Proposition 2.7, we have
By substituting them into above equation, we then
have
∗
∗
∗
This immediately implies the Slutsky equation.
The Slutsky equation says that when price
can be separated into two parts: the substitution effect
∗
∗
increases by one unit, the change
̅ ( , )
and the income effect
( , )
in
∗
∗
( , )
The substitution effect is the response of demand to a price change along the indifference
∗
curve. Since the expenditure on
is increased by approximately
∗
is
∗
if the price is increased by one unit, the expenditure
units. Hence, if the price increases one unit, then the pur-
chasing power is reduced by approximately
∗
units. Hence, the income effect is multiplied by
∗
These effects are shown in Figure 2.7, where the price goes up from
to
The income
level for the artificial AB line maintains the same utility level for the price change. To an increase in price, by the concavity of
in prices, the substitution effect is always negative;
but the income effect can be negative or positive. For a normal good, the income effect is
negative and thus the total change in the ordinary demand
∗
is negative; for an inferior good,
however, the income effect is positive and thus the total change in the ordinary demand can be
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either negative or positive depending on which effect is dominant. From this analysis, we find
that a Giffen good must be an inferior good.
x2
A
p1  p1¢
.
..
.
.
}
.
IE
I / p1¢
SE
B
I / p1
x1
Figure 2.7. Income and Substitution Effects
where
Example 2.4. Given utility function
we find the
ordinary demand functions:
∗
∗
Thus,
and
∗
One can solve
∗
for the expenditure function:
By Shephard’s Lemma, we get the compensated demand function:
Then,
We then have
Now we can verify the Slutsky equation:
∗
∗
∗
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where
is the substitution effect and
the income effect. For this case, both
substitution and income effects are negative. Therefore, the total change in
Example 2.5. Demand functions for a single good. Suppose that
represents all other goods. Let
be the price of
and
∗
is negative.
is the good in question, and
be the price of
The consumer’s
problem is
,
Since the ordinary demand
as
In practice,
is zero homogenous in prices, we can denote the demand
is some consumer price index, so that the demand is a function of
the real price and the real income. Consumer theory tells us that virtually any functional form
of
is consistent with utility maximization.16 We can therefore arbitrarily choose a
convenient functional form for
and it can be justified as a demand function derived
from utility maximization. Some popular functional forms are:
1. Linear demand:
2. Logarithmic demand:
3. Semi-logarithmic demand:
Example 2.6. The Linear expenditure system. Suppose that the utility function is
where
Here,
budget constraint
is the subsistence level of consumption. Utility maximization subject to
immediately implies the ordinary demand:
which is linear in parameters and thus can be used directly in econometric regression.
Example 2.7. A popular system of demand equations is the Almost Ideal Demand System
(AIDS). AIDS provides a first-order approximation to any arbitrary demand system, has desirable aggregation properties, and can “almost” be made consistent with classical demand theory through linear restrictions on the parameters of the system. AIDS has an expenditure function of the form:
∗
16 Essentially,
(2.8)
the only requirement is that the compensated own price effect is negative.
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Proposition 2.5 implies testable restrictions:
∗
∗
where
To see this, consider the expenditure share:
∗
∗
∗
The adding-up condition
∗
implies
The zero homogeneity condition
By definition,
∑
and thus
implies
and thus
which implies
There are also some inequality conditions that come from the concavity of the expenditure
function.
4. Aggregation
Suppose that there are
consumers with ordinary demand functions
The aggre-
gate demand is
Since available data in reality are often on aggregate demand and aggregate income
only,
we are interested in the following property:
(2.9)
for some function
If this holds, we can then apply our demand theory for individual
consumers to aggregate variables.
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Proposition 2.12. A necessary and sufficient condition for the existence of a function
such that (2.9) holds is that each consumer’s indirect utility function is linear in income with
a common multiplier
for income:
5. Integrability
Given a utility function, we can find the expenditure function; and symmetrically, given a
production function, we can find the cost function. Integrability asks the opposite question:
given an expenditure function or a cost function, can we respectively find the utility function
or the production function? This question is important because utility and production functions are unobservable while expenditure and cost functions are observable. The general question is how to recover unobservable variables from observable variables.
The answer is yes if the expenditure function and the cost function are differentiable and
satisfy the properties in Propositions 2.5 and 1.5, respectively. This means that given sufficient
differentiability of utility and production functions, the properties in Propositions 2.5 and 1.5
are sufficient and necessary conditions for a function to be an expenditure or cost function.
Here is a simple way to recover utility and production functions. Given an expenditure
function, we can find compensated demand functions using Shephard’s lemma, and we can
then find the utility function by eliminating the prices. Symmetrically, given a cost function,
we can find conditional demand functions using Shephard’s lemma, and we can then find the
production function by eliminating the prices. Since compensated and conditional demand
functions are zero homogenous in prices, we are basically guaranteed to be able to eliminate
the prices. We will use two examples to illustrate the process.
Example 2.8. Consider a cost function
By Shephard’s lemma, we have
These two expressions can be used jointly to eliminate the price ratio, which yields:
This is a production function with Cobb-Douglas technology. This production function can
generate the cost function.
This example is for usual production technologies. The Leontief technology is quite special. For it, we used a special method to solve for the cost function in Exercise 1.5 in Chapter 1.
It turns out that we also have to use a special method to recover the production function from
the cost function. This is illustrated in the following example.
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Example 2.9. Let
Thus, to produce
By Shephard’s lemma,
since the elasticity of substitution is
we need inputs
that is,
Hence, given
output
satisfies
implying that the maximum amount of output that
can produce is
which is by definition the production function.
6. Revealed Preferences
In reality, not only we do not observe consumer preferences directly, but we also often do
not have an expenditure function. What we do have is a set of observations
of consumption bundle
at price
The question that we would like to ask is:
Does this set of observations possibly come from utility-maximizing behavior? More specifically, what is the necessary and sufficient condition under which this set of observations can
be from a utility-maximizing consumer?17
We say that a utility function
for
where
∗
rationalizes the observed data set
∗
if
is the ordinary demand given the utility function. If so, when
we must have
In other words, one has to spend more in order to
get higher satisfaction. Conversely, by the same argument, if
then we must have
Hence,
(2.11)
Since we cannot observe
we try to use a relation such as
to reveal the prefer-
ences. This motivates the following definition.
Denote the relation
price
Similarly, denote
write
17 To
if there is a sequence
as
and say that
as
We say that
for
is directly preferred to
under
is indirectly preferred to
and
such that
rule out trivial cases, we have to require the utility function to be at least locally nonsatiated.
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Proposition 2.13.
1. If
are rationalized by
then
2. Both direct and indirect preferences are orders.
3. The direct preference is complete.
4. The direct preference implies the indirect preference:
If the data are generated by utility maximization, then we can easily show that
plies
Figure 2.8 shows that
implies
im-
Also, (2.10) and (2.11)
can be easily seen from Figure 2.8, especially the failure of the converse.
py ⋅ x ≥ py ⋅ y
px ⋅ x > px ⋅ y
.
.
.
y
x
y
px
.
x
py
Figure 2.8. Revealed Preferences
The necessary and sufficient condition for the data to be rationalized by a utility function
is the following.
Generalized Axiom of Revealed Preferences (GARP).
By (2.10),
implies
Then, by (2.11),
implies
Thus,
GARP is necessary. Is it sufficient? The following is the fundamental theorem about revealed
preferences. The proof can be found in Varian (1992, p.133–134).
Theorem 2.2 (Afriat). Let
be a finite set of observations. Then, the following
conditions are equivalent.
(1) The data satisfy GARP.
(2) There exists a locally nonsatiate utility function that rationalizes the data.
(3) There exists a locally nonsatiate, continuous, concave, monotonic utility function that
rationalizes the data.
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There are two more alternative conditions of revealed preferences:
Weak Axiom of Revealed Preferences (WARP).
(Figure
2.9).
Str0ng Axiom of Revealed Preferences (SARP).
x2
x2
py
px
.
x
.
y
. .
x
y
py
px
x1
x1
Figure 2.9. Preferences under WARP
Proposition 2.14. SARP implies WARP.
Proof.
ence, either
implies
or
By SARP, we have
By completeness of the direct prefer-
must be true. Since
cannot be true, we must have
By (2.10) and (2.11), we find that both WARP and SARP are necessary. Will they be sufficient? SARP and WARP require that there be a unique demanded bundle at each budget line,
while GARP allows for multiple demanded bundles. Both WARP and SARP are necessary
conditions for utility-maximizing behavior in the case of single-valued demand, but only SARP
is sufficient in this case. However, when there are only two goods, WARP is also sufficient in
the case of single-valued demand.
The following figure helps us understand Afriat’s theorem. For simplicity, assume that
there are two data points
and
only. With two data points, GARP is the same as
WARP. In the left figure, WARP is satisfied:
and
We can see that it will not
be difficult to find a utility function to justify
and
as coming from utility maxi-
mization in this case. That is, the two indifferent curves in the figure can be from the same
utility function. In the right figure, WARP fails:
and
We can see also that
the two indifferent curves in the figure cannot be possibly from the same utility function.
Indifferent curves of the same utility function can never intersect with each other.
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p1
p1
.
x2
. .
x1
x2
.
x1
p2
p2
Figure 2.10. Afriat’s Theorem
One interesting observation about Afriat’s theorem is that item (2) in the theorem implies
item (3). This suggests that market data cannot tell us whether the preferences are convex or
monotonic. We can produce a set of data
using a non-conventional utility function. By
the equivalence of items (2) and (3) in the theorem, we are guaranteed to find a nicely behaved
utility function that rationalizes the data. Therefore, from the data themselves, we do not
know whether the utility function that actually produces them is concave or not.
Notice that the difference between this section and the last section is that the demand
points are finite. In the last section, we recover the utility function from the compensated
demand, while in this section we recover the utility function from finite demand points. Given
a demand function, we can recover a unique utility function that rationalizes it uniquely up to
a strictly monotonic transformation. For finite demand points, there is enough freedom for
many more utility functions to rationalize the points.
7. Intertemporal Analysis
This section discusses some very basic concepts of intertemporal consumption together
with the idea of separation of production and consumption. Consider an economy with two
periods and one consumption good. The present consumption is denoted as
consumption is denoted as
are
and the future
For the main part, we assume that the spot prices of the good
for the two periods, and we will later drop this assumption.
We first assume
1. the individual has endowment
2. there is a capital market for intertemporal borrowing and lending at interest rate
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Then, the second-period consumption cannot exceed
plus the savings from the first period
plus the interest on saving
Therefore, the budget
constraint is
where
is the overall wealth. We see that the interest rate
acts as a discount
rate that converts future consumption into present consumption. By this explanation, is the
is the total consumption in
total wealth in terms of present consumption good,
terms of present consumption, and the budget constraint simply says that the total consumption should be less than or equal to the total wealth.
The consumer’s problem is
The solution to this problem can be illustrated by Figure 2.11.
c2
.
.
c2
1+ r
c1
c1
Figure 2.11. Consumer’s Problem, given Endowment and a Capital Market
Let us then consider a consumer like Robinson Crusoe, who is completely isolated from
any market. But, Crusoe can engage in production given an endowment
Suppose that
Crusoe has a production possibility set defined by
Here, a pro-
duction plan
means that if the firm invests an amount
good in the first period, it will produce an amount
period, that is,
with
and
produce given input
are net outputs and
Condition
of the consumption
of the consumption good in the second
defines the limit that the firm can
allows the agent to produce nothing. This
production setup is the same as that in our producer theory in Section 5 of Chapter 1 with
strict monotonicity and quasi-convexity on
In this case, we assume
1. the individual has endowment
2. there is a production possibility set defined by
with
for inter-
temporal saving and investment.
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The consumer’s problem is
In this case, savings can be made through the production process, but not through the market.
The solution of this problem can be illustrated by
c2
.
R*
c2
.
c1
G ( c1 - c1 , c2 - c2 ) = 0
c1
Figure 2.12. Robinson Crusoe’s Problem, given Endowment and a Production Frontier
The solution
∗
is called an “autarky” solution, meaning a solution without markets.
Finally, let us assume that the individual not only has a productive means but also has access to a capital market for borrowing and lending. That is, we assume
1. the individual has endowment
2. there is a production possibility set defined by
with
for inter-
temporal saving and investment;
3. there is a capital market for intertemporal borrowing and lending at interest rate
The consumer’s problem is
(2.13)
To solve this problem and show it graphically, we need the following theorem.
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Theorem 2.3 (The Fisher Separation Theorem). The individual maximization problem can
be divided into two problems: (1) profit maximization given the production frontier; (2)
utility maximization given the endowment and the profit from the profit maximization problem. That is, the individual’s problem (2.13) can be divided into two problems:
∗
,
(2.14)
and
,
(2.15)
∗
Proof. We can easily verify that problem (2.13) and dual problems (2.14) and (2.15) give the
same set of first-order conditions and the same set of budget constraints. Since these conditions and constraints determine the solutions, the two sets of problems thus give the same
solution.
∗
With Theorem 2.3, the solution
By the resource constraints
∗
of problem (2.13) can be illustrated by Figure 2.13.
and
the firm’s profit can be expressed in
terms of consumption variables:
(2.16)
Thus, the firm’s problem (2.14) can be expressed as
∗
,
which can be illustrated in the consumption space for
point is
as in Figure 2.13. The solution
∗
c2
Q*
.
R
c2
. .c
*
*
.
c1
p=0
p = p*
c1
Figure 2.13. Optimal Consumption and Investment Plan
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Given profit
∗
the consumer’s problem is
,
∗
The solution point is
∗
in the diagram. We can see that the consumer’s budget line is the same
∗
as the isoprofit line defined in (2.16) with
Let us use Figure 2.13 to explain the theorem. The Fisher Separation Theorem says that
∗
if
∗
is the solution of the consumer’s problem, then this
can also be obtained by the
following two separate problems: (1) the individual can first consider the optimal production
problem given the production frontier and pick up point
∗
which is totally independent of
the individual’s preferences; (2) he can then consider the utility maximization problem given
the budget constraint taking the optimal production
and he will pick up point
∗
∗
as the endowment point for the budget,
as his optimal consumption plan.
From Figure 2.13, we see that an improvement of the individual’s welfare (maximum
utility) from problem (2.12) to problem (2.13), and the individual consumption improves from
∗
to
∗
This improvement is because of the existence of a capital market. In Figure 2.14, we
can see how the individual manages production and financing in order to improve welfare.
c2
*
pay for debt Q
.
.c
*
output
.
c2
debt
input
c1
c1
Figure 2.14. Investment and Consumption
Notice that the Separation Theorem requires a perfect capital market (economic agents can
borrow and lend at the same interest rate). When the borrowing and lending rates are different, this Separation Theorem may fail. Let the lending rate be
Suppose that
and the borrowing rate be
By Figure 2.15, the profit maximization problem (2.14) will pick point
∗
and the utility maximization problem (2.15) using a budget constraint determined by the rates
and endowment
solution, since
∗
will pick point
As shown in Figure 2.15, this
is even worse than the autarky solution
tion must be better than
∗
∗
cannot be the optimal
(we know that the optimal solu-
). This failure is because of the difference between the profit line
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faced by a producer and the budget line faced by a consumer. This difference is due to the fact
that when the individual acts as a producer, the interest rate changes at
vidual acts as a consumer, the interest rate changes at
but when the indi-
∗
c2
Q*
.
c¢
. .
R*
.
c2
Profit line
budget line
c1
c1
Figure 2.15. Failure of Separation with Different Rates
We now drop the assumption of
Without it, the budget constraint is:
implying
Let
and call
and call
the inflation rate. Define
the real interest rate and, accordingly, call
by
the nominal interest rate. The nominal
interest rate tells us how many extra dollars we can get from savings, while the real interest
rate tells us how many extra units of goods we can get. We have
Then, the budget
constraint becomes
Hence, all the conclusions in this section are still valid when the prices are not constant; we
only need to replace
by
The essential message from the Fisher Separation Theorem is that, under perfect markets
(without distortions, transaction costs and monopolistic powers), we can assume that the
firm’s objective is always to maximize profits or expected profits. The owner(s) of the firm can
be treated as two persons: one maximizes profit subject to technology, market and resource
constraints, and the other is a typical consumer who maximizes utility subject to a budget
constraint. This is actually the approach taken in the standard general equilibrium theory (in
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Chapter 4) in which producers and consumers have distinct objectives. However, as shown in
Figure 2.15, this approach fails if markets are not perfect.
Notes
Materials in this chapter are fairly standard. Many books cover them. Good references are
Varian (1992), Jehle (2001), Dixit (1990), and Mas-Colell, Whinston and Green (1995). In
particular, a good reference for Section 4 is Mas-Colell et al. (1995, p.107), and a good reference for Section 7 is Hirshleifer (1984, Chapter 14).
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Chapter 3
Risk Theory
1. Introduction
So far, we have assumed a world under certainty. In the real world, however, choices are
often made under uncertainty. People’s preferences and choices are affected by uncertainty.
Risk is another dimension to an economic problem, in addition to the price ratio, elasticity
and substitutability. Hence, the attitude towards risk is an important issue. This chapter discusses behaviors under uncertainty.
Consider a game in which a coin is flipped repeatedly until the heads is up:
(3.1)
You cannot lose in this game, but you could become very rich from this game. How much will
you be willing to pay for participating in this game? The expected money value is
That is, you are expected to become very rich by playing this game. However, no one is willing
to pay a large amount of money to play this game. That means that people do not necessarily
care only about the expected money value. Common sense tells us that there is a large chance
that you will not be able get much out of this game and you do not like this risk. Hence, even
though you could become very rich at this game, your willingness to pay for this game is low.
In other words, you are risk averse. One explanation is that people assign their own personal
value to the prizes, which is not necessarily the monetary value of the prize. The personal
value of the prize is called the utility value.
For a person with utility function
ten as
the personal value for a game or lottery
If the person cares about the expected utility (we call this person an expected
utility maximizer), then he is willing to pay amount
where
is writ-
such that
is called the certainty equivalent, which is the equivalent amount in certainty for the
uncertain amount.
Suppose that a person has utility value
for monetary value
Then, the ex-
pected utility of the prize for the game in (3.1) is
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In this case,
That is, the person is willing to pay
to play this game.
Consider another game that flips a coin once:
(3.2)
This game can be written as
get
and with probability
lottery is written as
which means that with probability
you get
or
We call
Let
you
a lottery. The personal value over the
for monetary value
again. How
much is this person willing to pay for this game? The expected utility of the lottery is
By
we find
meaning that the person is willing to pay
to play the
game in (3.2).
2. Expected Utility
We shall imagine that the choices facing the consumer take the form of lotteries. A lottery
is denoted by a vector of the form:
for some
where
with probability
and
It is interpreted as: “the consumer receives prize
for any ”. The prize
may be money, bundles of goods, or even playing in
additional lotteries. Many situations involving behavior under risk can be considered in this
lottery framework.
In order to define the lottery space, we make three assumptions:
Axiom 10.
Axiom 11.
Axiom 12. Reduction of Compound Lottery (RCLA).
Axiom 1 says that getting a prize with probability one is the same as getting the prize for
certain. Axiom 2 says that the order of “ with probability ” and “ with probability
expressed in lottery
”
will not matter to the consumer. Axiom 3 is called the
reduction of compound lotteries axiom (RCLA), which says that the consumer’s perception of
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a lottery depends only on the net probabilities of receiving the prizes and the sequence of
events does not matter; see Figure 3.1.
1 − q
q
y
p
q (1 − p ) + 1 − q
pq
1 − p
x
x
y
y
Figure 3.1. Reduction of Compound Lotteries
The RCLA is equivalent to this gen-
Under Axioms 1-3, we can define the lottery space
eral form: for
we have
With RCLA, a lottery can be generally expressed as
For example,
The consumer is assumed to have a preference relation
better than
on
and
means that
is
As before, we will assume that the preference relation is a complete order.
Axiom 13. The preference relation
Axiom 14. (Continuity).
is a complete order.
and
are
closed sets in
Under minor additional assumptions, Theorem 2.1 on the existence of a utility function
representing a preference order on a consumption space can be applied to show the existence
of a utility function
find a function
representing this preference order
on the lottery space
We need to
such that
and
However, such a utility function is too general to be interesting. It cannot tell us much about
the special behaviors of the consumer under uncertainty. We want some kind of separation
between the prize and its probability in the utility function. One interesting utility representation with such a property is the so-called expected utility, namely the one that satisfies the
expected utility property:
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for all the lotteries in
It says that the utility of a lottery is the expected utility of its prizes. It
should be emphasized that, by Theorem 2.1, the existence of a utility function is not an issue;
any well-behaved preferences can be represented by a utility function. What is of interest here
is the existence of a utility that satisfies the expected utility property. The key feature in an
expected utility is the separation of probabilities from the valuation of prizes. For that, we
need an additional axiom; see Figure 3.2:
Axiom 15 (Independence).
p
x~y
x
~
1-p
z
p
1-p
y
z
Figure 3.2. Independence
We can expand a lottery by adding an additional prize and allocating a probability to it.
The independence axiom says that two expanded lotteries must be indifferent if their original
lotteries are indifferent, whatever the probability assigned to the additional prize is. This is a
form of separation between prizes and their probabilities. However, this axiom may not be
actually true for a typical person. For example, if
than getting
or
with uncertainty even though
then getting
and
for certain may be better
are indifferent in preferences under
certainty.
In order to avoid some technical details in the proof of existence, we will make two further
assumptions.
Axiom 16.
is bounded:
s.t.
Axiom 17.
Axiom 7 is purely for convenience. Axiom 8 can be derived from the other axioms. It says
that if one lottery about prizes
and
is better than another, this must be because there is a
higher probability of getting the best prize.
Under the above axioms, we present the following theorem about the existence of a utility
representation that satisfies the expected utility property.
Theorem 3.1 (Existence of Expected Utility). If
utility representation
satisfies Axioms 1-6, then there is a
that has the expected utility property.
Proof. For convenience, we will also include Axioms 7 and 8. For any
define
to
be the unique number that satisfies
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That is, the utility value of a lottery is the probability of getting the best prize
tion
For this func-
to be well defined, we have to ensure that such a number exists and is unique.
(1) Does such a number exist? By Axioms 5 and 7, sets
and
defined below are closed
and nonempty:
By Axiom 4, every
is in one of the above two sets (i.e.
is connected. If
and
then
and
which means that (by Axiom 5)
are nonempty open sets in
This contradicts with the fact that
by the definition of
(2) Is the number
. By Lemma 2.1,
and
is connected. Therefore,
and for any
and
unique? For any
suppose that
Then, by Axi-
om 8,
This contradicts the fact that both lotteries are indifferent to
Thus,
Similarly,
Therefore,
(3)
has the expected utility property. In the following derivation, the number above
indicates the corresponding axiom used. We have
Hence,
(4) Finally, we verify that
represents the preferences. We have
and
That is,
is a utility function over the lottery space.
Example 3.1. Given a lottery
where
are monetary values, the following
two utility functions have the expected utility property:
where
is some function. A risk-averse person will have a concave function
If the person
cares only about his expected monetary value, then his preferences can be represented by the
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second utility function. Under the assumption of perfect capital markets, as shown by the
separation theorem in Chapter 2, we can treat producers as this kind of individual.
We know that any strictly monotonic transform of utility function
is also a utility function on
on the lottery space
But, in general, such a transformation will not pre-
serve the expected utility property. Only when the transformation is linear will the expected
utility property be preserved. Specifically, given the mapping
is a monotonic linear transformation of
If
the mapping
if there are two constants
is an expected utility function on
such that
we can easily verify that
is also an expected
utility function on
We will show the converse in the following proposition: any strictly monotonic transform
of an expected utility that preserves the expected utility property must be a monotonic linear
transform.
Proposition 3.1 (Uniqueness of Expected Utility). The Expected utility function is unique up
to a monotonic linear transformation. That is, any two expected utility representations
and
of the same preferences must have a positive linear relationship
where
and
Proof. We have explained that a monotonic linear transformation of an expected utility function must be an expected utility function. Let us now show the converse.
Given two utility functions
monotonic function
both
and
and
on
by Proposition 2.2, there must be a strictly
such that
where
is the range
have the expected utility property, then, for arbitrary
of
If
we
have
or
Let
Then,
( )
( )
( )
where
and
Substi-
tuting this into the above equation, we have
This means that
is a monotonic linear transformation on interval
is a monotonic linear transformation on the range
Therefore,
of
An expected utility theory has now been well established. Under fairly normal conditions,
we are guaranteed to have an expected utility representation for preferences under uncertainty
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and this representation is unique up to a linear transformation. This utility representation is
also fairly convenient to use, due to the separation of the utility value from its probability in
each event. However, the expected utility has been found to be inconsistent with actual
individual behaviors in many cases. The following are two such examples. Consequently, an
extended utility theory, called the non-expected utility theory, is developed by scrutinizing the
axioms more carefully, especially the independence axiom. Such a utility representation is
more flexible than the expected utility property so that various paradoxes in individual choices
under uncertainty can be explained
Example 3.2 (Allais Paradox). Consider the following four lotteries with possible prizes US$5
million, US$1 million, or nothing. Each row is a lottery:
0.9-
0.1
where
can be any
5m
0
0
1m
1m
0
5m
0
1m
1m
1m
1m
If we look at only the first two columns, we have
When the third column is included and
many people have the following preferences:
(3.3)
The fact that the third column causes a reverse of preferences violates the independence axiom.
The explanation for the preference
is that one million US dollars is enough for life; if
you take a chance to go for lottery
and lose, you would kill yourself. However, the prefer-
ences in (3.3) are not consistent with the expected utility. To see this, suppose that the consumer has an expected utility function
with
Then, for
the latter choices
imply
These two inequalities contradict each other. Hence, the consumer’s preferences cannot possibly be representable by the expected utility function.
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Example 3.3 (Common Ratio Effect). Consider the following four lotteries:
Most people have the following preferences:
These preferences again violate the expected utility property. With an expected utility function
with
the preferences imply
These two inequalities contradict each other.
These two examples show that expected utility is an incorrect representation of individual
preferences under uncertainty.
3. Mean-Variance Utility
This section is based on Sargent (1987b, 154-155).
Proposition 3.2. If an agent has an exponential utility function of the form:
and
follows the normal distribution
then maximizing
is the same as max-
imizing the mean-variance:
Proof. We have
(
)
(
)
Another approach is to assume a quadratic utility function:
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The agent is risk averse
if
or risk loving
if
To guarantee
we restrict the domain of
(3.4)
and thus
See Figure 3.6. This implies
Risk Loving b>0
Risk Averse b < 0
-
a
2b
a
2b
Figure 3.3. The Utility Function
and
Let
. The expected utility is
This expected utility is expressed in terms of the expected value
and the standard variance
Other higher-order moments are not involved. This seems realistic since, in the real world,
people often focus on the expected value and the variance of income.
4. Measurement of Risk Aversion
In Section 1, we show in two examples that people are risk averse. Some may be more risk
averse than others. The degree of risk aversion may affect choices. In this section, we develop
some measures of risk aversion.
We say that an individual is
if his utility value of the expected income is greater
than his expected utility of the income; risk neutral if his utility value of the expected income is
equal to his expected utility of the income; and risk loving if his utility value of the expected
income is less than his expected utility of the income. That is, for any lottery
18
18
In reality, a person can be risk averse under one circumstance and risk loving in another. For example,
when a person buys insurance, he is risk averse; and when he buys a lottery, he is risk loving. The key in these two
cases is that, when the person buys insurance, he pays to move from a risky situation to a non-risky situation; and
when he buys a lottery, he pays to move from a non-risky situation to a risky situation. Our definition of risk
attitudes in (3.5) does not include such a person.
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(3.5)
Then, the first situation in (3.5) is illustrated in Figure 3.4. We
Let
can see that
Hence,
u [ E ( x ) ]
E [ u ( x ) ]
E ( x)
xl
xh
Figure 3.4. A Risk-Averse Individual
In Figure 3.4, we can further see that the more concave
is, the bigger the difference
is. That is, the more concave the expected utility function is, the more risk
averse the individual is. Thus,
must be closely related to risk aversion and we might think
that we could measure risk aversion by
However, this measure is not invariant to a linear transformation of the expected utility
function, although the utility function represents the same preferences after a linear transformation. We should therefore normalize
by dividing it by
this ratio is invariant to a linear
transformation. Therefore, we now have a sensible measure of risk aversion:
where the negative sign makes the measure a positive number.
There are two types of fluctuations: fluctuations in the level of income
tions in the percentage of income
where
and fluctua-
is the mean. The absolute risk aversion
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deals with the first case. The second case is dealt with by the so-called relative risk aversion,
which is defined as
To justify the above two definitions more rigorously, for any lottery
risk premium
let
Define
as
(3.6)
This risk premium is the amount that the individual is willing to pay for avoiding the risk; see
Figure 3.5. Here,
and
are indifferent in the expected utility. The key is that one has
risk but the other does not.
Eu
πa
x1
x2
E(~
x)
Figure 3.5. The Risk Premium
Define
Let us now find an approximated solution for
by
We have
By Taylor’s expansion,
Equalizing the above two formulae implies an approximated solution of
where
This approximation becomes exact when
This formula shows that
measures how much the individual is willing to pay for avoiding the risk. Note that
has nothing to do with individual preferences, while
Similarly, define the relative risk premium
and define
by
depends on preferences.
by
By the same method, we find
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We have therefore justified the definitions of the two risk aversion measures. Absolute
risk aversion measures the aversion for the risk contained in the absolute magnitude of a
lottery
Relative risk aversion measures the aversion for the risk contained in the relative
magnitude of a lottery
5. Demand for Insurance
When facing a potential loss, what is an individual’s demand for insurance? Specifically,
consider a consumer who faces a lottery
That is, the person has wealth
but he may lose an amount with probability
This consumer can buy insurance for a cost
from a competitive insurance company, which pays him
Here,
amount
dollars if the bad event happens.
is called the price of insurance. Given this price, the consumer decides an insurance
The consequences are shown below:
Probability
Before insurance
After insurance
What is the optimal amount
∗
for the individual? The consumer’s problem is:
The first-order condition (FOC) is
∗
∗
This equation determines the demand for insurance
∗
∗
In a competitive market, the
The expected profit for the company is:
expected profit will be driven to zero, implying
Substituting this into (3.7), we get
∗
If the consumer is strictly risk averse so that
ing
∗
(3.7)
∗
then
∗
∗
imply-
That is, the consumer will completely insure himself against the loss
This problem can be shown graphically. Let
(3.8)
which are the income levels in the two events. By eliminating
have the budget constraint:
from the two expressions, we
Then, the consumer’s problem becomes
,
This problem is illustrated in Figure 3.6.
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.
.
Figure 3.6. The Insurance Problem
The slope of the budget line is
(
)
( )
( )
At the optimal point
and the slope of the indifference curve at
∗
∗
is
the two slopes must be equal:
∗
(3.9)
∗
∗
or
∗
∗
which is on the
∘
-line. A point on the
∗
Hence, (3.9) implies
In a competitive insurance market, zero profit implies
∘
-line offers the same income in any
event. This situation is called full insurance.
In summary, when the insurance market is competitive
the consumer will de-
mand full insurance. However, when the market has more than one type of consumer and has
imperfect information, full insurance may not be an equilibrium solution for some consumers.
We will deal with this issue later.
6. Demand for Risky Assets
Since the absolute risk aversion
measures an individual’s attitude towards fluctuations
in the level of income, we would expect a risk-averse individual to become less risk averse to
this kind of uncertainty when his income rises. That is, we expect
creases. Since the relative risk aversion
to decrease as
in-
measures an individual’s attitude towards fluctua-
tions in a percentage of income, we would expect a risk-averse individual to become more risk
averse to this kind uncertainty when his income rises. That is, we expect
increases. Hence, we assume a decreasing
and an increasing
to increase as
in this section, and
we want to find out what they imply to an individual’s portfolio choices.
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Suppose that an economic agent has initial wealth
ey and the rest
He can invest an amount
in mon-
in a risky asset. The net return from the money is zero, and the net
return from the risky asset is
which is a random variable and can be negative. He consumes
nothing. Then, his final wealth is
As an expected utility maximizer who
cares about the expected utility value of future income, his problem is:
,
This is equivalent to:
∈[ ,
]
define
For
For an interior solution
∗
the first-order condition is:
∗
Since
∗
is a continuous function, it cannot change its sign without tak-
ing zero value. This means that either
We know that if
∗
Assuming that
for all
if
if
for all
is concave, we have
cave, which means that any
there exists a solution
∗
then
∗
That is,
satisfying
∗
such that
then
is con-
is an optimal solution. Therefore,
∗
if
∗
if
∗
and
∗
∗
Since
cavity of
if
is not equal to zero with probability
will guarantee the strict concavity of
We denote this optimal solution as
Taking a derivative w.r.t.
∗
the strict con-
and therefore the uniqueness of
∗
∗
on the FOC (3.10), we have
∗
∗
Thus,
∗
∗
∗
(3.11)
The following proposition shows that, for a risk-averse individual, a decreasing absolute risk
aversion
will imply an increasing
∗
meaning that the individual will invest more in
the risky asset if his initial income increases.
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∗
Proposition 3.3.
Proof. Decreasing
implies
∗
or
∗
∗
implying
∗
∗
implying
∗
∗
implying
∗
∗
∗
Using (3.10), we have
Using (3.11) and
∗
we therefore have
∗
∗
The income elasticity of demand for money is denoted by:
Money is a luxury
good if
Proposition 3.4.
∗
Proof. We have
∗
∗
∗
Increasing
∗
and
∗
Using (3.11),
∗
∗
(3.12)
implies that
∗
or
implying
implying
implying
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where (3.10) implies
From
and (3.12), we therefore have
7. Portfolio Analysis
We introduce the well-known Mean-Variance Analysis in this section. Assume that an
and
economic agent invests in two risky assets
rest in
His income is
We call
with the proportion
in
the portfolio [or call
and the
the portfo-
lio]. The agent’s problem is
∈[ ,
]
Assume that the utility function is quadratic:
The agent is risk averse
if
or risk loving
if
Let
and
. Then, the expected utility is
This convenient feature allows us to analyze behaviors under risk. We can transform the consumer’s problem of choosing
into a problem of choosing
Let
We now need to express the budget condition in terms of
. Taking an expectation and
yields
variance on the budget condition
(3.13)
They can be combined into one condition by eliminating
(3.14)
We call this the portfolio curve. We can thus treat the two assets
bundles
and
and
as two consumption
available in the market and the individual is to choose
The consumer’s problem becomes:
,
The consumer now chooses
rather than
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Because of the simplicity of the utility function, we can use a graph to analyze the solution.
For this, we need to draw the indifference curves and the portfolio curve on a diagram. First,
the indifference curve with utility level
is
(3.15)
Suppose that this indifference curve determines a function
w.r.t.
By taking the derivative
we have
(3.16)
In the space of
as in Figure 3.7, this means that the indifference curve is increasing in
the risk-averse case and decreasing in the risk-loving case. We also have
Therefore,
This means that the indifference curve is convex in the risk-averse case and concave in the
risk-loving case. We also know, in either case, that
This together with the concavity and the monotonicity of the indifference curves give us the
shapes of the indifference curves in Figure 3.7.
μ
μ
better
better
Risk loving b>0
Risk averse b<0
σ
σ
Figure 3.7. The Indifference Curves
We also need to find which direction the indifference curve shifts when the utility level
is
increased. For this, we have
(3.17)
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where
is because of the assumption in (3.4). Expressions in (3.17) tell us the
directions of increasing utility for the indifference curves as shown in Figure 3.7.
Second, let us now draw the portfolio curve. From elementary geometry, we know that
equation (3.14) in the space for
is a hyperbola if
and
are allowed to take negative
values. There are two branches in this hyperbola. We will show that our portfolio curve is one
of the branches. We know from (3.13) that, for any
left of that for
connecting
the portfolio curve for
Furthermore, the right-most portfolio curve is for
is on the
and is a straight line
and
Therefore, the portfolio curve for any given
axis (i.e.,
must be the branch that is bent towards the
on the curve). Besides that, for the left-most portfolio curve with
by
(3.13), we have
This portfolio curve is straight except at
it takes
if
These imply the
shape of the portfolio curves in Figure 3.8.
m
X
ρ = −1
ρ =0
ρ =1
Y
s
Figure 3.8. The Portfolio Curves
With this graphic illustration, we can now easily combine Figures 3.7 and 3.8 to analyze
portfolio behaviors of an investor under various conditions. For example, from Figure 3,9, we
can see that a risk-averse person tends to invest in both assets while a risk-loving person will
always invest all his money in one asset.
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m
m
.x
.x
.
.
Risk averse
y
.
Risk loving
y
s
s
Figure 3.9. Portfolio Decisions
Proposition 3.5. A risk-averse person tends to invest in both assets, while a risk-loving
person will always invest in one asset only.
8. Stochastic Dominance
We denote
and
respectively, as the accumulated distribution function and the densi-
ty function of random variable
such that
Assume that, for any distribution function
and
there exist
Let all the distributions in this section have finite
supports and the supports are bounded by
If
for any
we say that
stochastic dominance (FOSD), denoted as
more likely than
If
dominates
or
or
dominates
by first-order
It means that, for any
is
to have a value larger than
for any
we say that
dominates
by second-order stochastic dominance (SOSD), denoted as
or
or
dominates
SOSD is often
used in a comparison of risks in two assets.
FOSD is based on the idea of a higher probability of getting a higher value. SOSD is based
on the idea of relative riskiness or dispersion.
We first have two results that relate the concepts of stochastic dominance to preferences.
It turns out that the two concepts closely relate to the monotonicity and concavity of a typical
utility function
Theorem 3.2 (Hanoch-Levy 1969).
increasing
dominates
by FOSD iff
for any
Proof. We have19
19 In
the proofs of this section, we assume sufficient differentiability for
this is for convenience. Also, we use
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Then, the increasingness of
tells us whether or not we have
One immediate implication is that if
dominates
by FOSD, then
However,
the converse is obviously not true.
Theorem 3.3. Suppose that
and
have the same mean. Then,
dominates
by SOSD iff
for any concave
Proof. We have
We also have
Hence,
Therefore, the concavity of
determines whether or not we have
The rest of this section takes a different approach to understanding the concepts of stochastic dominance.
and
in places where
and
should be used; this leaves room for distribution functions with unbounded
support.
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Example 3.4. Consider a two-stage lottery that offers
second stage. The final lottery is
follows a distribution
stages; more precisely,
values of
and
where, for each realization
Here, the symbol
of
in the
in the first period,
means a sum of the two lotteries in two
is a two-stage lottery for which the value of
but the distribution of the second-stage lottery
tion of the first-stage lottery
tion
in the first stage and offers
Assume
for any
is a sum of realized
is conditional on a realiza-
Thus, for any increasing func-
we have
Hence,
dominates
by FOSD. That is, given a random variable
random variable such that
where
Conversely, given a random variable
is obtained from
we can construct another
by adding a positive shock
we can similarly construct another random variable
such that
Notice that risk does not play a role in this case, which reflects the fact that FOSD is concerned with a potential return only. However, adding a shock will add more risks. In particular,
when this shock does not contribute to an increase in expected return, risk will then become a
crucial issue. This is what the next example will discuss.
Example 3.5. Consider a two-stage lottery that offers
value
in the second stage, where
each realization
of
mean as
Hence,
but
is another random variable with
in the first period, and
We say in this case that
in the first stage and offers the final
follows a distribution
is a mean-preserving spread of
is more random (a larger variance) than
dominates
We have
which means that
has the same
is. Then, for any concave function
by SOSD. That is, given a random variable
random variable with the same mean such that
20
for
we can construct another
Conversely, given a random variable
we can similarly construct another random variable with the same mean such that
basic idea here is that
20
dominates
is dependent on each realization
by SOSD if
i.e., for each
is
The
with some noise.
there is an
And,
for any
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Example 3.5 basically proves the following theorem.
Theorem 3.4. For any two random variables
by
by SOSD iff
and
with the same mean,
is a mean-preserving spread of
Example 3.6. For two distribution functions
and
increase in risk from
by taking all the mass of
if
is generated from
and assigning it to the endpoints
and
we say that
̅
̅
constitutes an elementary
in an interval
in such a manner that the mean is preserved. This is
illustrated in the following figure. With finite support
3.10, we have
is dominated
for
and
as indicated in Figure
The two triangle areas indicated in Figure 3.10 are equal
in size. Since
̅
̅
the two random variables defined by
and
have equal means.
1
G ( x ), red broken curve
F ( x ), black curve
equal areas
a
b
x
Figure 3.10. An Elementary Increase in Risk
We can easily verify that
any
dominates
by SOSD by comparing
with
for
Thus, an elementary increase in risk is a mean-preserving spread.
Notes
Good references for Section 5 are Campbell (1987) and Laffont (1995), who offer detailed
discussions on insurance problems under various conditions. A good reference for Section 6 is
Huang–Litzenberger (1988, 20–25). A good reference for Section 7 is Huang–Litzenberger
(1988, Chapter 3). A good reference for Section 8 is Mas-Colell et al.
(1995, 194–199).
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Chapter 4
Equilibrium Theory
This chapter deals with three subjects:
(1) General Equilibrium
(2) Pareto Optimality
(3) Welfare Properties
1. The Equilibrium Concept
Previous chapters assume exogenous prices, that is, consumers and producers take prices
as given and choose their best actions based on their own budget/resource constraints. In this
chapter, all prices are endogenous and they adjust to clear all the markets. An equilibrium is
reached when all markets are clear and when no one wants to change anymore.
This equilibrium concept is applicable to an individual market, an industry, a sector, or
the whole economy. Given an economic model, we say that we have a general equilibrium if
there is an equilibrium in each of the individual markets and there is an overall equilibrium in
the whole economy.21 A general equilibrium takes into account the functioning of individual
markets as well as the interactions among all the markets. An overall equilibrium for the
whole economy is necessary since, although some prices can be determined by individual
markets, there are always some prices that must be determined by the whole economy. It is
the interactions of markets that affect the determination of these prices. An equilibrium in an
individual market is called a partial equilibrium. The crucial difference between a partial
equilibrium and a general equilibrium is that in a partial equilibrium some prices are given
while in a general equilibrium all prices are endogenously determined within the model.
The concept of goods considered here is very broad. Services, such as banking services,
are taken to be just another kind of good. Goods can be distinguished by time, location, and
the state of the world. This means that the value of a good is not only determined by its physi-
21 A
necessary condition for the whole economy to be in equilibrium is that each individual market is in
equilibrium. Hence, we can simply define a general equilibrium to be an equilibrium for the whole economy.
However, there is a purpose to mention individual markets here.
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cal characteristics, but also by the location where it is delivered and consumed, by the time
when it is produced and consumed, and by the environment in which it is made available. For
example, a good consumed today and the same good consumed in the future are considered as
two different goods. By this approach, a dynamic problem can be dealt with in a one-period
model.
The market is complete if there is a market for each good, more precisely, if any affordable
consumption bundle is available in the market. The market is perfect if there is no friction and
no distortion such as transaction costs, taxes, etc. Perfect competition is also assumed, by
which economic agents all take prices as given. An economy is the so-called Arrow-Debreu
world if the market is complete, perfect, and competitive.22
2. GE in a Pure Exchange Economy
A pure exchange economy consists of
(1)
commodities
(2)
consumers
(3) each consumer
has an endowment
utility function
of goods, a consumption space
representing his preferences
and a
23
The key feature of a pure exchange economy is that there is no production.
Definition 4.1. Any
for agent
Definition 4.2. An allocation
is feasible if
Denote
is an allocation.
as consumption of person for good
good
xi
j
person
If there are two goods
and two agents
then the economy can be conveniently
illustrated by the Edgeworth box; see Figure 4.1:
22 See
23 In
Campbell (1988) for a more precise definition.
this framework, everyone is concerned with only his own consumption, called private consumption. But,
in the real world, other people’s utility, consumption or income may affect your utility.
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x2
w1B
B
.
wB2
W
w A2
A
w1A
x1
Figure 4.1. The Edgeworth Box
All the points in the Edgeworth box satisfy
The Edgeworth box has two distinct features:
1. Each point represents a feasible allocation.
2. Given a price vector, the budget line for one person happens to be the budget line for
the other person. The budget line for person A is
By the feasibility conditions, this becomes
implying
which is the budget line for person B.
Because of these useful features, the Edgeworth box is often used to illustrate equilibria in
a two-person two-good pure exchange economy.
Suppose that there is a market for economic agents to buy and sell goods. The vector of
market prices for the
goods is
Each consumer takes prices as given and
chooses the most preferred bundle from his consumption set; that is, each consumer acts as
if he is solving the following problem:
(4.1)
The optimal solution to this problem is denoted by
which is the Marshallian de-
mand function. Under the assumption of strict convexity of preferences and by Proposition 2.3,
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the demand function is a well-defined continuous function. The equilibrium price
∗
is a solu-
tion of the following equation:
∗
Since the ordinary demand
∗
an equilibrium price vector, then
∗
(4.2)
is zero homogeneous in prices, it means that if
∗
for any
∗
is
is also an equilibrium price vector. This
means that (4.2) does not uniquely determine all the prices. In fact, one and only one of the
prices is completely free and the rest of the prices are generally uniquely determined.
It will be shown later that if all goods are desirable, then the above inequality becomes an
equality, that is, the equilibrium price clears all the markets. Equilibrium allocation
vector of the quantities demanded at the equilibrium prices:
∗
pair
∗
is called the general equilibrium. That is,
∗
∗
∗
∗
∗
∗
is the
And the
is a general equilibrium if it
satisfies (4.1) and (4.2) jointly. The following is the formal definition.
∗
Definition 4.3.
(1) For each
∗
∗
is a general equilibrium if
solves consumer i’s utility maximization problem:
∗
(2)
∗
∗
is feasible:
∗
The function
defines the so-called offer curve. We can draw the offer
curves in the Edgeworth box; see Figure 4.2. The general equilibrium occurs at the intersection point of the two offer curves. Each point on a consumer’s offer curve is the best consumption bundle for the consumer given the price vector. Only at the intersection point of the offer
curves do the two consumers choose the same consumption bundle, which means that the
allocation is feasible. It is a general equilibrium since the two conditions in Definition 4.3 are
satisfied.
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x2
B
x1
x*
.
B’s offer curve
uA
uB
slope =
p1*
p2*
.
ω
A’s offer curve
x1
A
x2
Figure 4.2. The General Equilibrium in the Edgeworth Box
We are going to show the existence of the general equilibrium. For that, we first need to
introduce a fixed-point theorem and prove a lemma. Let the aggregate excess demand function be:
Then, the equilibrium price vector
∗
is defined by
∗
Given the demand functions, the
existence of an equilibrium is equivalent to the existence of an equilibrium price vector, and
the equilibrium allocation is the demand at the equilibrium prices. For this reason, we often
simply call the equilibrium price vector
∗
the equilibrium.
Proposition 4.1 (Walras’s Law). If the preferences are strictly monotonic, then, for any
price
we have
i.e., the value of the excess demand is always zero.
Proof. By strict monotonicity of preferences, more is always better. Hence, the budget equation must be binding for the optimal consumption:
We thus have
Walras’s Law actually says something quite obvious: if each individual balances his budget, then the aggregate spending balances as well. An important implication of Walras’s Law is
this: if, at some
markets clear, then the other market must also clear. Technical-
ly, this means that condition
∗
only implies
independent equations; the other
equation is satisfied automatically. Since the demand function is zero homogenous, only price
need to be determined. Hence, the
equations are just enough to
ratios
determine
∗
The following fixed-point theorem can be found in Smart (1980).
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Lemma 4.1 (Brouwer). For any nonempty compact convex set
mapping
there is an
and any continuous
such that
Lemma 4.2. Let
If a mapping
satisfies
∗
then there is a
is continuous and
∗
such that
by
Proof. Define
where
By Brouwer’s fixed-point theorem, there is a
∗
that
∗
∗
∗
∗
Together with the fact
∗
by multiplying
∗
on both sides of the above equation, we have
∗
Since
∗
such that
this implies
∗
Since
∗
∗
∗
for any
∗
(4.3)
∗
(4.3) implies
∗
for any
Thus,
∗
Theorem 4.1 (Existence of Equilibrium). If preferences are strictly convex, strictly monotonic, and continuous, then an equilibrium exists.
is continuous. By Proposition 4.1,
Proof. By Proposition 2.3,
law. Hence, by Lemma 4.2,
satisfies Walras’s
has an equilibrium.
implies
A good is said to be desirable if
Proposition 4.2 (Market Clearing). Suppose that preferences are strictly monotonic. If
∗
good is desirable, then, in the general equilibrium, we have
good must clear:
and the market for that
∗
Proof. By Walras’s Law,
∗
∗
Since
Then, by the desirability, we cannot have
∗
∗
we must have
implying
∗
∗
for all
∗
Proposition 4.2 tells us that if preferences are strictly monotonic and if all goods are desirable, then, in general equilibrium, all markets clear.
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Theorem 4.2 (Uniqueness of Equilibrium). If the preferences are strictly monotonic, all
demand functions are differentiable, and all goods are desirable and substitutes in excess
demand (
for all
∗
tive multiplier (if
and
, then the equilibrium price vector is unique up to a posi-
are equilibrium prices, then there is
That is, equilibrium price ratios
∗
∗
∗
Proof. For any equilibrium price vector
∗
∗
since
∗
such that
are unique.
∗
is zero homogeneous, any price
is also an equilibrium price.
Conversely, given any two equilibrium price vectors
then we find a contradiction. By desirability,
∗
Suppose
∗
and
∗
and
We now lower each price
∗
to
Since
∗
and
∗
if
∗
∗
for some
∗
Let
∗
By the
are not proportional,
such that
and thus
Since at least one price,
is constant, by gross substitutability and
dicts the fact that
∗
∗
and
By zero homogeneity, we have
definition of
∗
∗
and
we have
goes down strictly
This contra-
is an equilibrium price vector and
Example 4.1. There are two goods and two consumers with:
Let us find the equilibrium. The ordinary demand functions are:
where the incomes are
and
We have
implying that the two goods are gross substitutes in excess demand. By Theorem 4.2, the
equilibrium price is unique. As explained in Theorem 4.2, only the ratio of the prices is determined by market clearing conditions. Let
ly,
and
or equivalent-
Then,
The market clearing condition for good
where
represent the price ratio:
and
is:
are the endowments of good
This condition implies that
∗
see Figure 4.3.
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y1
B
x2
ω
.
.
x*
uA
uB
slope =
p1* 8
=
p2* 7
x1
A
y2
Figure 4.3. A Unique Equilibrium
automatically clears. Therefore, a typical equilibrium
By Walras’s law, the market for good
price vector is
∗
∗
The equilibrium price is unique up to a positive
multiplier. By substituting the price ratio into the demand functions, we have the equilibrium
allocation:
∗
∗
∗
∗
Example 4.2. There are two goods and two consumers with:
Let us find the equilibria. Different from the last example, because of the specialty of the utility
functions, we are going to use the Edgeworth box to find the equilibria; see Figure 4.4. Notice
that the two goods in this case are gross complements and thus the uniqueness theorem is not
applicable.
B
y
2
u1
.
.
slope=p
u2
1
A
.W
x
Figure 4.4. General Equilibria
Let
which is the slope of the budget line going through the endowment point
can see that, when
We
demands from the two consumers are not the same point in the
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Edgeworth box. Only when
can demands from the two consumers be the same point.
Therefore, equilibria are those with price ratio
∗
and allocations on the interval
3. Pareto Optimality
Is such an equilibrium good for the society as a whole? Since a society includes many consumers, in what sense is an equilibrium a good solution for the society? Should we maximize a
joint utility function? It turns out that Pareto optimality is a minimum group optimization
criterion in the sense that any sensible criterion for group optimization will have it as a necessary condition.
Definition 4.4. A feasible allocation
is no other feasible allocation
is Pareto optimal (PO) or weakly Pareto optimal if there
such that
That is, we can no longer make every-
one better off.
Definition 4.5. A feasible allocation
allocation
is strongly Pareto optimal if there is no other feasible
such that
(1)
(2) there is a
s.t.
That is, we can no longer make anyone better off without hurting others.
This definition of PO is just like the definition of technological efficiency in producer theory, except that Pareto optimality is defined for preferences
while technological efficiency
is defined for vector inequality
Example 4.3. Suppose that there is only one good and two agents. Individual 1’s consumption
is
and individual 2’s consumption is
The allocation is a vector
The feasible set of allocations is the shaded area in Figure 4.5.
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x2
A
B
x1
Feasible set
C
D
Figure 4.5. One Good and Two Agents
is
Assume that both individuals’ preferences are strictly increasing. Then, the boundary
the set of weakly PO allocations and the
boundary is the set of strongly PO allocations.
Example 4.4. Consider utility functions
and
These two utility func-
tions are continuous but not strictly monotonic. Because of the failure of monotonicity, weakly
PO allocations may not be strongly PO.
2
y
x
1
Figure 4.6. Weak Pareto Optimality
Strong Pareto Optimality
As shown in Figure 4.6, for these two individuals, all the points in the Edgeworth box are
weakly Pareto optimal, but only point 2 is strongly Pareto optimal.
Proposition 4.3. A strongly Pareto optimal allocation is weakly Pareto optimal. Conversely, if the preferences for all the consumers are continuous and strictly monotonic,24 then a
weakly Pareto optimal allocation is strongly Pareto optimal.
24 Recall
that strict monotonicity means that
implies
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Proof. To show “weak PO => strong PO,” we need to show “not strong PO => not weak PO.”
Given an allocation
if it is not strong PO, there exists a feasible allocation
that makes at least one person, say
better off without hurting others. By
continuity, we can take a little bit of the goods away, say
We can equally distribute
we have
and yet maintain
to the rest of the population. By strict monotonicity,
That is, we find a feasible allocation
that makes everyone strictly better off than
Thus,
is not weakly PO. This finishes the proof.
Agent ’s marginal rate of substitution between two goods and
The
from
is:
is the slope of the indifference curve. It measures the substitutability between two
goods.
Proposition 4.4. Suppose that
Then, a feasible allocation
is differentiable, quasi-concave, and
for all
is Pareto optimal iff
(4.4)
Proof. Suppose
is a Pareto optimal allocation. By the definition of Pareto optimality,
is the
solution of the following problem:
(
,…,
)
(4.5)
The Lagrange function is
The FOCs are
implying
implying
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(4.6)
Conversely, given (4.6), let
Then, by (4.6), for each
That is,
Thus, (4.6) implies the FOCs. We have therefore proven the equivalence between the FOCs
and (4.6). By the quasi-concavity of all
the FOCs are necessary and sufficient for the solu-
tion of problem (4.5).
Proposition 4.4 can be easily illustrated by Figure 4.7. First, if two difference curves at an
allocation point are tangents, we can see from the following figure on the left that we can no
longer find another feasible allocation that makes everyone strictly better off. Thus, the alloca∗
tion
is PO. Conversely, if the two indifference curves intersect at
on the right, then we can find another allocation
∗
as shown in the figure
that makes everybody strictly better off.
x2
x2
x1
B
x1
B
uA
uA
uB
uB
.
x*
x
.
.
x*
x1
A
x1
A
x2
x2
Figure 4.7. PO Allocations
The contract curve is the set of Pareto optimal allocations. Figure 4.8 shows such a curve.
Example 4.5. For the agents in Example 4.1, find the contract curve. The feasible allocations
satisfy
We have
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Thus,
for
defines the so-called contract curve; see Figure 4.8.
y
B
x
ω
.
.
uA
uB
Contract curve
x
A
y
Figure 4.8. The Contract Curve
We have
That is, the contract curve is increasing and concave.
Example 4.6. For the agents in Example 4.2, let us find the contract curve. Since the utility
functions are not differentiable, to find Pareto optimal allocations, we have to use a graphical
means.
We can graphically determine a point in the Edgeworth box to be weakly PO or strongly
PO. In Figure 4.10, point
is a weakly PO point if the interior areas
have no common points. Point
is a strongly PO point if interior area
have no common points and also interior area
and
(better-off areas)
and indifference curve
and indifference curve
have no common
points.
2
2
A
.a
.a
u1
B
u1
u2
1
1
u2
Figure 4.9. Find PO Points Using Indifference Curves
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We thus find that the shaded area in Figure 4.11 is the set of Pareto optimal allocations.
2
y
P.O. points
w
x
1
.
Figure 4.10. Pareto Optimal Set
Furthermore, the weakly PO allocations are all strongly PO.
4. Welfare Theorems
We now present two welfare theorems on Pareto optimality of general equilibria.
Theorem 4.3 (The First Welfare Theorem). If
∗
∗
∗
is
Then, since
∗
is a general equilibrium, then
Pareto optimal.
be a feasible allocation that all agents prefer
Proof. Suppose not. Let
to
is the best choice among the consumption bundles inside the budget set,
∗
must be outside
the budget set:
∗
But, by
∗
∗
∗
we have
∗
∗
This is a contradiction.
The First Welfare Theorem indicates that the general equilibrium (GE) in Figure 4.3 must
be on the contract curve in Figure 4.8. This situation is shown in the following figure. If we
add strictly monotonicity of preferences, a general equilibrium allocation is strongly PO.
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x2
B
x1
ω
.
.
x*
uA
uB
slope =
p1* 8
=
p2* 7
Contract curve
x1
A
x2
Figure 4.11. The GE in on the Contract Curve
Similarly, the GEs in Figure 4.4 must be in the region of PO points in Figure 4.10. This situation is shown in the following figure.
B
y
u1
2
P.O. points
.
.
slope=p
.x W
u2
A
1
Figure 4.12. The GEs are in the Contract Region
The First Welfare Theorem tells us that a market equilibrium must be Pareto optimal. The
next theorem shows the converse.
Theorem 4.4 (The Second Welfare Theorem). Suppose that preferences are continuous,
strictly monotonic, and strictly convex. Then, any Pareto optimal allocation
∗
can become a
general equilibrium allocation by a redistribution of endowments.
∗
Proof. Given wealth levels
Since
∗
have
mality of
since
∗
∗
by Theorem 4.1, there exists an general equilibrium
satisfies the budget constraint
If there were such that
for each consumer, we must
∗
it would contradict with the Pareto opti-
(strong PO is the same as weak PO in this case). Therefore,
provides maximum utility on the budget line, so does
resource constraint. Hence,
∗
∗
∗
∗
Thus,
obviously satisfies the
is a general equilibrium.
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The second welfare theorem can be illustrated by Figure 4.13:
x2
B
x1
uA
uB
.
x*
Contract curve
.
w0
.w
x1
A
x2
Figure 4.13. The Second Welfare Theorem
For any point
∗
on the contract curve, if the indifference curves are convex and continuous,
we can then find a straight line with a positive slope going through point
∗
and separating the
two indifference curves. The slope of this straight line is the equilibrium price ratio. The original endowment point
point
may not happen to be on this straight line. We can arbitrarily pick a
on this straight line as the new endowment point. This straight line becomes a budget
line for the two consumers. By this,
dowment point from
to
∗
becomes an equilibrium allocation. Changing the en-
is what we mean by ‘a redistribution of endowments.’
Intuitively, why do we need a redistribution of endowments? Imagine a PO allocation
∗
∗
for both agents. If the original endowments are
will not be able to find a price vector that supports
∗
∗
and
we
since the second agent has nothing
to gain by trading with the first agent in a free market. Hence, to obtain certain allocation in
equilibrium, we need to have a proper allocation of endowment to start with.
In Figure 4.14, we have two indifference curves, one of which has an unusual shape. This
unusual shape means that consumer ’s preferences are not convex. As a result, we can see
from the figure that an equilibrium price does not exist no matter where the endowment point
is — we cannot find a straight line that separates the two indifference curves. Here, we see that
the convexity of preferences is crucial for the second welfare theorem.
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x2
B
x1
uB
.
x*
uA
x1
A
x2
Figure 4.14. Nonconvexity of Preferences
We have now established a close link between Pareto optimality and the general equilibrium. However, nothing is said about fairness of a market equilibrium. Since the outcome depends crucially on the original distribution of endowments, the market equilibrium may not
be “fair.”
Pareto optimality is only concerned with optimality of an allocation, but it has nothing to
say about the distribution of welfare. Even if we agree that we should have a Pareto optimal
allocation, we still do not know which one we should have. There are generally many Pareto
optimal allocations. In this section, we assume that there exists a social welfare function
that
aggregates
individual
utility
values
to
In other words, for each distribution
there is a social utility. As it should be, we will assume that
An allocation
∗
a
social
utility
of individual utilities,
is strictly increasing.
is said to be socially optimal if it solves:
{
}
That is, a socially optimal allocation maximizes the social utility subject to the resource constraint. Similar to the consideration of the relation between Pareto optimality and the general
equilibrium, we now investigate the relation between Pareto optimality and social optimality.
We will have two results parallel to the first and second welfare theorems.
∗
Proposition 4.5. If
is socially optimal, then
∗
is strongly Pareto optimal.
Proof. Suppose not. Then there would be some feasible allocation
and
∗
for some
By strict monotonicity of
∗
such that
this contradicts the fact that
∗
maximizes the social welfare function.
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Proposition 4.6. Suppose that preferences are continuous, strictly monotonic, and strictly
∗
convex. Then, for any Pareto optimal allocation
such that
∗
∗
with
there are weights
solves25
{
}
∗
Proof. By the second welfare theorem, there is a price vector
equilibrium given endowments
∗
∗
∗
∗
This implies that
∗
such that
∗
∗
∗
is a general
solves
∗
∗
∗
∗
By the Lagrange theorem, there exist
∗
such that
∗
∗
(4.8)
Consider
{
}
(4.9)
∗
and take
∗
∗
By (4.8), if we take
condition. Therefore, since
∗
satisfies the FOCs of maximizing
and the Kuhn-Tucker
is quasi-concave, by Theorem 3.11 in Wang (2008),
∗
solves
(4.9).
Figure 4.15 offers a graphic illustration of Proposition 4.6. Problem (4.7) can be transformed into the following problem. For any PO point on the curve AB, we can find positive
weights
and
such that the PO point is the maximum of
subject to the feasi-
bility condition.
(
,…,
)
∗
25
Perhaps we need disability of all goods to ensure
∗
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A
constant
.
Feasible set
C
B
Figure 4.15. One Good and Two Agents
Proposition 4.5 comes directly from the monotonicity of the social welfare function. For
Proposition 4.6, we know by the Envelope Theorem that
∗
i.e., the weights
∗
is the marginal utility of income:
∗
are the reciprocals of the marginal utilities of income. This makes sense. If
some agent has a large income at some Pareto optimal allocation, then his marginal utility of
income will be small and his weight in the social welfare function should be large, which
means that the competitive market tends to “favor” individuals with large incomes. Of course,
a socially optimal allocation is generally not a general equilibrium allocation. As shown by
Proposition 4.6, only with a linear social welfare function and special weights, a general equilibrium allocation is socially optimal.
Here is a brief summary of the relationships among general equilibria, Pareto optimal allocations, and social welfare maxima:
(1) general equilibria and social welfare maxima are always Pareto optimal;
(2) Pareto optimal allocations are general equilibria under convexity of preferences and endowment redistribution;
(3) Pareto optimal allocations are social welfare maxima under convexity of preferences and a
linear social welfare function with special weights.
5. General Equilibrium with Production
In this section, we allow the economy to have a production sector.
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There are
firms
These
be owned by
of firm
with production possibility sets defined by functions
satisfy Assumptions 1.1 and 1.2 in Chapter 1. The firms are assumed to
individuals
owned by agent
consumption bundle,
with endowments
with
Let
and
Let
be firm ’s net output bundle, and
be the share
be consumer ’s
be a price vector.
5.1. General Equilibrium with Production
Like the pure exchange economy in Section 2, we have parallel definitions for the various
concepts.
Definition 4.6. An allocation
∗
Definition 4.7.
(1) For each
∗
∗
∗
where
is a general equilibrium if
solves consumer i’s utility maximization problem:
∗
(2) For each
∗
is feasible if
∗
∗
∗
solves firm j’s profit maximization problem:
∗
(3)
∗
∗
is feasible:
∗
∗
Notice here that we have supposed that the consumer’s problem can be separated into
profit maximization and utility maximization, as indicated by Fischer’s separation theorem in
Chapter 2.
We can illustrate the equilibrium in an Edgeworth box. Suppose that there is only one
firm with net output vector
Since
by feasibility, letting
and
the firm’s problem can be equivalently written as
If the optimal solution from this problem is
then this
can be allocated to individuals. This
profit maximization problem can be shown in the consumption space in Figure 4.14. The
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figure is drawn in the consumption space for
tion
becomes
Since consumption is
Note that since we usually have
condithe endow-
ment point is on the production frontier (PPF).26
x2
MRS = MRT =
G ( x - w) = 0
slope = MRT =
.
uA
x*
p1*
p 2*
B
uB
.
p1*
p 2*
w
slope = MRS
.
Iso-profit line
p ⋅ x = p * + p* ⋅ w
A
x1
Figure 4.16. General Equilibrium
Example 4.7. We add a firm to the economy in Example 4.1. The firm inputs
production function
The two consumers share the firm equally. Let
to produce
by
and
We first consider the profit maximization problem:27
We find the solution:
We next consider the utility maximization problem. Given incomes
and
the individual
Since consumption is
condition
demand functions are:
We have
Hence,
26
The figure is drawn in the consumption space for
becomes
27 We
here have
Thus, the PPF is
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The feasible allocations are
The first feasibility condition implies
implying
implying
∗
Hence,
∗
∗
∗
Example 4.8. Add a firm to the economy in Example 4.2. The firm inputs
production function
to produce
by
The two consumers share the firm equally. We first consider the
profit maximization problem:
which implies
We next consider the utility maximization problem. Given prices
et equation is
at point
With the special utility function
where
and
the budg-
each individual must consume
By the budget condition, the individual demand functions are:
We have
Hence,
The feasibility conditions are
The first feasibility condition implies
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The solution is (it can be verified easily):
∗
From this, we can calculate the equilibrium allocation.
5.2. Efficiency of General Equilibrium
Definition 4.8. A feasible allocation
tion
is Pareto optimal (PO) if there is no feasible alloca-
such that
on the production frontier, the slope of the production frontier at
For a point
this point is called the marginal rate of transformation (MRT), which is defined in Section 5 of
Chapter 1. That is,
With two consumers
and
and one firm, we find that a feasible allocation
is
Pareto efficient iff
(4.10)
In other words,
(1) the two indifference curves at
are tangents of each other, and
(2) the slope of the two indifference curves at
is the same as the slope of the production
frontier at
This is intuitive. Given net output
divide the total supply
Pareto efficiency implies that the two individuals will
of goods so that
where
is the total
endowment. Further, if
(4.11)
then we can reduce the output of
units. Consumer
in
by one unit in exchange for an increase of output in
is willing to sacrifice one unit of
Thus, such a change in production plan will at least make
for an
by
unit increase
better off without hurting
Thus, the situation in (4.11) could not happen to a Pareto efficient allocation. Therefore, Pareto
efficiency must imply condition (4.10).
Conversely, if (4.10) holds for an allocation
no change can improve both; second, given
can improve both
then, first, given
by
by
’s welfare and the firm’s profitability; finally, given
no change
by
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no change can improve both ’s welfare and the firm’s profitability. Thus, such an
allocation must be Pareto efficient.
x2
MRS = MRT
G ( x - w) = 0
slope = MRT
.
uA
uB
B
.
.
x*
w
slope = MRS
A
x1
Figure 4.17. Pareto Efficiency
We can easily understand this with Figure 4.18. If
∗
defined by
we consider the curve
This curve will intersect with the
-indifference curve at
∗
Then, we can easily find another feasible point that improves consumer ’s welfare while
consumer
keeps
∗
x2
MRS A ≠ MRT
G ( x - w) = 0
slope = MRT
G ( x A + x - w) = 0
*
B
uA
.
x*
.
B
.
w
slope = MRS
A
x1
Figure 4.18. A Pareto Inefficient Allocation
In general, we have the following result. We can prove this result simply following the
above graphic illustration.
Proposition 4.7. Suppose that
and
is differentiable, quasi-concave, and
is differentiable, quasi-convex and
Then, a feasible allocation
for all
is
Pareto optimal iff
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(4.12)
Example 4.9. Add a firm to the economy in Example 4.1. The firm inputs
production function
to produce
by
The two consumers share the firm equally. Let us find the PO
allocations. We use index
for the firm. By (4.12), the condition for PO is
implying
(4.13)
The second equation of (4.13) implies
(4.14)
Then, using the first equation of (4.13), (4.14) becomes
implying
(4.15)
Thus, given any
we can get
from
from (4.15),
from (4.14), and
and
from the feasibility constraints. Such a solution is a PO allocation.
x2
G ( x - w) = 0
.
uA
uB
.
B
.
w
A
x1
Figure 4.19. Pareto Efficiency
Since
is totally free, all points on the PPF belong to PO allocations; but, by (4.15), for each
there is only one point
belong to the PO allocation.
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Example 4.10. Add a firm to the economy in Example 4.2. The firm inputs
production function
to produce
by
The two consumers share the firm equally. Let us find the PO
allocations. Given any
if the Edgeworth box defined by
is not a square box, then at any point
sumers will be willing to give up some units of
and
in the Edgeworth box, one of the conor
to the firm for free, as shown in Figure
4.20. Such an allocation cannot be PO.
y
y - 20 = 40 - x
MRT
o
45
.
20 + y 0
2
.
u2
.
w
u1
40 − x 0
1
x
Figure 4.20. A Pareto Inefficient Allocation
Thus, the only possible PO allocation must have a square Edgeworth box, implying
which determines a unique
satisfying
implying
y
y - 20 = 40 - x
45o
u2
1
.
2
24
.
.w
u1
24
x
Figure 4.21. A Pareto Efficient Allocation
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implying
and
Thus, the PO allocations
and
on the
are those with
-line of the
Edgeworth box; see Figure
4.21.
Similar to the discussions in Section 3, we also have first and second welfare theorems.
The proofs are also similar to the proofs of Theorems 4.3 and 4.4.
Theorem 4.5 (First Welfare Theorem). If
Pareto
∗
∗
∗
is a general equilibrium, then
∗
∗
is
optimal.28
To see this result, consider a simple case with two consumers, two goods and one firm. By
utility maximization, we have
Profit maximization implies
Hence, in a general equilibrium,
(4.16)
Then, as shown in (4.10), the equilibrium allocation must be Pareto efficient.
How about the converse of Theorem 4.5? That is, is any Pareto efficient allocation a general equilibrium allocation? The answer is yes. Given a Pareto efficiency allocation
shown in Figure 4.17, we can draw a straight line through
ence curves at
∗
∗
We can choose prices
∗
∗
∗
∗
∗
for
and
∗
as
that is tangent to the two indiffer-
such that
∗
and the firm maximizes its profit at
and an endowment point
29
∗
Thus, as shown in Figure 4.14, given this price vector
each individual maximizes his utility
assign profit shares
∗
∗
∗
The slope of the straight line is
and
∗
∗
such that
∗
∗
∗
Finally, we
∗
∗
Thus, we have shown that any Pareto efficient allocation can be a
general equilibrium allocation with a proper distribution of endowments and profit shares.
The following is a general result.
28 This
is the same as the allocative efficiency of a competitive market known in the industrial organization
literature.
29 There
are two additional conditions in this assignment:
∗
∗
∗
and
five equations for six variables. This situation is the same as that in Figure 4.12, where the choice of
We have
has one
degree of freedom.
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Theorem 4.6 (Second Welfare Theorem). If preferences are continuous, strictly monotonic,
and strictly convex, then any Pareto optimal allocation
∗
∗
is a general equilibrium
allocation with a proper distribution of profit shares and endowments.
The two results about social optimality in Section 4 still hold in this case.
Keep in mind that the number of firms is fixed in the Arrow-Debreu world. Due to this,
even though the firms take market prices as given, positive profits are possible in equilibrium.
Hence, calling a general equilibrium a general equilibrium is actually a bit misleading. In
Chapter 6, the number of firms will be endogenously determined in a long-run equilibrium in
a competitive industry; in this case, competitive firms will have zero profit in the long run.
6. General Equilibrium with Uncertainty
We now further allow uncertainty in a competitive economy. We will treat the same good
in different states of nature as different goods. We label a good in different states by index
Let
be the set of possible states of nature or realizations of random events. There are
ferent possible states:
There are
Each state
dif-
occurs with probability
goods, and these goods are different due to their differences in physical de-
scriptions, delivery dates, and locations, but not in terms of the state of nature in which a
transaction is made. Given
consumer i’s consumption bundle is a vector
In-
cluding different realizations of the state, consumer i’s consumption bundle is
from which consumer i gains utility
There are
individuals indexed by
have endowment
space
Each individual
When the utility function
thinks that event
If state
is realized, then individual will
has a utility function
over his consumption
has the expected utility property and individual
will occur with probability
his utility function is
But, in general, we do not require expected utility functions.
There are
firms
with production possibility sets defined by
The functions
owned by
individuals. Let
Let
satisfy Assumptions 1.1 and 1.2 in Chapter 1. The firms are
be the share of firm owned by agent
be firm ’s net output bundle when
be firm j’s net output bundle, and
with
and
occurs,
be the price vector.
In this economy, contracts are made before the events occur. In other words, consumers
buy contingent claims to goods in advance, and firms sell claims to their output in advance
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with deliveries contingent on the realization of the state of nature. Shareholders (the consumers) receive their share of the profits from contingent sales before the state of nature is known.
This economy is called the contingent contracts economy.30
Uncertainty
Markets
Consumption
0
1
Definition 4.9. An allocation
where
is feasible if
Definition 4.10. A feasible allocation
tion
is Pareto optimal (PO) if there is no feasible alloca-
such that
∗
Definition 4.11.
(1) For each
∗
∗
∗
is a general equilibrium if
solves consumer i’s utility maximization problem:
∗
(2) For each
∗
∗
∗
∗
solves firm j’s profit maximization problem:31
∗
(3)
∗
∗
is feasible:
∗
Let
∗
Then, the three conditions in Definition 4.11 are the same
as those in Definition 4.7. Hence, this model is actually a special case of that in the last section.
This model is more specific in that it specifically labels the difference in the state of nature.
The previous model allows this difference but mixes it with other differences in goods. Hence,
the first and second welfare theorems still hold; so do the social optimality theorems.
30 All
the contracts are enforceable and ex-post renegotiation is not allowed.
31 The
profits here are not realized profits. They are profits from contingent contracts; as are the consumer
expenditures.
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This contingent contracts economy requires
markets. This system is not informa-
tionally efficient: there is a much simpler way of establishing the same equilibrium, in which
economic agents need much less information. In the next chapter, by the introduction of
securities markets, the number of markets is reduced from
to
A contingent securi-
ty is a financial asset that pays one unit of money if a specified state is realized; otherwise it
pays nothing. We can show that the same equilibrium allocation can be achieved in an economy with
goods markets and
contingent securities markets.
Notes
Materials in this chapter are fairly standard. Many books cover them. Good references for
this chapter are Varian (1992), Jehle (2001), Dixit (1990), and Mas-Colell et al. (1995). In
particular, good references for Section 5 are Campbell (1987, 40-47) and Varian (1992, Chapter
18).
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Chapter 5
Financial Markets
In this chapter, we look at the financial market. With complete and perfect markets, the
models in this chapter are within the Arrow-Debreu world. However, to problems under uncertainty, the approach taken in this chapter is distinctly different from that in Chapter 4.
Specifically, we introduce security markets into the Arrow-Debreu world. Financial securities
allow economic agents to allocate their portfolios across time and across states of nature. We
first see how securities can be used to obtain the same efficient equilibrium allocations as in
the Arrow-Debreu world. We then try to price these securities under various circumstances.
1. Security Markets
1.1. Contingent Markets
We have
space is
ment
Here,
in state
uncertain states
×
with
and utility
and
states and
The consumption
commodities. There are agents
with endow-
Denote
is the price of commodity
and
commodities
in state
is the consumption of commodity
is the endowment of commodity
to agent in state
Thus,
by agent
is the price
vector of the contingent market for commodities
Consider a market for contingent contracts. Each contract delivers a particular commodity contingent on a particular state. There are a total of
such contracts. Contracts are settled
before the market is opened and the true state is revealed; then, contracted deliveries occur
and consumption ensues. In this market, the commodity prices
and consumption demand
are formed and determined before the actual commodity markets open; they are all determined in the contingent contracts.
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A contingent market equilibrium (CME) is a pair
where
solves
∈ℂ
for all
and
(5.1)
is feasible:
This is a competitive equilibrium in the pure exchange economy defined in Chapter 4. The
welfare theorems in Chapter 4 state that a competitive equilibrium is Pareto optimal and,
conversely, that a Pareto optimal allocation is a competitive equilibrium under some regularity
conditions and with a reallocation of endowments.
1.2. Security Markets
A security market is an effective alternative to the contingent market. In comparison, the
latter is very simplified. Any CME can be implemented by a security market equilibrium (SME)
in a complete market. As explained by Campbell (1987), the contingent market is informationally inefficient in the sense that the same equilibrium in a contingent market can be implemented in a security market using only
Suppose that there are
markets rather than
markets.
securities that are traded at prices
before
the uncertainty is resolved. The security dividend vector for each state is
where
is the dividend paid by the nth security in state
Denote the dividend matrix as
×
Spot markets for commodities are opened after uncertainty is resolved, with prices
where
×
is the price of good
where
in state
in the spot market. Agent ’s plan is
is a portfolio of securities
is the ex-ante price and
is the chosen number of the th security). Here,
is the ex-post price of a commodity. Also,
unit of measure, which is the same as the unit of measure for
called money or it can be one of the available goods. Similarly,
in the unit of one of the goods. In general,
and
and
are all in the same
This unit of measure can be
can be in the unit of money or
can be in different units of measure, but
are in the same unit of measure.
If
are linearly independent, the securities are called independent securities and
the financial market is said to be complete. Here, a complete market actually requires two
things: the number
of securities is larger or equal to the number
of states and among these
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securities there are
independent securities. 32 Securities allow consumers to move re-
sources across states. A complete market means that there are enough linearly independent
securities that allow consumers to move to any point in the consumption space. For example,
suppose that there are two states,
∗
∗
of the consumer’s income in
ble to the consumer. If
∗
exist an
and
∗
and
and with a
consumption point
are linearly independent and
is availais affordable, there must
such that
∗
∗
and a consumer wishes to consume at the point
Given two securities with dividend vectors
proportion
If
and
∗
∗
∗
is Pareto optimal (PO), then this complete market is efficient.
We assume that the security market opens before uncertainty resolves; after the uncertainty resolves, the spot market opens. Before the spot market opens, individuals do not have
any endowments. A security market equilibrium (SME) is
where
solves
(
,
)
(5.2)
for all
and
Here,
is feasible:
is the budget constraint for securities.33 Each individual is endowed with zero
units of securities; thus, the total expenditure on securities is less than or equal to the value of
the endowed securities. We need this constraint to determine the prices of the securities. We
can treat each security just like a good.
Proposition 5.1. Suppose that utility functions are quasi-concave. Given a CME
an arbitrary non-singular dividend matrix
let
for any
and
and
take
(5.3)
where superscript
represents the transpose of a vector or a matrix. Then,
is
an SME.
32 This
is implied by the fact that, for any matrix, the rank of its column vectors is the same as the rank of its
row vectors.
33 Suppose
get
that there are two assets and one asset is money and the other is bonds. If I issue 10
bonds, I
but lose 10 bond units. My budget for this transaction is
In this example,
and
where
indicates money and
indicates bonds.
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Proof. Since the utility functions are quasi-concave, we only need to verify that the optimal
allocation of a CME must satisfy the equations (FOCs, budget constraints and feasibility constraints) that determine an SME.
The Lagrangian function for the security market problem (5.2) is
where
and
are the Lagrangian multipliers. The FOCs are
(5.4)
We need to find the Lagrangian multipliers
the CME allocation
the pair
prices
and a portfolio
such that, for
satisfies the equations in (5.4) and the budget and feasi-
bility constraints in (5.2).
The first-order condition for (5.1) is
(5.5)
where
is the marginal utility of income. Besides (5.3), we also take
(5.6)
Then, the first equation in (5.4) is the same as (5.5):
The second equation in (5.4) is satisfied by:
Also, by the budget condition in (5.1), the first budget constraint in (5.2) is satisfied by:
By the definition of
in (5.3), we have
Hence, the second budget constraint in (5.2) is satisfied by:
The first feasibility condition is satisfied by:
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Finally, the second feasibility condition is already satisfied by the fact that
is a CME alloca-
tion. The proof is complete.
Proposition 5.1 immediately implies the following simpler version.
let
Corollary 5.1. Suppose that utility functions are quasi-concave. Given a CME
where
is the -dimension identity matrix, and, for any
and
take
(5.7)
Then,
is an SME.
Proposition 5.1 means that an allocation
in a CME can always be implemented in an
SME. We have a few remarks on this result.
First,
in (5.6) can be defined as
for any arbitrary constant
Since
This arbitrary
osition 5.1 is in units of the first good, we have
in Prop-
allows us to use
any other good as a unit of measure.
in (5.6), which will imply a different set
Second, we can choose a different set of
of expressions in (5.3). For example,
•
In Duffie (1988, p.4), by taking
•
If we take
and
and
for all
for an arbitrary
and
then
we have
and
There are many possible versions of SME. Our version can be conveniently used to derive
asset pricing formulas in the rest of this chapter.
Third, the converse of Proposition 5.1 holds. Conversely, any SME under a complete market can be supported by a CME. However, when the complete market assumption is dropped,
one can easily construct a counter example to show that this equivalence of the two markets
fails. This conclusion is summarized by the following proposition.
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Theorem 5.1. If the market is complete, then any CME can be supported by an SME, and
vice versa. Furthermore, the completeness of the market is a necessary and sufficient condition to guarantee the equivalence of the two equilibrium notions.
2. Static Asset Pricing
Consider a two-period model with one commodity and
where
second period. The consumption space is
units of consumption in the first period and
different states of nature in the
units in state
represents
of the second period,
Suppose that preferences are given by an expected utility:
(5.8)
where
is the probability of state
occurring and
is the time discount factor. Then, the
FOC for (5.1) is
(5.9)
which is equivalent to
That is,
(5.10)
Together with the budget and resource constraints, this determines a CME.
Suppose now that the market is complete with dividend matrix
×
5.1, we can convert the above CME into an SME in which the spot price is
34
Assuming that
By Proposition
for each
and in which the market value of the nth security is
35
by (5.10), for each agent
(5.11)
34 This
is implied from (5.3). Intuitively, since the equilibrium price vector is unique up to a positive multiplier
and since there is only one good, the price can be simply one.
35 This
means that the good
is used as the numeraire for
and
As we know, for CME prices, one of them
can be set to
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That is, the market value
of a security is the expected future dividends
marginal rate of substitution
with probability
discounted by the
between two periods. Since
(5.11) can be written
and
as36
(5.12)
This pricing formula (5.12) is a direct implication of Proposition 5.1 and it is the basis for all of
the available equilibrium asset pricing models, whether in discrete-time or continuous-time
settings.
3. Representative Agent Pricing
The pricing formulae in the last section are agent-specific. We now derive the corresponding formulae for a representative agent.
For the model outlined in Section 2, the pricing formula (5.12) depends on the agent. This
means that it can only be applied to a single-agent economy. We wonder if the pricing formula
(5.12) can be extended to a multi-agent economy by the construction of a representative agent
in a given equilibrium
We know that (5.12) is derived from (5.8) and (5.9). If we
can extend (5.8) and (5.9) to a multi-agent economy, then (5.12) is naturally extendable to a
multi-agent economy.
Define a utility function for the representative agent
as
(5.13)
,⋯,
Here,
for some
is the social welfare maximum function used in the welfare
theorems of the Arrow-Debreu world. We first generalize (5.9) to a multi-agent economy.
if
Proposition 5.2. For any given CME
utility of income defined in
(5.1)37
and
then
where
is the marginal
solves (5.13) for
and
(5.14)
is a solution of (5.13) for
Proof. By Proposition 4.6, since
weights
36 The
and
the Lagrangian function for (5.13) is
MRS
is there because of risk aversion. Without it, the price of the security will be the
discounted expected income.
37 Let
∈
Then,
( , )
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The FOCs of (5.13) are
The Envelope Theorem implies
Together with (5.9), we have
For the utility functions in (5.8), we can similarly define a utility function
for
the representative agent by
,⋯,
Then,
,…,
, ,…,
,…,
,…,
That is, the representative agent has a similar utility function as in (5.8):
(5.15)
We can now extend (5.12) to a multi-agent economy.
Theorem 5.2. If the market is complete, for a given equilibrium
if one chooses
then
(5.16)
where
and
are the aggregate endowments in the two periods. Further,
the completeness of the market is generically necessary.
Proof. With (5.15) replacing (5.8) and (5.14) replacing (5.9), we can immediately use (5.16) to
replace (5.12).
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The generic necessity of a complete market means that, if the financial market is not complete, the representative-agent pricing formula does not hold, except in pathological or extremely special cases.
4. The Capital Asset Pricing Model
Consider a set
of random variables (or lotteries) with finite variance on some probability
corresponds to the random payoff of some security. As in Section 2, there is
space. Each
only one commodity, i.e.,
variable in
The choice space for agents is
is an asset. The total endowment
as some riskless security paying value
Each random
is called the market asset. Denote
in all states. For simplicity, assume also that
is
finite-dimensional. Each agent ’s utility function is assumed to be strictly variance averse,
meaning that
By Theorem 3.3 in Chapter 3, under expected utility, any concave utility function is variance
averse.
is a CME for this economy. Each
Suppose that
variable. A linear functional
on
are not zero. We can show that
can be written as
can be viewed as a random
38
Assume that
can be represented by a unique
and
via the formula:
(5.17)
is called the state price of the asset since it is based on the state, while
For discrete time, we can easily find this
is an ex-ante price.
from Section 2, as shown in the following example.
For continuous time, we need the so-called Riez representation theorem to define this
Example 5.1. Assume
and
Given a CME
∗
∗
∗
for the model in Sec-
tion 2, by (5.10), define
∗
∗
Then, for any dividend bundle
Proposition 5.3. For a CME
we have
let
be the state price of
Then,
(5.18)
38If
there are only finite states
then we write
as a vector
For any
we have
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(
where
, )
( )
and
Proof. For the CME
consider the least-squares regression model:
By the definition of
and
in the proposition, we have
we have
which means that
39
By (5.17),
is affordable for agent
For asset
Thus,
By
we have
Hence,
we have
unless
is zero. Therefore,
inequality
unless
is zero. Since
cannot be true, implying
Theorem 5.3. Given a CME price
is optimal for agent
The proof is complete.
and assuming that
for any asset
we have
the Capital Asset Pricing Model (CAPM):
(5.19)
where
and
⋅
is the beta defined by
Proof. (5.18) implies
where
and
Since
we have
Then, for any asset
implying
Applying (5.20) to the two special assets
and
we find that
which can be used to eliminate the two parameters
and
in (5.20) and change it to
This then implies the CAPM.
39 Conversely,
these two conditions actually imply
and
as defined in the proposition.
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Equation (5.19) means that the expected excess return on any asset is the beta times the
expected excess return of the market asset. One interesting feature is that, from (5.19), any
asset whose return is uncorrelated with the market return is expected to have the riskfree rate
of return.
5. Dynamic Asset Pricing
We now extend our discussions of pricing models to the infinite horizon.
5.1. The Euler Equation
Consider a multiperiod model with one representative agent and one good. Given a probability space, let the consumption choice space be
Given a random process
the state at time is a realization
a first-order Markov process, in the sense that
at time
is
There are
securities with
whose th component is the payoff of the th security at state
Let the prices of the securities be
rity in state
Assume that
has a conditional distribution of the form
A security is defined by a dividend sequence in
dividend vector
of
whose th component is the price of the th secu-
In a recursive model with a state variable being a first-order Markov process,
such equilibrium security prices exist under mild conditions. We take the security prices to be
ex dividend, so that purchasing a portfolio
ment of
of securities in state
and yields a market value of
strictions are that, for any time
both
and
The informational remust depend only on observations of
or in technical terms, that there is a function
Given
and the plan
requires an invest-
such that
the wealth process
of the agent is defined by
Notice that there is no income endowment, i.e. no human capital. For simplicity, we require
and no short sales
for all
The supply of shares is
The representative agent has a utility function
{
}∈ℂ, {
for each security.
and his problem is:
}
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The Bellman Equation for this problem is (see Appendix B):
(
where
,
)∈ℝ ×ℝ
The problem can be simplified to
By the Envelope Theorem,
∗
The FOC is
∗
These two equations imply
∗
∗
or
∗
(5.21)
∗
A triple
dends
∗
∗
∗
is an equilibrium if
∗
∗
is an optimal plan given prices
∗
and divi-
and if the markets clear:
∗
∗
In equilibrium, (5.21) becomes
∗
∗
(5.22)
This is the multiperiod version of the pricing formula (5.12). It is called the Euler Equation.
From this, we can further derive the explicit solution for
fact that
for
∗
Using (5.22) recursively, by the
and assuming the transversality condition (as explained in Ap-
pendix B):
∗
→
we find the solution:
∗
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5.2. Dynamic CAPM
Formula (5.22) shows that the current price of a security that pays
the next period will cost
To extend this to any portfolio
define a linear operator
in
of securities, for each state
by
(5.23)
By (5.12),
is the market price for the securities that pay
In other words,
is the conditional beta of
at
Let40
relative to the aggregate consumption, analo-
gous with the static CAPM, where the market portfolio is in fact the aggregate consumption
since the model is static. Assuming for illustration purposes that
where
and
are two positive constants. Let
is quadratic:
Then, (5.23) becomes
implying
For convenience, let us drop the
Applying it to
and
temporarily. Then,
we have
Using them on (5.24) yields
That is,
This is the so-called Consumption-Based Capital Asset Pricing Model. The interesting feature
is that this CAPM has no cross-period component. That is, its components are based on the
current period only.
In a continuous-time model, one can extend this Consumption-Based CAPM to nonquadratic utility functions. Under regularity conditions, the increment of a differentiable
function can be approximated by the first two terms of its Taylor series expansion (which is a
quadratic function); this approximation becomes exact in the expected value as the time increment shrinks to zero when the uncertainty is generated by a Brownian Motion.
40 Notice
that
is the same as
in (5.19).
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6. Continuous-Time Stochastic Programming
6.1. Continuous-Time Random Variables
For simplicity of notation, we deal with real-valued variables in this section. The generalization to vector-valued variables is straightforward.
The continuous-time counterpart of discrete-time accumulated white noise is the standard Brownian motion. Formally, the standard Brownian motion is a stochastic process
on some probability space satisfying
and
(a) For any
is normally distributed with zero mean and variance
(b) For any
ent.
(c)
for
are statistically independ-
almost surely.
The three conditions basically say that (1)
and (2) for any partition
is white noise.
Brownian motion is very popular partly because of the following convenient rule:
(5.25)
Since
we have
formulas in (5.25). A term
with
A process
/
and
which justify the
is treated as zero.
is called a diffusion process41 if it follows the following stochastic differential
equation,
(5.26)
where
and
and
are bounded and Lipschitz. We may heuristically treat
as the instantaneous mean and variance of
The following is a very useful
result for the differentiation of continuous-time random variables.
Lemma 5.1 (Ito). If
is twice continuously differentiable and
is a diffusion process, then
or
where
41 This
definition is actually for a more general process.
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A more general version of Ito’s formula is
or if
is a vector,
If
is bounded, then
is a martingale defined by the condition:
It then follows that
→
That is, the expected derivative of
at any point
is
6.2. Continuous-Time Stochastic Programming
Consider a single-agent economy with uncertainty generated by a Brownian motion
There are two securities: a risky security and a riskless security. The risky security has (exdividend) price
which is a diffusion process:
(5.27)
This security pays dividend
We may think heuristically of
at any time
where
and
are strictly positive constants.
as the instantaneous expected return and
taneous variance of the return. The riskless security has a price that is always
dividends at a constant rate
(the interest rate). Here, we assume that
wise no one would hold the risky security. Let
time in the risky security and
as the instanand it pays
other-
be the proportion of total wealth invested at
be the consumption at time
Then, the wealth process
follows
which can be written as
(5.28)
Hence, the individual’s problem is
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( , )
where
is a discount rate, and
can depend only on the information available at time
The optimal solution
the wealth
is strictly increasing, differentiable, and strictly concave.
contains all relevant information at any time
Since
we may limit the choice of the
optimal consumption and portfolio to the case with
∗
where
and
∗
are some measurable functions. Then,
is a diffusion process:
(5.30)
and
For any
we can break the expression into two parts:
(
Notice here that we have assumed that
Take limits as
)
is a time-consistent solution. Then,
and use Ito’s Lemma to arrive at
(5.31)
where
where
and
are defined in (5.30), i.e.,
Then, by (5.31), the consumer’s problem of maximizing
becomes the following Bellman
equation:
( ), ( )
(5.32)
The first-order conditions are
which gives us the optimal solution:
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(5.33)
One difficulty with this solution is that function
use (5.31) to determine
to determine
is not explicitly given yet and we have to
That is, we need to substitute
into equation (5.31)
After that, we return to (5.33) to find the final solution for
then we must have
Example 5.2. If
verify this, for initial wealth
and
for some constant
To
by inspecting the budget constraint (5.28), we know that
satisfies the budget constraint. Also,
By taking
and
, we have
Thus,
maximizes the utility, and
for some
Then, by (5.33),
Hence, it is optimal to consume a fixed fraction of wealth and to hold fixed fractions of wealth
in assets. We now need to determine
By (5.31), we have
where, by (5.30),
implying
implying
Thus, the solution is
7. The Black-Scholes Pricing Formula
The classical example of pricing a derivative security is the Black-Scholes Option Pricing
Formula. Consider a single-agent economy with uncertainty generated by a Brownian motion
There are two independent securities: a stock and a bond. The stock has (ex-dividend)
price
which is a diffusion process:
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(5.34)
at any time
and pays dividend
where
the expected return per unit of time and
and
is
is the standard deviation of the stock return per
unit of time. Assume that there is no dividend,
riskless security (the bond) is
are strictly positive constants. Here,
in this section. The market value of the
for some
which means that
A derivative security is a security whose value depends on the values of other more basic
underlying securities. We want to evaluate a derivative security that pays a lump sum of
at a future time
where
is a sufficiently well-behaved function to justify the derivation in
this section. Suppose that the value of the derivative security at any time
Instead of holding one unit of the derivative with wealth
at time
is
the investor can
alternatively put his wealth in a stock-bond combination. Suppose that an investor decides to
hold a portfolio
of the stock and the bond at time
Then,
(5.35)
Assume that the stock and the bond are independent securities so that a return from any other
asset can be achieved by a portfolio of the stock and the bond. In this complete market, a noarbitrage condition is
(5.36)
We can see that the following portfolio satisfies the feasibility condition (5.35):
(5.37)
By Ito’s Lemma,
(5.38)
By the definition of
in (5.37),
Therefore, (5.36) and (5.38) imply
This means that
must satisfy the following partial differential equation:
(5.39)
The boundary condition is
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The solution is shown in the following theorem, which can be easily verified.
Theorem 5.4 (Derivative Pricing). Equation (5.39) has the solution:
(
)
(5.40)
where
Equation (5.40) can be solved numerically by standard Monte Carlo simulation and variance reduction methods.
As an example, let us apply (5.40) to the original Black-Scholes Option Pricing Formula.
We evaluate a European call derivative. A European call option gives the holder the right to
buy the underlying asset on a certain date for a certain price. The price is called the exercise
price and the date is called the exercise date or maturity. Let
Since the option is exercised only if
and in that case the net gain to the option holder is
the value of the option at maturity is
(5.41)
In this case, we have
Using Theorem 5.4, the option price
is found in the
following proposition.
Proposition 5.4 (The Black-Scholes Formula). The initial price of the call option is
(5.42)
where
is the cumulative standard normal distribution function and
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Notes
The materials in this chapter are quite standard and appear in many books. We have covered only very basic and elementary financial theory, much of which is on asset pricing. The
two best books on standard financial theory are Duffie (1988) and Duffie (1992).
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Chapter 6
Industrial Markets
This chapter introduces the standard theory of how industries behave in the marketplace.
We focus mainly on the output market and later cover briefly the input market.
All the firms are assumed to maximize profits. Given a firm’s revenue function
cost function
for a level of output
the optimal level of output
and
is determined by the
following well-known MR=MC equation:
∗
where
and
(6.1)
∗
are the marginal revenue and marginal cost functions, respectively.
This condition is applicable to any kind of firm. Here,
in the output market, and
is derived from market conditions
is derived from market conditions in the input market. When
we focus on the output market, we will derive
and take
the input market, we will derive
as given. The truth of condition (6.1) can
and take
as given; when we focus on
be easily seen in Figure 6.1.
MC
Surplus
.
MR
∗
Figure 6.1. Optimal Output
1. A Competitive Output Market
A (perfectly) competitive industry is an industry in which
•
There are many small firms (small in market shares): firms are independent of each other
in decision making.
Susheng Wang, HKUST
•
Each sells an identical product: each firm faces a horizontal demand curve at the market
price and therefore takes the price as given.
•
There is free entry: zero profit in made in the long run.
Under these conditions, no single firm can exert a significant influence on the market price of
a good. Firms in such a market behave as price takers. If there is a positive profit, other firms
can enter the industry, imitate the incumbent firms, and take a share of the profit. On the
other hand, if there is a loss, some firms will leave the industry. Hence, in the long run, the
profit is zero for all firms in such an industry.
A competitive firm is a firm that takes the market prices as given. Some caution should be
exercised here. The firm faces two markets: the output market and the input market. A firm
may be competitive in the output market but not so in the input market. In this section, we
deal with competitive firms in the output market. Note also that the competitiveness of a firm
is not an additional assumption on the firm. We only assume that the firm maximizes its profit
or expected profit. The competitiveness is implied/induced by the market conditions.
Since there are many firms in the market producing the same product, any downward deviation from the market price will attract all the consumers in the market, and any upward
deviation from the market price will drive away all the consumers. Therefore, any competitive
firm faces a demand curve that is perfectly elastic at the prevailing market price, as shown in
Figure 6.2.
P
p = MRi
Q
Figure 6.2. The Demand Curve for Each Firm
Hence, given a market price
supply a quantity
by the first-order and second-order conditions, firm
will
satisfying
(6.2)
These two conditions correspond to the increasing part of the MC curve. However, these two
conditions in (6.2) only guarantee the maximum profit; they do not guarantee profitability.
Since the firm is free to exit, when the price is below its average variable cost, the firm will stop
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production or exit. The fixed cost is
Then, the variable cost is
and the average variable cost is
This implies the supply curve shown in Figure
6.3.
MC
Supply curve
AC
AVC
y
Figure 6.3. The Firm’s Supply Curve
The industry supply curve is the horizontal summation of all individual firms’ supply
curves, as shown in Figure 6.4. Specifically, given any price
sum of individual firms’ supply functions
the industry supply
is the
at price
p
S ( p ) =  Si ( p )
y
Figure 6.4. Industry Supply Curve
When a cost function is convex, its MC curve is increasing. In this case, since each firm supplies the good based essentially on its MC curve, the industry supply curve is basically the
horizontal sum of individual firms’ MC curves, except when the price is too low.
The industry demand is the total quantity demanded by consumers for the product, which
is typically a downward sloping curve.
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Remarks:
•
The industry faces a typical downward sloping market demand and each firm faces a
horizontal demand curve.
•
The behavior of each firm in the input market is assumed away by the given cost function
•
The supply curves in both the short run and the long run are shown in Figure 6.3, except
that, in the long run,
By the LeChatelier Principle, the only difference between
the long run and the short run supply curves is that the long-run supply curve is more flat.
The equilibrium price is the price that equalizes the industry demand and the industry
supply, as shown in Figure 6.5.
P
AC i
MCi
P
S SR
AVCi
.
.
P
D
Firm
Q
Industry
Q
Figure 6.5. Short-Run Equilibrium
When the time horizon comes into play, we have short-run and long-run equilibria. The
above definition of equilibrium is for a short-run equilibrium. In the long run, the number of
firms may vary. Entry and exit may then affect the industry supply curve, which may result in
changes of the short-run equilibrium price over time. In this case, when the entry and exit stop
(or more precisely, when there is no potential entry and exit to come), we say that the industry
reaches a long-run equilibrium and the resulting price is called the long-run equilibrium price,
as shown in Figure 6.6.
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P
P
S LR
.
.
P
D
Firm
Q
Q
Industry
Figure 6.6. Long-Run Equilibrium
Within each firm, some inputs cannot be varied in the short run. Capital inputs, such as
equipment and buildings, may not be adjustable in the short run. In this case, a short-run
equilibrium is dependent on given fixed amounts of capital inputs. In the long run, all inputs
are variable and thus they can all be chosen optimally.
Let us now use the above framework to analyze a few situations.
Example 6.1. Suppose that there is a demand shock, which moves demand from
to
as
shown in Figure 6.7. What is the new long-run equilibrium?
P
P
AC i
S'
MC i
S
.
P0
. .
P1
q1 q 0
D'
Q
Firm
Q 2 Q1 Q 0
D
Q
Industry
Figure 6.7. A Decrease in Demand
As Figure 6.7 indicates, starting from the original industry equilibrium
brings the industry to a short-run equilibrium
equilibrium
the shock
and eventually leads to a long-run
Here, the industry supply curve shifts to the left since some firms leave
the industry due to negative long-run profit. The firms stop leaving when the price returns to
the original level. Hence, in the short run, the shock causes a temporary drop in price; in the
long run, the price remains the same, but there is a permanent reduction in quantity.
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Example 6.2. Suppose that the industry demand is
where
and
are positive constants. Suppose that the firms have identical cost functions and
the cost function is
Since
the supply curve for each firm is
If there are
firms in the industry, the industry supply curve is the horizontal sum of the
’s:
By equalizing the industry supply and demand, we find the equilibrium price:
∗
(6.3)
We can see that the equilibrium price is lower if the number of firms increases.
In the short run, at price
∗
a firm may have a positive profit or may have a loss. From
we find the break-even price for each firm
∗
Since we always have
all the existing firms are profitable and will produce in the short run.42 The short∗
run equilibrium price is
p
S (m )
S (m*)
SR
p
p LR
2
.
S(m * +1)
.
.
D
y
LR
y
Figure 6.8. Short-Run and Long-Run Equilibria
In the long run, from
we find that the break-even price for each firm is
This means that a firm will stay or enter the industry if the price is above
the price is below
42 Even
and exit if
in the long run. When new firms are entering the industry, they will drive
though a firm may decide to produce in the short run (when the firm is taking a short-term view), it
may decide to leave an industry in the long run (when the firm is taking a long-term view).
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down the price, as shown in (6.3). This process will continue until a negative profit is expected.
The long-run equilibrium price is the smallest price that satisfies
The largest
∗
satisfying the above two conditions, i.e., the equilibrium number of firms, can
be easily solved from the two conditions:
∗
where
represents the largest integer that is less than or equal to
The long-run equilibri-
um price is then
As shown in the figure, the firms in the industry will generally make a positive profit even
though the expected profit for a newcomer is negative. In fact, the firms in the industry will
make a positive profit in the long run if and only if
is not an integer.
Let us look at another example relating to the literature on public finance.
be the inverse industry demand curve and
Example 6.3. Let
be the inverse indus-
try supply curve.43 Suppose that the industry is originally in an equilibrium:
∗
∗
Now, the government imposes a sales tax of dollars per unit of output on the producers.
Originally, at output level
the firms need to be paid
for an additional unit of output.
After the tax is imposed, since the firms are paying per unit of output, the firms now need to
be paid
for an additional unit of output at
This means that the after-tax industry
supply curve is
The new equilibrium condition is therefore
If the time horizon is introduced, the above
serves as the short-run industry supply
curve. We know that the long-run supply curve will be more flat. For simplicity, assume that
the long-run MC curve (i.e., the long-run supply curve) for the industry is constant. The effects
of the tax can be easily seen in Figure 6.9.
43 A
demand function is typically written as
the price. We often write
meaning that the quantity demanded
to indicate it as a demand function. When this function
we can also express the demand relationship by
this the inverse demand function. Again, we often write
or
where
is dependent on
is strictly decreasing,
is the inverse function. We call
to indicate that it is a demand function.
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SMC '
p
SMC
.
C
p LR
..
p SR
S ' = LMC '
B
}t
p*
S = LMC
A
D
y
y LR y SR y *
Figure 6.9. The Effects of a Sales Tax
The original equilibrium is at
With the tax, the short-run equilibrium is now at
where the
consumers and the industry share the burden of taxes. The short-run price is increased less
than
The long-run equilibrium is at
at which point the consumers bear the tax alone. The
long-run price is increased exactly by
∗
This is expected: with zero profit for the
firms in the long run, the tax burden has to be passed on to the consumers.
2. A Monopoly
A monopoly is an industry in which
•
there is only one firm: the firm is the industry;
•
the firm faces a downward-sloping demand curve: it is able to manipulate both price and
quantity;
•
there is no entry: it is possible for the firm to maintain a positive profit in the long run.
There are three types of monopolies:
1. A single-price monopoly, which charges the same price for each and every unit of its output;
2. A price-discriminating monopoly, which charges different prices for the same good to
different people or for different quantities demanded.
3. If the monopoly doesn’t know each consumer’s willingness to pay, we have monopoly under
asymmetric information.
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2.1. The Single-Price Monopoly
A downward-sloping demand means that the price is a function of quantity:
The revenue is
By (6.1), the FOC in this case is
∗
∗
∗
(6.4)
∗
As shown in Figure 6.10, when the firm adds one more unit of output, it gets paid
unit, but since the price has been lowered for all units, it loses
in Figure 6.10, where
Hence,
for that
on all previous units.
is negative.
p
p ¢( y ) y
p ¢( y ) {
p ⋅ Dy
D
Δy = 1
0
y
Figure 6.10. Marginal Revenue
we have
Since
meaning that the MR curve is always below the de-
mand curve except at
monopoly produces
quantity
∗
∗
The monopoly’s problem is shown in Figure 6.11, in which the
and charges
∗
In the figure,
∗
is decided by condition (6.4). To sell
since the monopoly only uses one price for all units, the highest price is
∗
as
shown in Figure 6.11.
p
η >1
p*
Profit
. η =1
.
.
y
*
MC
AC
η <1
D
y
MR
Figure 6.11. Monopoly Pricing
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Discussion:
1. We have
where
is the price elasticity of demand. With
profit maximization implies that a
single-price monopoly will always produce in the elastic range
of the demand curve.
2. There is no guarantee that a monopoly will make a positive profit. A monopoly may make
zero profit or even a loss.
3. Unlike a competitive firm, a monopoly has no supply curve. The monopoly picks a combination of output and price using its marginal cost and marginal revenue curves. Therefore,
the monopoly only supplies at one single point.
where
Example 6.4. Consider a simple demand function:
and
are positive
constants. In this case, the MR curve is half way between the demand curve and the vertical
axis, as shown in Figure 6.12.
p
A
MC
A+ c
2
Profit
.
.
.
AC
MR
A− c A
2a 2a
D
A
a
y
Figure 6.12. A Single-Price Monopoly’s Profit-Maximizing Problem
2.2. The Price-Discriminating Monopoly
Price discrimination is the practice of charging some customers a higher price than
others for an identical good or of charging an individual customer a higher price on a purchase
of a few units of the good than on a purchase of many units of the good. Price discrimination
can be practiced in varying degrees. Perfect price discrimination occurs when a firm charges a
different price for each unit sold. Some people may pay less with price discrimination, but
others pay more. But, the firm can make more profit by price discrimination.
Given an output
charging multiple prices
the left side of Figure 6.13 shows that a monopoly will do better by
and
than a single price
for the total quantity
The right side
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of the figure shows that for a given quantity
by charging a different price for each different
unit, the monopoly achieves the maximum revenue for selling that amount
This is perfect
discrimination. The monopoly can alternatively charge a fee equal to the area abp and charge a
single price for the quantity
a
a
p1
c
p2
e
b
b
p
D
D
y
0
y
0
Figure 6.13. Price Discrimination
The right side of Figure 6.13 suggests that
44
In other words, the demand
curve is the MR curve. By (6.1), the profit is maximized when marginal revenue equals marginal cost. Therefore, a perfectly price-discriminating monopoly will produce its output at the
point where the marginal cost curve intersects with the demand curve, as shown in Figure 6.14.
p
MC
.
D
y
*
y
Figure 6.14. Perfect Price-Discriminating Monopoly
Discussion:
1. The monopoly price is generally higher than the competitive price.
2. The monopoly output is generally less than the competitive quantity.
3. The more perfectly the monopoly can price discriminate, the closer its output is to the
competitive output.
44 Rigorously,
the revenue function is
implying
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2.3. Monopoly Pricing under Asymmetric Information
We have so far assumed complete information (or symmetric information if there is disinformation): demand and supply functions are all public knowledge. Price discrimination
becomes difficult when the monopolist does not know the demand curve. Suppose now that
the monopolist does not know the demand curve of each consumer although it can practice
price discrimination. More specifically, the monopolist knows all existing types of consumers,
but it does not know which consumer belongs to which type. The information is asymmetric
since the consumers know their own types but the monopolist does not. What should the
monopoly do?
This problem belongs to the field of mechanism design, which is formally covered in
Chapter 9. In this section, we give a simple example to illustrate the solution.
Suppose that there are two consumers with the demand curves indicated in Figure 6.15,
where consumer 2’s demand curve is above consumer 1’s demand curve, with maximum demand
and
respectively, at zero price. Assume zero cost. We have labeled the areas as
and
As shown in Figure 6.15(a), with perfect price discrimination, the monopo-
list would like to sell
to consumer 1 for price
and sell
to consumer 2 for price
(the price here is the total charge rather than a unit price). However, since the monopolist
cannot distinguish the consumers, if these two packages are offered to the market, consumer 2
will choose package
by which he retains a surplus of
second package
instead of no surplus for the
The profit for the firm is
p
p
p
B¢
B¢
B
E
A
A¢
C
A¢
C
C¢
D
x1
(a )
x2
D
x2
x1
(b )
x 2*
x1*
(c )
Figure 6.15. Monopoly Price under Incomplete Information
To avoid the above problem, the monopolist can make two alternative price-quantity offers:
and
If so, both consumers will choose the package intended for them.
In particular, consumer 2 will choose
since the other yields the same surplus
This strategy yields a higher profit for the monopolist. The profit is now
Comparing this profit with the previous one, we realize that profit improves if the monopoly designs a set of offers so that each consumer will pick what is intended for him. Such a set
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of offers is called incentive compatible. In the following, we impose the condition of incentive
compatibility and try to find the profit-maximizing solution.
The above solution may not the best to the monopolist. For example, as shown in Figure
6.15(b), the monopolist can offer a lower
and
The monopolist still offers two packages:
Each consumer will still choose the package that is intended for him.
The monopolist’s profit is increased since she loses the small red area but gains the larger
green area. The profit is
which is better than before as long as
Hence, the condition for profit maximization is
i.e., the optimal
∗
is the one where the marginal benefit from one more unit of reduction in
equals the marginal loss. As shown in Figure 6.15 (c), the optimal offers are
∗
∗
and
which maximize the monopolist’s profit. Consumer 1 gets zero surplus and
consumer 2 gets surplus
The profit is
∗
We will revisit this problem again in Chapter 11 using a rigorous approach.
3. Allocative Efficiency
Given ordinary consumer demand
For consumption level
quantity
∗
suppose that its inverse function is
suppose that the price is fixed at
Then, the consumer surplus at
is
(6.5)
as shown in Figure 6.16. We can break this into two parts.
is the maximum price that the
consumer is willing to pay for one more unit of the good when the consumer already has
units of the good. Thus, the consumer surplus is the difference between the total amount that
the consumer is willing to pay for
Note that
and what the consumer actually pays:
is the surplus in terms of “money” value (as opposed to utility value). See
Varian (1992, 160–163) for the equivalence between this definition and a surplus in welfare
when the utility function is quasi-linear.
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MC
Consumer
Surplus
p
p
Producer
Surplus
D
x
0
y
0
Figure 6.16. Consumer Surplus & Producer Surplus
Producer surplus can also be defined similarly. Let
output level
be the firm’s marginal cost at
Suppose that the price at which the firm sells that quantity
the producer surplus at quantity
is fixed at
Then,
is
(6.6)
as shown in Figure 6.16. It is the gain without taking into account the fixed cost. We have
where
is the variable cost and
is the fixed cost. Hence, the profit is
The social welfare is defined as the sum of all participants’ surpluses. In particular, if only
consumers and producers are involved in a trade, the social welfare is:
A market equilibrium reaches allocative efficiency when the social welfare is maximized.45
Using this definition, we can now compare the allocative efficiency of a monopoly and that of a
competitive industry.
Under perfect competition, market equilibrium occurs where the industry supply curve
and the industry demand curve intersect, as shown by point
in Figure 6.17(a). Now, suppose
that the industry is taken over by a single firm. Assume that there are no changes in production techniques so that the new combined firm has the same cost structure as the cost structures of the separate firms. If it is a single-price monopoly, it will produce quantity
charge price
in Figure 6.17(b). We see that the monopoly produces less and charges a high-
er price. If it is a perfectly price-discriminating monopoly, it will produce quantity
45
and
and
In our definition, we implicitly assume that there are no external costs and external benefits. In this case,
the demand curve is the marginal social benefit (MSB) curve and the marginal cost curve is the marginal social cost
(MSC) curve. The social optimal point is where the MSB and MSC curves intersect.
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charge prices along the demand curve. The highest price will be
the lowest price will be
for the last unit sold.
P
P
PA
PA
Consumer
Surplus
MC
Consumer
Surplus
PC
F
0
for the first unit sold and
Producer
Surplus
.E
PC Producer
Surplus
D
C
(a) Perfect Competition
Q
F
0
MC
.H
PM
DWL
.G
.
E
MR
D
M
C
(b) Monopoly
Q
Figure 6.17. Welfare Implications
In a competitive equilibrium, as shown in Figure 6.17(a), the producer surplus is the area
the consumer surplus is
and the social welfare is the largest possible area
In a single-price monopoly, as shown in Figure 6.17(b), the consumer surplus drops to area
the producer surplus increases to area
The loss in social welfare is the area
and the social welfare drops to area
which is called the deadweight loss because
no one gets it. The size of the deadweight loss measures the loss in efficiency. The monopoly
captures some of the consumer surplus by charging a higher price. But it also eliminates some
producer surplus and consumer surplus by producing at an inefficient output level.
However, there is no deadweight loss if the monopoly practices perfect price discrimination. We see that this monopoly will produce at level
which means that the monopoly is as
efficient as a competitive industry. Thus, perfect price discrimination implies allocative efficiency. But in this case, the monopoly takes all the consumer surplus away. That is, with a
perfectly price discriminating monopoly, the consumer surplus is zero and the producer surplus is equal to the total social welfare, which is the area
The solution under incomplete information for the case in Figure 6.15 is inefficient. Compared with the solution under perfect information, the loss of social welfare due to incomplete
information is the size of
Since monopoly generally implies a positive profit, there is competition for monopoly
rights. If there is no barrier to this opportunity, in equilibrium, the resources used up in such
an activity may equal the monopoly’s profit. That is, competition for monopoly rights will
result in the use of resources to acquire those rights. Those resources may be equal in value to
the monopoly’s potential profit. As a consequence,
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However, there are gains from a monopoly such as economies of scale, economies of scope,
and the incentive to innovate. This is especially true in the case of natural monopolies.
Example 6.5. Let us now consider the welfare implications of Example 6.3.
SMC'
p
p
SMC
p SR
p*
C
. .A
B
E
.
S=LMC
S ' = LMC '
C
p LR
.
A
p*
G
S = LMC
D
D
F
y SR y *
y
y LR
y
y*
Figure 6.18. Welfare Implications of a Sales Tax
After a tax is imposed, in the short run, the consumer surplus drops from
in Figure 6.18, and the producer surplus drops from
∗
to
∗
to
The tax revenue is
The tax is shared by both the consumers and the producers. If the tax revenue is used as efficiently as it would be in the consumers and producers’ hands, then the net loss (the
deadweight loss) is
In the long run, the consumer surplus drops from
∗
to
The tax revenue is paid
by the consumers only. If the tax revenue is used as efficiently as it would be in the consumers’
hands, then the dead weight loss is
In both the long run and the short run, the tax creates a loss of efficiency. This is not surprising since the industry was competitive and hence efficient before the tax was levied. A
price distortion in the industry caused by the tax can only reduce its efficiency.
4. Monopolistic Competition
A monopolistically competitive industry is an industry in which
• there are many firms: firms are independent of each other in their decisions;
• product differentiation exists: each firm faces a downward-sloping demand curve and
therefore can choose its own price and quantity;
• there is free entry: zero profit is made in the long run.
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An important difference between monopoly and monopolistic competition is free entry. Because of free entry, profit cannot persist in the long run. Apart from this, each firm behaves
just like a monopoly.
Suppose that firm faces inverse demand
pends on firm ’s own output
The demand not only de-
but also on other firms’ outputs. Denote
Since there are many firms in the industry, we assume that each firm takes other firms’ actions
as given when it makes choices. That is, each firm follows a Nash strategy in output. Therefore,
firm ’s problem is
Then the first-order condition for firm is
∗
∗
An equilibrium
∗
∗
∗
∗
is the point where all the firms maximize their own profits. That is,
the equilibrium is determined by:
∗
∗
∗
∗
∗
∗
(6.7)
When a time horizon comes into play, the equilibrium defined by (6.7) is only a short-run
equilibrium. Under condition (6.7), the firms may make positive or negative profits. In the
long run, when exit and entry are feasible, some firms may enter and some firms may exit
until the profit is zero for every firm in the industry. This means that, in addition to condition
(6.7), the number of firms in the industry will be determined by the following zero-profit
conditions:
∗
∗
∗
∗
(6.8)
The short-run and long-run equilibria are illustrated in Figure 6.19. In the short run, since
no firm can enter, each firm acts just like a monopoly as shown in Figure 6.19(a). With profit
being earned in the short run (a loss is also possible), new firms enter the industry and take
some of the market away from the existing firms in the long run. As they do so, the demand
curve for each firm starts to shift to the left. As the demand curve continues to shift to the left,
each firm’s profit gradually falls. In the long run equilibrium, as shown in Figure 6.19(b), every
firm makes zero profit.
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p
p
MC
MC
.
.
.
p SR
Profit
AC
AC
.
.
p LR
D SR
MR SR
D SR
D LR
y
y SR
y
y SR
(b) Long run
(a) Short run
Figure 6.19. Monopolistic Competition
Discussion:
1. In the long-run equilibrium, as shown in Figure 6.19(b), the AC curve is tangent to the
demand curve at the optimal point for each firm. This can be easily shown. By (6.8), we
know that firm ’s AC curve cuts or touches its demand curve at the optimal point. From
(6.7), we have
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
That is, firm ’s AC curve and demand curve have the same slope at the optimal point.
Therefore, these two curves are tangent at the optimal point.
2. In the long run, the firm in monopolistic competition always has excess capacity in that its
output is lower than that at which the average total cost is minimum. This result arises
from the fact that the firm faces a downward-sloping demand curve. Product differentiation
causes a downward-sloping demand curve and thus excess capacity.
3. Allocative efficiency is achieved when the demand curve intersects with the MC curve.
Hence, monopolistic competition is allocatively inefficient. Even though there is zero profit
in the long-run equilibrium, a monopolistically competitive industry produces a level at
which price exceeds marginal cost.
4. Monopolistically competitive firms may attempt to differentiate the consumer’s perceptions
of the product, principally by advertising, which incurs costs.
5. The loss in allocative efficiency in monopolistic competition has to be weighed against the
gain from greater product variety and product innovation. Monopolistically competitive
firms are constantly seeking out new products that will provide them with a competitive
edge, even if only temporarily.
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5. Oligopoly
Oligopoly is an industry in which
•
there are a small number of firms: firms depend on each other;
•
there is a single product: firms jointly face a downward sloping industry demand curve;
•
there is no entry: long-run positive profits are possible.
Different from a monopolistically competitive industry, in an industry with a small number of
firms, the firms realize that they are dependent on each other. Since they jointly face a common demand curve, each firm’s sales depends not only on its own price but also on other firms’
prices. If one firm lowers its own price, its own sales increase, but the sales of the other firms
in the industry decrease. In such a situation, other firms will most likely lower their prices too.
Hence, before deciding to cut its own price, each firm may try to predict how other firms will
react and attempt to calculate the effects of those reactions on its own profit. To analyze the
behavior of the firms in such a market, game theory is needed.
Game Theory analyzes strategic interactions. Strategic interaction occurs when a firm
takes into account the expected behavior of other firms and their mutual recognition of interdependence. What kind of situation in such a game can be reasonably called an equilibrium/solution? A simple concept of equilibrium is called the Nash equilibrium. A player is said
to play a Nash strategy if he makes a choice by assuming that others will stay where they are. A
Nash equilibrium is when no one wants to change by assuming others will not change. In
other words, a Nash equilibrium is when everyone is playing an optimal Nash strategy. Firms
may play Nash strategies in quantities or in prices. The Cournot equilibrium is a Nash equilibrium when both firms play Nash strategies in quantities. The Bertrand equilibrium is a Nash
equilibrium when both firms play Nash strategies in prices.
We will focus on a simple case of oligopoly called duopoly. Duopoly is an oligopoly in
which there are only two firms.
A Nash game is a game in which all players move simultaneously. A game can also be
played dynamically, in which some players move ahead of others. In a duopoly, we call the
firm that moves first the leader and the firm that moves second the follower. A Stackelberg
equilibrium is when the leader makes her choice to maximize her own payoff by taking into
account the follower’s reactions and the follower makes his choice to maximize his own payoff
using a Nash strategy. We may think of a Nash game as simply a one-period game in which
both players take an action simultaneously and of a Stackelberg game as a two-period game in
which the leader moves in the first period and the follower moves in the second period.
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5.1. Bertrand Equilibrium
Consider a duopoly. The two firms compete on prices, i.e., they play Nash strategies using
their prices as strategic variables. Let
be the price vector and
be the market
demand for the product. Assume that the two firms have the same cost function and the cost
function has a constant marginal cost
and
The two firms simultaneously offer their prices
Sales for firm are then given by
,
The profit is
An equilibrium is said to be stable if, after a deviation from the equilibrium, the system is
able to move back to the original equilibrium by itself. Figure 6.21 illustrates such a situation.
Proposition 6.1 (Bertrand). In a duopoly with a constant marginal cost
∗
stable Bertrand equilibrium
Proof. Suppose that firm
∗
in which
offers price
∗
there is a unique
∗
Then, if firm 1 offers the same price
they share the market half-half, by which firm 1’s profit is
However, if firm 1 chooses a price slightly lower than
say
firm 1 takes over the
whole market and firm 1’s profit becomes (see Figure 6.20):
Since
as
as
we have
when
is small enough. Therefore, as long
firm 1 will offer a price lower than
p
p2
p1
c
D
x
Figure 6.20. Bertrand Equilibrium
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Symmetrically, as long as
firm 2 will offer a price lower than
Therefore, there is a
Nash equilibrium only when
is indeed a Nash equilibrium.
We can easily verify that the situation when
Also, by the way that we derive the equilibrium, it must be stable.
We can similarly show that when there are
identical firms with a constant marginal cost
the Bertrand equilibrium is still the outcome when
∗
∗
The striking implication of the Bertrand equilibrium is that a competitive outcome can be
obtained with two firms only. In fact, price competition in this case results in a horizontal
demand curve for each firm.
Is this consistent with reality? The answer is no. This suggests that the Nash equilibrium
concept is problematic. For example, in Figure 6.20, given
ing its price slightly below
when firm 1 thinks about lower-
it should realize that its action will cause the other firm’s reac-
tion, which is likely to lower
below
If so, firm 1’s action leads to zero profit eventually.
That is, if firm 1 takes into account a possible
Hence, firm 1 is better setting its price
reaction from firm 2 when it considers lowering its price, firm 1 may not choose to do so. If so,
the two parties may set a price above
and stays there. This is an equilibrium (not a Nash
equilibrium) since a possible reaction from the other party serves as a deterrent for a deviation.
5.2. Cournot Equilibrium
Consider the duopoly again, but the two firms now play Nash strategies in quantity. In a
Cournot game, firm 1’s problem is
firm 2’s problem is
the first-order conditions are
(6.10)
Hence, the Cournot equilibrium
∗
∗
∗
∗
∗
∗
∗
is determined by the two first-order conditions:
∗
∗
∗
∗
∗
∗
∗
We can also look at this equilibrium from another angle. The first-order condition for firm
1 determines a reaction function of firm 1:
and the first-order condition for firm 2 determines another reaction function of firm 2:
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The Cournot equilibrium is where the two reaction curves intersect, as shown in Figure 6.21.
y2
y2
f2
f1
f1
f2
.
y 2*
.
y 2*
y1*
y1
y1*
y1
Figure 6.21. The Stability of Cournot Equilibria
is steeper than
As shown in Figure 6.21, the Nash equilibrium is stable if
than
If
is steeper
it would be unstable. The left diagram means that the stability condition is
Of course, the intersection points may not be unique, implying the possibility of multiple
equilibria.
Proposition 6.2 (Cournot). In a Cournot equilibrium with constant marginal cost
for
both firms, we have
∗
where
∗
is the market price,
is the competitive price, and
is the monopoly price.
Proof. By (6.10), we have
∗
where
∗
∗
∗
∗
∗
∗
(6.11)
is firm ’s output in the Cournot equilibrium. Since
for any
we have
∗
Let
be the monopoly output. Given outputs
increase its output from
∗
to
the two firms is the monopoly profit
∗
∗
∗
∗
if
while firm 2 keeps at
∗
∗
then firm 1 can
. Then, the joint profit for
which cannot be less than the joint profit
∗
in the
Cournot equilibrium. Also, when the total output is increased, the price must fall, implying
that firm 2 must be worse off. With the joint profit larger than before and with firm 2 losing
profit, firm 1’s profit must increase by deviating from
∗
to
Thus,
∗
∗
could not be a
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∗
Nash equilibrium, which is a contradiction. Therefore, we must have
∗
(see Figure
6.22).
p
MC
.
pm
*
.
p
pc
MR
.
.
y
D
m
*
1
y +y
*
2
y
c
y
Figure 6.22. Cournot Equilibrium
In general, with
firms, condition (6.11) becomes
∗
where
∗
∗
∗
(6.12)
is the joint output in Cournot equilibrium. In one extreme, when
(6.12) gives
∗
us the monopoly solution
In the other extreme, when
since
is always less than
∗
we have
and thus (6.12) gives us the competitive solution
the competitive solution
Most firms in reality seem to use their prices, not their quantities, as their strategy variables, yet the reality tends to produce an outcome that is close to the Cournot solution. That is,
the Cournot model gives the right answer for the wrong reason. One possible explanation for
this is capacity constraints. We can think of a quantity choice in the Cournot model as a longrun choice of capacity, with the price being an outcome of short-run price competition.
What is the outcome in a model in which firms first choose their capacity levels and then
compete on prices? Kreps–Scheinkman (1983) show that the unique subgame perfect Nash
equilibrium is a two-stage model with short-run capacity constraints being the Cournot outcome. In other words, the competition of price in the Bertrand model can be thought as the
second-stage price competition. We can thus think of the Cournot quantity competition as
capturing long-run competition through a capacity choice, with price competition occurring in
the short run given the chosen level of capacity. See Wang–Zhu (2001) for such a model.
Example 6.6. Consider two identical firms with
where
and with industry demand
Then, the firms’ profits are
(6.13)
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If both play Nash strategies in quantity, the FOC for firm is
(6.14)
By symmetry, with
∗
∗
we find the Nash equilibrium solution:
5.3. The Stackelberg Equilibrium
The Stackelberg equilibrium is a two-period solution. By applying the dynamic optimality
principle, the equilibrium can be solved using the backward-solving method.
Suppose that firm 1 is the leader and firm 2 is the follower, which means that firm 1 moves
first by taking into account firm 2’s possible reactions and firm 2 follows based on its own
reaction function. Firm 2’s reaction function is
∗
determined by (6.10). Then, firm
1’s problem is
(6.15)
as a constant. If (6.15) gives a
Notice that this is different from (6.9) in that (6.9) treats
solution
∗∗
∗∗
then firm 2’s optimal choice is
∗∗
∗∗
and the Stackelberg equilibrium is
∗∗
Example 6.7. Consider the firms in Example 6.6. Suppose now that firm 1 is the leader and
firm 2 is the follower. By (6.14), firm 2’s reaction function is
Thus, firm 1’s prob-
lem is
implying the Stackelberg equilibrium solution:
∗
∗
In this case, the leader has a larger output share and a higher profit. But, this is not always so.
In some cases, a firm is better off to be a follower rather than a leader.
5.4. Cooperative Equilibrium
The Nash game and the Stackelberg game are noncooperative games in which the two
players do not cooperate in any way to enhance their common objectives. A cooperative game
is a game in which the players have a common objective even though they may act in their own
best interests.46 In the duopoly case, if the two firms agree to cooperate with each other and to
maximize their total profit, they consider the cost minimization problem:
46
The fact that each player acts in his own best interest does not automatically mean that the game is
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,
and then the profit maximization problem:
The first-order conditions of the two problems for the cooperative equilibrium are
∗
where
∗
∗
∗
∗
∗
∗
∗
∗
∗
(6.16)
This problem is the same as the problem of a single monopoly with two
plants.
MC1
p*
MC
MC 2
.
.
p*
c ¢( y1* )
p*
.
c ¢( y 2* )
D
y1*
Firm 1
y
y 2*
MR
y
Firm 2
y
*
y
Industry
Figure 6.23. The Cooperative Solution of a Duopoly
The difficulty in a cooperative game is how to share the benefit among the players, although equations (6.16) can generally tell us how much each firm should produce (not determined in Figure 6.23. A feasible sharing arrangement will at least induce players to accept this
cooperative solution based on their own best interests. They may bargain for it, for example.
When information is incomplete, determining the best incentives for all the players is a further
tricky matter. For example, in the figure, we can see that the price is higher than the marginal
costs, implying that each firm can do better by expanding its own output. Hence, when the
output is not observable or enforceable, this equilibrium is prone to cheating.
Example 6.8. Reconsider the firms in Example 6.6. If both choose cooperation, they will maximize
Assuming equal shares, we find the solution:
noncooperative. For example, the players may work as a team for the total surplus of a project, but they may
compete for or react to their shares of the surplus in their own best interests.
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5.5. Competition vs Cooperation
In the duopoly, we have seen three possible games that the two firms may play: Nash, cooperation, and Stackelberg. Which game will they play? This question is like the famous story
of the prisoners’ dilemma.
The prisoners’ dilemma goes as follows. Two prisoners who will be in jail for the next two
years are suspected of having committed another crime. Each prisoner is told separately that if
they both confess to that crime, they will be convicted for the crime and each will stay an
additional year in jail; if one prisoner confesses to the crime and the accomplice does not, they
will be convicted of the crime and the confessor will stay one year less in jail and his accomplice will spend another eight years in jail; and if neither of them confesses, they will not be
convicted of the crime and they will continue to stay in jail for two more years. This is a twoperson game. Each player has two strategies: confess or deny. There are four outcomes and
their payoffs are listed in the following table, where, for example,
oner 1 loses
years of freedom and prisoner 2 loses
means that pris-
year of freedom.
Prisoner 2
Confess
Confess
Deny
-3, -3
-1, -10
-10, -1
-2, -2
Prisoner 1
Deny
Figure 6.24. The Prisoner’s Dilemma
The Nash equilibrium of this game is that both choose to confess. Furthermore, this equilibrium is a stronger type of equilibria called a dominant strategy equilibria. A dominant
strategy is a strategy that is the best strategy regardless of actions taken by the other players. A
dominant strategy equilibrium occurs when every player is taking a dominant strategy in
equilibrium. The Nash equilibrium for the prisoner’s dilemma is a dominant strategy equilibrium.
Let us now apply this story to the duopoly case. Consider the firms in Example 6.6. They
have two strategies: produce at the cooperative output or cheat by playing Nash strategies on
quantity. For the two firms in Example 6.6, if one firm, say firm 1, produces at the cooperative
quantity,
and the other firm, firm 2, cheats by playing a Nash strategy. Firm 2 will maximize
and produce at
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The profits in this case are
where subscript
stands for the Nash strategy and
stands for the cooperative strategy. The
profits are put into the following matrix box, where the entries are the profits without the
constant term
Firm 2
Nash
Nash
1/9,
1/9
Coop
9/64, 3/32
Firm 1
Coop
3/32, 9/64
1/8,
1/8
Figure 6.25. The Duopoly’s Game on Games
It is just like the prisoners’ dilemma. The Nash equilibrium for this game is
i.e.,
both play the Nash strategy. This Nash equilibrium is also a dominant strategy equilibrium.
For this result, we predict that the cooperative solution is not sustainable and the two firms
will end up in the Nash equilibrium.
5.6. Cooperation in a Repeated Game
For the prisoners or the firms in the duopoly, the Nash equilibrium is not a Pareto optimal outcome. Isn’t there some way by which the better solution can be achieved?
The duopoly game is played once. But if the duopolists play the game more than once,
they might find some way to cooperate. For example, if a game is played repeatedly, called a
repeated game, one player will be able to penalize the other player for ‘bad’ behavior by a
trigger strategy. If one player cheats once, the other will then refuse to cooperate from now on.
This form of punishment is referred to as the trigger strategy. By this trigger strategy, a player
may find that the one-time gain from cheating may not cover the future losses. If each player
takes this into account, none of them may cheat. Thus, a cooperative solution may be sustainable under the trigger strategy.
For the game in Figure 6.25, suppose that the two firms are in a cooperative equilibrium
and the discount rate of time preference is
With the trigger strategy, the net gain from
cheating is
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Hence, the cooperative solution is sustainable if
but the Nash equilibrium will be the
outcome if
6. Production Differentiation by Location
Even if firms have the same product, price differentiation may result if firms are located
in different locations and consumers have to pay transportation costs to purchase the product.
Consider two identical firms, firms
and
located at
and
selling the same product with a constant marginal cost
formly on
respectively on interval
Consumers are located uni-
The total size of the consumers is normalized to
and each consumer buys
one and only one unit of the good. The total cost of buying the product from firm is
where
is the price charged by firm ,
is the distance between the consumer and the firm,
and is a positive constant representing transportation cost. See Figure 6.26.
p2 + t (1 − z )
p1 + tz
p2
p1
0
1
ẑ
Figure 6.26. Location Differentiation
Let
be the location of the consumer, called the neutral consumer, who is indifferent be-
tween the two firms, where
satisfies
implying
We can see that consumers on the left of
the demand for firm 1’s product is
will go to firm 1 and the rest will go to firm 2. Then,
assuming
Thus, given
firm 1’s problem is
The FOC is
Given the symmetry of the model, we look for a symmetric equilibrium. With
∗
∗
the
above reaction function immediately implies the symmetric equilibrium:
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∗
(6.17)
∗
One natural question to ask is: why not lower down the price further as in the case in
Proposition 6.1? The explanation is that when a firm lowers its price, it will attract some consumers but not all the consumers. Due to the transportation cost, some consumers will not
change their choices. Hence, the firm has to weigh between attracting more consumers and a
lower price on all its sales. Therefore, it is really a case of product differentiation, not by physical characteristics of the product, but by location.
7. Location Equilibrium
Have you ever wondered why some shops selling similar products gather in a single building? For example, most computer shops in Hong Kong gather in several buildings only
throughout Hong Kong; each building houses many dozens of computer shops selling similar
products. Why don’t they spread out? By spreading out, they can charge a higher price for the
same product, as shown in (6.17). In the previous section, the firms understand the effect of
location and may thus locate themselves in the best locations. How will firms locate themselves in equilibrium?
Consider the same model as in the last section. Now, the two firms play a two-stage game,
in which the firms decide locations first and then play a Bertrand game in the second stage.
Suppose that firm sits at
the two firms, assuming
and charges price
47
Consider first the case with
then everyone at
er moves along
at
Given prices
a consumer
and locations
of
will go to firm 1 iff
(as opposed to
If the neutral person is at
will be neutral. This is because, given prices, when a consum-
his preference will not change.48 Symmetrically, if the neutral person is
then everyone at
will be neutral. Thus, if there is a Nash equilibrium with
the neutral person can be assumed to be at
Denote the neutral person at
as
For this person, we must have
implying
47 There
48 This
is no loss of generality with this assumption as we can always name the firm on the left to be firm 1.
is because, for
if
then, for any
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x1
0
x2
ẑ
1
will go to firm 1 and the rest will go to firm 2. Hence, the profits
All the consumers in
are
Given the locations, consider the Bertrand equilibrium. The FOC for firm 1 is
implying firm 1’s reaction function:
(6.18)
Symmetrically, firm 2’s reaction function is
(6.19)
Substituting (6.19) into (6.18) yields the equilibrium prices:
∗
∗
The prices are generally different. We find that the neutral person is
Then, the profits are
∗
∗
∗
∗
We have
∗
∗
∗
∗
Therefore, given
and given the assumption that
can; symmetrically, given
rium, we must have
If
firm 1 will move as close to
as it
firm 2 will also move as close to firm 1 as it can. Thus, in equilibIn other words, the two firms will locate at the same location.
one firm can move slightly to the left without changing its price, and this
firm will capture all the consumers on the left although it loses all the consumers on the right
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of
This firm will be better off since the size of consumers on the left of
on the right of
is larger than that
This cannot be an equilibrium situation. Therefore, in equilibrium, both
firms must sit in the middle:
∗
∗
When the two firms do sit together, the price behavior derived above no longer applies. In fact,
once they both sit together, Proposition 6.1 applies and the Bertrand equilibrium must be
∗
∗
Remarks:
1. If there are more than two identical firms, there does not exist a Nash equilibrium. The
reason is intuitive. For example, if there are three firms, it is still the case that the only possible equilibrium is the situation in which all three firms sit in the middle. But this situation
cannot be an equilibrium since one of them can capture half of the market, instead of one
third, by moving slightly to one side.
2. What will happen if
identical firms are located on a circle? What will be the equilibrium
positions? It seems that the Nash equilibrium must be the situation in which all firms sit
around the circle at an equal distance from each other. For example, if there are two firms,
they must have the same price in the Nash equilibrium. If so, no matter how they locate
themselves, they will get the same market share. However, if they sit opposite to each other,
the consumers will have the lowest overall transportation cost. If so, the profit must be the
highest. Thus, the Nash equilibrium is one in which the two firms sit opposite to each other.49
3. In reality, consumers are distributed on a flat surface, rather than on a line or a circle. What
will happen? If there are no buildings to block a path, the behavior of the two firms should
be just like the two firms on a line. That is, the two firms should sit together in the middle
of the region in equilibrium. In this case, on a surface, a Nash equilibrium exists for an arbitrary
4. However, the non-existence of the equilibrium is due to the naiveness of the Nash equilibrium concept — each player acts on the assumption that others will not react to his move.
Under the reactive equilibrium concept,50 a player will not make a move if others’ reactions
can cause him to lose in the end. Under this equilibrium concept, the situation where all
firms sit in the middle is an equilibrium. In fact, any situation in which all the firms sit together is a reactive equilibrium. A reactive equilibrium is like a Stackelberg equilibrium but
49
In this case, we need to include a participation constraint: the consumer is willing to buy a unit from firm
where is the consumer’s utility value from consumption of the product. See Mas-Colell et al.
only if
(1995, p.399).
50 The
reactive equilibrium is defined by Riley (1979). Wilson (1977) defines the anticipated equilibrium,
which is very similar to the reactive equilibrium.
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for a simultaneous-move game in which each player treats other players as followers after
his move.
x2
x3
x1
0
1
5. There are two standard models of spatial competition in the literature: the linear city model
pioneered by Hotelling (1929), and the circular city model pioneered by Lerner and Singer
(1939), developed by Vickrey (1964) and made popular by Salop (1979).
8. Entry Barriers
A monopolist industry can be a result of entry barriers and entry costs. In this section, we
use a simple model to discuss the effect of entry cost on the formation of an industry.
Consider an industry in which
Stage 1. All potential firms simultaneously decide to be in or out. If a firm decides to be in, it
pays a setup cost
This
is a sunk cost.
Stage 2. All firms that have entered play a Cournot game.
We use simple cost and demand functions:
(6.20)
where
and
with
are two constants. In stage 2, suppose that there are
firms in
the industry. Then, each firm considers the following problem
The reaction function is
In the symmetric equilibrium with
∗
∗
the above reaction function implies
∗
which implies a profit of
∗
The total output
∗
∗
∗
approaches the competitive output
when
How many firms will the industry have in equilibrium? Firms will continue to enter the
industry until the expected profit is zero:
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which implies the equilibrium number of firms:51
∗
As
decreases,
∗
increases and the total output increases. Indeed, as
we have
∗
That is, a reduction in entry cost will increase the number of firms in equilibrium.
On the other hand, an increase in the intensity of competition may reduce the number of
firms in equilibrium. See Question 6.5 in the problem set for the result of the Bertrand competition in the second stage, by which more intense competition in stage 2 lowers the equilibrium level of competition in the market.
Let
denote the output of a firm in an -firm industry. All the firms are identical with
the same cost function
Social welfare is
Instead of a zero-profit condition, let
be the socially optimal number of firms. For the specific functions defined in (6.20), with
the socially optimal number of firm, determined by
is
/
(6.21)
/
Therefore,
/
∗
implying
∗
That is, there are too many firms in the industry. This is due to the loss of
social welfare
for each entry. Without it, the socially optimal number of firms is infinity.
The following proposition from Mankiw–Whinston (1986) is a general result on entry bias.
For the specific functions defined in (6.20), the three conditions in the proposition are all
satisfied.
Proposition 6.3. Suppose
and
Let
be the symmetric equilibrium
output. Assume
(1)
is increasing in
is decreasing in
(2)
for all
(3)
Then,
51 If
∗
an integer is needed, the equilibrium number of firms is
∗
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9. Strategic Deterrence Against Potential Entrants
Incumbent firms in an industry often make strategic investments to deter potential entrants. These investments include investments in cost reduction, capacity, and new-product
development.
Consider a two-stage duopoly model in which firm 1 is the incumbent and firm 2 is a new
entrant:
Stage 1. Firm 1 has the option to make a strategic investment
Stage 2. If firm 2 enters the industry, firms 1 and 2 play a Cournot game, choosing strategies
respectively, resulting in profits
and
respectively.
Let the reaction functions be
∗
Suppose that there is a unique Nash equilibrium
∗
in stage 2 satisfying the local
stability condition:
(6.22)
Suppose also that
(6.23)
These two are fairly natural assumptions. By differentiating the equilibrium condition
∗
∗
w.r.t.
we find that
∗
We also have
∗
∗
∗
∗
∗
∗
where we have used the Envelope Theorem or the FOC
∗
Hence, to discourage entry, by
(6.22) and (6.23), we need
(6.24)
implying a negative effect of
on
investment for firm 1. If there is a
Here, condition (6.24) means that
such that
∗
∗
is a productive
then entry can be prevent-
ed.
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However, investing in
larger
may be costly for firm 1. We also need to look at the effect of a
on firm 1’s profit. Similarly, using the Envelope Theorem and equation
∗
∗
we find that
∗
and
∗
∗
∗
∗
∗
∗
(6.25)
The first term on the right side of (6.25) is the direct effect, and the second term is the strategic effect, which depends on the strategic response of firm 2. There are two possible responses:
Under condition (6.24), if
is a strategic substitute of
the strategic effect is positive. If the
direct effect is positive or not very negative, firm 1 will expand
without any hesitation to
prevent entry.
Example 6.9. Suppose that firm 1 is the incumbent and firm 2 is a new entrant. Let
where
and
with
are constants,
and
The profit functions
are
where
is the entry cost. Conditions in (6.23) are satisfied. If the second-stage game is a
Cournot game, the reaction functions are
Condition (6.24) is satisfied, implying
the strategic effect on
∗
Also, since
and
are strategic substitutes,
is positive. The solution is shown in the following figure. When
increases, firm 1’s reaction curve shifts out, implying
∗
which in turn implies
∗
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y2
yˆ 1 = yˆ 1 ( y 2 , k )
k↑
. .
yˆ 2 = yˆ 2 ( y1 )
y1
Figure 6.27. Equilibrium under Strategic Deterrence
In fact, the equilibrium outputs are
∗
∗
and the profits are
∗
∗
∗
We can easily see that
can also have
∗
if
implying that an increase in
is negative enough, i.e., if
reduces the threat of entry. We
is very useful in cost reduction.
10. A Competitive Input Market
Factors of production can be divided into four broad categories: labor, capital, land and
raw materials. The owners of factors of production receive incomes from the firms that use
those factors as inputs. These incomes, which are the opportunity costs to firms of using those
input factors, are payments made for raw materials, wages paid for labor, rental rates paid for
capital, and rent paid for land. In the short run, usually labor and raw materials are variable
inputs while capital and land are fixed inputs.
Given revenue function
and cost function
define their counterparts for input
for output
and production function
52
The marginal revenue product is the increase in revenue resulting from one more unit of input:
The marginal cost product is the increase in cost resulting from one more
unit of input:
52 Here,
Then,
we can allow
notation, we treat
to be a vector of inputs,
All the results still hold. However, for simplicity of
as a scalar.
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where
is the marginal product of input. By (6.1), we find that maximum profit
means that the marginal revenue product
equals the marginal cost
∗
(6.26)
∗
Just like (6.1), this condition is applicable to any type of firm in the input market.
10.1. Demand and Supply
In a competitive input market, input prices are determined in input markets by demand
and supply, much the same way as output prices are determined in goods markets. A profitmaximizing firm that is a competitive company in the input market takes the market price for
input as given. This means that
(6.26), the optimal input
∗
where
is determined by
is the given market price for input
∗
By
This means that a competitive
firm’s demand curve for input is its MRP curve (see Figure 6.28):
(6.27)
w
w
.
S : MCP ( x )
MRP ( x )
xd
x
Figure 6.28. The Optimal Problem for a Competitive Buyer
In a competitive input market, the industry demand curve is the horizontal sum of all the
firms’ demand curves. The industry supply curve is taken as given and is typically a upward
sloping supply curve. The price and quantity traded for a factor of production are determined
in equilibrium by the intersection of the demand and supply curves. If time horizon is taken
into account, this is a short-run equilibrium. In a long-run equilibrium, the number of firms is
endogenously determined in equilibrium by the zero-profit condition.
10.2. Equilibrium and Welfare
Economic rent is the income received by the supplier over and above the amount required
to induce him to offer the input. Transfer earnings is the income required to induce the supply
of the input. These two together yield the total income of an input:
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w
S
.
w*
Economic
Rent
Transfer
Earnings
0
D
x
x*
Figure 6.29. Economic Rent and Transfer Earnings
In Figure 6.29, the income of the factor is the square area, which is split into two parts:
the transfer earnings and the economic rent. The supply curve shows the minimum price at
which an additional factor unit is willingly supplied. If the supplier receives only the minimum
amount required to induce him to supply each unit of the factor, he will be paid a different
price for each unit; those prices will trace the supply curve and the earnings received will be
the transfer earnings. The economic rent is similar to the consumer surplus, which is the
difference between what the supplier is willing to accept and what is actually paid to him. In
two extreme cases, when the supply is perfectly inelastic (e.g., land) and the supply curve is
vertical, the entire income is the economic rent, as shown in Figure 6.30; when the supply is
perfectly elastic (e.g., unskilled labor) and the supply curve is horizontal, the entire income is
the transfer earnings.
S
S
r
r
Transfer
Earnings
Economic
Rent
D
D
Land
Unskilled labor
Figure 6.30. Land and Unskilled Labor
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Example 6.10 (Housing Market). An equilibrium in the housing market is shown in Figure
6.31.
S
Sn
.
.
r
Dn
New Housing Flow
Flow Equilibrium
D
Housing Stock
Stock Equilibrium
Figure 6.31. Short-Run vs Long-Run Equilibria in the Housing Market
The left chart represents the newly built units, which are flows of demand and supply. The
right chart represents the stock of demand and supply, which determines the price or the
rental rate. The housing stock is a balance between newly built units and demolished units.
The housing stock is fixed in the short run. When new units exceed demolished units, the
housing stock will increase over time, and vice versa. For each equilibrium flow in the new
housing supply on the left chart, there is a balance between newly built units and demolished
units, which determines a corresponding housing stock on the right chart.
Three cases are presented here for discussion.
(1) An earthquake destroys a chunk of housing stock, which pushes up the rental rate immediately from
to
The short-run supply of new housing units increases and it is larger
than the original equilibrium supply of new housing, which leads to an accumulation of the
housing stock over time. In the end, the housing stock will return to its original level together
with the rental rate, as shown in Figure 6.32.
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S
Sn
.
r'
.
r
.
D
Dn
New Housing
Flow Equilibrium
Housing Stock
Stock Equilibrium
Figure 6.32. An Earthquake
that is lower than the mar-
(2) Suppose that the government imposes a rental ceiling
ket rate
With the policy, the demand curves are changed, as shown in Figure 6.33.53 It re-
duces the flow of new apartments. The housing stock thus decreases over time. In the long-run,
the ceiling rate will be the market rate, but the housing stock is reduced as a result.54
S
Sr
r
rmax
.
rmax
.
D
Dr
Housing Flow
Flow Equilibrium
Housing Stock
Stock Equilibrium
Figure 6.33. Rental Ceiling
(3) The Hong Kong Government restricts the supply of land, i.e., it pushes
up. It will
lead to a reduction of housing stock and a higher rental rate
53
Since a demand curve represents the maximum price (that the demander is willing to accept), a price ceiling
is imposed on the demand curve. Since a supply curve represents the minimum price (that the supplier is willing to
accept), a price floor is imposed on the supply curve.
54 Since
the rental rate is low, the landlords may have less incentive to maintain housing quality, which may
reduce demand further (a leftward shift of the demand curves). If this reduction is large enough, it may cause an
even lower equilibrium housing stock.
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S
Sn
..
..
r'
r
D
Dn
New Housing
Flow Equilibrium
Housing Stock
Stock Equilibrium
Figure 6.34. Restricting Land Supply
11. A Monopsony
If there is only one firm that demands a factor of production in the input market, then this
firm is called a monopsony. When the firm is a monopsony in the input market, the price of
input will be determined by the supply curve, i.e., the price
i.e.,
of the input will depend on input
Assuming a single-price monopsony (the firm pays a single price for all its
units of input, just like a single-price monopoly), the
is
(6.28)
The formula (6.28) can be understood by analysis of Figure 6.35. When you buy one more unit
of
you pay
for that unit; but, in addition, since the price for all units is increased by
you also pay
for the existing units
Hence, the total cost for that additional unit
is
w
ws ( x)
w' ( x )
{
w( x )
{
Δ C = w + x ⋅ Δw
x
Δx = 1
x
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Figure 6.35. MCP
The monopsony’s problem is
The FOC is:
∗
∗
∗
∗
This can be written as
∗
where
∗
is the price elasticity of supply. As expected, when
goes to infinity, the behavior of a
monopsony approaches that of a competitive firm in the input market.
is the marginal reve-
This equilibrium is illustrated in Figure 6.36, where
nue of input
defined by
is the marginal cost of input
55
∗
The condition
∗
and
is the supply curve
determines the optimal
is determined, the monopsony picks the lowest possible price
∗
Once
∗
to purchase the input. As
shown in the figure, the factor price and quantity in the monopsony equilibrium
are lower
than those in the competitive equilibrium
w
MCP
S : W ( x)
.
wC
wM
.M
.C
D : MRP
xM
xC
x
Figure 6.36. Monopsony
Both the monopoly and monopsony equilibria can be thought of as Stackelberg equilibria
in which the firm is the leader and the consumers or the suppliers of inputs are the followers.
Example 6.11 (Minimum Wage). Suppose that the government imposes a minimum wage
on the labor market. This means that the part of the supply curve where laborers are willing to
accept a lower wage becomes
55 We
the
can draw the
That is, the minimum wage condition replaces the
curve for a linear supply curve
to get some idea about the position of
curve in the figure.
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original upward-sloping labor supply curve
by the horizontal supply curve
for the lower part of the original supply curve.
The impact of this minimum wage on employment depends on the nature of the labor
market. If the labor market is competitive, as shown in Figure 6.37(a), the equilibrium employment of labor is reduced from
to
However, if the labor market is dominated by a
monopsonist, as shown in Figure 6.37(b), the employment will be increased from
to
In
both cases, the equilibrium wage rate will be the minimum wage, but the implication on employment is different.
MCP
MCP '
S
S' S
S'
wmin
wc
wmin
wm
D
lmin
D
lm l min
lc
( a ) Competitive
(b) Monopsony
Figure 6.37. Minimum Wage
In the case of monopsony, the new
but coincides with
when
intersect with the demand curve
is the broken curve that takes value
when
in Figure 6.37(b) and we can see that it does not
which means that the FOC will not be satisfied and the
solution must be a corner solution. At level
there are two possible corner solutions and
the lower one must be the one (the upper one is above the firm’s willingness to pay). We can
also look the monopsony’s surplus as shown in Figure 6.38 to see where the monopsony maximizes its profit/surplus.
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MCP
MCP '
S' S
Surplus
wmin
wm
D
lm l min
Figure 6.38. Monopsony’s Profit Maximization
For a government policy, why do we change the demand curve in Figure 6.33 but change
the supply curve in Figure 6.38? The reason is that a demand curve represents a price ceiling,
while a supply curve represents a price floor. Hence, when a policy represents a new price
ceiling, it is imposed on the demand curve; when a policy represents a new price floor, it is
imposed on the supply curve.
12. Vertical Relationships
So far, firms’ relations have been horizontal: firms interact with each other for the same
good in the same market (input or output market). This section considers an example in which
two firms interact with each other vertically: one firm is the supplier of the other.
Consider a situation in which there is a upstream firm that produces output
and there is downstream firm that inputs
to produce output for revenue
with cost
The
upstream firm is a monopolist in its output market (the downstream firm’s input market). The
downstream firm is a competitive firm in its input market. The nature of the upstream firm in
its input market is captured in its cost function
output market is captured in its revenue function
the nature of the downstream firm in its
Consider a simple case, as shown in
Figure 6.39, with
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Upstream firm
Cost c ( x ) = cx
x
Input Market
Monopolistic supplier
Competitive demander
x
Downstream firm
Revenue R(x) = (a - bx)x
Figure 6.39. Vertical Relationship
12.1. Independent Firms
Suppose that the two firms are independent. Since the downstream firm is a competitive
firm in the input market, it will take the input price
as given. Its problem is
implying
which implies a demand function for
Then, the upstream monopolist’s problem is
which implies
∗
(6.29)
12.2. An Integrated Firm
Suppose now that the two firms merge into one firm. The integrated firm has revenue
function
and cost function
The integrated firm’s problem is
implying
(6.30)
From (6.29) and (6.30), we can see that the integrated firm produces double of the independent firms.
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12.3. Explanation
The reason is as follows. Given revenue function
for the downstream firm, its prob-
lem is
implying
This is the demand function for the upstream firm. Then, the problem of the upstream firm is
That is, the revenue function for the upstream firm is
∗
(6.31)
∗
which is the marginal cost for the upstream firm. In Figure 6.40, the
where
upstream firm faces demand
and has marginal revenue
∗
Since it is a monopoly, according to
price
The FOC is
∗
cost
∗
and marginal cost
the upstream firm sells output
The downstream firm, on the other hand, has marginal revenue
∗
and marginal
∗
Since it is a competitive firm in the input market, its marginal cost is fixed at
∗
price
∗
Again, according to
∗
at
the downstream firm buys input
∗
at
∗
On the other hand, the problem for the integrated firm is
which implies
MCU
.
w*
MCU
MC D
MRU
.
.
D = MRD
x*
x
Figure 6.40. Independent Firms
MRD
x
x
Figure 6.41. Integrated Firm
(6.32)
As shown in Figure 6.41, by the condition
the integrated firm inputs at
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From Figure 6.40, since
is about half way below
∗
will be about half of
In
general, we can show that an integrated firm will always produce more than an upstreamdownstream pair of independent firms. With an upstream-downstream pair, the upstream
monopolist cuts its input and raises its price above its MC; the downstream competitive firm,
as a price taker, passively follows the upstream monopolist’s choices.
We can similarly consider other alternative situations. If both are competitive in their own
markets, the solution with separate firms is the same that under integration.
If the upstream firm is competitive in its output market and the downstream firm is a
monopsony in its input market. In this case, the upstream firm’s problem is
which implies the inverse supply function:
Then, the downstream firm’s problem is
which implies
This equation determines the optimal solution
a supply function
∗
We can also use backward induction. Given
the downstream firm’s problem is
which implies
(6.33)
The upstream firm’s problem is
which implies the inverse supply function:
That is, the supply function is
Substituting this into (6.33) yields
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MC D S = MCU
.
.
w*
MCU
.
MRD
x*
Figure 6.42. Independent Firms
x
MRD
x
x
Figure 6.43. Integrated Firm
If the downstream firm is competitive, it offers a demand curve for the other firm to manipulate, as shown in Figure 6.40; if the upstream is competitive, it offers a supply curve for
the other firm to manipulate, as shown in Figure 6.42. If both firms are competitive, one offers
a demand curve and other offers a supply curve; the intersection point is the solution. If both
firms are monopolistic in their own market, no one is to offer a demand or supply curve for the
other to manipulate; if so, the two firms have to bargain to settle for a trade.
Notes
Good references are Varian (1992, Chapters 13–16) and Mas-Colell et al. (1995, Chapter
12).
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Chapter 7
Imperfect-Information Games
In the last chapter, we saw that game theory is a powerful tool in dealing with the economic problems, especially when there are a small number of economic agents with conflicts
of interest. The focus of this chapter is on problems under imperfect information. We will
present several popular equilibrium concepts.
There are two kinds of disinformation: imperfect information and incomplete information
relating two pieces of information: knowledge of actions taken by other players and knowledge
of other players’ payoffs. When each player perfectly knows other players’ actions, it is a game
of perfect information; otherwise it is a game of imperfect information. If the players’ payoffs
are all common knowledge, it is a game of complete information; otherwise, it is a game of
incomplete information.
1. Two Game Forms
Games come in various sorts. Many can only be roughly described in words. However,
many relatively simple games can be clearly defined by the two standard forms: the extensive
form and the normal form. This section defines these two forms of games.
1.1.
The Extensive Form
The extensive form relies on a game tree. We first present two examples, by which we gain
some basic understanding of a game tree.
Example 7.1 (Matching Pennies under Perfect Information). There are two players 1 and 2, P1
and P2. Player 1 puts a penny down first. Then, after seeing player 1’s choice, player 2 puts her
penny down. If the sides of the two pennies match (heads or tails), player 1 pays $1 to player 2;
otherwise player 2 pays $1 to player 1.
Susheng Wang, HKUST
.
P1
H
P2
T
. P2
.
H
 − 1
 
 + 1
T
H’
 + 1  + 1
  
 − 1  − 1
T’
 − 1
 
 + 1
P1’s payoff
P2’s payoff
Figure 7.1. Matching Pennies under Perfect Information
The game starts at the initial decision node and ends at a terminal node. The terminal nodes
are assigned payoffs and the non-terminal nodes are decision points. The branches represent
possible moves/actions of each player.
Example 7.2 (Matching Pennies under Imperfect Information). This game is just like the
previous game except that when player 1 puts her penny down, she keeps it covered. Hence,
when player 2 moves, he does not know what player 1 has chosen.
.
P1
.
P1
H
.
P2
H
Information Set
.
.
Information Set
.
H
T
H
T
T
H
P2
T
T
 − 1
 
 + 1
 + 1  + 1
  
 − 1  − 1
 − 1
 
 + 1
P1’s payoff
P2’s payof
Figure 7.2. Matching Pennies under Imperfect Information
The game is presented in the game tree on the left. In the game tree, player 2’s two decision
nodes are contained in an information set. This means that, when it is time for player 2 to
make a choice, he does not know which decision node he is at, since he does not know which
choice player 1 has made. The only thing that player 2 knows is that the game has arrived at
his information set and he has to make a choice without knowing which decision node he is at.
Hence, player 2’s decision will be dependent on the knowledge of his own information set
rather than on the knowledge of one of his decision nodes. In order to indicate payoffs, we
draw two identical copies of P2’s actions, as indicated in the game tree on the right.
Note that the game tree is the same if the two players move simultaneously. As long as
they cannot observe each other’s choices, the timing of the moves is irrelevant in a simultaneous-move game.
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Notice that, in Example 7.1, player 2’s possible actions
following
from the possible actions
following
following
are different
However, in Example 7.2, the possible actions
are the same as those following
This is because the strategies in Example
7.2 are dependent on information sets rather than on decision nodes. This will be clear in the
following definition.
With observations of some typical game trees in the above examples, we now formally define a game tree.
Definition 7.1. A game in extensive form consists of the following items:
1. Sets. A set of nodes
a set of possible actions
tion of information sets
a set of players
and a collec-
56
2. Sequence. A game starts from a single node. Except the initial node, each node follows from
a single immediate predecessor node. The set of terminal nodes is
All other nodes in
are called decision nodes.
3. Information Structure. Each decision node belongs to one and only one information set.
Denote
as the information set that contains node
we often refer to
as node
When
is a singleton, i.e.,
Each information set is followed by a few
branches. Each branch represents a possible action taken by the player who is to make a
decision upon observing that information set, i.e., when the play reaches that information
set. For
let
move at an information set
all the branches following
we call
If it is player s turn to make a
player ’s information set. Each information set be-
longs to one and only one player, including a special player called nature.
4. Nature. Sometimes nature is included. Let
move. A function
be the information set where nature makes a
assigns probabilities to actions at information set
Nature is like a player in the model, except that it does not have a payoff function and it
does not optimize its choices.57
5. Payoffs. A collection of payoff functions
assigns utilities to the players
at each terminal node,
Thus, a game in extensive form is specified by the collection
56
We can actually allow infinite steps, infinite possible actions, and infinite players.
57
In game theory, if nature is involved in a game, nature typically moves first. In particular, in a game of
incomplete information, nature always moves first.
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Figure 7.3 shows a typical game tree.
.
r0¢ a0¢
a 0 r0
P1
a1
.
P1
x1
a1¢
x1¢
a1
a1¢
.
a2¢
A( x1 ) = {a1 , a1¢} , A( x1¢) = {a1 , a1¢}
.
x2¢
H
x2
a2
.
a2
P2
H = { x2 , x2¢ }
a2¢
Figure 7.3. A Game Tree
The payoffs of the players are actually not clearly defined in an extensive-form game. For
example, for the game in Figure 7.2, since P2 does not know P1’s action, P2 does not know the
payoffs of his actions. Only in two situations, each player know the payoffs of his actions. In a
Nash equilibrium where other players are taking equilibrium strategies, each player knows the
expected payoffs of his actions. In a Bayesian game where players form beliefs, each player
knows the expected payoffs of his actions.
A game is a perfect-information game if every information set is a singleton set; otherwise,
it is an imperfect-information game.
The structure of the game is common knowledge, in the sense that all players know the
structure of the game, know that their rivals know it, know their rivals know that they know it,
and so on.
We will only consider games of perfect recall, meaning that a player does not forget what
she once knew, including her own actions.
1.2. The Normal Form
A strategy is a complete contingency plan that specifies how a player will act in every possible distinguishable circumstance. The set of circumstances for a player is his set of information sets, with each information set representing a different distinguishable circumstance
in which he may need to make a move. Thus, a player’s strategy amounts to a complete specification of how he plans to move at each of his information set.
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Definition 7.2. Let
denote the collection of player ’s information sets,
actions in the game, and
the set of possible
the set of possible actions at information set
strategy for player is a function
the strategy space of player
such that
for all
A (pure)
Denote
as
which contains all the possible strategies of player
We motivate the definition of the two forms using two popular simple examples.
Example 7.3 (Matching Pennies under Perfect Information). Given his information set, player
1 has two strategies:
Player 2 has two information sets. Denote a typical strategy of Player 2 as
is an action if the information set on the left is reached and
where
is an action if the infor-
mation set on the right is reached. Hence, player 2 has four possible strategies:
Their strategy spaces are
and
Example 7.4 (Matching Pennies under Imperfect Information). Each player has one
information set. Each player has two strategies
and
Thus,
for both
players.
where
Denote a profile of strategies as
is a strategy from player
The normal form of a game is a specification of strategies and their associated payoffs.
Definition 7.3. For a game with
a set of strategies
players, the normal form of a game specifies for each player
and a payoff function
Formally, we write the game as
Example 7.5 (Matching Pennies under Perfect Information). The strategy sets are already
defined in Example 7.3. The payoff functions are
A more convenient way to present this game is the game box below:
P1\P2
-1, +1
-1, +1
+1, -1
+1, -1
+1, -1
-1, +1
+1, -1
-1, +1
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Example 7.6 (Matching Pennies under Imperfect Information). Given the strategies in Example 7.4, the normal form is
P1\P2
-1, +1
+1, -1
+1, -1
-1, +1
A game is usually first described by a game tree or the extensive form. We can view the extensive form as the original representation of a game and the normal form is a reduced form of
the extensive form. In the normal form, it is as if the players simultaneously pick a strategy
from the their own strategy set
Dynamic features in the original extensive form are no
longer obvious in the normal form. In an extensive-form game, there is a unique normal form;
the converse is however not true. For example, in Example 7.5, the normal form can also be
derived from the extensive-form game in Figure 7.4.
P1
s11
s21
P2
s12 s22 s32
−1 −1
   
+1 +1
s42
 + 1
 
 − 1
s12 s22 s32
 + 1  + 1 −1
     
 − 1  − 1 +1
 + 1
 
 − 1
s42
−1
 
+1
Figure 7.4. An Alternative Game Tree
Because of the condensed representation, the normal form generally omits some of the details
present in the extensive form. We wonder whether this omission is critical and whether the
normal form summarizes all of the strategically relevant information. This question has long
been a subject of discussion among game theorists. In my opinion, the answer depends on the
equilibrium concept. For some equilibrium concepts, these two forms are equivalent; but for
others, these two forms are not equivalent.
1.3. Mixed Strategy
Instead of following a certain strategy for sure, we now allow players to follow a strategy
with a certain probability. We call the probabilistic distribution of a player’s strategies a mixed
strategy and call the original strategies pure strategies.
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Definition 7.4. Given a normal-form game
mixed strategy as
mixed strategy
extension of
where
for
we denote a
is the probability that
is taken. That is, a
is a probability distribution over the pure strategies in
Denote the mixed
as
We sometimes denote
as
i.e.,
is the probability that the mixed strategy
assigns to the pure strategy
as a profile of pure strategies, where
We denote
player
Similarly, we denote
is a pure strategy from
as a profile of mixed strategies, where
is a
mixed strategy from player
There is another way that a player can randomize. Rather than randomizing over the potentially very large set of pure strategies in
she could randomize separately over the possi-
ble actions at each of her information sets
This is called a behavior strategy.
Definition 7.5. Given an extensive-form game
a behavior strategy for player specifies, for
every information set
with
a probability distribution
over the actions in
and
For games with perfect recall, the two types of randomization are equivalent. Because of
this, we typically use behavior strategies for extensive-form games and mixed strategies for
normal-form games. In fact, we will refer to behavior strategies as mixed strategies as well.
Since a player’s pure strategy is a special mixed strategy, for example,
we
can write a mixed strategy as
This representation of a mixed strategy is convenient in some occasions.
We also need to extend payoffs to include payoffs over mixed strategies. We introduce
some notation first. Denote
We have fined payoffs for pure strategies, but we have not defined payoffs for mixed strategies.
We use expected utility to define payoffs for mixed strategies. Given a profile of mixed strategies
the payoff is
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(7.1)
,
ℕ
is the probability that player follows his strategy
where
players
with strategy sets
For example, for two
and
∈
we have
,
,
We will now introduce equilibrium concepts to the two forms of game. We consider the
normal form first in Section 2 and then the extensive form in Section 3.
2. Equilibria in Normal-Form Games
We study simultaneous-move games using the normal form. We define three equilibrium
concepts: the Nash equilibrium, the dominant-strategy equilibrium, and the trembling-hand
Nash equilibrium.
2.1. Nash Equilibrium
if, for every
∗
∗
Definition 7.6. A mixed strategy profile
∗
we have
∗
is a Nash equilibrium (NE) of
∗
∗
for all
If an NE consists of pure strategies only, we call it a pure-strategy NE; otherwise it is
called a mixed-strategy NE.
Proposition 7.1. The strategy profile
if and only if, for each
58
∗
∗
∗
is a Nash equilibrium in
58
∗
∗
∗
∗
(7.2)
∗
∗
∗
∗
(7.3)
Condition (7.2) is usually enough to determine a mixed strategy NE. However, condition (7.3) cannot be
deleted. There are cases in which one player strictly prefers one strategy no matter what the other player does; in
this case, (7.2) never happens. That is, one player plays a strategy with probability
in equilibrium. In this case, we
need (7.3) to identify the equilibrium.
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∗
Proof. Necessity.
is the solution of the following problem:
∗
,…,
By introducing the Lagrangian multiplier
the problem is equivalent to
∗
(
,…,
∗
)
Since this objective function is linear in
we immediately have the following three necessary
conditions: for all
∗
∗
∗
∗
∗
∗
These three conditions immediately imply (7.2)–(7.3).59 Notice that if
∗
for some
(7.2)
never happens.
∗
Sufficiency. Given conditions (7.2) and (7.3), if
some player i who has a strategy
strategy
all
∗
under
∗
with
∗
for which
∗
∗
∗
∗
If so, there must exist a pure
∗
Since by (7.2)
∗
∗
for
we find that
∗
Note that
is not a Nash equilibrium, then there is
∗
cannot be empty. If
under
∗
(7.4)
(7.4) contradicts (7.2); if
(7.4) contradicts (7.3). Either way, we have a contradiction. Thus,
∗
under
∗
must be a Nash equilibri-
um.
∗
Proposition 7.2. If a strategy profile
be a NE in
∗
is a NE in game
it must
In other words, if a pure strategy is a NE strategy from
the pure-strategy set, it must be a NE strategy from the mixed-strategy set.
Proof. Suppose
∗
∗
∗
is a Nash equilibrium in game
Then, there is a mixed strategy
so, there must exist
59
∗
such that
but not in
such that
∗
∗
∗
∗
∗
If
This contradicts the fact that
We can also prove the necessity in the following way. If either (7.2) or (7.3) does not hold, then there are
strategies
and
such that
whenever he is supposed to play
∗
∗
If so, player
would do better by playing
in the original Nash equilibrium, implying that the original equilibrium is not
an equilibrium.
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∗
∗
is a Nash equilibrium in
um in
Thus,
∗
∗
must be a Nash equilibri-
60
we use Propo-
In summary, to find pure strategy Nash equilibria in
sition 7.2 to restrict the game to
To find mixed strategy Nash equilibria
(when at least one player takes a totally mixed strategy), we use the two NE conditions in
Proposition 7.1.
Example 7.7. In the following game, among the pure strategies in
there are obviously two pure-strategy NEs:
they are also NEs in mixed-strategy spaces
and
and
By Proposition 7.2,
and
P1\P2
2, 3
-2, 2
-2, 3
4, 5
Do we have mixed-strategy NEs? Assume that P1 mixes his two strategies. By (7.2), P1 must be
indifferent between his two pure strategies, i.e.,
∗
implying
∗
∗
∗
∗
Conversely, since P2 mixes her two strategies, P2 must be indifferent
between her two pure strategies, i.e.,
∗
implying
∗
∗
∗
∗
This result is consistent with our initial assumption that P1 mixes his two
strategies. Hence, we have a unique mixed-strategy NE with
∗
and
Example 7.8. In the following game, there are two pure-strategy NEs:
P1\P2
L2
R2
-2, -1
1, 0
-1, 2
1, 1
∗
and
Assume that P1 mixes his two pure strategies. Let P2’s strategy be
for any
Since P1 mixes his two pure strategies, he must be indifferent between his two pure
strategies, implying
implying
∗
Conversely, let P1’s strategy be
taken with zero probability
∗
and
for any
Since
is
is taken with positive probability (actually with
probability ), according to Proposition 7.1 (the second condition for NE), we have
60
We can alternatively use condition (7.1) to show this. It is straightforward.
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implying
∗
For
, P1 mixes his two strategies, which is consistent with our
initial assumption. Hence, we have a mixed-strategy NE:
Assume that P2 mixes her two pure strategies. Then,
NEs are obviously not unique. Can we guaranteed the existence of at least one NE? The
existence of a mixed-strategy NE is guaranteed for a finite game. The existence of a purestrategy NE can also be guaranteed only under certain conditions. The proofs of the following
two existence results can be found in Mas-Colell et al. (1995, p.260).
Proposition 7.3 (Existence). If
are finite sets, there exists a Nash equilibrium
in
If, for all
Proposition 7.4 (Existence). A Nash equilibrium exists in
(1)
(2)
is nonempty, convex, and compact subset of some Euclidean space
is continuous in
and quasiconcave in each
2.2. Dominant-Strategy Equilibrium
Definition 7.7. A strategy
for player
if there exists another strategy
all
In this case, we say that strategy
is strictly dominated in game
such that
strictly dominates strategy
is a strictly dominant strategy for player in game
every other strategy in
another strategy
Similarly, a strategy
such that
for
A strategy
if it strictly dominates
is weakly dominated if there exists
for all
and with strict
inequality for some
In words, a strategy is a strictly dominant strategy for player if it the best strategy for
him no matter what strategies his rivals may play.
In the prisoner’s dilemma mentioned in Chapter 6, the Nash equilibrium is that both confess and this is a dominant strategy equilibrium. It is a paradigmatic example of self-interested
rational behavior that leads to a socially inferior result for the players.
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Proposition 7.5. Player i’s strategy
iff
is strictly dominated by
in game
for all
we have
Proof. By (7.1), given any
∈
∈
Hence,
∈
This expression is positive for all
iff
is positive for all
This proposition indicates that when we test whether a strategy
another strategy
for player
is strictly dominated by
we only need to test these strategies against the pure strategies
of the opponents.
∗
Definition 7.8. A mixed strategy profile
(DSE) in
∗
∗
is a dominant-strategy equilibrium
∗
if for every
for all
and
Proposition 7.7.6. A DSE is a NE in which each player’s equilibrium strategy is a dominant
strategy.
The following proposition suggests that, to reduce the complicity of finding NEs, we can
first eliminate some dominated pure strategies since they will never be played in a NE.
if a pure strategy
Proposition 7.7. For player
that never uses
(assigns zero probability to
(assigns a positive probability to
is strictly dominated by a mixed strategy
then every mixed strategy that uses
is strictly dominated by a mixed strategy that never uses
for all
Proof. Suppose
with
That is, for all
(7.5)
̅
Suppose that
is a mixed strategy of player that assigns a positive probability to
can design another mixed strategy
nothing to
that assigns
to any
Then, we
but
i.e.,
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The condition
(7.6)
is equivalent to
̅
∈
or
̅
for all
This inequality is implied by (7.5). Thus, (7.6) holds. That is,
is strictly
dominated in
Mas-Colell et al. (1995) offer the following corollary. We provide a general result in Proposition 7.7, which is much more useful. Some pure strategies can be dominated by a mixed
strategy even though they cannot be dominated by a pure strategy.
Corollary 7.1. If, for player i, the pure strategy
is strictly dominated in
then so is every mixed strategy in
that assigns a positive probability to
this strategy.
With Proposition 7.7, we can iteratively eliminate strictly dominated strategies when we
try to find Nash equilibria in a normal-form game. We can eliminate not only strictly dominated strategies and strategies that are strictly dominated after the first deletion of strategies but
also strategies that are strictly dominated after the next deletion of strategies, and so on. One
feature of this process of iteratively eliminating strictly dominated strategies is that the order
of deletion does not affect the set of strategies that remain in the end. That is, if at any given
point several strategies are strictly dominated, then we can eliminate them all at once or in any
sequence without changing the set of strategies that we ultimately end up with. However, we
cannot use the iteratively eliminating process for weakly dominated strategies since the final
set of strategies depends on the order of deletion.
Example 7.9. Consider the following game:
P1\P2
We can first eliminate
2, 3
-2, 2
5, 2
-2, 3
4, 5
2, 3
1, 4
-3, -1
8, 1
since it is strictly dominated by
ed, we can further eliminate
since it is strictly dominated by
Once
is eliminat-
The game is now reduced
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the one in Example 7.7, from which we know all the NEs of the reduced game. These NEs are
all the NEs in the original game.
2.3. Trembling-Hand Nash Equilibrium
Trembling-hand perfection is a term given to consideration of the robustness of Nash
equilibria. In particular, it is concerned with the possibility that players may deviate slightly
from their Nash strategies by mistakes. If so, will the Nash equilibrium be destroyed?
Example 7.10. There are situations in which a player is indifferent between two alternative
strategies, one of which is the equilibrium strategy. This player has no incentive to deviate if
other players do not make any mistakes. However, the situation changes if possible mistakes
by other players are taken into account. Consider the following simple game:
P1\P2
There two NEs:
and
1, 2
0, 2
0, 1
3, 3
In
given
player 2 is indifferent between
However, if player 1 may make some mistakes by taking
matter how small
is, player 2 will strictly prefer
equilibrium, while
to
with probability
Thus,
and
no
is not an error-proof
is.
We now formally model an error-proof equilibrium concept. A totally mixed strategy is a
mixed strategy in which every pure strategy receives a probability in
∗
Definition 7.9. A NE
in
is trembling-hand perfect (THP) if there is a
sequence of totally mixed strategies
•
•
such that
∗
→
For each and when
By definition,
∗
small mistakes since
is large enough,
∗
is the best response to
is the best response to
∗
∗
is the best response to
By THP, other players are allowed to make
∗
when
Example 7.11. Reconsider the above example. Show that
For
we consider totally mixed strategies
is THP while
is not.
and
P1\P2
1, 2
0, 2
0, 1
3, 3
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We suppose that
∗
choose
and
and
Under this circumstance, player 1 will still
iff
which holds when
∗
player 2 will continue to choose
holds. Hence,
iff
is close to
Also,
which always
is trembling-hand perfect.
consider totally mixed strategies
For
and
P1\P2
We suppose that
choose
∗
and
1, 2
0, 2
0, 1
3, 3
and
Under this circumstance, player 1 will still
iff
which holds when
∗
player 2 will continue to choose
hold. Hence,
iff
is close to
Also,
which does not
is not trembling-hand perfect.
The following proposition dramatically simplifies the issue on trembling-hand perfection.
Proposition 7.8. When
a NE is trembling-hand perfect iff none of its equilibrium
strategies is weakly dominated.61
This result is consistent with the conclusion in Example 7.11. The proof of this proposition
is actually fairly simple. To understand the result better, we illustrate the essence of the proof
by the following example.
Example 7.12. Consider the following two very similar games A and B.
Game A
Game B
P1\P2
P1\P2
1, 2
0, 2
1, 2
0, 2
0, 1
3, 3
0, 1
3, 3
0, 1
4, 0
0, 1
4, 1
According to Proposition 7.8, NE
and
mixed strategies
which hold when
is THP in game A. To see this, consider two totally
is close to
For P2 to keep
For P1 to keep
∗
∗
we need
we need
(7.7)
61
This result does not hold when
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which holds if
Let
Then, (7.7) holds and
and
as
Hence,
The situation changes in game B. First, P1 keeps
for P2 to keep
∗
where
is THP.
∗
when
is close to
However,
we need
which can never hold. The reason for this failure is that strategy
strategy
is an integer,
is weakly dominated by
In comparison with game A, we can see that this weak dominance is the key. Also,
P1 maintains his equilibrium strategy since this strategy is not dominated by the other two
strategies.
The proof of Proposition 7.8 follows the same steps as in this example. Although we have
shown using pure strategies only, the same approach is obviously applicable to mixed-strategy
NEs also.
Using Proposition 7.8, we can easily prove the following proposition.
in which each
Proposition 7.9 (Existence). Every game
is a finite
set has a THP NE.
2.4. Reactive Equilibrium
Example 7.13 (Meeting in an Airport). Mr. Wang and Ms. Yang are to meet in an airport.
However, they do not know whether they are to meet at door A or door B. It is better for them
to meet at door A since it is closer to a parking lot. The payoffs are specified in the following
normal form game:
Wang\Yang
A
B
A
B
20, 20
0, 0
0, 0
10, 10
There are two pure-strategy Nash equilibria with payoffs
7.2, the two pure-strategy NEs with payoffs
and
and
By Proposition
are indeed NEs in the mixed
strategy game; they are not just NEs in the pure strategy game. Unfortunately, these two equilibria do not tell us where they meet and what they should do.
Let us see if a mixed-strategy NE can tell us more. To find a mixed-strategy NE, suppose
that Ms. Yang chooses
with probability
and Mr. Wang chooses
with probability
Then, the condition for Mr. Wang to be indifferent between his two strategies is
(7.8)
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implying
By symmetry, Mr. Wang also chooses
mixed-strategy NE, each player chooses
with probability
with
Therefore, in the
Furthermore, if Mr. Wang is
playing a mixed strategy, by (7.8), Ms. Yang must be playing a mixed strategy too. By symmetry, if Ms. Yang is playing a mixed strategy, Mr. Wang must be playing a mixed strategy too.
Thus, we do not have an equilibrium where one player is following a pure strategy and the
other player follows a mixed strategy. However, this mixed-strategy NE does not make much
sense. While they both prefer
they both are more likely to choose
One problem with a mixed-strategy NE is that it sometimes does not make sense. In fact,
a mixed strategy NE is a situation in which all players do not care what they do. It is precisely
this indifference that determines a mixed-strategy NE. In the above example, why do the
players go to door
with a higher probability? The answer is that, when a player does this, the
other player will then be indifferent between the two doors, and only in this situation do we
have an equilibrium. In the real world, when a player faces such a choice problem, he would
usually take a guess on what the other player is likely to do and decide what he himself should
do. For example, Ms. Yang may assume that Mr. Wang is likely to go to door
i.e.,
since it offers a higher payoff. If so, what should Ms. Yang do? She will go to door
With Ms. Yang’s belief of
the same reasoning and go to door
she will go to door
as well if
Similarly, Mr. Wang will follow
This outcome is the best result and it is consistent with
their beliefs. This approach is the so-called Bayesian approach, which will be discussed later.
In a NE, each player is dumb in the sense that when he considers his move, he assumes
that other players will never react to his move. Under this assumption, each player chooses his
best move. A NE is reached when every player’s move is the best for himself given others’
moves. In this subsection, we introduce a different equilibrium concept, by which each player
is smart enough to realize that other players may react to his move. In this case, when a player
chooses his best move, he will take into account other players’ reactions to his move. A situation in which each player chooses his best move after taking into account other players’ reactions is called a reactive equilibrium.
Definition 7.10. A reactive equilibrium (RE) is a situation in which each player plays a
Stackelberg strategy, by which the player takes into account possible reactions of others one
step ahead.
Consider two players with utility functions
and
In a Nash equilibrium,
player 1’s problem is
which implies
Player 2’s problem is
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which implies
The two reaction function satisfy the following two equations, respec-
tively,
,
,
The intersection point of these two reaction functions is a Nash equilibrium. Hence, a NE is
the solution of the following set of equations:
,
,
In a reactive equilibrium, player 1’s problem is
Let the solution be
∗∗
and let the solution be
Player 2’s problem is
∗∗
The pair
∗∗
∗∗
is the RE, which is the solution of the following
set of equations:
,
,
,
,
Example 7.14. Given utility functions:
to derive the reaction functions, we have
implying
(7.9)
Symmetrically,
(7.10)
Substituting (7.10) into (7.9) yields:
∗
Symmetrically,
∗
∗
∗
is the NE.
For the reactive equilibrium, we first find
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The FOC of maximizing this payoff is
implying
∗∗
Symmetrically,
∗∗
We can see that the RE is different from the NE.
We have
∗∗
∗∗
We can see that
∗∗
∗∗
This is expected since player 2 is not supposed to choose his
best action under the assumption that player 1 will not react to his action.
Example 7.15. Consider the following game. There are two pure-strategy NEs:
∗
∗
To reach a RE, P1 first figures out P2’s reactions: if P1 takes
P2 takes
∗
∗
and
P2 takes
if P1 takes
Taking these reactions into consideration, P1’s optimal choice is
Symmetri-
cally, P2 figures out P1’s reactions: if P2 takes
P1 takes
if P2 takes
ing these reactions into consideration, P2’s optimal choice is
P1 takes
Tak-
Hence, the unique RE is
This RE happens to be one of the NEs.
P1\P2
2, 3
-2, 2
-2, 3
4, 5
More formally, the reaction strategies of P2 are
Then, from
and
, we find
Then, from
and
, we find
strategy RE:
∗∗
∗∗
The reaction strategies of P1 are
Hence, there is only one pure-
Notice that I have focused on pure strategies only.
This example shows that strategies in a RE are like strategies of a chess game.
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In the airport example, there is only one RE, which is
Question: Will the RE hold if the opponent does not react perfectly rationally?
3. Equilibria in Extensive-Form Games
We study dynamic games using the extensive form. We define four equilibrium concepts:
the Nash equilibrium (NE), the subgame perfect Nash equilibrium (SPNE), the Bayesian
equilibrium (BE), and the sequential equilibrium (SE).
3.1. Nash Equilibrium
A game is usually initially defined by the extensive form. When there is a need, it is then
converted into the normal form. This conversion is unique. For the normal form, the NE concept is well defined. We can thus easily define the NE concept for an extensive form.
Definition 7.11. A Nash equilibrium (NE) of an extensive-form game
is a NE for the normal-
form game of
The NE concept ignores the timing and sequence of actions and assumes that everyone is
acting simultaneously. Also, NE assumes that when a player is choosing his actions, he supposes that others will not react to whatever actions chosen. In particular, if he deviates from
his equilibrium strategy, he still supposes that others will not react accordingly. This is a
strong assumption. Although it is optimal for a player to stick to his equilibrium strategy
assuming that others will never change their strategies, if others do change their strategies
accordingly when he changes his strategy, it may be better for him to change. Let us see an
example in which a problematic equilibrium occurs due to the simplicity of the Nash equilibrium concept.
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Example 7.16. In the game shown in , the normal form has two pure-strategy NEs:
and
.
P1
L
æ 0ö
çç ÷÷
çè3÷ø
R
L
æ-2ö
çç ÷÷
çè-1÷ø
.P2
P1\P2
R
(0, 3)
0, 3
-2, -1
(1, 2)
æ1 ö
çç ÷÷
èç 2÷ø
Figure 7.5. Game Tree and NEs
However,
P2 is to choose
is not a sensible equilibrium. In this equilibrium, P1 decides to choose
However, once P1 has chosen
P2 prefers
Hence, P2’s choice of
since
is not
credible. Here, the problem is that the NE concept assumes that when a player chooses a move,
he assumes that others will stick to their equilibrium strategies no matter what he does. This is
the weakness of the NE concept: a player always takes his opponents’ equilibrium strategies as
given without asking whether it is rational for the opponents not to change their strategies.
We can also use the RE concept to explain this game. P2 can react, but P1 cannot. Hence,
P1 should take into account P2’s reactions. If so, P1 will choose .
As extensive-form games are dynamic games, we will naturally apply dynamic optimality
to rule out unreasonable NEs. In particular, to rule out NEs such as
in Example 7.16, we
impose the principle of sequential rationality.
Sequential rationality under perfect information. Under perfect information, the
players’ strategies imply an optimal action at every decision node, i.e., rationality at
every decision node.
This rationality at each decision node takes into account possible reactions by subsequent
players. Since the extensive-form game in Example 7.16 belongs to the class of finite games of
perfect information, we can use a procedure called backward induction to find the NE. Such a
solution satisfies the principle of sequential rationality. We first look at the subgame after
player 1 has chosen
Player 2 will choose
see Figure 7.14 on the left. Once this is done, we
can then determine player 1’s optimal choice given the anticipation of what will happen after
his choice. This second step is accomplished by considering a reduced extensive form game
where player 2’s decision point is replaced by the payoffs that will result from player 2’s optimal decision; see Figure 7.14 on the right. We can see that player 1’s optimal decision is to play
from this reduced game. Therefore, we find the sequentially rational NE
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. Player 2
.
Player 1
æ1 ö
çç ÷÷
èç2÷ø
æ-2ö
çç ÷÷
èç-1÷ø
R
L
R
L
æ1 ö
çç ÷÷
çè 2÷ø
æ 0ö
çç ÷÷
çè3÷ø
Figure 7.6. Sequential Rationality
Proposition 7.10. Under perfect information, SR solutions are NEs.
Proof. A NE requires that given that others have taken the equilibrium actions, any player’s
equilibrium action is optimal. A SR solution satisfies this condition and is hence a NE.
The above example shows that the procedure of backward induction for a finite game with
perfect information proceeds as follows. We start by determining the optimal actions at the
final decision nodes in the tree. Then, given that these will be the actions taken at the final
decision nodes, we can proceed to the next-to-last decision nodes and determine the optimal
actions to be taken by the players there, and so on backward through the game tree.
Example 7.17. Consider the following game tree in Figure 7.7.
.
P3
L3
.
P1
L1
R1
.
L2
R3
.
P2
R2
.
P3
L3
R3
L̂3
P3
R̂3
Figure 7.7. Backward Induction under Perfect Information
This is a finite game with perfect information. By backward induction, on the left of Figure 7.8
is the reduced game formed by replacing the final decision nodes by the payoffs that result
from the optimal play once these nodes have been reached. On the right of Figure 7.8 is the
reduced game derived in the next stage of the backward induction procedure when the final
decision nodes of the reduced game on the left are replaced by the payoffs arising from optimal play at these nodes.
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.
.
Player 1
Player 1
L1
R1
.
L2
L1
R1
Player 2
R2
Figure 7.8. Backward Induction
By this backward induction, we find the Nash equilibrium
∗
∗
∗
that satisfies the principle
of sequential rationality:
∗
∗
∗
with payoff vector
3.2. Subgame Perfect Nash Equilibrium
To apply backward induction to finite games with imperfect information, we need the
concept of the subgame.
Definition 7.12. A subgame of an extensive-form game
is a subset of a game having the
following properties:
(a) It begins with an information set that contains a single decision node, contains all the
decision nodes that are successors (both immediate and later) of this node, and contains
only these nodes.
(b) If a decision node
of an information set
is in the subgame, then every
is
also. That is, there are no broken information sets.
Note that the game as a whole is a subgame by definition. A subgame that is not the
whole game is called a real subgame. Generally, a subgame starts from a single decision node
and proceeds to include everything following this point. In finite games with perfect information, every decision node initiates a subgame. We denote
as the subgame that starts
at decision node
We say that a strategy profile
in a subgame
of
∗
in an extensive-form game
if the strategies in
∗
induces a Nash equilibrium
constitute a Nash equilibrium when this sub-
game is considered in isolation.
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Definition 7.13. A strategy profile
∗
∗
∗
in an extensive-form game
is a subgame
perfect Nash equilibrium (SPNE or PNE) if it induces a NE in every subgame of
Sequential rationality under imperfect information. Under imperfect information,
the players’ strategies imply an NE equilibrium in every subgame, i.e., rationality in
every subgame.
The rationality at each subgame takes into account possible reactions by subsequent players. To identify SPNEs in a finite extensive-form game
we use the generalized backward
induction procedure, which derives NEs backwards a subgame at a time, instead of step by
step. Specifically,
1. Start at the end of the game tree and identify NEs of each of the final subgames.
2. Select one NE in each of these final subgames and derive the reduced extensive-form game
where these final subgames are replaced by the payoffs that result in these subgames when
players use equilibrium strategies.
3. Repeat steps 1 and 2 for the reduced game. Continue the procedure until every move in
determined. This collection of moves at the various information sets of
is
constitutes a pro-
file of SPNE strategies.
4. If multiple equilibria are never encountered in any step, this strategy profile is the unique
SPNE. If multiple equilibria are encountered, the full set of SPNEs is identified by repeating
the procedure for each possible equilibrium that could occur for the subgames in question.
For an infinite game, the definition of subgame perfection remains the same. However,
backward induction can no longer be used to find SPNEs. Instead, we impose a recursive
structure on a game.
Example 7.18. Consider the following game in Figure 7.9:
.
P1
R1
L1
x
æ0ö
çç ÷÷
çè0ø÷
.
L̂1
.
P1
R̂1
.
P2
L2
R2
L2
R2
æ-1ö
çç ÷÷
çè-1ø÷
æ1ö
çç ÷÷
èç2÷ø
æ-1ö
çç ÷÷
èç 0 ÷ø
æ1ö
çç ÷÷
èç1÷ø
Figure 7.9. Subgame Perfect under Imperfect Information
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This game has two subgames: the whole game and
as shown in Error! Reference
source not found..
x
.
P1
R̂1
L̂1
.
.
L2
R2
æ-1ö
çç ÷÷
èç-1÷ø
P2
L2
R2
-1, -1
(1, 2)
-1, 0
(1, 1)
æ1ö
çç ÷÷
èç1÷ø
æ-1ö
çç ÷÷
èç 0 ÷ø
æ1ö
çç ÷÷
èç2÷ø
P1\P2
Figure 7.10. Subgame
has a two NEs
and
For each pair of the resulting payoffs, after replacing
this subgame with the payoffs, the reduced game implies P1’s optimal choice, implying one
SPNE. The two NEs in the subgame lead to two SPNEs. The SPNEs are
In fact, in
choose
∗
∗
∗
since P1 is indifferent between
(7.11)
and
when P2 chooses
for certain no matter what P1 chooses, for each
strategy NE
will take
∗
in
with probability
there is a mixed-
where the mixed strategy
and
with probability
and P2 will
means that P1
Each of the NE yields a SPNE.
Hence,
∗
∗
These two SPNEs are also REs, in which P2 is allowed to react while P1 cannot.
On the other hand, the normal form of the whole game is
P1\P2
(0, 0)
0, 0
(0, 0)
0, 0
-1, -1
(1, 2)
-1, 0
(1, 1)
There are total four pure-strategy NEs:
NE 1:
∗
∗
NE 2:
∗
∗
NE 3:
∗
∗
NE 4:
∗
∗
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The first two NEs are unreasonable and they are not SPNEs.
However, the SPNE concept has a few problems.
Example 7.19. We first show a problem with the concept of subgame perfection when there are
multiple equilibria in a subgame. Consider the following game in Figure 7.11:
.
P1
R1
L1
x
æ1ö÷
çç ÷
èç1÷ø
.
L2
æ0ö÷
çç ÷
çè0÷ø
.
P1
L̂1
R̂1
R2
æ-1ö
çç ÷÷
çè-1÷ø
.
P2
L2
R2
æ-1ö÷
çç ÷
èç-1÷ø
æ2ö÷
çç ÷
èç2÷ø
Figure 7.11. A Problem with Subgame Perfection
obviously has a two NEs
and
The SPNEs are
∗
∗
∗
∗
(7.12)
However, this example shows a serious problem of the definition of SPNE when there are
multiple NEs in a subgame. For example, for SPNE1 in (7.12), when P1 is making a decision at
the starting node, P1 must assume that the NE in
different choice. If P1 expects
expectation), than P1 will choose
is
to have a better chance than
otherwise P1 may make a
(which is a sensible
at the start. Hence, SPNE1 may not be a sensible solution.
In practice, we cannot rule out the possibility that a player makes expectations on which NEs
in a subgame are more likely to happen and such expectations affect the player’s choice. SPNE
does not take into account this possibility.
The following example shows another problem of SPNE.
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Example 7.20. We modify the game in Example 7.18 slightly so that there are no real subgames, as shown in Figure 7.12.
.
P1
L1
M1
.
æ0ö÷
çç ÷
èç0÷ø
L2
æ-1ö
çç ÷÷
çè-1÷ø
R1
.
H
L2
R2
æ1ö
çç ÷÷
çè2÷ø
P2
R2
æ ö
çç-1÷÷
çè 0 ÷ø
æ1ö
çç ÷÷
çè1÷ø
Figure 7.12. A Game with a Bad SPNE
As in Example 7.18, the normal form is
P1\P2
(0, 0)
0, 0
-1, -1
(1, 2)
-1, 0
(1, 1)
There are three pure-strategy NEs:
and
Since there is no real sub-
game, all these NEs are SPNEs.62
Again, the first NE is not reasonable. Once P1 decides to go to information set
definitely choose
Even though P2 does not know whether it is
always strictly better than
P1 should understand this and, if so,
or
P2 will
the choice
is
is an inferior choice.
Unfortunately, this unreasonable NE cannot be ruled out by subgame perfection. This failure
calls for a different equilibrium concept for extensive-form games.
3.3. Bayesian Equilibrium
How can we eliminate the unreasonable equilibrium? We find that if we allow players to
form beliefs about his opponent’s strategies, we can then eliminate the unreasonable equilibrium.
62
Again, there are mixed-strategy NEs in which P1 mixes
1
and
1
arbitrarily and P2 chooses
2
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Example 7.21. Reconsider Example 7.13. The game tree is shown in Figure 7.13.
.
Wang
A
B
m1
m2
.
A
æ20ö
çç ÷÷
çè20ø÷
B
A
æ 0ö
çç ÷÷
çè0÷ø
æ 0ö
çç ÷÷
çè0÷ø
.
Yang
B
æ10ö
çç ÷÷
çè10÷ø
Mr. Wang's Payoff
Ms. Yang's Payoff
Figure 7.13. Bayesian Solution to a Meeting Game
When Ms. Yang is to make a move, she does not know what Mr. Wang has chosen. However,
Ms. Yang can take a guess and assign a probabilistic distribution over the two possible choices
by Mr. Wang. Suppose that the probabilities are
and
for the two possible choices as
shown in the diagram. Given the beliefs, for Ms. Yang,
That is, if Ms. Yang believes that Mr. Wang will go to gate
she will choose gate
with a probability higher than
Here, since Mr. Wang prefers to meet at gate
belief makes sense. Interestingly, if Ms. Yang chooses
should choose
than gate
such a
with probability one, Mr. Wang
with probability one as well, which is consistent with the condition
In other words, there is an equilibrium where
and
and this belief is consistent
with the equilibrium strategies derived from this belief. This solution seems much more acceptable than other solutions in Example 7.13.63
Definition 7.14. A system of beliefs
ability
in an extensive-form game
for each decision node
in
such that
is a specification of a prob-
∈
for all information
sets
A system of beliefs specifies, for each information set, a probabilistic assessment
by the
player who moves at that point on the relative likelihood of being at each of the information
set’s various decision nodes, conditional upon play having reached that information set.
Definition 7.15. Given a belief system
her information set
denote
if she uses strategy
as player ’s expected utility at
and her rivals use strategies
Denote
as
63 One problem with a belief system is that it can be quite arbitrary, which can lead to some unreasonable sothen we have a solution in which both Yang and Wang go to
lutions. For example, if for some reason,
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the player who moves at information set
(SR) at information set
given a belief system
∗
( )
( )
for all
( )
( )
∗
A strategy profile
∗
A strategy profile
( )
in
is sequentially rational
if
( )
∗
in
( )
∗
(7.13)
( )
is sequentially rational given belief system
if it satisfies (7.13) for all the information sets in the game.
Sequential rationality in Bayesian games. In a Bayesian game, the players’ strategies
imply an optimal action at every information set, i.e., rationality at every information set.
The rationality at each information set takes into account possible reactions by subsequent players. In other words, a strategy profile
∗
is sequentially rational if no player finds it
worthwhile, once one of her information sets has been reached, to revise her strategy given her
rivals’ strategies and her beliefs about the future (as embodied in
Hence, sequential ration-
ality is very much like the Nash equilibrium concept except that it is based on a belief system.
Notice that sequential rationality is different from the principle of dynamic optimization.
Since the payoffs in our games are at the end of a game, our games should be considered as
one-period games. Sequentially in our games refers to a sequence of actions from the players,
rather than a sequence of payoffs over multiple periods. In other words, the principle of dynamic optimization applies to one individual in multiple periods, while sequential rationality
applies to multiple individuals in ‘one period.’
A key problem with NEs is that they may not be SR. The following example shows that
imposing SR can rule out some unreasonable NEs.
Example 7.22. For the game in Example 7.20, given a belief system
∗
find a SR strategy profile
The game tree is redrawn in the following game.
.
L1
æ0ö
çç ÷÷
èç0÷ø
L2
æ-1ö
çç ÷÷
èç-1÷ø
P1
M1
.
m1
R2
æ1 ö
çç ÷÷
èç2÷ø
R1
.
H
m2 P2
L2
R2
æ1ö
çç ÷÷
çè1÷ø
æ-1ö
çç ÷÷
çè 0 ÷ø
Figure 7.14. A SR strategy profile
Given arbitrary beliefs
chooses
over
and
for the information set
with
and
P2
iff
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This is completely impossible. Hence, no matter what belief system P2 has, he will always
choose
once information set
chooses
fact,
for sure,
and
has been reached. That is,
is a rational choice. As P2
is an inferior choice for P1. Hence, P1 will choose either
are indifferent to P1. Thus,
or
In
is a rational strategy, where
Therefore, given an arbitrary belief system
the sequentially rational strategies are
(7.14)
The unreasonable SPNE is not sequentially rational.
Example 7.23. For the game in the following figure, given a belief system
profile
find a SR strategy
∗
.
P1
L1
æ0ö
çç ÷÷
çè 0 ÷ø
.
R1
M1
H
m1
L2
L2
R2
æ2ö
çç ÷÷
çè- 1÷ø
æ1ö
çç ÷÷
çè1÷ø
.
m 2 P2
R2
æ1ö
çç ÷÷
çè1÷ø
æ 3ö
çç ÷÷
çè 0 ÷ø
Figure 7.15. A SR strategy profile
First, P2 chooses
iff
If so, P1 will choose
Symmetrically, if
tional, where
iff
Also, when
Thus, if
P2 will play
for sure.
for certain.
P2 will choose
P2 is indifferent between
If
over
i.e.,
It is obvious that
for sure. If so, P1 will choose
and
for certain.
That is, any strategy
is ra-
is an inferior choice for P1. Then, P1 chooses
which is
P1 is indifferent between
Symmetrically, P1 chooses
iff
and
Therefore, in summary, the SR strategies are
(7.15)
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With a belief system, we can now derive optimal strategies backward step by step, instead
of subgame by subgame. This process can go on repeatedly from the last step until we reach
the starting point of the game. Such a solution from backward induction will be sequentially
rational. Here, the belief system is assumed to be public knowledge so that each player can
figure out what future moves will imply.
When a player takes a certain mixed strategy to go to a node, this node will be reached
with certain probability. This is the objective probability that the node will be reached. At the
same time, there is a belief that this node would be reached with certain probability; this is a
subjective probability. The objective probability actually depends on this subjective probability
(as embodied in the sequentially rational strategies). These two probabilities should be consistent with each other in some way. Hence, besides sequential rationality, a Bayesian equilibrium requires certain consistencies in the equilibrium path.
SR and NE do not imply each other. Under perfect information, SR solutions are NEs.
However, under imperfect information, SR solutions may not be NEs; it depends on the belief
system. For example, in Example 7.23, although there are many SR solutions, there is only one
NE:
∗
∗
Conversely, NEs may not be SR. For example, in Example 7.22,
imposing SR rules out some unreasonable NEs.
A path of a game is defined as a sequence of decision points by which the game goes from
the initial point to a terminal point. Here, mixed strategies are allowed. Notice that each decision point belongs to one and only one information set and each information set belongs to
one and only one player. We say that an equilibrium, or equivalently an equilibrium strategy
profile, is rational along a path if every player behaves rationally at his/her information sets
along the path. Rationality may be based on mixed strategies or beliefs. When rationality is
based on beliefs, we call it sequential rationality. With a belief system, perfect information
games are special cases of imperfect information games. Hence, we can generally say that
rationality is based on beliefs.
We call a strategy profile
an equilibrium
∗
∗
a BE if there exists a belief
we say that an information set
∗
such that
∗
∗
is on the equilibrium path if
is off-equilibrium or is on an off-equilibrium path if
is a BE. Given
and
A Bayesian equilibrium re-
quires that at any point in the game, a player’s strategy prescribes optimal actions from that
point on, given a consistent belief system along the equilibrium path. A strategy is a series of
planned actions by a player. Those planned actions in equilibrium strategies (planned to be
taken with positive probability) are called equilibrium actions. Those equilibrium actions on
the equilibrium path are called realized equilibrium actions, and those equilibrium actions on
off-equilibrium paths are called unrealized equilibrium actions.
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Definition 7.16. A strategy-belief pair
∗
∗
is a Bayesian equilibrium (BE) [formally, a weak
perfect Bayesian equilibrium (weak PBE)] in
(a) Sequential Rationality: Strategy profile
if it has the following properties:
∗
is sequentially rational given the belief system
∗ 64
(b) Equilibrium-Path consistency: The belief system
∗
and nature’s odds
with
for
∗
∗
is consistent with the strategy profile
through the Bayes rule along the equilibrium path: for any
∗
, we have
65
In the above definition, the required consistency condition between strategies and beliefs
is
∗
Pr
given the fact that
where
is the (objective) probability that node
has been reached under the equilibrium strategy profile
is not on the equilibrium path, then the probability that any node
Given the requirement of
to hold for this
sets
will be reached
∗
∈
in
∗
However, if
is reached is zero.
for a belief system, there is no way for
∗
Hence, the consistency condition applies only to those information
that are on the equilibrium path. Since there are no restrictions on those beliefs on off-
equilibrium paths, they can be quite arbitrary.
Example 7.24. Let us find all the BEs in the airport game in Figure 7.13.
.
Wang
.
A
B
m1
m2
A
B
æ 20 ö
çç ÷÷
çè 20÷ø
Ms. Yang will go to gate
∗
æ0 ö
çç ÷÷
çè0 ÷ø
iff
.
A
B
æ0 ö
çç ÷÷
çè0 ÷ø
æ10 ö
çç ÷÷
èç10÷ø
Mr. Wang's Payoff
Ms. Yang's Payoff
Then, Wang goes to
which can be satisfied. Hence, one BE is:
64
Yang
∗
∗
and
By this, we must have
∗
Here, sequential rationality needs to be satisfied for all information sets, including those on off-equilibrium
paths. This is important to know since, even with a proper belief system, some Nash equilibria do not satisfy
sequential rationality on off-equilibrium paths. By Proposition 7.11, this happens when a NE is not a BE.
65
The words “whenever applicable” in Gibbons’ (1992) definition of this equilibrium are vague. A strategy is
only a plan. Even for an equilibrium strategy, a planned action may never be taken in equilibrium. For example,
given the equilibrium strategy profile
∗
∗
We can either say that
∗
∗
defined by (7.16), for point in Figure 7.20, we have
is a BE since
is an off-equilibrium path or say that
∗
But,
is not a BE since
The weak version of BE in this book takes the former definition, while the strong version of BE in Gibbons (1992)
takes the latter definition.
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Susheng Wang, HKUST
Ms. Yang will go to gate
∗
iff
Then, Wang goes to
which can be satisfied. Hence, another BE is:
and
Ms. Yang is indifferent between
Then, for
iff
∗
and
∗
Let Yang’s strategy be
to hold, by the consistency condition, Wang must have
Wang must be indifferent between
∗
implying
∗
By this, we must have
and
.
∗
If so,
which implies
∗
Hence, we have a third BE:
∗
∗
Example 7.25. Let us find BEs for the game in Example 7.22. By the SR solution in (7.14),
condition (a) in Definition 7.16 is already satisfied. For condition (b), we only need to impose
Hence, the BEs are
∗
∗
∗
Alternatively, we can also solve for BEs directly without finding the SR strategies first. For
arbitrary beliefs
and
to the information set
iff
over
fact,
and
P2 chooses
This is completely impossible. Thus, no matter what belief
system P2 has, he will always choose
chooses
with
for sure,
and
once information set
is an inferior choice for P1. Hence, P1 will choose either
∗
are indifferent to P1. Thus,
consistency condition, we have
∗
has been reached. As P2
and
∗
where
or
In
If so, by the
Hence, the BEs are:
∗
The unreasonable SPNE is ruled out.
The reason that the BE concept can solve the above problem is that a belief system gives
us the opportunity to derive an equilibrium backward one step at a time. A SPNE is derived
backward one subgame at a time but this backward approach does not apply within a subgame.
A subgame itself may contain some unreasonable NEs, as shown in Example 7.20. A belief
system in Example 7.22 has effectively transformed the game into the following game in Figure 7.16, in which the unreasonable SPNE is ruled out.
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.
P1
L1
R1
.
P1
R̂1
L̂1
.
æ0 ö
çç ÷÷
èç0 ÷ø
H
m1
L2
m2
R2
æ- 1ö
çç ÷÷
çè- 1÷ø
æ1 ö
çç ÷÷
çè 2 ÷ø
.
P2
L2
R2
æ- 1ö
çç ÷÷
çè 0 ÷ø
æ1ö
çç ÷÷
çè1÷ø
Figure 7.16. An Alternative Game Tree for Figure 7.14
Example 7.26. Let us find BEs for the game in Example 7.23. By the SR solution in (7.15),
condition (a) in Definition 7.16 is already satisfied. For condition (b), we only need to impose
This is possible only when
and
That is, we find one and only one
BE:
∗
∗
∗
Alternatively, we can also solve for BEs directly without finding the SR strategies first.
First, P2 chooses
iff
If so, P1 will choose
i.e.,
Thus, if
for certain, implying
P2 will play
which contradicts
for sure.
Hence, there
is no such BE.
If
P2 will choose
which contradicts
If
for sure. If so, P1 will choose
for certain, implying
Hence, there is no such a BE either.
P2 is indifferent between
ities for P1: either
and
Let
is strictly better than the other two or
former case,
can be any value. However,
have chosen
with probability
There are two possibiland
are indifferent. If the
is obviously been dominated. Hence, P1 must
, implying that P1 must be indifferent between
and
i.e.,
implying
Then, by choosing either
Since P1’s expected profits from
chosen.
66
1
and
or
1
P1’s payoff is66
are the same, we can simply assume that only
1
is
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Hence,
Then, by the Bayes rule,
and
That is, we find
one and only one BE:
∗
∗
∗
Let us find all the NEs in the game. The normal form is:
P1\P2
Since P1’s strategy
0, 0
0, 0
1, 1
2, -1
3, 0
1, 1
is strictly dominated by his strategy
we only need to consider the
following normal form:
P1\P2
1, 1
2, -1
3, 0
1, 1
This game has no pure-strategy NE and, by (7.2), it has one mixed-strategy NE, which is:
∗
∗
This mixed-strategy NE is the same as the BE.
Here are some further discussions about the belief system. The key feature of a belief system is that we can now derive an equilibrium by backward induction one step at a time (pretending that every information set will be reached). The belief system is assumed to be public
information and all the players will use it to determine their strategies in equilibrium.
Given a strategy profile , for
by the Bayes rule, we have
∈
which requires
Consistency between a mixed strategy profile and a belief system is
through the Bayes rule. However, the Bayes rule can no longer be used at
fact, when
if
In
we are allowed to assign any beliefs to such an information set without
violating the Bayes rule. In this case, there is no direct way to ensure consistency between
and
which may result in some unreasonable BEs. As a result, some BEs are not sensible. We
will show two problems with BEs in Example 7.28 and Example 7.29.
For example, in Figure 7.17, if
where
the Bayes rule implies
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.
x0 P1
m1 + m2 = 1
L1
s11
M 1 s 21
.
m1
æu11 ö
çç ÷÷
çèu12 ÷÷ø
x1
R1
H
s31
x2 m 2
L2
R2
s12
s 22
æu21 ö
çç ÷÷
çèu22 ÷÷ø
æ u31 ö
çç ÷÷
çèu32 ÷÷ø
.
P2
L2
s12
R2
s 22
æ u51 ö
çç ÷÷
çèu52 ÷÷ø
æ u41 ö
çç ÷÷
çèu42 ÷÷ø
Figure 7.17. Consistency in a Bayesian Equilibrium
, we have
For another example, in Figure 7.18, if
Consistency means
.
.
.
. .
.
Figure 7.18. BE with Intuitive Criterion
There is an implicit assumption: once a belief system is introduced, the calculation of the
expected utility is based on the belief system whenever possible, rather than on the mixed
strategy profile. That is, instead of
For example, in the above figure, at information set
action
is
where the mixed strategies
calculation. Given player 2’s mixed strategy
ing
is
we now use
player 2’s expected payoff for taking
and
are not used in the
player 1’s expected payoff for tak-
However, with a sequentially rational strategy profile
∗
∗
and
∗
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are to be derived using the beliefs, implying that both players’ expected payoffs will be based
on the belief system. If there is no belief system, the calculation will be based on the mixed
strategies. The belief system consists of subjective probabilities, while the mixed strategy
consists of actual probabilities. Hence, consistency between strategies and beliefs is crucial.
Unfortunately, such consistency cannot be easily imposed on off-equilibrium paths, which can
result in various problems. In the following section, we point out some problems with BEs and
introduce new equilibrium concepts that impose certain consistency between strategies and
beliefs on off-equilibrium paths.
Although BE can be used to deal with some problems that SPNE cannot deal with, some
problems remain. The following three examples show two problems with BE.
Example 7.27. One problem with the BE concept is that the belief system need not make sense.
In the following game, given
so, player 1 will choose
player 2 will play
iff
In this case,
i.e.,
hence condition (b) in Definition 7.16
with
needn’t be checked. Since the SR condition is satisfied, any
and
If
is a BE.
.
Nature
1
2
1
2
μ 1 = 0. 5
L1
 2
 
 10 
.
H1
R1
m2
.
L2
0
 
5
.
P1
.
H 2 P2
R2
 5
 
2
1 − μ1 = 0.5
R1
1- m2
L2
R2
0
 
 5
 5
 
 10 
L1
 2
 
 10 
is not sensible. If we allow P1 to make mistakes by deviating from
But
times, we should have
∗
some-
What are the NEs in this game? The normal form is67
P1\P2
(2, 10)
2, 10
0, 5
(5, 6)
We can see that the above BE is nevertheless a NE or a SPNE.
Example 7.28. One more example of a senseless belief system. Consider the following game
tree in Figure 7.19:
67 We
have expected payoffs in the cells since the nature offers a half-half chance. This NE is actually a BNE.
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Susheng Wang, HKUST
.
P1
L1
R1
.
Nature
0.5
0.5
.m
æ 0ö
çç ÷÷
çè0÷ø
H
1
L2
æ-2ö
çç ÷÷
çè-1÷ø
.
m2 P2
R2
L2
R2
æ1ö
çç ÷÷
çè-2÷ø
æ-1ö
çç ÷÷
çè 1 ÷ø
æ2ö
çç ÷÷
çè 3÷ø
Figure 7.19. Problem with Off-Equilibrium Paths
In this case, P2 will choose
choose
∗
for certain iff
or
for certain. Hence, a belief system with
and payoff pair
If so, P1 will
supports a BE with
∗
and
This belief system does not make sense (we should have
yet, it can support a BE. The reason is that consistency between strategies and nature’s
odds are not required on off-equilibrium paths. A sensible belief system should have
which supports a different BE with
∗
and
∗
and payoff pair
Example 7.29. Another problem with the BE concept is that a BE need not be subgame perfect.
Consider the following game in Figure 7.20. Given
or
of
P1 chooses
If so, P1 chooses
Then, since choosing
at the beginning. Hence, any belief system with
BE that leads to the payoff pair
∗
with payoff pair
at node
P2 chooses
iff
means a payoff
can support a
In particular, we have the following BE:
∗
∗
Here, since the information set
∗
(7.16)
is off the equilibrium path, there is no
need for consistency between the beliefs and the strategies.
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.
P1
L1
R1
x
. P1
R̂1
L̂1
.m
æ0ö
çç ÷÷
çè 0÷ø
.y
H P2
m2
1
L2
æ- 2 ö
çç ÷÷
çè- 1 ÷ø
R2
L2
R2
æ 1 ö
çç ÷÷
çè- 2 ÷ø
æ- 1ö
çç ÷÷
èç 1 ÷ø
æ2ö
çç ÷÷
èç 3÷ø
Figure 7.20. BE May Not Be Subgame Perfect
is
What are the SPNEs? The normal form for
P1\P2
The only pure-strategy NE is
shown by the fact that
∗
and
∗
-2, -1
1, -2
-1, 1
(2, 3)
There are no mixed-strategy NEs, which can be easily
is the dominant strategy for player 1. Hence, the only SPNE is:
with payoff pair
Therefore, the BE above is not subgame per-
fect.
The problem with the BE comes from the fact that consistency between strategies and
beliefs is not ensured off the equilibrium paths. For the BE in (7.16), P2 chooses
she firmly believes that P1 will choose
rather than to P1’s strategies. Since
choose
at node
information set
because
P2’s choice is a rational response to her own belief
dominates
P2 should firmly believe that P1 will
Due to the inconsistency between P2’s beliefs and P1’s strategies at the
on an off-equilibrium path, the BE strategies within
are not NE
strategies. Therefore, the BE is not a SPNE.
In fact, BE and SPNE do not imply each other. In Example 7.25, a SPNE is not a BE; in
Example 7.29, a BE is not a SPNE. These problems call for a stronger equilibrium notion that
can unite the two concepts. In a BE, players react optimally based on beliefs; in a NE, players
react optimally based on strategies. Hence, consistency between strategies and beliefs is crucial. However, in a BE, this consistency is not guaranteed on off-equilibrium paths. One naturally looks for a suitable condition for off-equilibrium paths. The following proposition
strengthens the view that what happens on off-equilibrium paths is crucial.
To consider stronger versions of BE, let us first put NE in the context of a Bayesian game.
Nash equilibrium is based on rationality. What is its rationality? It is not based on the ration-
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ality at every information or subgame. Instead, in a Nash equilibrium, each person is rational
on the equilibrium path when all others take their equilibrium strategies.
∗
Proposition 7.11. A strategy profile
is a NE of
iff there exists a belief system
∗
such
that68
(a) The strategy profile
sets
∗
is sequentially rational given belief system
∗
at those information
∗
with
(b) The belief system
∗
is derived from the strategy profile
and nature’s odds
through
∗
the Bayes rule whenever
Proof. Necessity. Given a NE
∗
∗
we can easily assign beliefs using
Specifically, on the equilibrium path, we assign beliefs
using
∗
∗
such that (b) is satisfied.
based on the Bayes rule; on
off-equilibrium paths, we can assign arbitrary beliefs. Since the expected utility on an offequilibrium path is zero, the players in a NE are guaranteed to be rational only on the equilibrium path against
∗
Further, since
∗
and
are consistent along the equilibrium path, the
players must be rational on the equilibrium path against
That is, (a) holds.
Sufficiency. Conversely, by (a), each player acts rationally against the beliefs on the equilibrium path. By (b), the beliefs are consistent with the strategies on the equilibrium path.
Thus, each player is acting rationally against the strategies on the equilibrium path. Since the
NE concept requires rationality on the equilibrium path only, the strategies must be a NE.69
In fact, differences among the three equilibrium concepts, NE, SPNE and BE, are due to
differences in the required conditions for off-equilibrium paths. The difference between the
NE concept and sequential rationality is that NE requires rationality along the equilibrium
path, while sequential rationality requires rationality at any information set. Proposition 7.11
indicates that to be a NE, we need sequential rationality only on the equilibrium path where
Condition (a) in Definition 7.16 is stronger than condition (a) in Proposition 7.11.
Hence, a BE is a NE.
68
It is from Mas-Colell et al. (1995, p.285).
69
Those decision makers whose information sets are on off-equilibrium paths, given that everyone else is
staying on the equilibrium path, have no need to deviate from their equilibrium strategies (since what they do does
not matter assuming that others will stick to their current strategies). Thus, for a NE, we only need to verify rationality for those information sets on the equilibrium path. In other words, if rationality is guaranteed on the equilibrium path, we have a NE. Since we do not solve a NE by backward induction, there is no guarantee that a NE is
rational on off-equilibrium paths. In Example 7.16, for NE
∗
the initial decision node is on the equilibrium
path and the decision maker there is indeed rational at that node. But the second decision node is on an offequilibrium path and the decision maker there is not rational. In fact, given player 1 choosing
player 2 is indiffer-
ent among his choices. Player 2 simply reasons that since the equilibrium path will not pass through his information set, whatever he decides to do does not matter and he has no need to deviate from the equilibrium strategy.
The situation in Example 7.18 for NE
∗
is precisely the same.
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However, not every NE is a BE. The key difference between NE and BE is due to sequential rationality at those information sets on off-equilibrium paths where
In a NE,
sequential rationality is satisfied at any information set along the equilibrium path, but there
is no guarantee on off-equilibrium paths.
arrive at
and thus the player
literally means that the game will never
does not need to ensure rationality at
7.16 and Example 7.20, the NEs with payoffs
and
in NE. In Example
respectively violate sequential
rationality at an off-equilibrium information set.
The difference between NE and SPNE is also due to restrictions on off-equilibrium paths.
Since we solve for a SPNE backward, the SPNE is rational in every subgame, including those
subgames on off-equilibrium paths. However, a NE guarantees rationality only along the
equilibrium path; in particular, rationality may fail in an off-equilibrium subgame. Only when
for all information sets
is NE
∗
a SPNE.
The difference between BE and SPNE is again due to restrictions on off-equilibrium paths.
In a BE, each player reacts rationally at his information sets via beliefs. In a SPNE, each player
reacts rationally at his information sets via strategies along the equilibrium path within each
subgame (including those subgames on off-equilibrium paths). If the beliefs and strategies are
inconsistent, the players in a BE may not act rationally via strategies. This means that for a BE,
on an off-equilibrium path, even though part of it is on the equilibrium path within a subgame,
the players may not act rationally against strategies. Thus, a BE may not be a SPNE. Only if
beliefs at any
with
∗
In Example 7.29, BE
are also consistent with the strategies, can BEs be SPNEs.
is not rational via strategies in
simply because the
beliefs are not consistent with the NE strategies.
4. Refinements of Bayesian Equilibrium
In this section, we offer three kinds of refinements to BEs. NEs have three kinds of refinements: subgame perfection, trembling hand perfection, and dominance. We will also offer
these three kinds of refinements to BEs and they are called respectively perfect BEs, sequential
equilibria, and equilibrium dominant BEs. The focus of the refinements on BEs is to strengthen the consistency condition.
4.1. BE under Subgame Consistency: Perfect Bayesian
Equilibrium
This part mainly follows Gibbons (1992, p.183–244). It is traditionally called perfect
Bayesian equilibrium (PBE) and sometimes called a strongly perfect Bayesian equilibrium
(strong PBE). It adds one more condition to the definition of the weak PBE in Definition 7.16.
This definition is first provided by Gibbons (1992, p.180). Fudenberg and Tirole (1991) provide
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a formal definition of PBE for a broad class of dynamic games of incomplete information. Our
definition here is based on Gibbons (1992). However, Gibbons (1992, p.180) does not give a
precise definition; he uses words “where possible.” I interpret it as meaning “all subgames,” as
defined in the following definition. From the examples in Gibbons (1992, 180–183), he seems
to mean this.
One defining feature of a subgame is that there is only one branch/action that connects to
it. Hence, if the subgame is on the equilibrium path, the mixed strategy/probability on this
action must be positive. Hence, a BE always induces a BE in any subgame on its equilibrium
path. However, this property does not hold on off-equilibrium paths. This leads to an idea that
one way to strengthen consistency between strategies and beliefs is to require that a BE induces a BE in any subgame, especially subgames on off-equilibrium paths.
Given a BE
∗
∗
∗
if the mixed strategy
in
an equilibrium action. Within a subgame
∗
for taking an action
although the subgame may or may not be
on the equilibrium path, we say that an information set
path in
∗
∗
if
is positive, we call
in the subgame is on the equilibrium
We can still discuss whether the induced strategy-belief pair
∗
in this subgame from
∗
is a BE or not. It is obviously sequentially rational within
this subgame. The question is whether
∗
∗
is consistent along the equilibrium path in this
∗
subgame. If it is consistent in this subgame, we say that
∗
has subgame consistency in
this subgame.
in the following figure. The total
To explain in more detail, consider the information
probability of reaching
in the subgame is
Hence, on
consistency
requires
x
r3
r2
r1
H1
d1
a1
s1
m1
H2
d2
a2
s2
a3
s3
m2
m3
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Definition 7.17. A perfect Bayesian equilibrium (PBE) (a strong PBE or a subgame perfect BE)
is a pair
∗
∗
∗
of strategies
∗
(a) Sequential Rationality:
(b) Subgame Consistency:
∗
∗
and beliefs
that satisfy the following conditions:
∗
is sequentially rational given the belief system
∗
is consistent with
through the Bayes rule in all subgames.
There are several versions of PBE in the literature. Two well-known versions are:
 A BE
∗
∗
is a PBE if
∗
 A BE
∗
∗
is a PBE if it is a BE in every subgame (Albert Banal-Estanol).
is a NE in every subgame (Giacomo Bonanno).
Bonanno’s version is based on subgame perfection and is weaker than ours, while BanalEstanol’s version is equivalent to ours, which is based on subgame consistency. Some versions
of PBE are based on continuation games instead of subgames. At the end of this chapter, we
list the alternative versions.
To explain the definition of PBE, we use the following game.
.
P1
A
D
.
x P2
R
L
.
L¢
H
m
R¢
.
1- m
L¢
P3
R¢
We find a BE by solving it backward. P3’s prob-
This game has only one real subgame
lem is
Hence, if
P3 will choose
P2 will choose
and P1 will choose
information set
is not on the equilibrium path, there is no restriction on
In this case, since
Hence, we have a
BE:
(7.17)
To have a PBE, we apply the Bayes rule on
cannot satisfy the requirement of
This means that we must have
which
in (7.17). Hence, the BE in (7.17) is not a PBE. In
the normal form is
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P2\P3
2, 1
(3, 3)
1, 2
1, 1
We can see that there is a unique NE is this subgame, which is
and
Hence,
there is a unique SPNE:
This means that the BE in (7.17) is not subgame perfect.
If
P3 will choose
formation set
P2 will choose
and P1 will choose
In this case, since in-
is on the equilibrium path, we require consistency, which means
satisfies
This
Hence, we have another BE:
(7.18)
Since there is no subgame on an off-equilibrium path, consistency on off-equilibrium paths is
automatically satisfied. Hence this BE is a PBE. Notice that this PBE is subgame perfect. In
fact, according to the following proposition, any PBE is a SPNE.
Proposition 7.12. For any PBE
∗
1.
2.
∗
∗
∗
∗
of an extensive-form game
is a BE.
is a SPNE.
The converse is false.
Proof. Since a PBE ensures it is a BE in every subgame, it ensures that it is a NE in every subgame. Hence, a PBE must be a SPNE.
The converse is false. That is, a BE that is also a SPNE may not be a PBE. Since a NE may
not be a BE, if a BE induces a NE but not a BE in a subgame, this BE is not a PBE.
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Example 7.30. Reconsider Example 7.28. The game is redrawn in Figure 7.21. The strategies
∗
and
∗
∗
with beliefs
∗
consist of a PBE. In fact, this is the unique PBE.
.
P1
L1
R1
.
x P1
r1
0.5
0.5
.m
æ 0ö
çç ÷÷
çè0ø÷
r2
.
H
m2 P2
1
L2
æ-2ö
çç ÷÷
èç-1÷ø
R2
L2
R2
æ1ö
çç ÷÷
çè-2÷ø
æ ö
çç-1÷÷
çè 1 ø÷
æ 2ö
çç ÷÷
çè 3÷ø
Figure 7.21. PBE
Example 7.31. Reconsider Example 7.29. The game is redrawn in Figure 7.22. It cannot be a
PBE when P2 chooses
P1 chooses
at node
Hence, we consider
and
by which P2 choose
in the starting point. The SR needs
for sure. If so,
to ensure subgame
consistency. Hence, we find a PBE:
∗
∗
This unique PBE is the unique SPNE.
.
P1
L1
R1
.
x P1
R̂1
L̂1
æ0ö
çç ÷÷
çè0÷ø
.m
H
1
L2
æ-2 ö
çç ÷÷
èç-1÷ø
.
m2 P2
R2
L2
R2
æ 1 ö
çç ÷÷
çè-2÷ø
æ-1ö
çç ÷÷
çè 1 ÷ø
æ2ö
çç ÷÷
çè 3÷ø
Figure 7.22. PBE
PBE is stronger than BE. In the next section, we introduce another stronger equilibrium
concept, which is slightly stronger than PBE. Fudenberg and Tirole (1991) give a formal definition of perfect Bayesian equilibrium for a broad class of dynamic games of incomplete information and provide conditions under which their perfect Bayesian equilibrium is equivalent to
Kreps and Wilson’s (1982) sequential equilibrium.
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Susheng Wang, HKUST
4.2. Sequential Equilibrium
Recall that we used trembling-hand perfection to strength NEs. We now use tremblinghand perfection to strength BEs. This work is done by Kreps and Wilson (1982), who propose
a stronger equilibrium concept called sequential equilibrium. They impose certain consisten∗
cies between strategies and beliefs to restrict
strategy profiles
to be the limit of a sequence of totally mixed
For each of the mixed strategy profiles
arrived with a positive probability:
for
. Thus, given such a
→
∗
under
∗
although
will be
the Bayes rule can
over all the information sets. If
be used to derive a consistent belief system
∗
any information set
can still happen at some information set
there is some consistency between
∗
and
∗
for all the information sets. Such con-
sistency allows us to rule out many unreasonable equilibria. This leads to a stronger equilibrium concept called sequential equilibrium.
Definition 7.18. A strategy-belief pair
hand perfect
BE70
∗
∗
is a sequential equilibrium (SE) or trembling-
of an extensive-form game
(1) Strategy profile
∗
if
is sequentially rational given belief system
∗
with beliefs
(2) There exists a sequence of totally mixed strategy profiles
from
derived
and nature’s odds using the Bayes rule such that71
∗
∗
→
We can think of SE as a kind of stable BE, just like a trembling-hand perfect NE. We can
call a SE a trembling-hand perfect BE. The SE concept requires beliefs to be justifiable as
coming from a sequence of totally mixed strategies that are close to the equilibrium strategy
profile
∗
Using the Bayes rule, condition (b) makes sure that off-equilibrium beliefs are
sensible, which prevents unreasonable off-equilibrium beliefs from producing a BE that is not
a SPNE.
Since condition (b) in the definition of SE is stronger than condition (b) in the definition
of BE, a SE must be a BE. But, the reverse is generally not true.
Proposition 7.13. Any SE is a PBE.
Proof. 1. Since
and
for an information set,
70
are consistent and
∗
and
∗
relating to
∗
and
∗
when
under
∗
must still be consistent. For example, if
A trembling-perfect BE is different from Selton’s (1975) trembling-perfect NE for an extensive-form game.
NE doesn’t involve beliefs.
71
Nature’s odds are always strictly positive in a game. That is, if
tion over an information set, we should have
is nature probability distribu-
for all
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we will still have
⋯
∗
∗
∗
after taking the limit with
∗
⋯
Thus, a
SE must be a BE.
∗
2. Given a SE
strategies
∗
∗
suppose that
∗
∗
is not a SPNE. Then,
is strictly irrational w.r.t.
along a subgame’s equilibrium path.72 That is, there exists an information set
along the subgame’s equilibrium path at which player
implying that
is irrational w.r.t. strategies
∗
will strictly prefer a non-equilibrium strategy within this subgame if all
∗
others choose their equilibrium strategies. If we replace
by
when
is large enough,
will still strictly prefer the non-equilibrium strategy within this subgame if all others choose
their corresponding strategies defined in
probability distribution of actions from
change. Since
∗
Since
is totally mixed, when we switch the
to the belief system
the preference will not
∗
will still strictly prefer the non-equilibrium strategy w.r.t.
with-
in this subgame if all others choose their equilibrium strategies. This contradicts the fact that
∗
is rational w.r.t.
∗
at any information set. Hence, a SE must be a SPNE.
Proposition 7.14. In any finite game, there exists a SE.
Since a SE is a PBE, in any finite game, there always exists a PBE.
The following two examples show that a SE avoids the problems in the above two examples. In particular, Example 7.34 gives a good illustration of the proof for Proposition 7.13.
Example 7.32. Reconsider Example 7.27. Consider the first PBE:
∗
Given any totally mixed strategies
∗
∗
and
∗
, by nature’s odds,
; since
where the beliefs are derived by the Bayes rule. For this PBE, the requirements of SE
are
∗
Obviously, since
72
∗
it is impossible to have
Hence, this PBE is not a SE.
Being on the equilibrium path means that that action (or actions in a mixed strategy) will be taken with a
positive probability. Hence, the Bayes rule is applicable. However, an equilibrium path within a subgame is not
necessarily a part of the equilibrium path of the whole game; see the equilibrium
which the equilibrium path within the real subgame is
2
2
rather than
∗
in Example 7.16 for
(which is part of the equilibrium path
for the whole game).
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.
Nature
1
2
1
2
μ 1 = 0. 5
.
H1
L1
R1
m2
 2
 
 10 
.
L2
 5
 
2
1 − μ1 = 0.5
.
H 2 P2
R2
0
 
5
.
P1
L1
R1
1- m2
L2
R2
0
 
 5
 5
 
 10 
 2
 
 10 
Figure 7.23. SE
Consider the second pure-strategy PBE:
∗
∗
∗
∗
For this PBE, the requirements of SE are
(7.19)
For
define
Then, by the Bayes rule and nature’s odds,
and
They satisfy the conditions
in (7.19). Hence, this PBE is a SE.
BE1 in Example 7.32 shows that PBE is strictly weaker than SE. This example also shows
that PBE defined by continuation games instead of subgames is also strictly weaker than SE.
Example 7.33. Reconsider Example 7.28. The game is redrawn in Figure 7.24. Given the BE
with
∗
∗
and
∗
for total mixed strategies
and
with
(by the
Bayes rule), to support the BE as a SE, we need
∗
Since
it is impossible to have
∗
Hence, this BE is not a SE.
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Susheng Wang, HKUST
.
P1
L1
R1
s1
.
x P1
0.5
0.5
.m
æ0ö
çç ÷÷
èç0÷ø
. P2
H
1
L2
R2
s2
æ-2ö
çç ÷÷
çè-1÷ø
L2
R2
æ-1ö
çç ÷÷
çè 1 ÷ø
æ 2ö
çç ÷÷
çè 3÷ø
s2
æ1ö
çç ÷÷
çè-2÷ø
Figure 7.24. Sequential Equilibrium
∗
Consider the second BE:
∗
and
∗
Define totally mixed strategies and
beliefs as
where
is an integer,
We have
∗
Hence, the second BE is a SE. In fact, this is the unique SE.
Example 7.34. Reconsider Example 7.29. The game is redrawn in Figure 7.25. We now show
that the unique SE is the unique SPNE.
.
P1
L1
R1
s1
.
x P1
L̂1
æ0ö
çç ÷÷
çè0÷ø
R̂1
ŝ1
.m
.
H
1
L2
R2
s2
æ-2ö
çç ÷÷
çè-1÷ø
P2
L2
R2
æ-1ö
çç ÷÷
çè 1 ÷ø
æ2ö
çç ÷÷
çè 3÷ø
s2
æ1ö
çç ÷÷
çè-2÷ø
Figure 7.25. Sequential Equilibrium
Consider the first BE, BE1:
and
and
∗
∗
and
∗
Define totally mixed strategies
(by the Bayes rule). By the requirements of SE, we need
∗
However,
and
∗
cannot hold together. Hence, this BE1 is not a SE.
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Susheng Wang, HKUST
Consider the second BE, BE2:
egies
and
where
is an integer,
and
∗
∗
and
∗
Define totally mixed strat-
as:
We have
∗
Hence, the second BE is a SE.
It is actually very easy to judge whether or not a BE is a SE. We can easily identify ‘inconsistency’ between strategies and beliefs without relying on the Bayes rule. For example, in a BE
∗
with
in this example, P1 is taking
∗
for certain, but P2 believes
Such in-
consistency cannot happen in a SE even though the information set is not on the equilibrium
path. We have to have
∗
in a SE.
Intuitively, the BE and SE concepts require rationality via beliefs at every information set
including off-equilibrium paths. The SPNE concept requires rationality via strategies on the
equilibrium path within each subgame. Hence, when beliefs are always consistent with strategies at every information set, especially in every subgame, BE implies SPNE. In a SE, such
consistency is guaranteed at every information set and thus it implies SPNE.
A SPNE is a NE within each subgame; hence, rationality is guaranteed along each subgame’s equilibrium path, but rationality may not be guaranteed on off-equilibrium paths
within a subgame. Conversely, NE is rational along the whole game’s equilibrium path, but it
may not be rational along a subgame’s equilibrium path; of course, it may not be rational on
off-equilibrium paths within a subgame or the whole game. Hence, a SPNE must be a NE, but
the converse is not true. Example 7.18 is a good example. For SPNE
rium path for the whole game is
Information set

 (1,2), where

the equilibmeans the payoff cell.
is on the equilibrium path. First, the players are rational via strategies along
the equilibrium path. Second, the equilibrium path within
is
and
is
on the equilibrium path. The players are again rational via strategies along this equilibrium
path. Let’s now look at the NE
and the equilibrium path for
The equilibrium path for the whole game is
is
In this case,
the equilibrium path in the whole game, but it is on any equilibrium path within
is not on
The
players are rational via strategies on the equilibrium path for the whole game, but P2’s choice
of
is not rational within the subgame w.r.t. strategies in
given P1’s choice of
Hence,
this NE is not a SPNE.
Example 7.20 is also interesting. NE
since
is always worse than
is a SPNE. However, P2 at
The equilibrium path is
and
is not rational
is not on the equi-
librium path. That is, SPNE may not be rational on off-equilibrium paths within each subgame,
although it is rational on the equilibrium path within every subgame.
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The following game shows that SEs are stronger than PBEs. The figure shows a BE with
∗
and
∗
We can easily assign payoffs so that the choices shown in the
figure are sequentially rational for some beliefs
and
This SR solution is a BE. Since this it
has no real subgames, it is a PBE. However, if it is a SE, it must have
if
or
and
. Hence,
this PBE is not a SE.
.
.
.
.
I1
p
q
. .
Figure 7.26. PBE
I2
.
SE
In summary, we have introduced five equilibrium concepts for an extensive-form game.
Their relations are shown in Figure 7.27.73
SE
PBE
BE
SPNE
NE
Figure 7.27. Five Equilibrium Concepts
SE strengthens BE by trembling-hand perfection; PBE strengthens BE by subgame perfection; and SPNE strengthens NE by subgame perfection. These equilibrium concepts requires different conditions as described verbally in the following:
1. NE requires rationality via strategies at those information sets along the whole game’s
equilibrium path.
73
Albert Banal-Estanol shows the relationships. Albert calls BE as WPBE.
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2. SPNE requires rationality via strategies at information sets along every subgame’s equilibrium path.
3. BE requires rationality via beliefs at every information set and consistency between beliefs
and strategies along the whole game’s equilibrium path.
4. SE requires rationality via beliefs at every information set and consistency (defined by
trembling-hand perfection) between beliefs and strategies at every information set.
However, the SE concept also has its problems. In fact, the SE concept seems too strong in
some aspects but too weak in other aspects. For example, it implies that any two players with
the same information must have exactly the same beliefs regarding the deviations by other
players that have caused play to reach a given part of the game tree. Also, a SE may not be
trembling-hand perfect, as shown in Question 7.14 in the Problem Set. Due to these problems,
there are further refinements of equilibrium concepts in the literature. However, for applications under imperfect information, depending on the issues and the particular model specifications, researchers generally use either the SPNE concept or the BE concept. In fact, from
Example 7.28 and Example 7.29, we see that the problems with BE occur only if some beliefs
are obviously not sensible. Thus, as long as beliefs look sensible, a BE will generally be a SE.
A PBE is slightly weaker than a SE. For example, for the game in Figure 7.23, there is a BE
with
. Since this game has no real subgame, this BE is also PBE. However, this PBE is
not a SE
In many games, especially games of incomplete information, there are no real subgames.
For these games, subgame perfection cannot help to rule out senseless equilibria. In the following, we introduce two dominance concepts for such games. These two dominance concepts
help us eliminate some unreasonable BEs in incomplete-information games when subgame
perfection is not helpful.
4.3. BE under Complete Dominance Criterion: CDBE
We have another two strong versions of BE, which are based on the concept of dominance
in actions. These concepts are completely independent of the concepts based on subgames and
equilibrium paths.
Some PBEs are unreasonable. The following example is from Gibbons (1992, p.233). Note
that, in an incomplete-information game, each type is treated as a separate player when we
discuss PBEs.
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Example 7.35. Consider
.
P1
L1
P2
.
.
M1
m
L2¢
R2¢
æ4ö
çç ÷÷
èç1 ÷ø
æ2ö
çç ÷÷
èç2÷ø
L2
æ3ö
çç ÷÷
çè1 ÷ø
R1
.
1- m
H
P2
R2
L2
R2
æ0ö
çç ÷÷
çè0÷ø
æ1 ö
çç ÷÷
èç0÷ø
æ0ö
çç ÷÷
èç 1÷ø
Figure 7.28. Unreasonable PBEs
iff
For P2,
That is, P2 chooses
if
Hence, we have a pure-strategy
PBE:
P2 chooses
if
Hence, we have another pure-strategy PBE:
P2 is indifferent between
and P1 obviously won’t take
and
if
In this case, P2 takes a mixed strategy
even if
. Since
equilibrium path. Hence, P1 must choose
That is, when
P1 will choose
However, for P1,
In order for P1 to choose
cannot be on the
we need
Hence, we have the third PBE:
is completely dominated by
P2 to have
the information
Hence, we should have
in payoffs. Hence, it is not reasonable for
Thus, BE2 and BE3 are not reasonable. CD
criterion defined below intends to rule out such BEs.
For a strategy
That is,
and an action
consists of all actions in
An action
at
in
at an information set, denote
except action
.
from player is completely dominated if there is another action
such that ’s worst possible payoff under
at
is better than ’s best possible payoff under
whichever player ’s other actions are and whatever other players do:74
74
See the definition in Gibbons (1992, p.236). For games of incomplete information, each type is treated as a
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,
i.e., there is an action
An action
,
that is always strictly better than
for whatever other players do.
of player ’s is completely dominated if there is another action
same information set such that ’s worst possible payoff under
possible payoff under
of i’s at the
is strictly better than ’s best
whatever subsequent players do:75
Complete dominance (CD) criterion: Zero belief should be placed on a node of an information set
if the immediate action leading to this node is completely dominated. However,
in an all-dominance case when all the actions immediate leading to this information set
are
completely dominated, no additional restriction should be placed on beliefs over this information set.76
The following figure illustrates an example. The focus is whether we should impose
at information set
. The question is whether action
preceding the node is completely
dominated. For this game, the question is whether action
should impose
completely dominated
cannot impose
if so, we
unless we encounter an all-dominance situation. The all-dominance
situation relates to the question of whether action
action
completely dominates
is also completely dominated, or whether
. If indeed both
and
are completely dominated, we
.
.
.
.
P1
.
.
.
P2
P3
Figure 7.29. Complete Dominance
The CD criterion argues that a player will never take a completely dominated action and
hence other players should play zero belief on the node immediately following this action.
separate player in the definition of complete dominance.
75
See the definition in Gibbons (1992, p.236), called strict dominance. For games of incomplete information,
each type is treated as a separate player in the definition of complete dominance.
76
This condition is Requirement 5 in Gibbons (1992, p.235).
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However, if all the nodes in an information set
would have zero belief by this principle, since
zero belief on all nodes of an information is not possible (the total sum of beliefs on any information is 1), we cannot apply this principle to this
and hence any beliefs over
are ac-
ceptable.
A BE under the CD criterion is called a CDBE or a completely dominant BE.
If we replace payoff 3 in Figure 7.28 by 1.5, PBEs are:
Since both
and
are strictly dominated by
we cannot apply CD criterion on
Hence,
all these three PBEs are CDBEs. This is the special case mentioned in the CD definition. In the
meantime, this example also shows that the CD criterion cannot be used to rule out all unreasonable PBEs. This leads to the intuitive criterion by Cho and Kreps (1987) on equilibrium
dominance in the next subsection.
Note that SE is not stronger than CDBE. For example, for the game in Figure 7.28, BE2 is
SE, but it is not a CDBE.
4.4. BE under
EDBE
Given a BE
∗
equilibrium payoff
∗
Equilibrium
in a subgame, an action
∗
∗
Dominance
of player ’s is equilibrium-dominated if ’s
from the equilibrium action
strictly greater than ’s highest payoff by taking
∗
Criterion:
∗
of i’s at the same information is
whatever subsequent players do:
∗
(7.20)
Equilibrium dominance at an information set of player i means that, given the equilibrium strategy, there is no reason for player to deviate from it even if other players may follow
suit by deviating from their equilibrium strategies. Notice that we obviously have
∗
the key is that (7.20) holds for any
∗
∗
∗
See the definition in Gibbons (1992, p.239). For games of incomplete information, each
type is treated as a separate information set in the definition of equilibrium dominance. Note
that at each information, at least one action is an equilibrium action, although the information
set may not be on the equilibrium path. In particular,
∗
in (7.20) is an equilibrium action, but
it may not be on the equilibrium path (an action of an information set on the equilibrium
path).
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Equilibrium dominance (ED) criterion: Given a BE, in any subgame, if a node in an
information set
is from an equilibrium-dominated immediate predecessor action, players
should place zero belief on this node. In an all-dominance case when all the nodes in
equilibrium-dominated, any beliefs over
are
are acceptable.
The ED criterion argues that, given an equilibrium, a player will never take an equilibrium-dominated action and hence other players should play zero belief on the node immediately
following this action. However, if all the nodes in an information set
would have zero belief
by this principle, since zero belief on all nodes of an information is not possible (the total sum
of beliefs on any information is 1), we cannot apply this principle to this
beliefs over
and hence any
are acceptable.
Note that there are equilibrium actions both along the equilibrium path and on offequilibrium paths. Our definition of the ED criterion applies to all the equilibrium actions.
A BE under the ED criterion is called an EDBE or an equilibrium dominant BE.
Proposition 7.15. A completely dominated action is an equilibrium-dominated action.
Hence, an EDBE is generally a CDBE, except the special case in which all actions preceding
an information are equilibrium dominated.
We use the following figure to illustrate the proof of Proposition 7.15. Suppose that action
is an equilibrium action and action
the worst outcome of
completely dominates
must be better than that of
must be better than the worst outcome of
is,
Hence,
Since
Since
completely dominates
is an equilibrium action,
must also equilibrium-dominate
That
is equilibrium-dominated. In other words, if an action is completely dominated, it is
generally equilibrium-dominated. Hence, if a node is given zero belief by the CD criterion, it is
generally given zero belief by the ED criterion. Hence, the ED criterion generally imposes zero
belief on more nodes than the CD criterion. Therefore, a EDBE is generally a CDBE. BE1 is
Example 7.35 is consistent with this explanation.
.
P1
L
P2
.
.
R
M
H
.
P2
Figure 7.30. CD dominance implies ED dominance
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However, even if complete dominance imposes zero belief on a node, equilibrium dominance may not. Suppose
completely dominates
but not
in the above game, implying
by complete dominance. However, it is possible that
and
implying no restriction on
implies
equilibrium-dominates both
by equilibrium dominance. That is, even if CD dominance
ED dominance may not. This leads to the possibility that a BE is not a CDBE
but is an EDBE. In other words, if a node is given zero belief by the CD criterion, it is generally
given zero belief by the ED criterion, except the special case in which all actions preceding the
information set are equilibrium-dominated.
Will a BE under the CD criterion be a PBE? Will a BE under the ED criterion be a PBE?
The answer to both questions is No. We can easily find counter examples.
In the following, we present an example for the newly defined concepts. We find PBEs
first and then use the ED criterion to eliminate some PBEs.
Example 7.36. Consider the following game.
Nature
0.9
0.1
t2
t1
.
surly
wimpy
Quiche
.
I1
q
. .
dual
dual
not
dual
æ1ö
çç ÷÷
çè1÷÷ø
æ3ö
çç ÷÷
çè0÷÷ø
æ 0ö
çç ÷÷
çè1 ÷÷ø
æsender's payoff ö
÷÷ :
çç
èç receiver's payoff ÷ø÷
Sender
Quiche
Beer
p
.
.
I2
Receiver
not
not
æ2 öæ0 ö
çç ÷÷çç ÷÷
çè0 ÷÷øèç-1÷÷ø
Beer
dual
æ2ö
çç ÷÷
çè0 ÷÷ø
æ1 ö
çç ÷÷
çè-1÷÷ø
not
æ 3ö
çç ÷÷
çè0÷÷ø
Figure 7.31. BE with Intuitive Criterion
We now try to find a PBE that satisfies ED criterion. At
At
the Receiver considers
We consider four possible cases. First, if
both
the Receiver considers
and
i.e.,
and
and
equilibrium-path consistency requires
then the Receiver picks dual at
This cannot be a BE since
and
Second, if
and
then
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and
Equilibrium-path consistency then requires
which can be satisfied. Hence, we have a BE:
Notice that the wimpy sender is not truthful in BE1. Third, if
and
and
then
Equilibrium-path consistency then requires
which can be satisfied. Hence, we have another BE:
Notice that the surly sender is not truthful in BE2. Fourth, if
and
and
Consistency then requires
then
and
which cannot be satisfied. Hence, we have total two pure-strategy BEs.
Since at each of the Sender’s information set, none of the Sender’s actions is completely
dominated, all the BEs satisfy CD criterion.
Will these two BEs satisfy ED criterion? For BE1, since the Surly’s action Quiche is equilibrium-dominated, ED criterion is satisfied if
Hence, the following BE satisfied ED
criterion:
For BE2, the wimpy’s action Beer is equilibrium-dominated. Hence, ED criterion requires
But, BE2 does not allow this. Hence, BE2 is not an EDBE. Therefore, BE1’ is the only BE
with ED criterion.
Example 7.37. Find the PBE in Figure 7.28 that satisfies the ED criterion.
The dominance concept is completely independent of the concepts based on subgames
and equilibrium paths. Consequently, SE is not stronger than CDBE and vice versa. For example, BE1 in Example 7.34 is not a SE but is a CDBE, and BE2 in Example 7.35 is a SE but not a
CDBE. Also, SE is not stronger than EDBE and vice versa. For example, BE1 in Example 7.34
is not a SE but is an EDBE, and BE2 in Example 7.35 is a SE but not an EDBE.
Notice that for BE1 in Example 7.34 the equilibrium path is to take
when we consider possible beliefs for information set
path
action

we should consider the equilibrium
 exit. By taking this equilibrium path, P1 gets
P1 gets either
cannot impose
or
and exit. However,
while taking the alternative
Hence, the equilibrium path doesn’t dominate
by which we
Therefore, BE1 is an EDBE.
Our definitions of the CD and ED criteria are based on Gibbons (1992) but are more general. We define these two concepts for any game, while Gibbons defines them only for games
of incomplete information with two players. We define these two concepts on BEs, while Gibbons defines them on PBEs.
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At the end of this chapter, we now briefly mention the alternative versions of PBE in the
literature.
Definition 7.19. Given an extensive-form game , starting from an information set , a continuation game starting from is the part of the
that
− includes and all the subsequent nodes;
− no information set is broken.
If including all the subsequent nodes after an information set causes some information
sets to be broken, then there is no continuation game starting from that information set .
The definition of a continuation game is almost the same as that of a subgame, except that
a subgame starts from a singleton information set. A subgame is a game itself, but a continuation game may not be a game. Even though a continuation game may not be a game, BE can be
similarly defined on it. Alternatively, we may convert a continuation game into a game and
then define BE on it. The conversion is done by replacing the starting information set by nature. The following diagram illustrates this conversion.
.
Nature
.
.
.
p
I1
. .
q
I2
.
.
.
.
p
I1
. .
q
I2
.
Figure 7.32. A continuation game is converted to a game
In the literature, there are four versions of PBE:
1) A BE is a PBE if it is a NE in any subgame. Call it PBE(1).
2) A BE is a PBE if it is a BE in any subgame. Call it PBE(2).
3) A BE is a PBE if it is a BE in any continuation game. Call it PBE(3).
4) A BE is a PBE if it is a PBE( ) and satisfies either the CD criterion or the ED criterion,
where
or . Call it PBE(4).
All four versions of PBE are SPNEs. Among these four versions, the earlier versions are weaker
than the later versions.
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Notes
Good references for this chapter are Mas-Colell et al. (1995, Chapters 7–9), Gibbons (1992)
and Kreps (1990).
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Chapter 8
Incomplete-Information Games
The focus of this chapter is on problems under incomplete information. We will present
several popular equilibrium concepts.
There are two kinds of disinformation: imperfect information and incomplete information,
which result in four kinds of games as indicated in the following table. When a player in a
game does not perfectly know what other players’ actions, it is a game of imperfect information. If the players’ payoffs are all common knowledge in a game, it is a game of complete
information; otherwise, it is a game of incomplete information.77
Imperfect information
Incomplete information
Normal Form
NE
BNE
Extensive Form
NE, SPNE, BE, PBE, SE
BE, PBE, CDBE, EDBE
We typically use the Bayesian model for a game of incomplete information. If we treat a
player’s each possible type as an information set of this player, then a game of incomplete
information becomes a game of imperfect information and all equilibrium concepts for games
of imperfect information can carry through. By this, there are only two types of games: games
of perfect information and games of imperfect information. However, since incompleteinformation games has many special applications, we often focus on some special issues with
them, instead of treating them as imperfect-information games.
1. Bayesian Nash Equilibrium
There are situations in which some players have private information. A player’s private information is the information that the player knows but others don’t. In the Bayesian approach,
other players simply treat this unknown information as uncertainty. 78 That is, the player
knows the information precisely while others only know the distribution function of the information. This private information is typically called the player’s type.
77
There are more general definitions of incomplete information. For the types of games covered in this book,
this definition is appropriate and sufficient.
78
The fact that the unknown information is treated as uncertainty is referred to as awareness, i.e., other play-
ers are aware of the existence of the information.
Susheng Wang, HKUST
According to Myerson (1991, p.67), a game is a game of incomplete information if it has
asymmetric information. That is, when some players in a game have private information that
others don’t know, it is a game of incomplete information. More specifically, when at least one
player has type (private information) in a game, it is a game of incomplete information; otherwise it is a game of complete information. The presence of incomplete information raises the
possibility that we may need to consider a player’s beliefs about other players’ preferences, his
beliefs about their beliefs about his preferences, and so on, much in the spirit of rationalizability. Fortunately, there is a widely used approach to this problem, originated by Harsanyi (1967,
1968), that makes this unnecessary. In this approach, one imagines that each player’s preferences are determined by the realization of a random variable. Through this formulation, the
game of incomplete information becomes a game of imperfect information: Nature makes the
first move, choosing realizations of the random variable that determine each player’s preference type. Each agent then knows her type, but others only know the distribution function of
her type. A game of this sort is known as a Bayesian game.
Formally, there are
players
each with a type parameter
The type
is random ex ante and is chosen by nature and its realization is observed only by player
The joint density function is
knowledge. Each player
preference order
function
by agent
over
for
which is common
has a strategy space
Let
Each player
or equivalently a payoff function
for
is also common knowledge, but each specific value of
A Bayesian model is represented by
79
has a
The utility
is observed only
Thus, in a Bayesian game, we
can think of each player as being a separate player who maximizes his payoff given his conditional probability distribution over his rivals’ strategy choices.80
A mapping
is called a strategy of agent
the strategy mappings
where
type
Player ’s strategy set
Player ’s payoff for a profile of strategies
is the mathematical expectation over
other players don’t know player ’s type
is
Here, although each player knows his own
we need to calculate the strategy function
79
is the set of all
for all possible value of
and they need to figure out
We can actually allow an agent’s utility function to take the form
since
for all possible
rather than
All the
concepts on Bayesian implementation are readily extendable to this case.
Given the density function
conditional on the knowledge of is
80
The expectation operator
|
for the whole population, the conditional density function
of
uses this conditional density function.
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If we view each type as an information set, then an incomplete information game is an
imperfect information game and a strategy in an incomplete information game is defined
exactly the same as a strategy in the corresponding imperfect information game.
Definition 8.1. A Bayesian Nash equilibrium (BNE) in the Bayesian model
a strategy profile
∗
∗
is
that constitutes a Nash equilibrium of the game
∗
That is, for each
solves
∗
(8.1)
∈
One crucial different difference between incomplete information and imperfect information is asymmetric information under incomplete information. A player knows his own
type while others don’t under incomplete information; each player has the same information
(being either perfect or imperfect) as others under imperfect information. Consequently,
under incomplete information, a player’s strategy is not only a contingent plan on all his turns
to move but also a function of his type, while under imperfect information a player’s strategy is
simply a plan on all his turns to move.
The procedure for deriving a BNE is as follows. Given a game defined by an extensive
form, we first convert it to the normal form, using the given density function
By this, we
transforms the game into a simultaneous game. The payoffs are generally uncertain and this
uncertainty is determined by a joint density function
which is common knowledge. This
density function is already given and it defines the special player called nature in the game.
Note that this density function is not from the players’ beliefs. In a simultaneous game (a
normal-form game), there are no beliefs; instead, the density function represents true probabilities. Since we do not have beliefs in a BNE, we do not need to discuss whether or not the
subjective beliefs are consistent with the objective probabilities.
The following proposition shows that the strategies are time consistent in the sense that
the players play according to their plans after they find out their own types.
∗
Proposition 8.1. A strategy profile
iff, for all and
and
is a BNE in Bayesian model
occurring with a positive probability,
|
∈
for any
∗
where
|
∗
solves
∗
is the expectation operator over
conditional on
Proof. Problem (8.1) can be written as
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∗
∈
∗
∈
∗
∈
( )
where
(
)
By the Pontryagin Theorem, 81 the above problem is equivalent to
problem (8.2), where the Hamiltonian function is
∗
Example 8.1 (Battle of the Sexes). Consider an example from Gibbons (1992, p.152), in which
the two players, Chris and Pat, do not know each other’s payoffs very well. The payoffs are
listed in the following normal-form game, where
vately known to Pat. The beliefs are that both
distributed on
and
is privately known to Chris and
is pri-
are independent and each is uniformly
where
Chris\Pat
Opera
Boxing
Opera
Boxing
We try to find a pure-strategy BNE in which Chris chooses Opera if
otherwise and Pat chooses Boxing if
and chooses Boxing
and chooses Opera otherwise, where
and
are
some positive constants. Let us first calculate Chris’ payoffs in the two choices:
Then,
Hence,
(8.3)
Similarly, Pat’s payoffs in the two choices are:
Then,
Hence,
81
See Wang (2008, Theorem 4.8).
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Susheng Wang, HKUST
(8.4)
and
From (8.3) and (8.4), we find
which implies
∗
We find that, as
∗
probability
the ex ante probability
∗
that Chris chooses Opera and the ex ante
that Pat chooses Boxing approach
This limit is the mixed-strategy NE of the
which is
complete-information game when
Chris\Pat
Opera
Boxing
Opera
Boxing
This example shows that a mixed-strategy NE of a complete-information game can be the limit
of a pure-strategy BNE of an incomplete-information game as the incompleteness of information disappears.
Example 8.2. Consider a buyer-seller problem from Gibbons (1992, p.158–163) under doublesided private information. A buyer and a seller is making a trade on an item for a price
of them knows his/her own valuation,
seller names an asking price
for the seller and
for the buyer, on the item. The
and the buyer simultaneously names an offer price
then trade occurs at price
buyer’s strategy is
Each
if
and the seller’s strategy is
If
then no trade. In a BNE, the
A pair of
is a BNE
if the buyer solves the following problem:
(
(8.5)
)
and the seller solves the following problem:
(
where
is the density function of both
bution on
(8.6)
)
and
Assume that
follows the uniform distri-
There may be many BNEs. Let us find the BNE with linear pricing strategies:
(8.7)
Then, the FOC of (8.5) is
(
)
which implies
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(8.8)
The FOC of (8.6) is
which implies
(8.9)
Matching (8.7) with (8.8) and (8.9), we find
which implies a linear BNE:
One question is: can any other BNE do better? Myerson & Satterthwaite (1983) show that,
under the uniform distributions, the linear BNE is the most efficient.
We have considered normal-form games in the above two examples. The following example is an extensive-form game from Example 7.36. For an extensive-form game, to find BNEs,
we first convert it to the normal form, using the given density function .
Example 8.3. Find BNEs for the following game.
.
Nature
0.9
0.1
t1
Quiche
.
.
t2
. .
dual
æsender's payoff ö
÷÷ :
çç
çè receiver's payoff ÷÷ø
dual
not
dual
æ1ö
çç ÷÷
çè1÷÷ø
æ3ö
çç ÷÷
çè0÷÷ø
æ 0ö
çç ÷÷
çè1 ÷÷ø
.
Quiche
Beer
I1 q
p
surly
wimpy
Beer
.
I2
Receiver
not
not
æ2 öæ0 ö
çç ÷÷çç ÷÷
çè0 ÷÷øçè-1÷÷ø
Sender
dual
æ2ö
çç ÷÷
çè0 ÷÷ø
æ1 ö
çç ÷÷
çè-1÷÷ø
not
æ 3ö
çç ÷÷
çè0÷÷ø
Figure 8.1. BNE
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Susheng Wang, HKUST
Its normal form is
Sender\Receiver
<D,D>
<D,N>
<N,D>
<N,N>
<Q,Q>
0.1(1,1)+0.9(0,-1) 0.1(1,1)+0.9(0,-1) 0.1(3,0)+0.9(2,0) 0.1(3,0)+0.9(2,0)
<Q,B>
0.1(1,1)+0.9(1,-1)
<B,Q>
0.1(0,1)+0.9(0,-1) 0.1(2,0)+0.9(0,-1) 0.1(0,1)+0.9(2,0) 0.1(2,0)+0.9(2,0)
<B,B>
0.1(0,1)+0.9(1,-1) 0.1(2,0)+0.9(3,0) 0.1(0,1)+0.9(1,-1) 0.1(2,0)+0.9(3,0)
0.1(1,1)+0.9(3,0) 0.1(3,0)+0.9(1,-1) 0.1(3,0)+0.9(3,0)
or
Sender\Receiver
<D,D>
<D,N>
<N,D>
<N,N>
<Q,Q>
0.1, -0.8
0.1, -0.8
(2.1, 0)
2.1, 0
<Q,B>
1, -0.8
2.8, 0.1
1.2, -0.9
3, 0
<B,Q>
0, -0.8
0.2, -0.9
1.8, 0.1
2, 0
<B,B>
0.9, -0.8
(2.9, 0)
0.9, -0.8
2.9, 0
We have two BNEs: (<B,B>, <D,N>) and (<Q,Q>, <N,D>). BEs must be NEs. Indeed, the
strategies of the two BEs in Example 7.6 are our BNEs.
BNEs are actually NEs when an incomplete information game treated as an imperfect information game where a type corresponds to an information set. For example, if we treat the
game in Figure 8.1 as an imperfect information game, the NEs are exactly the same as the
BNEs. That is, BNEs for an incomplete information game are exactly the same as NEs when
we view the game as an imperfect information game.
2. Signalling Games
In extensive-form games of incomplete information, players form beliefs over other players’ types. With beliefs, signals can play an important role. Since BEs allow beliefs, BEs are
adopted as the equilibrium concept in signalling games. In this section, signalling games are
just like typical games, except that one of the players focuses on sending messages. Also, in
signalling games, we simply treat each different type a different agent. For the convenience of
discussing signalling games, for several cases, we restate the definition of BEs in the case of
signalling games.82
2.1. Pure Strategies in Signalling
As in Gibbons (1992), we will focus on two-player signalling games, in which there is one
message sender
82
and one message receiver
Here is a description of a signalling game: given
A good reference of this section is the work of Gibbons (1992, 183–218).
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Susheng Wang, HKUST
a set
of possible messages, a set
and payoffs
of possible actions, a set
and
of possible types, nature’s odds
, a signalling game
 Nature decides the sender’s type
involves:
based on a probability distribution
 The sender observes his type and then decides to send a message
 The receiver cannot observe the type , but she can observe the message
a belief that the probability/density of type is
She then forms
and takes an action
 The payoffs for the sender and receiver are respectively
Definition 8.2. A pure-strategy BE in a signalling game is
and
∗
∗
∗
that satisfies
the following conditions:
(a) For each
the receiver takes the following strategy:
∗
∗
∈
(b) For each
the sender takes the following strategy:
∗
∗
∈
(c) Equilibrium consistency: For each
for type who sends
if there exists
such that
∗
then
we require:
∗
∗(
)
Condition (c) requires consistency holds for those who send
∗(
where
)
In the following game, suppose
for all the types. If
is chosen by one of the types,
then condition (c) in the above definition means that the equilibrium-path consistency requires
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Susheng Wang, HKUST
.
Nature
p(t1 )
.
.
L
p ( t2 )
R
I1
p
p(t3 )
.
. .
Sender
L
R
q
I2
.
Receiver
Figure 8.2. Pure-Strategy BEs in a Signalling Game
This definition is only a special case of the definition of BE for a one-sender one-receiver
signalling game. Hence, we can solve for a PBE with signalling by the same way as we solve for
a BE. If a player knows his type when he moves, we treat each possible type as a separate
player. If the player doesn’t knows his type when he moves, we treat the type as uncertainty
and use the expected payoff to determine the player’s moves.
The above definition deals with a pure-strategy signalling game since the sender has a
pure strategy. In the following, we allow the sender to take a mixed strategy.
2.2. Mixed Strategies in Signalling
Following Crawford and Sobel (1982), we extend the above pure-strategy BEs to mixedstrategy BEs. A signalling game is:
 Nature decides the sender’s type
based on a probability distribution
 The sender observes his type and then decides to take a mixed strategy:
the probability/density of sending message
 The receiver cannot observe
by type
with
for any
but she can observe the message
She then forms a belief
that the probability/density of type is
and takes an action
with
for any
83
 The payoffs for the sender and receiver are respectively
83
which is
and
We may further relax this pure strategy to a mixed strategy. But, I have not seen this extension in the litera-
ture.
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Susheng Wang, HKUST
Definition 8.3. A mixed-strategy BE with mixed messages in a signalling game is
∗
∗
∗
that satisfies the following conditions:
the receiver takes the following strategy:
(a) For each
∗
∗
∈
(b) For each
∗
the sender’s mixed strategy
satisfies the condition: if
∗
∗
then
84
∗
∗
∈
(c) Equilibrium consistency: For each
if
such that
∗
then
∗
∗
∗
For condition (a), since sequential rationality is for all information sets, for each message,
no matter whether it is on the equilibrium path or not, rationality is required. For condition
(b), if there is a unique optimal message for a type, the sender will have a pure strategy; if
there are multiple optimal messages for a type, the sender will be happy to mix these optimal
messages arbitrarily. This mix will be pinned down by the consistency condition.
An explanation to condition (b) in Definition 8.3 is as follows. If (8.11) has a unique solu∗
tion
∗
, then type will choose
happy to mix these
∗
for sure. If (8.11) has multiple solutions of
∗
by arbitrary probabilities
∗
, then type is
These probabilities become equilib-
rium strategies if all three conditions in Definition 8.3 are satisfied.
Definition 8.4. A mixed-strategy BE with both mixed actions and messages in a signalling
∗
game is
∗
∗
that satisfies the following conditions:
the receiver’s mixed strategy
(a) For each
∗
∗
∗
satisfies the condition: if
then
∗
∗
∈
(b) For each
∗
the sender’s mixed strategy
satisfies the condition: if
∗
∗
then
∗
∗
∈
(c) Equilibrium consistency: For each
∗
84
∗
if
such that
∗
then
∗
∗
This requirement follows Crawford and Sobel (1982). An alternative in Matthews (1989) is
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Susheng Wang, HKUST
Again, this definition is only a special case of the definition of BE for a one-sender onereceiver signalling game. Hence, we can solve for a PBE in signalling by the same way as we
solve for a typical BE.
Note that, if we use
the density of
to denote density, i.e.,
conditional on
is the density of
and
is
etc., then, by the Bayes rule,
This implies the above equilibrium consistency condition. In many papers, authors do not
instead, they use the consistency condition to replace
mention a belief
( ) (
| )
( ) (
| )
by
in the receiver’s rationality condition. By this, the consistency condition is not
needed.
Note also that, if
∗
∗
is taken with positive probability
better than or indifferent from any other alternative message
∗
better by moving the assigned probability
∗
from
∗
∗
, it must be strictly
, otherwise the sender can do
to
. This explains condition
(8.11).
for all the types. If
In the following game, suppose
is chosen by one of the
types, then condition (c) in the above definition means that the equilibrium-path consistency
requires
In the special case when only type
ever sends
i.e.,
the above implies
.
Nature
p(t1 )
.
s ( L | t2 )
.
p
p ( t2 )
p(t3 )
.
s ( R | t2 )
s ( L | t3 )
I1
. .
Sender
s ( R | t3 )
q
I2
.
Receiver
Figure 8.3. Consistency
We can also consider mixed actions. If there are multiple solutions from (8.10), the receiver will be happy to mix these solutions, which leads to an equilibrium with a mixed action.
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Example 8.4 (BEs in Signaling Games). In Gibbons (1992), signaling games are treated as a
special type of games, called dynamic games of incomplete and imperfect information. We
treat them as typical games of imperfect information. This example illustrates our point. Consider the following game from Gibbons (1992, p.189):
.
Nature
0.5
0.5
t1
.
t2
R
L
.
æsender's payoff ö
÷÷ :
çç
èç receiver's payoff ÷ø÷
. .
I1
p
.
Sender
L
R
.
q
u
d
u¢
æ1 ö
çç ÷÷
çè3÷÷ø
æ4ö
çç ÷÷
çè0 ÷÷ø
æ2ö
çç ÷÷
çè1 ÷÷ø
d¢
I2
Receiver
u
d
u¢
d¢
æ0ö æ 2 ö
çç ÷÷ çç ÷÷
çè0÷÷ø çè 4÷÷ø
æ 0ö
çç ÷÷
çè1 ÷÷ø
æ1 ö
çç ÷÷
çè0÷÷ø
æ1 ö
çç ÷÷
çè 2÷÷ø
Figure 8.4. BEs in a Signaling Game
We solve it backward. For the receiver, for
which holds for any
For
,
,
which holds iff
then
If
and sender
takes
and sender
We now impose EP consistency. Since both information sets
librium path. Then, EP consistency requires
and
and
takes
i.e.,
are on the equi-
which can satisfy
Hence,
we have a BE:
If
then
and
equilibrium path. EP consistency requires
Then, only information set
which can satisfy
is on the
Hence, we have
a second BE:
Obviously, BE1 is a separating BE, since the sender’s action reveals her type; BE2 is a pooling
BE, since the sender’s action doesn’t reveal her type.
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If
then
For type
,
for
iff 2
. For type
. Hence,
if
if
for
. If
,
iff
, and
or
if
.
, and
consistency requires
which can
be satisfied. Hence, we have a second BE:
If
consistency requires
consistency requires
and if
and
.
.
, which cannot be satisfied. If
and
.
consistency requires
and if
which cannot be satisfied. If
, which can be satisfied and leads to BE3.
A PBE also requires consistency on off-equilibrium paths whenever possible. However,
since there are no real subgames, there are no additional requirements on the beliefs. Hence,
the three BEs are also PBEs.
Since none of the sender’s actions is completely dominated, the BEs are CDBEs. However,
only BE1 is an EDBE. Since for BE2 and BE3 the equilibrium path
dominates
the ED criterion requires
equilibrium-
This condition cannot be satisfied by BE2 and
BE3. Hence, BE2 and BE3 are not EDBEs.
Example 8.5 (Gibbons 1992, Exercise 4.3). Find all the PBEs in the following game:
.
Nature
t1
1/ 3
1/ 3
.
t2
L
.
R
. .
p1
L
.
.
p2
q1
.
t3 Sender
L
R
.
1/ 3
q2
p3
R
.
q3 Receiver
u
d
u
d
u
d
u
d
u
d
u
d
æ1ö
çç ÷÷
çè1÷ø
æ1 ö
çç ÷÷
çè0÷ø
æ0ö
çç ÷÷
çè1 ÷ø
æ0ö
çç ÷÷
çè0÷ø
æ2ö
çç ÷÷
çè1 ÷ø
æ0ö
çç ÷÷
çè0÷ø
æ1ö
çç ÷÷
çè1÷ø
æ1 ö
çç ÷÷
çè0÷ø
æ1ö
çç ÷÷
çè1÷ø
æ0ö
çç ÷÷
çè0÷ø
æ0ö
çç ÷÷
çè0÷ø
æ2ö
çç ÷÷
çè1 ÷ø
Figure 8.5. PBEs
We solve backwards. Consider the receiver’s problem first. Given
Given
the receiver considers
the receiver considers
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Next, consider the sender’s problem. If
Consistency requires
If
, implying
which can be satisfied. Hence, we have
then
and
then
implying
Consistency requires
which can be satisfied. Hence, we have
Since there is no a real subgame, these BEs are PBEs. BE1 is a pooling BE, while BE2 is a
partially pooling BE.
Also, for type
action
is completely dominated. No complete dominance for other
types. Hence, we should have
in a CDBE. Imposing
on the two BEs, we find two
CDBEs:
Further, given BE1, for types
and
, action
is equilibrium-dominated, implying
which cannot be satisfied. Hence, BE1 cannot be an EDBE. Given BE2, for types
and
, action
type
action
is equilibrium-dominated, implying
is equilibrium-dominated, implying
which is satisfied; also, for
which is also satisfied. Hence,
BE2 is an EDBE.
2.3. Cheap Talk
We now consider a special set of signalling games, called cheap-talk games. Crawford and
Sobel (1982) are the first to discuss cheap-talk games. In a cheap-talk game, the message
sender’s messages have no cost and no benefit to anyone. That is, when messages have no
effect on any player’s payoffs, the signalling game is called a cheap-talk game.
Given a set
nature’s odds
of possible messages, a set
payoffs
and
of possible actions, a set
of possible types,
a cheap-talk game
in-
volves:
 Nature decides the sender’s type
based on a probability distribution
 The sender observes his and then decides to send a message
85
 The receiver cannot observe the type , but she can observe a message
message
she forms a belief
After observing a
that the probability/density of type is
She
then takes an action
 The payoffs for the sender and receiver are respectively
85
and
This is a pure strategy. A mixed strategy can also be considered following Definition 8.3.
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 A pure-strategy BE in cheap talk is a BE in signalling games:
∗
∗
Definition 8.5. A pure-strategy BE in a cheap-talk game is
∗
∗
∗
∗
that satisfies
the following conditions:
the receiver takes the following strategy:
(a) For each
∗
∗
∈
(b) For each
the sender takes the following strategy:
∗
∗
∈
(c) Equilibrium consistency: For each
if there exists
∗
such that
then 86
∗
∗(
)
One unique feature of a cheap-talk game is that the payoffs of all players do not depend on
the sender’s messages. The message is cheap talk since the message has no effect on any player’s payoff. We often assume that the set
this,
of messages is the same as the set
of types. By
is rich enough to signal what needs to be signaled.
Since messages have no direct effect on payoffs, a pooling BE always exists. In fact, for
if action
any message
be a solution of
∈
then
is a pooling equilibrium, in which all types of the sender send
receiver takes action
only. In a pooling BE, the belief must be
∗
and the
since the
message reveals nothing.
Proposition 8.2. For a cheap-talk game, for any message
∗
∗
∗
∗
∗
where
for all
, there is a pooling BE
maximizes the receiver’s expected payoff, and
∗
and
.
Proof. For an arbitrary
, define
∗
and
∗
for all
and
∗
∈
where
∗
is independent of
. Then, we can easily verify that
∗
∗
∗
satisfies the
three conditions in Definition 8.5, which confirms it to be a BE.
86
Note that
( )
∗( )
( )
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Definition 8.6. A mixed-strategy BE with both mixed actions and messages in a cheap-talk
∗
game is
∗
∗
the receiver’s mixed strategy
(a) For each
∗
∗
that satisfies the following conditions:
∗
satisfies the condition: if
then
∗
∗
∈
(b) For each
87
the sender’s mixed strategy
∗
satisfies the condition: if
∗
∗
then
∗
∗
∈
(c) Equilibrium consistency: For each
∗
if there exist
such that
∗
then
∗
∗
An interesting question in a cheap-talk game is whether nonpooling equilibria exist. The
answer is yes. In the following, we show two examples, in which separating and partially pooling equilibria exist.
Example 8.6 (Gibbons 1992, p.214). There are two types
and two actions
and
two messages
Nature decides the type with
Assume
The payoffs are defined in the following table:
The game is indicated more clearly by the following tree. Here,
stands for the sender and
stands for the receiver. The first payoff value is the sender’s and the second payoff value is the
receiver’s.
87
This requirement follows Crawford-Sobel (1982). An alternative in Matthews (1989) is
∗
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.
Nature
.
.
.
.
Sender
.
.
Receiver
Figure 8.6. A Cheap-Talk Game
We now try to find pure-strategy BEs. We solve the game backward. We have
Then, first,
We consider pure-strategy BEs only. If
then we need
we need
and
If
and
then we need
Hence, if
If
If
then
then we need
we have two BEs:
r
r
If
no pure-strategy BE in this case. Second,
If
then we need
we need
and
If
then we need
If
then we need
and
If
Hence, if
then
we
have two BEs:
r
r
If
We need
no pure-strategy BE in this case. Third,
and
which can be satisfied. Hence, we have a BE:
r
Fourth,
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We need
and
which can be satisfied. Hence, we have a BE:
r
We have got all the pure-strategy BEs. Since there is no real subgame, each BE is a PBE. Since
there is no complete dominance, all the BEs are CDBEs. Also, since it is not possible to have
equilibrium dominance (one of the sender’s payoff is the same as the equilibrium payoff in
every case), all the BEs are EDBEs.
In the above example, a separating equilibrium appears only when the receiver reacts differently to different messages. However, BE1 to BE4 are trivial solutions since we know the
existence of such pooling solutions in cheap-talk games.
BEs in cheap talk games are always PBEs and EDBEs.
Example 8.7. This example is from Crawford & Sobel (1982, Econometrica) and Gibbons
(1992, p.214–218). The sender’s type in uniformly distributed on
The sender’s payoff is
action space is
where
preferences – when
The receiver’s
and the receiver’s payoff is
is a positive constant. Here,
measures the similarity of the players’
is closer to zero, the players’ interests are more closely aligned. We do
not define a message space
; here we are free to use whatever space necessary.
Under complete information when is publically known, the receiver’s optimal action is
∗∗
but the sender wishes the receiver to take
The parameter
measures how
closely these two players’ preferences are aligned.
Suppose that there is
while those in
such that all the types in
send another, say
After receiving the messages, the receiver believes
that the sender’s type is uniformly distributed on
ing
send one message, say
and
respectively. After receiv-
the receiver considers
(8.13)
which implies
∗
After receiving
the receiver considers
(8.14)
which implies
∗
In (8.13) and (8.14), we have used the true conditional probabilities
instead of beliefs. Given these actions, the sender’s payoffs are respectively
∗
To justify that type
send message
∗
will choose to send message
and type
will choose to
we need
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This implies
(8.17)
which implies
Condition (8.15) is equivalent to
With (8.18), this condition is equivalent to
∗
∗
iff
Since
∗
. Hence,
has to be positive, we need
∗
iff
and
Hence, when
the sender and receiver’s preferences are too dissimilar to allow a nonpooling solution.
Crawford and Sobel consider mixed strategies. All types in
domly, according to the uniform distribution on
choose a message ran-
all types in
randomly, according to the uniform distribution on
choose a message
If we assume that
, all mes-
sages are sent in equilibrium. Hence, equilibrium consistency by a BE implies that the receiver’s belief after observing any message from
is that
is uniformly distribution on
, and the receiver’s belief after observing any message from
distribution on
is that is uniformly
.
Now, suppose there are
steps:
with
Consider any two nearby intervals of type:
and
As in (8.13) and (8.14),
after receiving messages from the types in these two intervals (one message from each interval), the receiver’s actions are respectively
∗
∗
Then, as in (8.17), we have
implying
implying
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implying
implying
where
Then,
implying
implying
Therefore, if
there exists an n-step partially pooling equilibrium. Hence, the
largest possible number of steps
∗
is the largest value of
such that
which
is
∗
where
in
denotes as the largest integer less than or equal to
∗
We have
if
and
∗
if
We can see that
∗
decreases
Perfect communication occurs when
the players’ preferences are perfectly aligned.
Cheap talk games are special. Given each type
the sender has a wishful action
from
∈
We call
the ideal solution for the sender. If the receiver knows the type, she would take
from
∈
We call
the complete-information solution. If
for all
then the sender can
provide completely separating messages to fully reveal his type, by which the receiver will take
the complete-information solution. The complete-information solution serves the best interest
of the sender in this case when the ideal solution is consistent with the complete-information
solution. Example 8.6 is such an example, in which the sender with type
type
prefers
prefers
and the
BE5 and BE6 offer the ideal solution. If the complete-information solution
is different from the ideal solution, how should the sender send her messages? The following is
such an example.
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Example 8.8. This example is from Gibbons (1992, Exercise 4.8).
two possible messages
There are three types
and three possible actions
Nature decides the type with equal probabilities. The payoffs are defined in the following table:
0, 1
1, 0
0, 0
0, 0
1, 2
0, 0
0, 0
1, 0
2, 1
The game is indicated more clearly by the following tree. Here,
means the sender and
means the receiver. The first value in each payoff column is the sender’s and the second value
is the receiver’s.
.
Nature
t1
.
t2
L
.
.
æ1 ö
çç ÷÷
çè0÷ø
æ0ö÷
çç ÷
çè0ø÷
t3
æ1 ö
çç ÷÷
èç0÷ø
R
.
a1 a 2 a3
æ0ö
çç ÷÷
èç1 ÷ø
.
æ0ö
çç ÷÷
èç0÷ø
æ0 ö
çç ÷÷
çè0÷ø
æ1 ÷ö
çç ÷
çè2÷ø
æ0 ö
çç ÷÷
çè0÷ø
æ1 ö
çç ÷÷
çè 2 ÷ø
æ0ö÷
çç ÷
çè0÷ø
a1 a2 a3
æ0ö÷
çç ÷
çè0÷ø
.
q3
p3
a1 a 2 a 3
æ0ö÷
çç ÷
çè0÷ø
R
.
q2
a1 a2 a3
.
L
.
p2
q1
a1 a 2 a3
1/ 3
L
R
p1
æ0ö÷
çç ÷
çè1 ÷ø
1/ 3
1/ 3
a1 a2 a3
æ2ö÷
çç ÷
çè1 ø÷
æ1 ö÷
çç ÷
çè0÷ø
æ0ö÷
çç ÷
çè0÷ø
æ1 ö÷
çç ÷
çè0÷ø
æ2ö÷
çç ÷
çè1 ÷ø
Figure 8.7. A Cheap-Talk Game
When receiving message
the receiver forms a belief
; when receiving message
Given message
with
the receiver forms a belief
and
with
and
the receiver will have the preferences:
(8.19)
Similarly, given message
the receiver will have the preferences:
(8.20)
The sender
prefers
the sender
also prefers
and the sender
prefers
i.e.,
Given the preferences in (8.19) and (8.20), the sender try to choose his message so that his
preference is satisfied. We find that the following strategy will induce the receiver to choose
what the sender wants:
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Given this strategy, equilibrium-path consistency implies
by (8.19) and (8.20), the receiver will take
sees
when she sees
and
If so,
and she will take
when she
Hence, the sender’s wishes are satisfied. Therefore, we have a BE in cheap talk, which
is
s
r
Given only two possible messages, there are only three possible combinations: (a) the first
two types pool; (b) the first and third types pool; and (3) the second and third types pool. In
and
case (b), assuming the two pooling types send
receiver will choose either
will choose to send
or
when seeing
when seeing
Then, type
instead, which is a deviation. Hence, it cannot be a BE. In case (c), as-
suming the two pooling types send
choose
and choose
If so, by (8.19), the
when seeing
and choose
and
when seeing
If so, by (8.19), the receiver will
Then, type
will send
which is a
deviation. Hence, it cannot be a BE.
Although we take a guess to identify the ideal solution in the above case, there is indeed
some rules to apply. Since
type
since
send
or
the same message should be sent by these two types;
should send a different message. One question: should
and
Suppose
Then, EP consistency requires
and
By (8.19) and (8.20), we find
This solution is also ideal for the sender. Hence, BEs that imply the ideal solution are
not unique. This example suggests that messages themselves are not important; what is important is that different messages should be sent for different ideal actions and the same
message should be sent for the same ideal action.
Example 8.9. Reconsider Example 8.6. We look for those BEs that offer the ideal solution.
Given message
the receiver will have the preferences:
(8.21)
Similarly, given message
the receiver will have the preferences:
(8.22)
The sender
prefers
and the sender
prefers
i.e.,
which turns out to be the same as the complete-information solution. Given the preferences in
(8.21) and (8.22), the sender try to choose his message so that his preference is satisfied.
Consider the following strategy of the sender:
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Given this strategy, equilibrium-path consistency implies
(8.22),
and
If so, by (8.21) and
, which implies the ideal solution. The sender’s wishes are satisfied.
Therefore, we have a BE that implies the ideal solution, which is
Similarly, if the sender’s strategy is
consistency implies
and
, imply-
. The corresponding BE is
ing
This BE also implies the ideal solution.
Notes
Good references for this chapter are Gibbons (1992) and Kreps (1990).
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Chapter 9
Cooperative Games
For a group of players, it is possible that they cooperate in some way to improve their individual welfare. They may find a need for cooperation on a common objective, but they may
bargain over the sharing of benefits (bargaining solutions). They may form coalitions to cooperate within a group but compete between groups, just like political parties (the core and the
Shapley value). They may also seek third-party coordination, such as arbitration and government social programs.
Here, the term ‘cooperation’ does not mean that the individuals have to set aside their
own interests and work for the joint goal such as with a collective utility function. This is not
what cooperative games must imply. Instead, in cooperative games, each player is driven by
self-interest and a rational decision maker whose behavior is still determined by maximizing
his own expected utility/payoff. In other words, players in a cooperative game are driven by
self-interest but they realize that certain forms of cooperation can serve their individual interests well.
This chapter discusses several notions of cooperative solutions under complete information.
1. The Nash Bargaining Solution
Consider a situation in which two players bargain for a share of a fixed pie. Since this pie
is fixed, it is a zero-sum game: one player’s gain is the other player’s loss. Information is perfect: everything is public knowledge. What is a sensible solution in such a situation? The Nash
bargaining solution is a simple and popular solution.
8.1. The Nash Solution
There are two players
Let
be a set of potential bargaining outcomes and
the outcome if the bargaining fails. We call
and
and there exists
be
the disagreement outcome. Assume that, for all
such that
for both
The Nash bargaining solution is defined for multiple players, not limited to two players
only. An agreement
∗
is the Nash bargaining solution for
players
if
Susheng Wang, HKUST
∗
(9.1)
∗
The Nash bargaining concept can be explained in the following way. Player makes the argument: “I demand the outcome
∗
rather than
negotiation with probability
I back up this demand by threatening to quit
a threat that is credible since it is better for me to do so.”
Player objects to this argument with the counter argument: “Even if you may quit with probability
∗
it is still better for me to insist on
rather than to agree to
”
Remark 9.1. Two weak points about the Nash solution. First, the two players are equal in
bargaining power since each has the veto power. Second, the solution is not a limit point of a
bargaining process. In other words, the solution is more like a stalemate rather than a final
compromise. A bargaining process in reality seems to be a process of revealing and discovering each other’s bottom lines. Such a process is difficult to model in a bargaining model with
perfect information. The alternating-offer bargaining concept in Section 9 has some flavor of
such a process.
Proposition 9.1 (Nash Solution). There exists a unique Nash solution. And, an agreement
∗
is a Nash solution iff it is the solution of the following problem:
(9.2)
∈
where
is the expected utility representation of
We first prove that (9.2) is sufficient. Let
Proof. For simplicity, assume
solution of (9.2). Since there is
for all
If there is
such that
∗
∗
for some
∗
for all and
∗
for some
for
∗
be a
∗
then
∗
is indeed a Nash solution.
defined by it, for any
for any
For any
∗
∗
That is, condition (9.1) is verified and
Conversely, given condition (9.1) and
have
we must have
∗
such that
∗
implying
for
such that
consider
such that
∗
by (9.1), we
defined by
∗
∗
Since
for this
we have
∗
Letting
implies
∗
That is,
∗
∗
∗
is indeed a solution of (9.2).
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Example 9.1. Consider two risk-neutral players with
tion value of player
come
where
Assume that the size of the pie is
with
is called the reserva-
called the revenue. Then, for out-
the utility values are
and
By Proposi-
tion 9.1, the Nash solution is from:
∈[ ,
]
The solution is
∗
That is, the Nash solution gives each player his reservation value
plus half of the surplus
from the trade.
A generalized Nash solution is
∗
where
and
represents the bargaining power of player
extend this formula to a case with
Obviously, we can
individuals. However, this formula is not obtainable from
(9.2), since the Nash bargaining concept treats every player as equal in bargaining power.
Instead, this generalized solution is the solution of the following problem:
∈[ ,
]
That is, when the players have rank-dependent utility functions, the generalized Nash solution
is derived from this maximization problem.
Corollary 9.1 (Risk Aversion). Suppose
is less risk averse than
Nash solutions of the bargaining problems
and
Let
and
be the
respectively. Then,
This corollary suggests that risk aversion causes a player to accept a less favorable deal.
An alternative mechanism to this allocation problem under perfect information is the
general equilibrium approach. In the GE, the price will be
and any feasible allocation is a GE
allocation. This GE allocation is uninteresting.
8.2. Implementation of the Nash Solution
The Nash solution means that, if two players somehow end up in such a situation, they
cannot possibly deviate from it and hence it must be the solution. However, it does not show
how such a solution can be reached. We now present a game tree by which the Nash bargaining solution is a SPNE.
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Given a set
of potential bargaining outcomes and a disagreement outcome
find a simple procedure to implement the Nash solution
bargaining solution
∗
∗
for a pair
we want to
Given the Nash
consider an extensive-form game with perfect information and chance
moves consisting of the following steps:
• Player 2 chooses an alternative
with
and with probability
• Player 1 accepts
and
with probability
that game ends
that it continues.
∗
or chooses the lottery
this choice is the outcome.
Proposition 9.2 (Implementation). The above game implements the Nash solution as a
SPNE.
∗
Proof. Given the Nash bargaining solution
on
∗
∗
since
player 1 will insist
in the third step. See Figure 9.1. Moving backward one step, player 2 is indeed justified
to propose
and to threaten to quit with probability
∗
since
.
Player 1
x*
x
A repeated game:
.
x
1− p
Player 2
p
x
D
Figure 9.1. A Repeated Game
Going back one step further, player 1 will choose
sequential outcome of the above
∗
since
∗
That is,
∗
is a
game.88
The Nash solution can also be implemented by alternating-offer bargaining.
88 What
is actually shown is that the choice of
player 2 will insist on
∗
is a consistent choice for player 1 since he will insist on it. But,
too — a stalemate.
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Proposition 9.3 (Implementation). The solution of the alternating-offer bargaining game
with a probability
of breakdown after each rejection converges to the Nash solution as
9. The Alternating-Offer Bargaining Solution
9.1. The Alternating-Offer Solution
Two players P1 and P2 with time preference discount factors
gain alternately over a pie of size
and
respectively, bar-
The bargaining ends if one player accepts an offer from the
other. More specifically, P1 makes an offer first. If P2 accepts the offer, the game ends; if not,
P2 makes an offer in the next round, and P1 decides to accept or reject, and so on.
What is the SPNE? For each player, the future is always the same at any point of time.
Thus, if P1 thinks that
is his best share in period 1, he will still think that
is his best share in
is her best
period 3 when it is his turn to make an offer again; similarly, if P2 thinks that
share in period 2, she will still think that
is her best share in period 4 when it is her turn to
make an offer again. Thus, given the best shares
will accept the offer
make
and
for the two players, respectively, P2
from P1 in period 1 if
P1 understands this; he will thus
as large as possible such that the equality holds, i.e.,
Symmetrically, P1 will accept the offer
this; she will thus make
from P2 in period 2 if
P2 understands
as large as possible such that the equality holds, i.e.,
The solution from the two equations is
∗
In particular, when
(9.3)
∗
we have
∗
∗
The above analysis is based on risk-neutral players. For a general case, let
come space and suppose that player ‘s expected utility for outcome
discount factor is
for which
An alternating-offer bargaining solution is a pair
is proposed by P1 and
be the out-
is
and the
of proposals,
is proposed by P2. The following is the general result on
alternating-offer bargaining, which can be proven by the same approach as above.
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Proposition 9.4 (The Alternating-Offer Bargaining Solution). An alternating-offer bargaining game satisfying certain axioms (Osborne and Rubinstein, p.122) has a SPNE. Let
be the unique pair in
∗
pose
∗
∗
∗
satisfying
∗
Then,
∗
∗
∗
is the SPNE, for which P1 will propose
∗
∗
(9.4)
and P2 will accept, and P2 will pro-
and P1 will accept.
Example 9.2. When both
∗
∗
and
∗
are risk neutral, we have
∗
∗
∗
∗
∗
which imply (9.3).
There are three problems with this theory. First, there is no bargaining in equilibrium. It
cannot explain why people bargain in practice. Second, P1 makes an offer first and hence gets
a larger share. However, are you eager to make an offer first when you bargain with a seller?
The answer is probably no. Third, since a bargaining process may take only 3 seconds to complete, a time discount does not make sense. In such a time span, people are not likely to discount time. If so, the theory predicts a trivial outcome:
∗
∗
9.2. The Outside Option Principle
In the Nash bargaining game, there is a disagreement outcome
which plays an im-
portant role in the Nash bargaining solution. However, the alternating-offer bargaining solution in Proposition 9.4 has nothing to do with a failure option. This is not very ideal since a
realistic bargaining process often involves a threat of a disagreement and this threat often
plays a crucial role in reaching an agreement.
We now consider an extension of the alternating-offer game by allowing one or both players, at various points in the game, to opt out (without requiring the approval of the other
player). We allow a player to opt out only when the player is responding to an offer.89 We call
this option an outside option. This opt-out option is actually an IR constraint, and in this case
(9.4) are the IC constraints.
89 That
is, if player 1 is making an offer this period, player 2 has the option to opt out right away in this period;
player 2 is not allowed to opt out in the next period when it is player 2’s turn to make an offer.
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Proposition 9.5 (The Outside Option Principle). Suppose that P2 can opt out when responding to an offer. Let
∗
∗
be the unique SPNE outcome without an outside option, and
∗
be P2’s outside option. If
SPNE is a pair
the SPNE is
∗
∗
∗
If
satisfying
for which P1 will propose
and P2 will accept, and P2 will propose
and P1 will accept.
denote an outcome that the game ends periods later with payoff vector
Proof. Let
the
Let
be the outcome by which player 2 choose to opt out periods later.
∗
Let
∗
∗
be the unique SPNE. If
∗
ble. In this case, P1 proposes
P2’s threat of opting out is not credi∗
and P2 will accept. If
In this case, if it is P1’s turn to offer, P1 should offer
offer
satisfies
P2’s threat is credible.
such that
The efficient
P2 will accept it since continuing the bargaining cannot lead to a
better solution. If it is P2’s turn, P2 will offer
acceptance. The efficient offer
such that
satisfies
in order to induce
which P1 will accept. Thus, the pair
is a SPNE.
Hence, the ability of P2 to exercise an outside option ensures the bargaining outcome to
be no worse than the no-trade outcome for P2.
9.3. A Risk of Breakdown
Instead of discount factors
probability
we now assume that there is a chance of breakdown with
at the end of each period. Let the event of breakdown be
Proposition 9.6. Suppose
for all
and
and
Let
∗
∗
be the
unique pair of efficient agreements satisfying
∗
∗
Then,
pose
∗
∗
∗
∗
is the SPNE, for which P1 will propose
∗
∗
and P2 will accept, and P2 will pro-
and P1 will accept.
The proof of Proposition 9.6 is the same as the proof of Proposition 9.4, which is in turn
just like the derivation for (9.3). In fact, by interpreting
and
as probabilities, Proposition
9.4 immediately implies Proposition 9.6.
One problem in using the alternating-offer bargaining solution is to justify who should be
the first to make an offer. It is often difficult to justify who should be the first mover in an
applied bargaining problem, while the outcome clearly favors the first mover. The Nash bargaining solution does not pose such a difficult problem to a modeler. This is one reason that
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the Nash bargaining solution is used more often in applications than the alternating-offer
bargaining solution.
10. The Core
The Walrasian equilibrium uses the price mechanism to allocate resources. We consider
an alternative mechanism in which economic agents freely form coalitions. Each coalition is a
cooperative unit in which resources are pooled and shared among the members. The question
is: without a price system, what kind of equilibria are we going to get?
Assume that there is no production sector. As in Chapter 4, a pure exchange economy
consists of
commodities and
agents in
of goods, a consumption space
preferences
Each agent
and a utility function
A consumption plan
allocation
has an endowment
representing his
for agent
is an allocation. An
is feasible if
Definition 9.1. A group of agents
is a coalition. We say that coalition
allocation
such that
if there is some allocation
•
is feasible for
•
dominates
∈
blocks a given
∈
for
If an allocation
can be blocked, then there is some group of agents that can do better by
trading among themselves and hence
will not be implemented. A feasible allocation
the core of the economy, denoted as
core, if it cannot be blocked by any coalition. Hence,
is in
an allocation in the core is an implementable allocation to the whole society since no group
can block it.
Proposition 9.7.
Proof. If
core
is Pareto optimal and
for all
is not Pareto optimal, then it can be blocked by the coalition consisting of all the
agents. Therefore, any allocation in the core must be Pareto optimal. Furthermore, if
for some
then agent himself can block the allocation. Thus, we must have
Corollary 9.2. In a two-person economy,
core
is PO and
For the 2-agent 2-good case, the Edgeworth box in Figure 9.2 clearly illustrates the core.
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B
x2
contract curve
.w
Core
A
x1
Figure 9.2. The Contract Curve
As indicated by the above figure, there are many allocations in the core, including the
competitive equilibrium allocation. However, if we allow the number of agents in the economy
to grow, we will have more possible coalitions and hence more opportunities to rule out some
allocations from the core. Hence, we hope that the core can shrink to one point as the number
of agents goes to infinity, and we hope that this single allocation is the competitive equilibrium
allocation.
To increase the number of agents in a tractable way, we expand the economy by repeatedly replicating it. We say that two agents are of the same type if they have the same preferences
and endowments. We say that one economy is an r-replica of another if there are
times as
many agents of each type in one economy as in the other. The core of the r-replica of an economy is called the r-core of the economy.
The following proposition shows that the competitive equilibrium allocation is always in
an r-core and any allocation that is not a competitive equilibrium allocation must eventually
not be in the r-core of the economy when the economy gets larger and larger. Hence, only the
competitive allocation can stay in the core forever as an economy expands.
Proposition 9.8. Suppose that utility functions are continuous, strictly monotonic, and
strictly concave.
∗
(1) If
∗
is a competitive equilibrium, then
(2) If a feasible allocation
∗
r-core, for
is not a competitive equilibrium, there exists integer
such that
r-core, for
Proof. (1) An agent of type in the
will be denoted as
If
∗
Then, there is some coalition
∗
replica is denoted as
and his consumption bundle
is a competitive equilibrium, suppose
and some allocation
∗
r-core for some
90
such that
∗
90
∗
is itself in the 1-replica. Here, we show that
∗
cannot be blocked by any coalition in a replica economy.
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(9.6)
∈
∈
must not be afforda-
But, by the definition of competitive equilibrium, for any agent
ble. That is,
Hence,
∈
∈
which contradicts (9.6).
(2) For simplicity, we assume that there are only two individuals
and
allocation
and
and two goods
so that we can use an Edgeworth box in Figure .2 to illustrate our proof. If a feasible
is not a competitive equilibrium allocation, it must be in a situation like the one in
the following Edgeworth box, for which we cannot find a straight line going through the endowment point and separating the two indifference curves going through
A
y2
.w
d
D
.
xA
e
.
x B = xˆ
E
B
y1
Figure 9.3. A Coalition Blocks an Allocation
Consider an
and
-replica of the original economy. Construct a coalition
B-type agents,
and an allocation
to A-type agents, where
by agent
and
that gives
is defined in the figure. Denote
as excess supply of good
from agent
with
A-type agents
to B-type agents and
as excess demand for good
Then, as shown in the figure, the
excess demand and supply are
For the allocation
that is,
to be feasible for coalition
implying
This means that, for any point
matter how large
we must have
with
for some rational number,
is, we can find a coalition to distribute
to A-type agents and
no
to B-type
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agents. When the rational number
is close to
is close to
are continuous, we can always find a rational number
coalition
consisting of these
Of course, this implies that
and
Since the indifference curves
such that
agents will block the allocation
That is, the
Therefore,
-core.
-core, for any
The following result is expected.
Proposition 9.9 (Equal Treatment in the Core). Suppose that utility functions are continuous, strictly monotonic, and strictly concave. If
is an allocation in the r-core of a given
economy, then any two agents of the same type must receive the same consumption bundle.
Proof. Given
defined in the proposition, let
Then, by the concavity of
the utility value of the average is better than the average of the
utility values:
(9.7)
is not true, then, by strict concavity of
If, for some type
strict inequality for
For convenience, suppose
(9.7) is a
Given this
assumption, consider coalition
Using (9.7), we have
and
The allocation
for coalition
This means that coalition
can block allocation
is also feasible:
proposition, PO allocations are strongly PO within
(note that, by the given conditions in the
Therefore, we must have
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Since any agent of the same type receives the same consumption bundle, we can still use
an Edgeworth box to examine the core of a replicated two-agent two-good economy. Instead of
an allocation
in the core representing how much
gets and how much
gets, we think of
as telling us how much each agent of type A gets and how much each agent of type B gets.
11. The Shapley Value
11.1. The Balanced Contributions Property
The Shapley value is a value system that efficiently allocates the total obtainable value/welfare of a society to each member of the society. One advantage of the Shapley value is
that it has a clearly defined value system for each cooperative game. This makes it very easy to
use in applications.
There are
players
denote the number of members in
Definition 9.2. A coalitional game
A nonempty subset
of
is called a coalition and we
by
with transferable utility91 consists of
• a finite set
of players,
• a function
(characteristic function) that assigns every coalition
of
a real number
(the value of
is a value system if (1) it assigns value
, and (2)
to each coalition
so that every
gets
∈ℕ
In contrast to the core, the solution concept here restricts the way that an objecting coalition may deviate, by requiring that each possible deviation be balanced by a counterdeviation.
For the core, any feasible deviation is the end of the story and ignores the fact that a deviation
may trigger a reaction that leads to a different final outcome.
A pair of objection by player and counterobjection by player to a division of
may
take one of the following two forms.
(1) Threat to leave.
1. Objection by
only
91 Transferable
give me more otherwise I will leave the coalition, causing you to obtain
rather than the larger utility
so that you lose
utility means that all the agents’ utility values are comparable. We can then represent the total
amount of utility available to the members of a coalition
by a number
For example, money is a kind of
transferable utility that can be shared among individuals.
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2. Counterobjection by
it is true that if you leave then I will lose, but if I leave, then you
will lose more:
(2) Threat to exclude.
1. Objection by
give me more otherwise I will persuade others to exclude you from the
coalition, causing me to obtain
rather than the smaller utility
so that I
will gain
2. Counterobjection by
it is true that if you exclude me then you will gain, but if I ex-
clude you, then I will gain more:
Definition 9.3. A value system
satisfies the balanced contributions property if
The balanced contributions property in some sense suggests that the allocation is fair, a
balance of bargaining powers, or a balance of contributions. The Shapley value satisfies this
property by which every objection is balanced by a counterobjection: for every objection of any
player against any other player
there is a counterobjection of player
And, either of the
two forms of objection-counterobjection is balanced, where “balance” means that both the
objection and counterobjection are equally effective/convincing.
11.2. The Shapley Value
We now proceed to define the Shapley value system.
Definition 9.4. The marginal contribution of player to coalition
Let
be the number of players in . Let
be the set of all
with
permutations
be the set of players who precede in permutation
,
is
92
of
, and
The Shapley value
is defined as
(9.9)
,
∈
Suppose that all the permutations are equally likely to happen. Then,
is the ex-
pected marginal contribution of player to the society:
92
indicates the position of agent in permutation
For example, for
we have
and
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(9.10)
⊂ℕ\{ }
Here, given each coalition
players in this
is the number of permutations for which the
are all preceding and the rest are behind
93
sents one possibility of objection-counterobjection, where
Each such permutation repre-
players form a coalition
with-
out player and the remaining players form another coalition without player . Player can
contribute to any of these coalitions
by joining it. Here,
coalitions, and, given ,
is the total number of all possible
is the number of coalitions that player is not in-
volved. Alternatively, by assuming that all the orderings are equally likely,
pected marginal contribution of player to the society. We indeed have
is the ex-
∈ℕ
Proposition 9.10 (The Shapley Value). The Shapley value satisfies the balanced contributions property, and it is the only value system that satisfies the property.
Proof. Step 1. We first show that there is at most one value system that satisfies the balanced
contributions property. We use mathematical induction.
First, since
when
∈ℕ
implying
for any two value systems
since
for any coalition
and
in
we have
(in this case,
Hence, it is true when
or less. Let
Suppose now that it is true when the number of players is
game with
players. For any two value systems
they are identical for all games with less than
in
except for
and
be a
satisfying the property, suppose that
players. That is,
for any coalition
Since
by (9.8), we have
By summing over
Again, we have
we find
since
for any coalition
Step 2. We now verify that the Shapley value
Given a game
93 For
each such coalition
in
satisfies (9.8).
we have (9.10). Similarly, for game
agent has to be at position
positions. With agent standing at position
possible combinations] and the rest of the agents
we have
so that the group
the group
can all fill up the preceding
pick up positions preceding [which has
then pick up positions behind [which has
possible combinations]. Thus, the total number of combinations with group
preceding and the rest behind
is
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⊂ℕ\{ , }
Since is either behind or ahead of in a permutation, we have
⊂ℕ\{ , }
⊂ℕ\{ , }
where the first term on the right is for the cases when is behind and the second term is for
the cases when is ahead of
the positions of and
We have a similar formula for
as above when we reverse
Subtracting these two formulas implies
⊂ℕ\{ , }
(9.12)
⊂ℕ\{ , }
Also, by reversing the positions of and for the formula in (9.11), we have a similar formula
for
as in (9.11). Hence,
(9.13)
⊂ℕ\{ , }
Obviously, by definition, we have
and
Then, (9.12) and (9.13) imply
that is, (9.8) holds for
Example 9.3 (Cost Sharing). Let
How should
where
be the cost of providing some service to the community
be shared among the members? One answer is given by the Shapley value
is the share of cost by member
Example 9.4 (Glove Game). Let
, where players 1 and 2 have right hand gloves and
player 3 has a left hand glove. A coalition has value
if there is a match of gloves, and
oth-
erwise. That is,
Then,
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,
,
,
1, 2, 3
1, 3, 2
2, 1, 3
2, 3, 1
3, 1, 2
3, 2, 1
Hence,
. Symmetrically,
!
we have
. Since
.
workers
Example 9.5. A firm has
Each worker contributes
Then,
and
Each permutation
of
and an owner
who provides necessary capital.
to the total profit; without the firm, the workers yield nothing.
is equivalent to assigning the
consists of two parts: a permutation
location in the ordering. Given
players to
of
locations. Each
and a designation of the owner’s
, the owner can be assigned to one of the
the front, in the back, or between two workers. For example, if
obtained by assignig
, or
ly
is at the second location is also
assigned to the remaining
tion is still
,
since the
can be
,
is at the first location is obvious-
workers can be arbitrarily assigned to the remaining
of those ’s in which
locations: in
, then
to one of the 4 locations, implying
. The number of those ’s in which
since the
∈ℕ
since the
locations; the number
workers can be arbitrarily
locations; the number of those ’s in which
is at the third loca-
workers can be arbitrarily assigned to the remaining
locations, and
so on. Hence,
,
This implies that
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,
∈
Since
and
and the workers’
for
must have
should all be the same, we
. That is, the Shapley value is:
for
Alternatively, we can also calculate a worker’s Shapley value by the following approach.
For a worker , we assign him first to one of the
locations, then assign the owner, and
then assign the rest. If is at the first location, then
then
,
,
. If is at the second location,
if the owner is at the first location, otherwise
assigned to the remaining
third location, then
locations; there are
,
; the rest can be
such possibilities. If is at the
only if the owner is at one of the two locations in front of and
,
the rest can be assigned to the remaining
ties. If is at the fourth location, then
locations; there are
,
such possibili-
only if the owner is at one of the three loca-
tions in front of and the rest can be assigned to the remaining
locations; there are
such possibilities, and so on. This implies that
,
∈
Notes
Good references for Section 1 are Myerson (1991, Chapter 8) and Osborne and Rubinstein
(1994, Chapters 7 and 15). Good references for Section 9 are Osborne and Rubinstein (1994,
p.118–130) and Mas-Colell et al. (1995, p.296–299). A good reference for Section 10 is Varian
(1992). A good reference for Section 11 is Osborne and Rubinstein (1994, p.289–293). See also
Hokari (2000) for extensions of the Shapley value.
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Chapter 10
Market Information
Observed market failures and inefficiencies such as those in medical insurance and unemployment insurance are well known. Possible causes include incomplete information, incomplete markets, asymmetric information, and incentives. This and the next chapters will
focus on asymmetric information as a possible cause for market failures. We deal with competitive firms in this chapter and monopolies in the next chapter.
12. Akerlof Model: the Used Car Market
Akerlof (1970) illustrates a market failure caused by adverse selection by the following argument:
1. In the used-car market, an individual is more likely to sell her car when she knows that it is
not in good condition.
2. Uninformed buyers are wary of this behavior and their willingness to pay is low.
3. This fact further exacerbates the adverse selection problem: if the price of a used car is low,
only those sellers with bad cars will offer them for sale.
4. As a result, there may be no market for used cars.
1.1.
The Used Car Market with Asymmetric Information
There are two groups of people, buyers and sellers. In this section, we assume that the
sellers know the quality of their own cars, but the buyers do not know the quality of any car in
the market.
The Buyer’s Decision
Each buyer decides to buy one or no car, and his utility function is
Susheng Wang, HKUST
where
or
depending on whether or not the buyer buys a car,
other goods, and
is the quality of the car. Given income
is the consumption of
this individual faces the following
budget constraint:
where
is the price of the car. Here, we have assumed that the price of other goods is
in
other words, the price of other goods is taken as the numeraire. The buyer does not know the
quality
of the car; so he will form expectations
on the quality of a car. Let
Using the budget constraint, the buyer’s expected utility is
Thus, the buyer will buy the car,
iff
(10.1)
Notice that since the quality of the car is unknown, the willingness to pay does not depend on
the quality of a car.
The Seller’s Decision
The seller has a similar utility function and budget constraint. Her utility function is
and given income
the budget constraint is
Again, the seller will try to maximize her own utility. Given the budget constraint, her utility
function is
Thus, since she knows the quality of her own car, she will sell her car,
iff
(10.2)
Equilibrium
Assume that the quality of the car is uniformly distributed on
cision, only those cars with quality less than
will be on the market, as shown in Figure 10.1.
Thus, the average quality of cars on the market is
94
Assume that the buyer’s expected
car quality is the same as the average of the cars in the market, i.e.,
94
Due to the seller’s de-
Then, the buyer will buy
It is
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which can only be true iff
the car iff the price of a car satisfies
For this price,
only those sellers with the worst cars are willing to sell. In other words, only the worst cars
will be traded on the market.
with
1
p
1
2
0
m=
p
2
p
2
q
Figure 10.1. The distribution of the quality of cars
Discussion
A buyer’s marginal utility
of a car and a seller’s marginal utility
of the same
car satisfy
indicating a potential gain for trade. If the quality is known for both groups, there will be a
trade for any car, which can benefit both. However, since there is asymmetric information
about the quality of a car, the market fails.
The intuition for the market failure is this. Since the quality of a car is not distinguishable
to buyers, good and bad (lemons) cars will all trade at the same price. Hence, it becomes difficult to sell a good car at a fair price. Faced with this situation, the owners of the best used cars
may decide that it is not worth selling their cars. The average quality of used cars on the market will accordingly be lower as a result, and so will the price. Given this, the owners of second-best used cars may also decide not to sell their cars, so the average quality and price will
be lower still, and so on for the third best, fourth best, etc. The fewer good-quality cars there
are on the market, the lower the average quality, and so the lower the price will be for used
cars. This sort of process might lead to the market unraveling entirely; or, somewhat more
plausibly, the system may find an equilibrium in which the only viable market that remains is
quite literally the market for lemons – only the poorest quality cars are traded.
The sellers do not put a random sample of cars on the market; they sell only those cars
with quality
Because of the information advantage, the sellers are biased in selecting
their cars for sale. This is so-called adverse selection, a selection that results in a bad outcome.
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The market price plays a dual role here: it determines the average quality
of the
cars on the market, and it serves to equilibrate the demand and supply. This is rather like
having to hit two targets with one bullet. One price cannot simultaneously serve two roles well.
In equilibrium, the price is positively correlated with the average quality
Indeed,
it is the average quality of used cars on the market that determines the price.
Under asymmetric information, a high-quality car will raise the average quality of cars on
the market slightly and thus benefit all sellers (from a higher price), not just the lucky buyer.
This is so-called externality, which is caused by asymmetric information in this case.
The situation for medical insurance is the same. An insurance company may not know
each individual’s health condition, so that it has to offer insurance based on some indicators.
Since this company cannot distinguish one individual from another, it has to offer a uniform
medical insurance policy for everyone. Since the price of the insurance is based on an average
person in the whole population, the price may be too high for healthy individuals. Hence,
healthy individuals may choose not to buy the insurance. As a result, the average cost for an
insured person will be higher than the original estimate, which forces the company to raise the
price. This may cause less-healthy individuals to abandon the insurance and thus cause the
price to go up further. This process may go on until the insurance company is forced to abandon any insurance plan.
Supervision of the market becomes necessary. Under asymmetric information, bad goods
can drive out good goods. Due to this, many governments impose quality controls on exports.
Also, licenses for doctors, lawyers, taxi drivers and accountants are issued to ensure a high
level of average quality.
1.2. The Used Car Market with Perfect Information
What would happen if the quality of a car is known to everyone? One immediate answer is
that the market will be efficient and the price of a car will be dependent on the quality of the
car. The decision for the seller is still the same; she will sell the car iff
For the buyer,
since his utility function is now
he will buy the car iff
Hence, any car can be traded for a price
and the
settling price of a car is up to the bargaining skills of the two traders. Any car can be sold. The
price of a car will generally be higher for high-quality cars. The quality distribution of cars is
no longer relevant.
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1.3. The Used Car Market with Symmetric Information
Assume now that the sellers are as uninformed as the buyers. Then,
where
which is the average quality for all cars. Thus, the seller will sell her car iff
The buyer will still buy iff
i.e.,
Hence, any car can be traded for a price
In this case, the price is much higher than that under asymmetric information. Hence, it
is indeed asymmetric information that causes adverse selection, which in turn causes the price
to fall. The sellers who have the information advantage also suffer. There is thus an incentive
for the sellers with high-quality cars to provide information to the buyers. However, the buyers may not trust the sellers since there is also an incentive for the sellers to cheat. It is a dilemma.
1.4. Discussions
How about applying Akerlof’s model to the new car market? First, the dealers do not have
an information advantage on the quality of individual cars. Since no one has driven the car,
the dealer knows as little about the car’s quality as the buyer knows. Thus, it is a problem
under symmetric information. Both the dealers and buyers will trade based on the average
quality of cars. Adverse selection does not appear in this case. Second, there is a warranty on
new cars. The manufacturer will suffer if a bad car is sold.
Will the market failure go away if there is a money-back guarantee?
Suppose that the buyers can guarantee a minimum quality of a car by inspection and a
test drive. Will adverse selection disappear? We find that it is possible to have a range of cars
with different qualities to be traded in this case. See Question 8.1 in the Problem Set.
Will market efficiency improve dramatically if there is a car rental market for new cars?
The policy of car rental solves two problems. First, it solves the problem of asymmetric information on quality. Second, it ensures that the buyer will care about the car since the car may
be his at the end of the rental period. A working paper on this policy shows that the car rental
policy is efficient.
Grossman (1981) proposes a warranty in solving Akerlof’s lemons problem. He shows that
pooling with an optimally designed warranty contract is optimal if the working performance of
the good is public information. However, Lutz (1989) argues that such a warranty contract
may not be seen in practice, since buyers of the good may undetectably damage the good in
order to obtain a large warranty payment.
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2. The RS Model: Adverse Selection in the Insurance Market
Insurance is a way for a society to share risks. Insurance companies are particularly oriented toward handling independent individual risks that have no effect on society’s aggregate
resources but can seriously affect the welfare of the unfortunate individuals.
We discuss Rothschild-Stiglitz (1976) in this section. Consider an exchange economy with
individuals and a single good. An individual has wealth
amount
All the individuals have the same wealth
same potential loss
with probability
of losing an
the same utility function
and the
Thus, the endowment of any individual is stochastic and is
with probability
with probability
We consider two scenarios. In the first scenario, all individuals have the same probability
the second scenario, one group of individuals has a lower risk
risk
Since
In
and the rest have a higher
is the only thing that distinguishes the individuals, we call
the type of an
individual.
When there is only a single type of individuals, what insurance does is to shift the risk
from risk-averse individuals to risk-neutral companies. If there are multiple types of individuals, cross-subsidy between different types becomes possible under incomplete information.
2.1. Insurance with Symmetric Information: A Contingent Market
Assume that there is only one type of individual and that information is symmetric in the
sense that all available information is public knowledge. Hence, everyone has the same information and no one has an information advantage over others.
We first consider an economy with contingent markets. There are
For a system of complete markets, we need
’s consumption of the good in state
where
Let
states of nature.
markets for contingent goods. Let
be the probability of state
is the number of agents having an accident in state
be agent
occurring:
Let
be the contingent price
An Arrow-Debreu competitive equilibrium is a system of prices
and quanti-
for the good at state
ties
for
such that
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1.
solves
where
if doesn’t have an accident in state
and
otherwise.
2. Equilibrium conditions:
We know that such a competitive equilibrium exists under fairly general conditions and that it
is ex-ante Pareto optimal.
When
is large enough, approximately
proportion of agents has accidents. Thus, the to-
tal wealth in the economy is
implying a per capita endowment of
Since we have identical individuals, all the indi-
viduals should have the same consumption in equilibrium, implying
for any
That is, the consumption is independent of the state, meaning that the consumers will completely insure themselves in a large economy. In other words, the limiting case of the ArrowDebreu model has full insurance.
2.2. Insurance with Symmetric Information: Insurance Market
We can actually obtain the competitive equilibrium allocation in the above contingent
market with a much simpler market structure. Consider a perfectly competitive insurance
market in which a typical insurance company offers compensation
individual pays
to the company upfront and the company pays
for a price
That is, an
to the individual when he
has an accident. With insurance, the income of the individual becomes
if he does not have an accident,
if he has an accident.
The individual’s problem is:
The FOC is
The insurance company’s profit is
Zero profit implies
Then, the FOC implies
In other words, the individual will insure himself completely.
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We can illustrate this solution in a diagram. For this purpose, we transform the problem
into another equivalent problem. The individual’s incomes in the two states are:
implying
In fact, the conditions for
are
Hence, the budget conditions for
is
We can thus present the individual’s problem in a different way:95
,
This problem is conveniently illustrated in Figure 10.2.
I2
π
45 -line
q
slope = 1−qq
slope= MRS
.
A
u0
(1 − q) z
w−L
qz
.
o
I1
w
Figure 10.2. An Insurance Problem
The FOC of the problem is
(10.3)
which states that the MRS equals the slope of the budget constraint.
A policy
pair
on the line labeled
with profit
in Figure 10.2 corresponds to an compensation-price
This line is called the -line. When
the -line. The company makes a profit on the -line iff
95
Choosing
freely is equivalent to choosing
the line is then called
i.e., iff the -line is on the left of
with the constraint
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the -line. When
the budget line is the break-even line for the company; in this case, by
condition (10.3), the solution point
must be on
-line.
to the company in exchange for
In this insurance trade, each consumer gives
Hence, the firm’s expected profit from each consumer is
which can be shown to be
The following lemma is very convenient in a graphic analy-
sis of the insurance problem.
Lemma 10.1. If a firm has a linear profit function
are constants,
then
| (
√
,
)|
where
at any point
to the zero-profit line defined by
and
equals the minimum distance of
as shown in Figure 10.3.
y
| P( x0 , y0 ) |
a 2 + b2
( x0 , y0 )
( x* , y * )
x
P( x, y ) = 0
Figure 10.3. A Linear Profit Function
consider its minimum distance to the line defined by
Proof. For a given point
( , )
The Lagrangian function is
The FOCs are
These two equations indicate that the minimum distance is the orthogonal distance of the
point to the line. The solution is
∗
∗
By the budget constraint,
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implying
implying
∗
∗
Thus, the minimum distance is
∗
∗
2.3. Insurance with Asymmetric Information
In the last section, we assumed that all the agents were identical so that the insurance
company could easily know the probability of having an accident that is common among all
agents. The problem becomes more complicated if the individuals have different probabilities
of having accidents.
Suppose now that there are two types of agents, with probabilities of having accidents
and
We assume that the agents know their own types, but that the insurance
company only knows the existence of two types and their probabilities but not what type each
agent is. The key is that there is asymmetric information instead of symmetric information.
We study subgame perfect Nash equilibria (SPNEs) of the following two-stage game:
Step 1. Firms simultaneously announce contract offers. Each firm may announce any number
of contracts.
Step 2. Given the offers, individuals decide whether or not and which contract to accept.
The company proposes compensation-price policies or, equivalently, income policies of
the form
Suppose that a number of policies are offered to the market. Since there are
only two types of individuals and individuals of the same type will choose the same policy, at
most two policies can survive in the market. We therefore have only two possible (purestrategy) solutions: either one single policy is accepted by all the individuals in the market, or
two policies are accepted separately by the two types of individuals. The first case is called a
pooling equilibrium and the second case is called a separating equilibrium.
There is a crucial difference in information revelation between these two types of equilibria. In a separating equilibrium, since the two types of agents accept different policies, when
an agent accepts a particular policy, the company will be able to determine his type. There is
no such information revelation in a pooling equilibrium.
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The Pooling Equilibrium
Let us first consider the possibility of a pooling equilibrium. In this case, the acceptance of
a policy by a consumer does not reveal his type. Only one policy
agents. Let
is offered for all
denote “accident”. Then, the probability of having an accident for the pooled
population is
where
is the population share of the low-risk type. That is, the probability of having accident
As explained in Section 2.2, zero-profit
for the whole population is
means
The corresponding zero-profit line is labeled
zero-profit line the
-line because
for
respectively. We define the
and
curves for type
and type
in Figure 10.4. We call this
on this line. Similarly, the
-curve and
-line and
-line are
-curve as the indifference
respectively.
Zero profit in equilibrium means that a pooling equilibrium has to be on the
However, for any point on the
-line, a company can offer another policy that is better for
type L and is worse for type H. That policy can be the point
the
-line.
in Figure 10.4; since it is below
-line, the opportunist company makes a positive profit. The pooling equilibrium thus
cannot be sustained. There is thus no pooling equilibrium.
I2
pooling
πP
πL
45° line
A
C
πH
. .
B
uH
uL
.o
I1
Figure 10.4. The Pooling Equilibrium
The Separating Equilibrium
Consider now a separating equilibrium, where two policies
are offered, one for
type L and the other for type H. These two policies can separate the two types since, by accepting a policy, a consumer’s type is revealed to the company. The pair
of policies on the
left of Figure 10.5 can separate the two types. But they are not an equilibrium yet since another company can easily offer a policy to attract only the low-risk individuals. A policy in the
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green area can do just that. Thus, a necessary condition for
of policies is for
to be on the
to be an equilibrium pair
-line and at the intersection point of the
-curve and the
-line, as shown in the right figure.
is still not in equilibrium yet since the company will be losing
However, the pair
money. Thus,
must be on the
-line. Furthermore, if policy
high-risk individual’s indifference curve will cut the
offer a policy to beat
um is the pair
∗
∗
Thus,
must be on the
is not on the
-line, the
-line and, if so, another company can
-line. Hence, the only possible equilibri-
in Figure 10.6.
I2
I2
πL
πH
uH
.
CH
πL
°
45° line
45 line
πH
.
C
L
.o
uH
.
CH
.
CL
uL
.o
uL
I1
I1
Figure 10.5. The Separating Contracts
However, this equilibrium may be beaten by a policy in the shaded area under certain
conditions. If so, no equilibrium exists. If
line does not intersect the
is small enough such that the pooling zero-profit
-line, then a policy such as
in Figure 10.6 cannot exist and thus
the separating equilibrium cannot be beaten.
I2
pooling
πP
πH
πL
45° line
.
A
uH
.
D
.
C
*
H
.C
*
L
.o
uL
I1
Figure 10.6. The Separating Equilibrium
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More conditions are necessary to rule out other profitable opportunities. In the following
figure, the pair
is a separating pair, which may be profitable. This pair itself cannot be
an equilibrium, but it may be profitable. If this pair is profitable, then the separating pair
∗
∗
cannot be an equilibrium. In fact, we can show that the pair
∗
under certain conditions. That is, in order for
∗
is profitable
to survive as an equilibrium, additional
conditions are necessary to rule out separating pairs such as
to be profitable. Finally,
according to Rothschild-Stiglitz (1976), if the two situations in Figures 10.7 and 10.8 do not
happen, the pair
∗
∗
survives and it is the only separating equilibrium.
..
..
∗
∗
.o
Figure 10.7. More Profitable Opportunities
Discussion
The full information solution is the pair
∗
of policies in Figure 10.6.
We see that it is type L who suffers from asymmetric information. It is thus in type L’s
self-interest to reveal themselves to the companies. It is also in the insurance company’s selfinterest to attract type L and leave type H to other companies. One issue is how type L individuals can credibly reveal themselves to the companies.
To deal with the non-existence problem, Riley (1979) proposes a notion of a reactive equilibrium, which takes into account possible reactions from opponents to any deviation from the
equilibrium. The idea is to allow only the introduction of those policies that will yield profit
after all money-losing policies are removed. This makes
∗
∗
in Figure 10.6 an equilibrium
without an additional condition. When a company proposes an alternative policy
pair
∗
∗
against the
it is indeed myopic since it should realize that other companies will propose the
policy in Figure 10.4 that will render this pooling policy nonviable. Hence, policies such as
will not be proposed, so that
∗
∗
can survive as an equilibrium. By the same argument,
any pooling policy is an equilibrium. When a company proposes an alternative policy
against
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a pooling policy in Figure 10.4 by attracting only type L, it should realize that other companies
will have to abandon their offers so that all agents will go to this opportunistic company and in
the end this company loses. Hence, this company should not offer the alternative policy. If so,
the pooling policy survives as an equilibrium.
Similar to Riley’s reactive equilibrium, Wilson (1997) proposed the concept of the anticipatory equilibrium, in which a policy offer will not be made if a loss is expected after the new
offer effectively forces the existing offers to be withdrawn. Each company is assumed to be
able to anticipate the implications of a new offer. Offers that become unprofitable as a result of
a new offer are simply withdrawn. A company contemplating his new offer must ask whether it
will remain profitable in the face of such a withdrawal. If not, the new offer will not be made.
By this behavior, Wilson shows the survival of any pooling equilibrium and the separating
equilibrium.
Various equilibria have been based on ad hoc nonequilibrium expectations. Hellwig (1987)
modeled in a more precise manner the communication of information between insurance
companies and the insured and showed the high sensitivity of the results to the extensive form
of the game used for modeling.
2.4. Extensions
How about a two-period game? In the separating equilibrium, the insurance company can
figure out the types of the agents; it can thus take advantage of this information in the next
stage. If so, the high-risk type may not pick
∗
in the initial policy. How will the agents behave
in a two-stage game? If the high-risk agent picks
low risk agent and hence is offered
∗
in the first period, he could be viewed as a
is the second period. To prevent this, the company is to
make a better offer to the high-risk agent. Our possible solution is a separating solution as
shown in the following figure. In the first period, the policies are
period the policies are
∗
∗
∗
∗
and in the second
In the second period, there is complete information. See
Hosios and Peters (1989), who offers a solution to this problem.
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I2
I2
πL
πH
πL
45° line
45° line
C2* L
uH
.
*
1H
πH
C
.
*
C1L
.
.
.
C2*H
uL
.
I1
I1
First Period
Second Period
Figure 10.8. A Two-Period Insurance Model
Is it possible that the full information outcome
∗
in Figure 10.6 is achievable in an
infinitely repeated environment under adverse selection? The key in a repeated environment
is that hidden information can eventually be recovered by observing the choices that economic
agents make over time. An agent may be able to hide his private information in finite periods,
but he may not be able or it may be too costly to do in an infinitely repeated situation. Thus,
intuitively, when the discount rate of time preferences is large, the model is approximately a
one-period or two-period model and we know that inefficiency will exist; however, when the
discount rate is small, it is possible to achieve efficiency. See Fudenberg and Maskin (1986).
If insurance is run by a government that maximizes social welfare as opposed to profit,
what should the government do?
If both the government and commercial companies are allowed to provide insurance,
what will be the outcome? The outcome could be that some agents choose a government plan,
some choose a commercial plan, and some choose both plans. What are the welfare implications?
What would be the outcome if there are three types? What is the separating equilibrium?
Is it possible that two types may have a pooling policy and the other type chooses a separating
policy? Is the separating equilibrium more likely to survive with three types than with two
types?
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3. Job Market without Signals
This section is from MWG (1995, Chapter 13).
3.1. The Model
By relaxing some of the restrictive assumptions in Akerlof model (e.g., identical consumers), we yields further insights into market processes with asymmetric information.
Consider a labor market in which there are many potential firms that can hire workers.
Each firm produces the same output using an identical CRS technology in which labor is the
only input. The firms maximize their profits and are perfectly competitive.
There are two types of workers: low productivity
Here
Assume
is the number of units of output a worker can produce per hour. The propor-
tion of high-productivity workers is
or
and high productivity
or
and the proportion of low-productivity workers is
The reservation values of the two workers are respectively
and
with
(they are minimum acceptable wage rates).
Since
is not observable, the wage rate cannot be dependent on it. Hence, there is only
one wage rate for all. There are only three possible situations: (1) everyone is employed when
the market wage rate is high; (2) only low-productivity workers are employed when the market
wage rate is medium; and (3) no one is employed when the market wage rate is low. Hence, if
is the wage rate, then the expected productivity among employed workers is
(10.4)
We denote
as the mean productivity among employed workers, and
as the unconditional mean of productivity among all workers. Similarly, denote
3.2. Bayesian Equilibrium
Firms form expectations on the average productivity of workers who accept their jobs. Let
be the expected productivity of employed workers. As a standard assumption, assume that
all the firms have the same expectation. Market competition leads to zero profit, which implies
that the firms will offer wage
96
96
In Bayesian equilibrium (BE), we require the firms to
Notice here that there are no signals to help firms form their expectations.
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have the correct expectation, i.e.,
Combining conditions in (10.4) and
we find
three cases of BE:97
∗
∗
(10.5)
∗
ensures that all workers will accept the offer
In the first case, condition
∗
Firms make zero profits in this case. Firms expect the productivity to be
when they make the offer, which turns out to be correct. It is hence a BE.
In the second case, since
Since
∗
∗
the high-productivity workers will not accept
∗
the low-productivity workers will accept
∗
expectations by offering
The firms are making correct
Also, the firms have zero profit. Hence, it is a BE.
∗
In the last case, if
∗
no worker will accept the offer from the firm and hence the firm
make zero profit. If so, the firm has no way to find out whether its expectation is correct.
Hence, it is a BE. If
∗
a low-productivity worker may or may not accept the offer from
the firm and the firm still makes zero profit. If some low-productivity workers accept the offer,
the firms’ expectation is correct; if no workers accept the offer, the firm has no way to know if
its expectation is correct. Either way, the solution is a BE.
An efficient solution is the one in which a worker with
with
should work and a worker
should not work. For example, under perfect information, an efficient solution is:
(10 6)
where
is for a low-productivity worker and
is for a high-productivity worker. This solu-
tion requires perfect information to implement. A BE may not be efficient. For example, the
market may fail completely. When
one solution from (10.5) is
∗
with no one
working. This solution is inefficient since in an efficient solution every worker should work.
For a second example, if
is, if
and
(10.5) has the solution
there is BE in which everyone works. But, if
∗
That
the low-ability workers
should not work. Hence, the solution may not be efficient. For this BE, social welfare is
A social welfare-improvement solution is in (10.6), by which only high-productivity
workers will work, even though this solution needs perfect information to implement. Its
social welfare is
97
which is higher than
when
The unique features of BEs are (1) firms take actions based on their beliefs; (2) they have the same beliefs (a
standard assumption); and (3) the are many BEs. With the same beliefs, the firms take the same actions, by which
an equilibrium is easy to form. In contrast, without the restriction of following the same beliefs, SPNEs typically to
have fewer equilibria.
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A third example is when
but
∗
In this case, for the BE
only low-ability workers work. Social welfare is
This solution is inefficient. The
solution in (10.6) under perfect information yields higher social welfare
Hence, the BE is
inefficient.
3.3. Subgame Perfect Nash Equilibrium
In contrast to BE, in a NE, firms are not constrained by common beliefs and can hence
compete by choosing different choices. For a NE, consider a two-stage game:
Stage 1. Firms simultaneously announce their wage offers to maximize profits;
Stage 2. Workers decide whether to work for a firm and, if so, which one (if indifferent, choose
each one with probability
).
We consider subgame perfect Nash equilibria (SPNE) from this two-stage game.
At a SPNE, we must have zero profit. Hence, a SPNE must be a solution in (10.5). For the
first BE with
∗
when
no firm can attract a worker by offering a lower wage.
No firm can do better by offering a higher wage
∗
either while other firms keep at
since it will attract all the workers, maintaining an average productivity at
∗
and this firm will lose. Hence,
Consider the second BE with
is a SPNE when
∗
when
First, we need
wise a firm can do better by offering
other firms keep at
∗
with
for some
and it is higher than the wage. Hence, we
Second, since
need to impose
a firm will not try to attract the high-
productivity workers by offering a higher wage since such a higher wage must be
but the average productivity remains at
That is, if
for some
and
∗
Third, if
To prevent this, we need
∗
we have a SPNE:
obviously,
∗
one firm can choose
accept this wage. Also, since
and
∗
cannot be a SPNE. If all other firms
with
and
higher wage while other firms keep at
ing with the above, if
∗
to attract workers and make a
Consider any
profit. Hence, we need to assume
or higher
then some firms can do better by
such that
For the last case in (10.5), if
choose
while
This alternative wage will attract all the workers and the firm will
make a profit since the average productivity is
deviating to
other-
∗
Obviously no worker will
no firm can do better by deviating to a
Hence, in this case,
any
∗
∗
is a SPNE. Combin-
is a SPNE.
In summary, we have the following SPNEs:
∗
∗
(10.7)
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The solutions in (10.7) indicate that a SPNE wage is always the highest BE wage under the
same condition. See MWG (1995, Chapter 13) for a general conclusion.
3.4. Constrained Pareto Optimum
Suppose that an equally uninformed government tries to maximize social welfare by central planning. Assume that the government can only observe employment and unemployment.
Due to this, the government’s wage package is
ployed and
employment is
where
is the wage rate for the em-
is the wage rate for the unemployed. With this, the payoff of a worker under
and the payoff under unemployment is
There are only three possible situations: (1) everyone is employed; (2) only lowproductivity workers are employed; and (3) no one is employed. We can easily find the conditions for these three cases and their corresponding social welfare. Taking into account budget
balance, social welfare is
The government is subject to budget balance. In the above specification, the first condition is a
worker’s incentive condition to accept the offer and the second condition is the budget condition.
What are the social optimal solutions? Obviously, if conditions allow, the first case when
all workers are employed is the best. Under conditions, all three cases can be social optimal.
We can easily find the social optimal solution:
∗
∗
In the first case, we must have
by the budget condition. As long as
pair of
by which all the workers will accept.
such that
In the second case, with
we can find a
a high-productivity worker needs more than
to accept
the job, but the budget condition does not allow this. Hence, the government can at most
attract the low-productivity workers. We need two conditions to attract the low-productivity
workers and satisfy the budget constraint:
They are equivalent to
The condition to guarantee such a pair is
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In the last case, the budget condition doesn’t allow the government to hire any worker.
Hence, the government simply provides a wage package that no worker will accept.
The solutions in (10.8) indicate that a constrained Pareto optimal wage
∗
is always the
highest BE wage. See MWG (1995, Chapter 13) for a general conclusion.
4. Spence’s Model: Job Market Signalling
Given the problems under asymmetric information, we might expect mechanisms to develop in the marketplace to help firms distinguish between various types of workers. In fact,
both firms and some groups of individuals have incentives to do so. The mechanism that we
examine in this section is called signalling and it is based on Spence (1973). The basic idea is
that high-quality workers may take measures to distinguish themselves from low-quality
workers, while low-quality workers try to hide their identities. In this section, we present a
model in which an instrument is available to transmit credibly private information from the
informed to the uninformed via the use of publicly observable signals.
Suppose that employers are recruiting workers. The employers assess the likely productivity of each applicant and offer each a wage based on their assessment. The problem is: how can
employers know the quality of workers? There are some observable characteristics of each
worker, say the education level. Able people can get additional education at a low cost, which
can serve as a signal. Although low-quality workers can also try to get additional education,
the education cost may be very high for them and such a signal may be too expensive. Thus,
the presence of differences in these signalling costs of different quality workers may make the
signal of education credible. In other words, signalling activities may serve to generate information for employers as an endogenous market process.
We use the same model as in Spence (1973), except that we replace some specific values in
his paper with arbitrary parameters. The model involves the following three steps:
1. Workers decide on how much education they wish to invest in and they pay the cost.
2. Firms cannot observe a worker’s productivity, but they can observe the signal. They form
probabilistic beliefs about the relationship between the observed signal and unobserved
productivity. Assume that all the firms have the same beliefs.
3. With their beliefs, the firms make wage offers, and then workers respond with certain
signals.
For a tractable solution, we need some simplifying assumptions:
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(1) Firms and workers are all risk neutral.98 The labor market is perfectly competitive, implying zero profit for firms.99
(2) Workers are of two possible types:
where
Type L:
productivity
population proportion
Type H:
productivity
population proportion
Thus,
is the probability of a worker being of high quality. The firms
know this population distribution. Here, the productivity for each worker is assumed to
be fixed and the education level has no effect on productivity.100
(3) The costs for education level
are respectively
and
for the two types, where
Here, the cost and the marginal cost of education are lower for high-quality
workers.
(4) The utility functions are
We study a two-stage game:
Step 1. Given a belief system, firms simultaneously offer a wage schedule
Step 2. Given the offers, workers decide to choose an education level.
The model can be illustrated in the following figure.
.
Nature
.
.
Firm 1
Firm 2
.
.
. . . .
.
Applicant
.
Firm 1
. .
Firm 2
Figure 10.9. A Signalling Model
98
More precisely, they are risk neutral in income, but their cost functions can be arbitrary convex cost
functions. We will use linear cost functions for simplicity, but we will draw the same conclusions if we use general
cost functions.
99
We need to have two or more firms to achieve zero profit in equilibrium.
100
Spence (1974) and Kreps (1990) allow education to affect productivity.
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Here, nature decides the worker’s type first. The worker knows his type and then decides on a
choice of
The firms then decide on their offers based on the observed choices of
but with-
out the knowledge of each worker’s type.
and a belief system
The model consists of a strategy
where
is
the firms’ common probability assessment that a worker is of high quality after observing
This pair
is a Bayesian equilibrium (BE) if
(i) For each
the firms’ wage offers constitute a Nash equilibrium.
(ii) The workers’ strategy
is optimal given the firm’s strategies.
(iii) There is consistency between the belief and the actual probability:
(10.9)
In plain words, we have a signalling model in which four steps are involved:
We now analyze the model. First, the firms’ wage offer is determined by a Bertrand equilibrium. In a Bertrand equilibrium, zero profit means that the firms’ wage offers must equal
the expected productivity of the worker:
(10.10)
we can recover the belief system
Conversely, given
( )
is
by which
from (10.10), which
is the Nash equilibrium wage rate. Hence, we can
ignore the belief system from now on, since the wage function satisfying
al-
ways matches uniquely with an underlying belief system.
Second, what about consistency condition (10.9)? Since there are only two types of workers, there are only two possible equilibrium outcomes: either every worker chooses the same
education level — a pooling equilibrium, or one type chooses one education level and the other
type chooses another — a separating equilibrium. In a pooling equilibrium, consistency condi∗
tion (10.9) requires
for the equilibrium education level
∗
which in turn by (10.10)
means
∗
for everyone, where
type chooses
∗
∗
and
In a separating equilibrium, suppose that the low
and the high type chooses
∗
∗
Then, consistency condition (10.9) requires
which in turn by (10.10) means
∗
∗
(10.12)
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Third, we now turn to the worker’s optimal strategy. Given the wage function
a
worker’s problem is
,
Given type and reservation utility
the indifference equation is a line defined by
We can illustrate the problem in the following diagram.
w
u = w - ci e
θH
.
A
w (e )
θL
e
Figure 10.10. The Worker’s Problem
In the figure, an arbitrary wage curve is given, and the worker tries to push the indifference
deter-
curve up and to the left until it touches the wage curve by only one point. That point
mines the optimal choice of
and a corresponding wage
from the firms. Each such wage
curve can be justified by a belief system through (10.10). We have an equilibrium if consistency condition (10.9) is satisfied. Since we have only two possible types of equilibrium, the consistency means either (10.11) or (10.12).
For each type
Since
the slope of the indifference curve
type ’s indifference curve is steeper than is type
these two difference curves the
-line and the
is the marginal cost
’s. For convenience, we call
-line, as shown in Figure 10.11.
Education here serves as a signal of the unobservable worker’s productivity. The education signal may or may not effectively separate the two groups. In a pooling equilibrium, the
education signal does not distinguish the two types and hence everyone is paid the same wage.
In a separating equilibrium, firms can correctly use the education signal to distinguish the
workers and pay them different wages.
4.1. The Perfect Information Solution
When firms can observe each worker’s productivity, a Bertrand equilibrium implies
∗∗
∗∗
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That is, each firm will pay workers for their productivity. Since education does not enhance
productivity, the optimal education levels are
∗∗
∗∗
4.2. The No Signalling Solution
Suppose that education is banned so that
The firms cannot distinguish between workers and they have to pay everyone the same wage.
To satisfy the zero-profit condition, the wage must be
4.3. Separating Equilibria
We now consider the situation of asymmetric information, in which job applicants know
their own types but employers can only infer the types from education signals. There are two
types of equilibria: separating equilibria and pooling equilibria. When we have a separating
equilibrium, the signal can effectively separate the two types; when we have a pooling equilibrium, the signal cannot separate the two types.
Let
∗
be the worker’s equilibrium education choice and
∗
be the firms’ equilibri-
um wage offer. In a separating equilibrium, since the firms can tell the workers’ types by observing their choices of education, the firms’ wage offers depend on the education level. By the
zero-profit condition, the wage offers must be:
∗
∗
∗
∗
(10.13)
Since having an education will incur costs but will not change the pay
∗
for the low type,
the low type will choose zero education:
∗
(10.14)
Thus, one possible separating equilibrium is where a low-ability worker chooses
receives
∗
∗
and a high-ability worker chooses
∗
and receives
∗
∗
∗
and
where
∗
will be determined later.
We now need to find a belief system that supports this equilibrium. By (10.10), this is
equivalent to finding a wage function. We have drawn an arbitrary wage curve
∗
in the
following figure:
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w
uL
uH
.
θH
w* ( e)
θL
.
e L*
e
e
e H*
Figure 10.11. Separating Equilibrium
As shown in Figure 10.11, this wage schedule supports the choices
∗
and
∗
as optimal
choices of the high type and low type, respectively, and the corresponding wages are
respectively. Since
∗
and
by (10.10), it can be supported by a belief system. Further,
the consistency condition (10.12) for a separating equilibrium is also satisfied. Therefore, the
wage schedule
∗
in Figure 10.11 and the belief system
∗
implied by (10.10) constitute a
Bayesian equilibrium. In this equilibrium, the two types of job applicants are distinguishable
by their different choices of education.
Note that, with wage offers in (10.13), the firms must have beliefs
∗
∗
∗
and
These beliefs are indeed consistent with the choices made by the individuals in
equilibrium. However, for off-equilibrium education levels, there is no consistency requirement in a BE and the firms’ beliefs can be quite arbitrary. Since there is so much freedom to
choose beliefs, many wage schedules can arise that support different education choices. For
example, Figure 10.12 shows a simple wage schedule that supports a separating equilibrium.
The corresponding belief system is as follows: the firms believe that a worker with education
is of low productivity and a worker with education
is defined by the intersection of the
-line with the
is of high productivity, where
-line.
Note also that, in a Bayesian equilibrium, each firm’s wage offer (strategy) is derived from
its beliefs in a Nash equilibrium and all the firms are assumed to have the same beliefs. Hence,
all the firms make the same wage offer; alternative offers from other firms are not allowed.
This is different from the Rothschild–Stiglitz model in which different firms can make different wage offers; see, for example, Figure 10.6, in which a different firm can make an alternative offer
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w
uH
uL
θH
w* (e)
θL
o
e
e
Figure 10.12. Separating Equilibrium
Furthermore, workers may have many equilibrium choices. Type
∗
as shown in (10.14), but type
choose
equilibrium. If
above
has many alternatives. For example, type
as shown in Figure 10.13. In fact, any
equilibrium. But, any
will always choose
may
can be supported in a separating
cannot be an education level for the high type in a separating
is below
the low type will choose
the high type will choose
to pretend to be the high type; if
is
even though she may be viewed as the low type.
w
.
θH
uL
.
uH
w* (e)
θL
.
e
e
e
Figure 10.13. Separating Equilibrium
The separating equilibria can be Pareto ranked. In any equilibrium, firms earn zero profit
and firms are always indifferent between different equilibria. The low type always chooses
∗
and is paid
in a separating equilibrium. Hence, the low type is also indifferent be-
tween different equilibria. Only the high type has different utility values in different equilibria.
Among all the separating equilibria, the equilibrium in which
∗
Pareto-dominates all
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others since it has the lowest education cost for the high type. This is the most efficient separating equilibrium.
Discussion:
1. The signal is informative and correct. It separates the two types.
2. The equilibria are not unique. There are a continuum number of separating equilibria.
3. The equilibria can be ranked by the Pareto criterion. The closer
∗
is to
the better off is
type H, while type L is unaffected. The Pareto-efficient separating equilibrium is where
∗
4. The results do not depend on the population share
5. Comparing with the alternative solution under perfect information, a separating equilibrium is inefficient even though it also has perfect information in equilibrium. The perfectinformation equilibrium Pareto-dominates all incomplete-information separating equilibria.
6. The no-signalling equilibrium can Pareto-dominate some of the separating equilibria.
Without signalling, everyone gets
and
∗
∗
As shown in Figure 10.14, type
L will be better off without signalling. Type H can also be better off without signalling iff
where
is defined in Figure 10.14. The cost of education can make type H
worse off with signalling.
w
θH
.
E (θ )
.
uL
uH
qˆH
θL
.
e
e
Figure 10.14. No-Signalling Equilibrium
7. Education results in negative externalities, since there is a cost to education but it has no
benefit on productivity. Type H may benefit from it by separating themselves from the rest.
That is, education is unproductive, but people still invest in it.
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Example 10.1. Let
and
and the cost functions be
That is, it costs half for type H to obtain the same education level as type L does.
w
uL
CH
2
uH
.
θH
1.5
.
1
θL
CL
e
1
2
Figure 10.15. A Separating Signalling Equilibrium
0
suppose that the employers believe that only applicants with the
Given some constant
education level
or more are of high productivity. With this belief, the wage offer will be101
With this offer, each worker will choose an education level of either
conditions under which type L chooses
ing equilibrium. Type L chooses
and type H chooses
or
Let us find the
implying an separat-
iff
which is equivalent to
Type H chooses
iff
which is equivalent to
The employers’ belief is indeed correct at
and at
beliefs above, a separating equilibrium holds as long as
101
Here,
∗
. Therefore, given the employers’
as shown in Figure 10.15.
may convey some information about the company to the applicants.
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In this case, with
type L finds that the cost of education is too high so that they volun-
tarily separate themselves from type H by choosing no education.
4.4. Pooling Equilibria
In a pooling equilibrium, both types choose the same education level:
∗
∗
∗
Since
the firms cannot distinguish between the two types, the only consistent belief in equilibrium is
∗
Thus, the equilibrium wage offer must be
∗
∗
The consistency condition (10.11) is satisfied.
What will be the education level in an pooling equilibrium? It can be easily seen from Figure 10.16 that any education level
pooling equilibrium, where
can be supported by a wage schedule leading to a
is defined in Figure 10.16. Any education level
cannot be
supported since the low type would rather set
w
uL
.
A
θH
.
E (θ )
θL
uH
w* (e)
.
e
e * ê
Figure 10.16. Pooling Equilibrium
The most efficient pooling equilibrium is where both types choose
which is the
same as the no-signalling equilibrium. Thus, the no-signalling equilibrium Pareto-dominates a
pooling equilibrium.
Discussion:
1. All workers send the same signal so that the signal loses its role.
2. There is a continuum number of pooling equilibria.
3. The equilibria can be Pareto ranked. Higher values of
most Pareto-efficient pooling equilibrium is where
4. The equilibria depend on
∗
impose costs on both types. The
∗
The wage rate will be higher if
is higher.
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5. The full information solution does not Pareto-dominate the pooling equilibria, and vice
versa. Type L is better off under pooling while type H is worse off.
6. The no-signalling solution Pareto-dominates any pooling equilibrium.
7. Once again, there is a divergence between the social and private benefits of education. In
this case, education does not convey information.
Example 10.2. For the model in Example 10.1, suppose now that the firms hold the following
belief:
With the zero-profit condition, this belief implies the following pay scheme:
The job applicants will choose either
L, the applicant will choose
or
(no point to choose other levels). For type
iff
i.e.,
(10.15)
iff
For type H, the applicant will choose
i.e.,
(10.16)
w
uL
uH
θH
2
1+ λ
.
C
θL
1
0
e* λ
1
2
e
Figure 10.17. A Pooling Equilibrium
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Hence, if
all the applicants will choose
And in this case, the employers’ belief is
correct. This is thus an equilibrium, a pooling equilibrium, where the signal does not reveal
the identity of any worker, as shown in Figure 10.17.
4.5. Partial Pooling or Partial Separating Equilibrium
In a mixed-strategy equilibrium, at least one type chooses a mixed strategy/signal. Suppose that the high type chooses
and
with probabilities
and
ity that type
and
i.e., choosing
and , respectively. Since the low type mixes the two points
he must be indifferent between these two choices. Let
takes
be the probabil-
. Type ’s problem is
∈[ ,
If the optimal
for sure, but the low type mixes
]
is an interior solution, we must have the indifference condition:
(10.17)
The consistency condition is
By zero profit,
Given an arbitrary
an arbitrary
, (10.18) and (10.17) determine
and
. As shown in Figure 10.18, given
, we an easily find a wage curve to form a BE as described above.
uL
uH
θH
θL
.
.
eL
eH
Figure 10.18. A Hybrid Equilibrium
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Technically speaking, in each partial separating equilibrium, there are three variables to
determine: ,
and
We have only two equations, (10.17) and (10.18), which can deter-
mine two of the three variables, leaving one of them free.
and
Similarly, given arbitrary
which the low type takes
, with
, we can define hybrid equilibria in
for sure and the high type mixes
with
. We again have two
conditions as in (10.17) and (10.18). The indifference condition is
(10.19)
The consistency condition:
where
is the probability that type
Given arbitrary
and
, with
takes
. The zero profit condition is
, equations (10.19) and (10.20) determine
and
.
This case is shown in Figure 10.19.
.
θH
.
θL
eL
eH
Figure 10.19. A Hybrid Equilibrium
4.6. Government Intervention
Government intervention may improve efficiency under incomplete information. For example, when the best separating equilibrium is Pareto-dominated by the no-signalling equilibrium, the government can impose a ban on signalling activity to achieve a Pareto improvement.
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Furthermore, it may be possible for a Pareto improvement even when
ure 10.20, by offering a low-ability worker the wage
both types are better off at
and
In Fig-
and a high-ability worker the wage
respectively.
w
uL
( wˆ H , eˆ H )
uH
θH
qˆH
E (θ )
ŵ L
θL
e
e
Figure 10.20. Government Intervention
The central authority can achieve this outcome by mandating that workers with education
levels below
wage
receive wage
and that workers with education levels of at least
If so, the low type would choose
∗
and the high type would choose
receive
∗
As
long as the firms can break even on average under this scheme, this is an sustainable equilibrium and it achieves a Pareto improvement.
The key to this Pareto improvement is that the central authority introduces crosssubsidization, where high-ability workers subsidize low-ability workers. In particular, the nosignalling equilibrium is an extreme case of cross-subsidization. Of course, this solution is not
feasible in a free and competitive labor market. Opportunistic behaviors will prevent this
cross-subsidization from happening.
Note that in a Bayesian equilibrium, since the firms’ wage offers are derived from their
common beliefs, they won’t offer
to
and offer
to
.
4.7. Equilibrium Refinement
The multiplicity of equilibria stems from the great freedom that we have to choose beliefs
off the equilibrium path, although on the equilibrium path the beliefs are determined by either
(10.11) or (10.12). A great deal of research has focused on finding reasonable further restrictions on beliefs to reduce the number of equilibria.
We can use the dominance criteria to eliminate many BEs. Let us apply the complete
dominance criterion first. As shown in Figure 10.21, under any circumstance, no matter where
is the equilibrium point is, the low type would never take an
with
By the complete324/418
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dominance principle, the firms should believe anyone with
and pays
If so, a wage for an applicant with an
∗
only separating CDBE is the separating pair
satisfying
such as
∗
is the high type
must be
. Hence, the
102
any pooling
How about pooling equilibrium under CD consistency? If
∗
as
shown in Figure 10.21 is impossible under complete-dominance consistency. Since
by CD consistency, the high type will choose
instead. Therefore, this
under CD consistency. Therefore, if
a pooling
∗
cannot be a PBE
there is no pooling CDBEs. However, if
∗
is possible. As shown in Figure 10.21, each
w
is a CDBE.
w
uL
uH
uL
θH
qˆH
E (θ )
∗
θH
E (θ )
qˆH
.
θL
uH
.
θL
e*
e
e¢
e1
e
e * e2
e
e¢
e1
e
Figure 10.21. CDBEs
How about BEs under equilibrium-dominance consistency? If
pooling CDBE, there is no pooling EDBE. If
dominated by the equilibrium
∗
ciple, we should have
type will take this
instead of
as shown in Figure 10.22, any
is
for the low type; hence, by the equilibrium-dominance prinIf so, since this
∗
since there is no
is better than
∗
for the high type, the high
That is, under ED consistency, all pooling BEs are ruled out.
Therefore, there is no pooling EDBE. Therefore, there is only one EDBE with or without
In sum, the intuitive criterion or the equilibrium-dominance principle predicts the
most efficient separating equilibrium as the unique outcome.103
102
Any
is a completely dominated signal for both types. Hence, we cannot impose any restriction on
the belief. However, since such an education level cannot be an equilibrium education level, we don’t need to worry
about it.
103
This conclusion is for a case with two types. Stronger conditions are needed to generate a unique
equilibrium when there are more types; see Cho-Kreps (1987). See also Kreps (1990) and Gibbons (1992) for more
detailed discussions.
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w
uL
uH
θH
E (θ )
qˆ
.
H
θL
e * e1 e
e
e
Figure 10.22. EDBEs
In summary, the welfare effect of signalling is generally ambiguous. The prediction by
Akerlof’s model holds if there is no way of credibly transmitting information from the informed to the uninformed. Spence (1973) investigates the possibility of information revelation
by signaling. He finds that signalling can have rather surprising effects. Much of the signalling
literature involves refinements that reject unreasonable beliefs. A strong notion of refinement
such as the intuitive criterion implies a unique equilibrium, which is the Pareto-efficient separating equilibrium.
4.8. Questions
What if education does affect productivity? Since there will be a reward for education itself, can a pooling equilibrium still exist? How about efficiency?
In Spence (1973), all the employers have the same belief. What will happen if employers
have different beliefs?
In reality, a person’s education level at primary school, high school, BA, MS and Ph.D has
discrete values. How will a five-value education system affect our conclusions?
In our analysis, beliefs are given. How are beliefs formed? In a dynamic setting, beliefs
may be formed endogenously by a learning process over time.
Finally, Spence uses the BE concept. How about using the SPNE concept? Such a model
will have two stages:
•
Given a wage schedule in the market, the applicant chooses an education level.
•
Firms compete on wage offers.
The next section deals precisely with this issue.
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5. Job Market Screening
In Spence (1973), the signal is chosen by the applicants, while in Rothschild-Stiglitz
(1976), it is the firms that specify an education level in a contract. That is, instead of offering a
wage rate
level
based on their beliefs, the firms specify a wage rate
together with an education
in the contract. In other words, while the education level in Spence (1973) is a signal
from the applicants, it is used to screen the applicants in Rothschild-Stiglitz (1976). Screening
is the ability of the uninformed to use a public signal to screen economic agents.
We again use the same model setup as in Spence (1973), with two types of workers of
productivities
and
where the fraction of high-ability workers is
Now, instead of specifying a wage rate
in a contract, firms offer a pair
with the wage
rate matched to the education level in the contract. Multiple contracts may be offered by a firm.
To each worker, instead of a wage curve
in the marketplace, he faces a number of choices
However, we can also describe the wage curve as a set of
Hence, the real
difference between Spence (1973) and Rothschild-Stiglitz (1976) is the definition of equilibrium.
Again, as before, education has nothing to do with productivity. Utility functions are still
where
We study subgame perfect Nash equilibria (SPNEs) of the following two-stage game:
Step 1. firms simultaneously announce contract offers. Each firm may announce any number
of contracts.
Step 2. Given the offers, workers decide whether or not and which contract to accept.
The crucial difference between the Spence and RS models is the commitment by the companies when they observe a signal. In the RS model, the company commits to a certain wage
offer for a certain signal (no belief is needed), while in the Spence model, a company’s wage
offer is dependent on the belief when it observes a signal. A change of commitment may lead
to a different solution.
Again, we may have two kinds of equilibria: separating equilibria, in which the two types
accept different contracts, and pooling equilibria, in which both types accept the same contract.
5.1. The Pooling Equilibrium
Does there exist a pooling equilibrium? Let
be a pooling equilibrium contract. Ze-
ro profit means that it must lie on the pooled break-even line at
Suppose that a firm
offers such a contract on the pooled break-even line. As shown in Figure 10.23, another firm
can easily pick up a contract in the shaded area that attracts only high-ability workers. If so,
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will attract low-ability workers only and the firm will lose money. Therefore,
cannot be an equilibrium. Hence, there is no pooling equilibrium.
w
uL
θH
.
~ , e~ )
(w
.
E (θ )
uH
(w p , e p )
θL
e
Figure 10.23. Pooling Equilibrium
5.2. The Separating Equilibrium
We now try to find a separating equilibrium. In Figure 10.24, the pair of contracts
and
can separate the two types. However, this pair of contracts cannot be an equilib-
rium since another contract
can attract the high type and make a profit.
w
uL
θH
.
.
uH
~ , e~ )
(w
( wH , eH )
.
( wL , eL )
θL
e
Figure 10.24. Separating Contracts
Thus, given
contract
has to be in the position shown in Figure 10.25. Again,
this pair cannot be an equilibrium since the company is making a loss. The zero-profit condition implies that
must be on the
-line.
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w
uL
uH
.
θH
( wH , eH )
.
( wL , eL )
θL
e
Figure 10.25. Separating Contracts
However, the pair in Figure 10.26 is still not an equilibrium since some company can offer a
contract
to attract the low type and make a profit.
w
uL
.
θH
uH
( wH , eH )
~ , e~ )
(w
θL
..
( wL , eL )
e
Figure 10.26. Separating Contracts
Hence, the only possible separating equilibrium is the pair in Figure 10.27.
w
uL
.
θH
θL
uH
( wH* , eH* )
.
( wL* , eL* )
e
Figure 10.27. Separating Contracts
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Can this separating equilibrium survive? The answer depends on the value of
as shown in Figure 10.28, we can easily find a contract
If
that attracts both
types and yet makes a profit. In this case, this separating equilibrium is not sustainable. Hence,
there is no separating equilibrium when
w
uH
uL
θH
( w H* , e H* )
.
E (θ )
qˆH
θL
.
~ , e~ )
(w
.
( w L* , e L* )
e
e
Figure 10.28. Nonexistence of Separating Equilibrium
Even when
we may find an alternative profitable pair of contracts. For exam-
ple, a pair of contracts
and
in Figure 10.29 may be profitable. If so, the sepa-
rating equilibrium is not sustainable.
w
~ , e~ )
(w
H
H
θH
. .
uL
uH
( w H* , e H* )
.
.
~ , e~ )
(w
L
L
θL
( w L* , e L* )
e
e
Figure 10.29. Profitable Contracts
In summary, an equilibrium exists if the situations in Figure 10.28 and Figure 10.29 do
not happen, i.e., when
and there are no such profitable pair
and
in
Figure 10.29. This equilibrium is the separating equilibrium.
5.3. Discussion
The separating equilibrium is the same as that implied by the intuitive criterion of ChoKreps (1990).
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By the reactive equilibrium concept of Riley (1979), the separating equilibrium survives
even if
Also, any pooling equilibrium survives as a reactive equilibrium. See the
discussions in Section 2.3.
As in the signalling model, low-ability workers are always worse off with screening. Highability workers are better off with screening.104
The separating equilibrium is Pareto dominated by the full-information outcome.
When an equilibrium does exist, the situation in Figure 10.29 cannot happen. Thus, the
equilibrium is constrained Pareto optimal: a uninformed government can no longer make a
Pareto improvement. See Section 4.6 on government intervention.
Notes
Besides the cited papers, good references are Mas-Colell et al. (1995, Chapter 13) and Laffont (1995).
104
The conditions that ensure the survival of the separating equilibrium guarantee that high-ability workers
will be better off with screening.
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Chapter 11
Mechanism Design
The previous chapter focuses on competitive markets under asymmetric information. This
chapter focuses on monopoly pricing under asymmetric information. The basic approach to
monopoly pricing under asymmetric information is the revelation principle, by which the
uninformed is compelled to provide incentives for the informed to reveal their types.
1. A Story of Mechanism Design
To motivate the approach, reconsider the insurance problem of the RS model in the previous chapter. There are two types of individuals who differ in probabilities
accident. We call
of having an
the type of the individual. Each individual knows his own type but the
company does not. Instead of a competitive insurance market, we now assume that there is a
single monopoly company. Under incomplete information, what kind of contracts should this
company offer?
1.1.
Market Mechanism vs Direct Mechanism
Market Mechanism
An insurance company in the real world typically offers contracts of the form
market, where
is the amount of coverage in an accident and
risk premium). Given initial wealth
to the
is the price of insurance (the
and a possible loss of an amount
of that wealth, with
the insurance, an individual’s income becomes
without an accident,
with an accident.
Since the company does not know the types of individuals, the contracts cannot be based on
individual types. Instead, as in the car insurance market, a company typically provides a noclaim bonus (NCB) to individuals. Let
denote the choices of claims: claim or not
claim after an accident. The company’s contract offers are
offers, each individual then decides on a strategy
for
Given the contract
by which he receives
as the set of types. Each individual’s strategy is a mapping of
Denote
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In this case, the company has no direct control over who chooses which contract. In fact,
the individuals play strategies against the offers and an individual may not seek compensation
from the insurance company even if he has had an accident. This is the market mechanism.
Direct Mechanism
We now consider an artificial mechanism, called a direct mechanism, in which the company simply asks each individual to report his type and is then offered a contract based on his
reported type. Since the company does not observe the type, an incentive has to be provided in
order for each individual to report his type truthfully.
With two types of individuals in the market, at most two contracts are needed:
with
intended for type
and
and
If these two contracts happen to be the
same, only one contract is accepted by insurees. Given the two contracts, each individual
decides which contract to choose. Type
precisely, type
will choose
Symmetrically, type
will choose
only if it is better for him. More
iff
will choose
iff
We call these two conditions the incentive compatibility (IC) conditions. By these IC conditions, each type will voluntarily choose the contract intended for him in the direct mechanism.
If so, when they are asked to report their own types, they report the true types.
1.2. The Optimal Allocation: A Graphic Illustration
To illustrate the optimal solution of the direct mechanism in a diagram of the monopolistic insurance market, we transform
Here,
and
to
where
are the incomes in the two states. Then, given
the available choices are on the
-line are:
In this case, each individual gives the company his endowment
in exchange for
Thus, the company’s expected profit from an individual with probability
The company makes a profit from type iff
line, where the
-line is the -line when
is
i.e., iff the contract is on the left of the
-
and it is actual the zero-profit line for the
company.
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We can always start with a pair of contracts. If the optimal pair turns out to be one contract, we have a pooling solution, otherwise we have a separating solution. In a pooling solution the acceptance of a contract by a consumer does not reveal the consumer’s type, while in a
separating solution, the choice of a contract by a consumer will reveal the consumer’s type.105
that separates the types in Figure 11.1.
Consider an arbitrary pair of contracts
Type L will choose
and type H will choose
I2
πL
45° line
uL
πH
uH
.
CH
.
CL
.O
I1
Figure 11.1. Separating Policies
the company can increase profits by moving
First, using Lemma 10.1, given
-line until the new
same
-curve passes through
Thus, we can assume
and
towards the
to be on the
-curve, as shown in Figure 11.2.
I2
πL
45° line
uL
uH
πH
.
CH
.
CL
.O
I1
Figure 11.2. Separating Policies
Second, given the fact that
moving
105
away from the
and
are on the same
-line (and move
-curve, the company can do better by
towards the
-line) until the
-curve passes
It generally pays for the company to separate the two types and exploit the differences. Hence, we generally
expect a separating solution.
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through the initial point
which is the limit that the
must be the final position for the
-curve can move to the left and thus
-curve, as shown in Figure 11.3.
I2
πL
45° line
uL
uH
πH
. .C
CH
L
.
O
I1
Figure 11.3. Separating Equilibrium
We now know that the position of the
However, the final position of the
we want to have an
is as far from the
-curve is tricky. Its position will affect both
that is as close to the
and
and
-line as possible (from the right) and a
that
-line as possible. If we move the
may move closer to the
distance of
-curve, which must go through the initial point
to the
-curve closer to the
-line. Thus, the optimal position of the
-line and the distance of
for the optimal pair
to the
-line, point
-curve has to balance the
-line. The precise position of
can be known only by actually solving the mechanism design
problem. The solution is called the optimal direct mechanism. If the optimal pair turns out to
be one contract, we have a pooling solution, otherwise we have a separating solution.
1.3. The Optimal Allocation: Mathematical Presentation
We now set up the mechanism design problem. Each consumer gives the company
in exchange for
Thus, the firm’s expected profit from a consumer with
is
Although the consumer has no budget constraint under this setting, he does have an individual rationality (IR) condition:
Given two contracts
type
,
and
,
one intended for type
and the other for
the company’s per capita profit is
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where
is the population share of the low-risk individuals. Here, for simplicity, we assume
that both individuals have the same initial endowment
and the same utility function
Then, the company’s problem is
,
,
,
(11.1)
That is, the company minimizes the expenditure subject to the two IC conditions and two IR
conditions. The two IC conditions ensure that the low-risk individuals prefer
high-risk individuals prefer
to
to
and the
and the two IR conditions ensure that they both will
participate.
In problem (11.1), the two IC conditions induce the two types to truthfully report their
own types. Thus, an optimal solution under asymmetric information is the one that provides
the incentives for truthful reporting of types and the incentives for participation.
By the above graphic analysis, we know that
and
will not be binding and
and
will be binding. Thus, the problem becomes
,
,
,
The Lagrangian function is
The FOCs are
(11.2)
(11.3)
(11.4)
(11.5)
In particular, by (11.4)–(11.5), we find that
indicating that the high-risk type has full
insurance.
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1.4. The Optimal Allocation: A General Case
More generally, consider an insurance company dealing with a population of individuals
each of whom has probability
of having an accident that costs him
where
and
is a
set of parameters. Instead of two possible types, the individuals have a continuum of possible
types
For simplicity, assume that the loss
individuals have the same initial wealth
mechanism
is the same for all individuals and that all
In exchange for
that specifies a net income
in the bad state for a reported type
an agent reports the
Given his type
the company proposes a
in the good state and a net income
from an agent.
from
(11.6)
The FOC for the optimal
is
In order for the individual to reveal himself, the company must design
way that
and
in such a
In other words, we need to have
(11.7)
We also need the SOC for (11.6):
By (11.7), this SOC becomes
Using (11.7) again, we find
Therefore, the SOC for truthful reporting
is satisfied if
(11.8)
and
We have thus derived two conditions (11.7)–(11.8) for
policy
for
under which the insurance
is a truth reporting allocation scheme. In other words, with such
a policy, an individual of type
will truthfully report his type as
and thus will be offered the
policy.
What is then the optimal allocation scheme
∗
∗
Let
be the population dis-
tribution. Given a fixed revenue, the insurance company’s cost-minimizing problem is:
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(⋅),
(⋅)
1.5. Market Mechanisms vs Direct Mechanism
In the market mechanism
where
is a set of rules, each agent follows a strategy
and the resulting allocation scheme is
a special case of this general form, in which
Given a reported
The direct mechanism
is only
is a special set of rules,
the contract offer is
and
in the direct mechanism. One
natural question is: Can the market mechanism do better than the optimal direct mechanism
for the insurance company? Can any other mechanism do better? By the revelation principle,
as shown later, if the direct mechanism provides IC conditions, no other mechanism can possibly do better than the optimal direct mechanism.
2. The Revelation Principle
The presence of incomplete information raises the possibility that we may need to consider a player’s beliefs about other players’ preferences, his beliefs about their beliefs about his
preferences, and so on, much in the spirit of rationalizability. Fortunately, there is a widely
used approach to this problem, originated by Harsanyi (1967–68), that makes this unnecessary. In this approach, one imagines that each player’s preferences are determined by the
realization of a random variable. Through this formulation, the game of incomplete information becomes a game of imperfect information: Nature makes the first move, choosing
realizations of the random variable that determine each player’s preference type. A game of
this sort is known as a Bayesian game.
Formally, there are
players
each with a type parameter
The type
is random ex ante and is chosen by nature and its realization is observed only by player
The joint density function is
for
knowledge. There is a feasible collection
has a preference order
over
which is common
of choices, called the allocation set. Each player
or equivalently a payoff function
for
utility function
is also common knowledge, but each specific value of
served only by agent
A Bayesian model is represented by
106
We can actually allow an agent’s utility function to take the form
106
The
is ob-
Thus, in a Bayesian
rather than
All the
concepts on Bayesian implementation are readily extendable to this case.
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game, we can think of each player as being a separate player who maximizes his payoff given
his conditional probability distribution over his rivals’ strategy choices.
Individuals know their own types. But we need to calculate the strategy function
since other players don’t know individual
possible
’s type and they need to figure out
for all
107
In our allocation problem, a mapping
the knowledge of the agents’ types
is called an allocation scheme. Assuming
an allocation scheme assigns a feasible allocation
in
to the agents. Our problem is to find an optimal allocation scheme that maximizes a given
objective. An objective can be the revenue from a piece of art, social welfare from building a
bridge, or surplus from trade, etc. One difficulty involved in this problem is that none of the
players, especially the player who chooses an optimal allocation scheme, has full knowledge of
all the agents’ types.
What set of allocation schemes should we look into? Naturally, one desirable feature of an
allocation scheme is ex-post efficiency.
Definition 11.1. An allocation scheme
any
that is, there is no
is ex-post efficient if
and
such that
is Pareto optimal for
for all
and
for some
That is, an allocation scheme
is ex-post efficient if, all for
is Pareto opti-
mal. Ex-post efficiency requires full knowledge of each individual’s type. In other words, expost efficiency means Pareto optimality under complete information. That is, the maximum
efficiency achievable ex post is called ex-post efficiency. In contrast, when a planner can only
form expectations of the types, she can only achieve ex-ante efficiency (which will be defined
later).
To implement an allocation scheme, we need a mechanism. A mechanism is simply a set
of rules, including strategy variables, the allocation in equilibrium and other rules. Specifically,
a mechanism defines a Bayesian game of incomplete information that consists of a set of rules,
a strategy space for each agent and an allocation in equilibrium. With a mechanism, agents
interact through an arrangement or institution defined by a set of rules governing actions and
payoffs. Their interactions eventually reach an equilibrium and then the mechanism allocates
payoffs in equilibrium. We may have many possible mechanisms to implement an allocation
scheme. In a special mechanism, called the direct mechanism, each agent is asked to make an
announcement of his own type
107
and then
Although each player knows his own
know your
is allocated.
we need to consider all
The reason is that other players do not
and they have to consider all possible
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Definition 11.2. A mechanism
is a set of rules, a collection of strategy sets
and an outcome function
is called a strategy of agent
A mapping
by
and the allocation is given by
The allowed actions are summarized
in equilibrium. Players determine their strategies
before nature moves; after nature moves, the players play according to their strategies.
player ’s strategy set
Given a mechanism
mappings
where
is the set of all the strategy
Player ’s payoff for a profile of strategies
is
is the mathematical expectation operator over randomness defined by
Definition 11.3. A Bayesian Nash equilibrium (BNE) in the Bayesian model
under mechanism
is a strategy profile
a Nash equilibrium of the game
∗
∗
That is, for each
that constitutes
∗
solves
∗
(11.9)
∈
The following proposition shows that the strategies/plans are time consistent in the sense
that the players play according to their plans after they find out their own types.
Proposition 11.1. A strategy profile
under mechanism
∗
∗
∗
is a BNE in Bayesian model
iff, for all and
occurring with a positive probability,
solves
∈
for any
and
where
∗
|
(11.10)
is the expectation operator over
|
conditional on
Proof. Problem (11.9) can be written as
∗
∈
∗
∈
|
∈
where
( )
(
)
∗
Then, by the Pontryagin Theorem, problem (11.9) is equivalent to
problem (11.10), since the Hamiltonian function is
∗
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Definition 11.4. A mechanism
there is a BNE
can implement an allocation scheme
∗
∗
∗
mechanism exists, we say that
of
such that
∗
for all
in BNE if
If such a
is implementable
That is, we say that a mechanism
implements an allocation scheme
equilibrium induced by the mechanism that yields the same outcome as
if there is an
for all
in which each player’s strategy is to
Definition 11.5. The special mechanism
report his type in his own best interest (which may or may not be the true type), is called the
direct mechanism.
In the direct mechanism, each player’s strategy is to report
cated. The reporting strategy can be written as
each player derives his report
and then
is allo-
Given expectations on others’ types,
of his type from his utility maximization problem:
|
(11.11)
Definition 11.6. An allocation scheme
is truthfully implementable in BNE or incentive
∈
compatible (IC) if each player’s strategy is to report his true type in his own best interest in the
direct mechanism
That is, for each
and
the player will always report his true
type
To find the optimal allocation scheme, we first need to identify all implementable allocation schemes. The identification of all implementable allocation schemes may seem like a
daunting task since there are so many possible mechanisms. Fortunately, the revelation principle tells us that we can simply restrict our attention to the simplest mechanism, the direct
mechanism, in which each agent is asked to report his type
and, given the announcements
is allocated. Here, the strategy of the player with type
which is dependent on his type
is his report
The revelation principle indicates that an allocation scheme
is implementable if and only if it is truthfully implementable.
Proposition 11.2 (The Revelation Principle). If an allocation scheme
is implementable
iff it is truthfully implementable.
Proof. If an allocation is truthfully implementable, it is obviously implementable since it is
implementable by the direct mechanism.
implements the allocation
Conversely, suppose that a mechanism
scheme
∗
in BNE. Then, there exists a profile of strategies
for all
where
∗
∗
∗
∗
such that
solves the following problem:
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∗
|
∈
(11.12)
be the solution of the following problem:
Let
∗
Since
for all
∗
|
∈
∗
(11.13)
is the solution of (11.12), by (11.13), we must have
problem (11.13) can be rewritten as
Hence,
(11.14)
|
∈
is the solution of (11.14). That is,
is truthfully implementable in BNE.
The explanation of the revelation principle is simple: if
ship between type
∗
∗
Since
and strategy
is equivalent to reporting
wants
given
In
given
player
∗
offers a one-to-one relation-
, given others’ strategies, choosing the optimal strategy
. “
is implementable” means that, given
can choose
player can choose
to get
∗
to get
player
in the direct mechanism,
The proof follows this explanation.
The revelation principle can be illustrated by the following diagram: in a typical mechanism, you map your type to a strategy and then the mechanism maps a strategy profile to an
allocation. However, if the strategy profile and the mechanism are combined into one operation, which is the direct mechanism, then you map your type directly to an allocation. In the
latter one-step process, you have no incentive to cheat on yourself. That is, if everyone is doing
‘the right thing’ by choosing
(which is equivalent to choosing
original mechanism), you should choose
the original
∗
under the
(which is equivalent to choosing
∗
under
mechanism).108
Direct Mechanism
Qi
si*
f (q )
G
Si
The advantage of this revelation principle is that we have no need to know the actual
mechanism
particularly, the functions
∗
and
If our interest
is to find implementable allocation schemes, we only need to find those allocation schemes
108
This depends crucially on the nature of Nash equilibrium. In a Nash equilibrium, you assume that others
will play the right strategies whatever you do; hence, if you cheat, you cheat on yourself only.
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that are truthfully implementable. Hence, to identify the optimal allocation scheme under incomplete information, we only need to search among those allocation schemes that are
incentive compatible under the direct mechanisms.
Note that, in addition to incentive-compatibility conditions, certain participation conditions or individual rationality (IR) conditions must also be satisfied. We will ignore the IR
conditions until the last section.
The significance of the revelation principle is that to identify the set of implementable allocation schemes, we need only identify those that are truthfully implementable. To identify
truthfully implementable allocation schemes
that lead to the solution
we only need to identify those
from problem (11.11). We replace problem
(11.11) by its FOC and SOC conditions. The FOC for (11.11) is
(11.15)
|
for all
and
This FOC provides a set of equations for identifying those
We
may also impose the SOC:
|
Since (11.15) holds for all
by taking a derivative w.r.t.
in (11.15), we find
|
Hence, the SOC is
|
for all
and
(11.16)
Conditions (11.15) and (11.16) ensure an arbitrary allocation scheme
to be truthfully implementable. In other words, conditions (11.15) and (11.16) define the set of
all implementable allocation schemes. This is the set of allocation schemes from which we can
conduct a search for the optimal allocation scheme.
3. Examples of Allocation Schemes
We now give a few examples of allocation schemes and some real-world mechanisms.
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Example 11.1. The Buyer-Seller Model
Consider a trade of a private good between a buyer with utility function
and a seller with utility function
We impose typical assumptions on the func-
tions:
where the condition
means that the marginal social benefit
there is an improvement in the quality
and
where
respectively. Condition
of the product. For example, a special case is when
and
are standard utility and cost functions,
means that the quality has been so high that a further in-
crease in quality is not social desirable. We expect
for simplicity in this example, we impose
solution
the quality
is lower when
to hold in equilibrium; however,
to hold for any pair
maximizes social welfare, then
as in (11.17). The seller knows
of his product, but the buyer does not know this. Let
be the price of the trade. Let
Notice that if a
be the quantity traded and
be the solution of the ex-post social welfare maximization:
or
(11.17)
Define an allocation scheme by
where
tion scheme is ex-post efficient by the definition of
is some function. This allocaIs this
be implementable? To
answer this, we look at the seller’s problem when the seller is asked to report her type (i.e.,
product quality):
The FOC is
Hence, for truth reporting, we need
(11.18)
The SOC is
By (11.18), the SOC becomes
which is satisfied if
By (11.17), we find
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By condition
we have
that is, the SOC is satisfied. Therefore, the alloca-
tion scheme is truthfully implementable if and only if (11.18) holds, which defines
where
can take any value. Therefore, the allocation scheme
post efficient if
is defined by (11.17) and
:
is implementable and ex-
is defined by (11.19).
Note that we have imposed the requirement of ex-post efficiency on our design of the
above allocation scheme, i.e.,
is determined by social welfare maximization. But, an
optimal allocation scheme in terms of maximizing the buyer’s expected utility may not be expost efficient. Thus, the above allocation scheme may not be optimal for any
.
The next three examples show that ex-post efficiency is difficult to reach under incomplete information.
Example 11.2. The Price Mechanism
How about the so-called price mechanism defined by the general equilibrium? Consider a
pure exchange economy with
tion
consumption set
Allocation scheme
goods and
consumers, in which consumer has utility func-
and endowment
The set of feasible allocations is
is defined to be the GE allocation. We know that
is ex-post effi-
cient under usual conditions. Can this allocation scheme be implemented by some mechanism?
The answer is no. To see this, Figure 11.4 shows three offer curves in an Edgeworth box. The
intersection points are the GE allocation.
.
.
.
Figure 11.4. Walrasian Equilibria
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We can see from the figure that, if consumer 1 tells his true type, consumer 2 will generally not
tell his true type. By claiming
consumer 2 can obtain
to consumer 2. This means that
which is better than
is not implementable by the direct mechanism.
By the revelation principle, this means that
cannot possibly be implementable by any
mechanism, including the price mechanism. Akerlof (1970) uses the same mechanism and
yields a similar conclusion.
Example 11.3. A Public Project
Consider a situation in which
agents decide whether to undertake a public project, say
building a bridge, whose cost must be funded by the group themselves. An outcome is a vector
where
or
monetary payment from agent
indicates whether or not to build the bridge and
The cost of the project is
is a
Hence, the set of feasible
allocations is
Agent ’s utility function is
Here,
can be interpreted as the agent’s willingness to pay for the bridge. An allocation
scheme
is ex-post efficient if, for all
109
Given the decision to build determined by the above rule, how do the participants allocate the
cost properly? Consider first an equal-sharing rule:
This rule cannot work under the
direct mechanism. Assume that the reports are publicly observable. For example, if
then agent
does not want the project even though we may have
to discourage taking the project is to report
One way for agent
This problem is due to equal cost sharing
but unequal benefits. Agent 1’s decision has a positive externality on others, but he fails to
internalize this effect.
One better solution may be a proportional sharing rule:
∑
Under this rule, an
agent who benefits more pays a proportionally larger cost. By this, when
every agent
is willing to pay his share of the cost to build the bridge. However, even in this case, due to
incomplete information, an agent may not tell his true type. Let us see if agent
109
will report his
We can define a social welfare function as the sum of the individual’s utility functions. The choice of
maximizes social welfare.
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true type when others report their true types. If the project is worth to build, agent ’s problem
is110
This implies
Hence, if
general, any
agent 1 will under-report his type. In
satisfying
is a Nash equilibrium of the report-
ing game. This solution is ex-post efficient but agents generally do not report their true types.
Example 11.4. An Indivisible Item
There is a single indivisible item to be sold by a seller to one of
vector
where
agents. The outcome is a
if agent gets the good,
otherwise, and
the monetary payment from agent
111
Again,
is agent ’s willingness to pay for the item. Let
where
The seller’s revenue is
Then, the feasible allocation set is
An allocation scheme
is ex-post efficient if the highest type
Assume
gets the item.112
and suppose that both buyers’ valuations
the uniform distribution on
is
are drawn independently from
and this fact is common knowledge among the agents. An
allocation scheme
is defined by
It is ex-post efficient. Can we implement this allocation scheme by the direct mechanism?
Specifically: if buyer 2 always announces his true type, will buyer 1 do the same? If the reports
are observable, then, if buyer 2 reports his true type
and
buyer 1 will report
which is generally not the true value. Alternatively, suppose that the two buyers announce
their types simultaneously, or equivalently, they submit sealed reports of their types and the
reports are then opened after all the reports have arrived. Thus, given the density function
of
buyer 1 chooses his announcement
from
110
If the reports are not publicly observable, we need to take expectations over
.
111
This is the auction model, in which there are many competitive buyers, a single seller and a single
indivisible item for sale. This model is popular in the literature of asymmetric information since it is the simplest
case of monopoly pricing under asymmetric information.
112
For each
it is Pareto optimal since it maximizes social welfare
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implying
In other words, if buyer 2 tells the truth, buyer 1 will report only half of his true valuation.
Here, a buyer has an incentive to understate his valuation so as to lower his payment for the
item, even though this reduces his chance of getting the item.
Let us consider an alternative allocation scheme
that satisfies (11.20) but changes
(11.21) to
which is still ex-post efficient. That is, the winner pays the amount equal to his opponent’s
valuation (the second highest valuation). If the reported types are observable to all, given an
announcement
by buyer
if
then buyer
should get the item and he can do so by
announcing
if
announcing
Thus, truth telling is a BNE, in which each buyer’s optimal strategy is to
then buyer
should not get the item and he can do so by again
reveal their own true type. If it is a sealed bid like before, given
buyer 1 considers his
problem:
The FOC is
tion scheme
implying
Again, truth telling is a BNE. Therefore, the alloca-
is truthfully implementable.
The following two examples show two indirect implementations of the problem in Example 11.4 through the design of some institutions in which the agents interact. In other words,
Example 11.5 and Example 11.6 show some real-world indirect implementations through the
design of institutions in which the agents interact.
Example 11.5. First-Price Sealed-Bid Auction
Consider the auction setting in Example 11.4 again. In a first-price sealed-bid auction,
each potential buyer is allowed to submit a sealed bid
The bids are then opened and
the buyer with the highest bid gets the good and pays an amount equal to his bid to the seller.
Assume again
and both buyers’ valuations
are drawn independently from the uniform
distribution on
We look for an equilibrium in which each buyer’s strategy
the form
for
Given
takes
buyer 1 considers his problem:
/
(
)
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implying
∗
∗
Thus,
∗
is a BNE. Each agent balances between paying too
much and getting the item; it turns out that everyone reporting half of the willingness to pay is
a BNE. This first-price sealed-bid auction implements the allocation scheme
with113
(11.22)
(11.23)
Example 11.6. Second-Price Sealed-Bid Auction
Consider the auction setting in Example 11.4 again. The arrangement is similar to Example 11.5, except that the winner pays the second-highest bid.114 Since it is a sealed bid, buyer 1
should consider
(
Then, given
)
the problem becomes
/
The FOC is
implying
Thus, the strategy
for all is
a BNE. Again, each agent balances between paying too much and getting the item; it turns out
that everyone offering his
is a BNE. He may want to get the item when he has a high
he may not want to get the item if
is too high. Therefore, when
but
the second-price
with
sealed-bid auction implements the allocation scheme
Example 11.7. Mechanism
The first-price sealed-bid auction is a mechanism in which
for all and
∗
with
113
See Gibbons (1992, p.157) on other PBE solutions of this problem.
114
If the bids in the auction are not sealed, given an announcement
the item and he can do so by announcing
again announcing
Thus, the strategy
if
agent
by agent
if
agent
should get
should not get the item and he can do so by
for all is a weakly dominant strategy for each player.
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The second-price sealed-bid auction is also a mechanism in which
∗
and
Example 11.8. The Revelation Principle
Consider Example 11.5 again. We know that the allocation
in (11.22)–(11.23) comes
from a mechanism, i.e., the allocation is implementable. We want to verify that the agents will
tell the truth in the direct mechanism, just as we are told by the revelation principle. When
facing the direct mechanism, buyer 1’s optimal announcement
is determined by
|
implying
Symmetrically,
Thus, the allocation scheme implemented by the
first-price sealed-bid auction in a BNE can also be truthfully implemented in a BNE by the
direct mechanism. This confirms the revelation principle.
In mechanism design, we ask three sets of questions:
 Given an allocation scheme
is it ex-post efficient and truthfully implementable?
 Given a mechanism, what is the BNE and the allocation scheme in equilibrium?
 Given an objective function, what is the optimal implementable allocation scheme?
We have discussed the first two sets of questions in the examples. In the rest of this chapter,
we will focus on finding the optimal allocation scheme.
4. IC Conditions in Linear Environments
Consider a simple utility function form for an agent:
Given allocation scheme
|
denote
|
Proposition 11.3 (IC Conditions). An allocation scheme
(11.24)
is
incentive compatible iff, for all
(11.25)
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(11.26)
Proof. (1) Necessity. Agent ’s announcement
is from
It implies
(11.27)
(11.28)
Again, by (11.27), we have
By (11.27), we immediately have
Then, by (11.28),
(2) Sufficiency. By condition (11.26), we have
No matter whether
or
by condition (11.25), we always have
Hence, we have
Therefore, truth telling is an optimal strategy, i.e.,
is incentive compatible.
Conditions (11.25)–(11.26) impose two restrictions on the allocation schemes. Given utility functions, we can easily identify all the allocation schemes that satisfy these two conditions.
These are all the incentive-compatible allocation schemes. To recover an allocation scheme,
we first choose an arbitrary increasing function of
Given an arbitrary value for
for all
Then, by condition (11.26), we have
this formula then defines a function
we can then identify functions
and
Given functions
from (11.24); there are many such
functions. We can then define an allocation scheme
Proposition 11.4 (The Revenue Equivalence Theorem). In an auction with
with
with
(1)
for all
buyers and
and buyers’ types being statistically independent
if two auction procedures generate BNEs satisfying
are the same in the two procedures for all
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(2)
are the same in the two procedures,
then these two procedures yield the same expected revenue for the seller.115
As an example of Proposition 11.4, consider Example 11.5 and Example 11.6. The two conditions in Proposition 11.4 are satisfied: in both auctions, the buyer with the highest valuation
always gets the good and a buyer with a zero valuation has an expected utility of zero. Thus,
Proposition 11.4 tells us that the seller receives exactly the same expected revenue in these two
auctions. We will later show in Section 6.4 that the resulting allocation schemes are optimal in
maximizing the seller’s expected revenue.
5. IR Conditions and Efficiency Criteria
There are typically three time points in a problem under incomplete information:
1. Ex ante: the time point before nature moves, when no one knows his own and other players’ types.
2. Interim: the time point just after nature has moved, when each player knows his own type,
but not others.
3. Ex post: the time point at which everyone knows everyone’s type.
q1 , , qn random
Ex ante
θ i known, θ −i random
q1 , , qn known
Interim
Ex post
Figure 11.5. The Time Line
5.1. Individual Rationality Conditions
So far, we have implicitly assumed that each agent has no choice but to participate in any
mechanism. In many applications, however, agents’ participation is voluntary. Thus, an implementable allocation scheme must not only be incentive compatible but it must also satisfy
certain participation or individual rationality constraints.
Let
be the alternative utility value that agent i with type
can obtain elsewhere af-
ter he declines to participate in the mechanism in question. There are three types of IR constraints. The ex-ante IR constraint is to induce agent i to take part in a mechanism ex ante:
(11.29)
115
The proof of this proposition is in the problem set.
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The interim IR constraint is to induce him to take part in a mechanism after knowing his own
type:
(11.30)
|
The ex-post IR constraint is to induce him to take part in a mechanism ex post:
(11.31)
Proposition 11.5 (Myerson-Satterthwaite). Consider a bilateral trade in which
the valuations
all
and
are independently drawn from
with
for
There is no ex-post efficient allocation scheme that is incen-
tive compatible and individually rational in the interim.
That is, whenever gains from trade are possible but not certain,116 there is no ex-post efficient allocation scheme that is incentive compatible and individually rational in the interim.
This result is not surprising. By Akerlof (1970), we already know that a market outcome under
asymmetric information can be inefficient. A typical insurance problem also indicates inefficiency under asymmetric information.
5.2.Efficiency Criteria
This section defines several notions of efficiency under asymmetric information. With
these efficiency notions, we can then discuss optimal implementable allocation schemes.
When agents’ types are known with certainty, ex-post efficiency (Pareto efficiency) provides a
minimal requirement for an optimal outcome. When agents’ types are uncertain, we have
several efficiency notions. Similar to the definition of the three kinds of IR conditions, we also
have three kinds of efficiency.
Definition 11.7. Given a set of feasible allocation schemes
there is no
such that
for all and
such that
if
for some
Definition 11.8. Given a set of feasible allocation schemes
there is no
is ex-ante efficient in
for all and
is interim efficient in
and
if
for
some and
Definition 11.9. Given a set of feasible allocation schemes
there is no
such that
for all
is ex-post efficient in
and
if
and
for some and
116
For example, the case with
in which it is certain that a trade can benefit both, is ruled out.
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The ex-post efficiency in Definition 11.9 is the same as the ex-post efficiency in Definition
11.1 when
contains all possible mappings from
to
i.e.,
In other words,
Definition 11.9 is a general definition, while Definition 11.1 is a special case when
possible allocation schemes. For example, if
11.1, then there exists
for some
and
is not ex-post efficient according to Definition
such that
for all
and
Then, if we define
we have
for all and
some . That is,
contains all
and
for
and
with continuous
and
is not ex-post efficient according to Definition 11.9.
Proposition 11.6. Given a set of feasible allocation schemes
for
an ex-ante efficient allocation scheme must be interim efficient, and an
interim efficient allocation scheme must be ex-post efficient.
Proof. Suppose that
is not interim efficient. Then, there exists
for all and
tion
and
for some and
on the inequalities gives us
Taking expecta-
for all and
the strict inequality is due to continuity of
such that
and
for some
for all
Thus,
where
is not ex-ante
efficient.
Suppose that
is not ex-post efficient. Then, there exists
for all
Taking expectation
|
and
on the inequalities gives us
for some and
for all
and
Thus,
such that
for some
and
for all and
where the strict inequality is due to continuity of
and
and
is not interim efficient.
Comparing Definition 11.7 with Definition 11.9, we can see that condition
is easier to satisfy than condition
Hence, it is easier to find a Pareto improving
for an allocation scheme
ex ante than ex post, implying that it is harder
to be ex ante efficient than ex post efficient. This explains why an ex
ante efficient allocation must be ex post efficient.
Although we have three time points for IR conditions, there is only one time point for IC
conditions. IC conditions are based on Definition 11.6, which is at the interim stage. In other
words, when it is time for an agent to report his type, he is always assumed to know his own
type but not others’ types.
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6. Optimal Allocation Schemes
Given an allocation problem, there are generally many allocation schemes that are incentive compatible and individually rational under the direct mechanism. This section discusses
the choice of the best among incentive compatible and individually rational allocation schemes.
We consider four models under different scenarios.
6.1. Monopoly Pricing
Suppose that there are two consumers with the demand curves indicated in Figure 11.6,
where consumer 2’s demand curve is above consumer 1’s demand curve, with maximum demand
and
respectively, at zero price. Assume zero cost. We have labeled the areas as
and
As shown in Figure 11.6(a), with perfect price discrimination, the monopolist
would like to sell
to consumer 1 for price
and sell
to consumer 2 for price
(the price here is the total charge rather than a unit price). However, since the monopolist
cannot distinguish the consumers, if these two packages are offered to the market, consumer 2
will choose package
by which he retains a surplus of
second package
instead of no surplus for the
The profit for the firm is
p
p
p
B¢
B¢
B
E
A¢
C
A
x1
A¢
C
C¢
D
x2
D
(a )
x1
x2
(b)
x1*
x2
(c )
Figure 11.6. Monopoly Price under Incomplete Information
To avoid the above problem, the monopolist can make two alternative price-quantity offers:
and
If so, both consumers will choose the package intended for them.
In particular, consumer 2 will choose
since the other yields the same surplus
This strategy yields a higher profit for the monopolist. The profit is now
However, this is not the best the monopolist can do. For example, as shown in Figure
11.6(b), the monopolist can offer a lower
and
The monopolist still offers two packages:
Each consumer will still choose the package that is intended for him.
The monopolist’s profit is increased since she loses the small red area but gains the larger
green area. The profit is
which is better than before as long as
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The optimal
∗
is the one where the marginal benefit from one more unit of reduction in
equals the marginal loss. As shown in Figure 11.6(c), the optimal offers are
∗
and
which maximize the monopolist’s profit. Consumer 1 gets zero surplus and
consumer 2 gets surplus
∗
The profit is
Is the optimal solution efficient? Under complete information, social welfare is
Under incomplete information, with the optimal solution, social welfare is
∗
Hence, the optimal solution is inefficient.
We now derive the solution rigorously. Suppose that the two demand functions are
The firm’s problem is to solve for
,
∗
∗
∗
from
,
and make offers:117
To satisfy the IC condition, the firm chooses
Hence, the problem
becomes
,
The FOCs are
They imply
∗
∗
and
∗
∗
Under complete information, social welfare is
∗∗
̅
̅
With the optimal solution, social welfare is
∗
̅
∗
∗
∗
̅
∗
Hence, the optimal solution is inefficient and the dead weight loss is area D:
117
For simplicity, I have here severely narrowed down the firm’s admissible package offers.
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∗∗
̅
∗
∗
6.2. A Buyer-Seller Model with Linear Utility Functions
Suppose that a seller is selling a product to a buyer. The seller has type
which the buyer
does not know. The buyer only knows that the type follows the uniform distribution on
The buyer knows that the relationship between investment and quality
the seller of type
of the product for
is
The cost of investment for the seller is
For
the buyer’s value from the product of quality
The buyer can observe and verify investment
Hence, the buyer can offer a deal
seller. The direct mechanism is: the seller of type
the price
reports his type
for the product and demands investment
Given a contract
is
the seller’s surplus is
to the
and then the buyer pays
for the production of the product.
and the buyer’s surplus is
We will ignore an IR condition. Instead, we impose the following boundary conditions:
where we take
and
for convenience.
A contract between the buyer and the seller is an allocation scheme. By the revelation
principle, the buyer can confine her search for an optimal contract to the set of incentivecompatible allocation schemes
The buyer’s problem is
(⋅) [ (⋅), (⋅)]
As a comparison, we will use two approaches to solve this problem for the optimal contract
∗
∗
Approach 1
Given an offer
the FOC for reporting
the value function for the seller is
Then,
is
(11.32)
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The SOC is
Taking a derivative on the truth reporting condition
yields
Then, the SOC becomes:
(11.33)
Hence, the buyer’s problem is
( ), ( )
(11.34)
We ignore the SOC for the time being and consider the following problem:
( ), ( )
Define the Hamiltonian function:
where
is the Lagrangian multiplier, dependent on
The Euler equations are
(11.35)
(11.36)
Equation (11.35) implies
(11.37)
where
is a constant. Equation (11.36) implies
By (11.35) and (11.37), this equation becomes
(11.38)
implying
Since
we have
implying
∗
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This solution also satisfies the SOC. Hence, this is indeed the solution of problem (11.34).
∗
We now solve for
By (11.32), we have
implying
∗
∗
The optimal contract
∗
is solved.
Approach 2
Since we have a linear environment, we can use Proposition 11.3 to solve the problem. The
value function for the seller is
Hence,
for the
in Proposition 11.3. The value function for the buyer is
By Proposition 11.3, the buyer’s problem is
(⋅), (⋅)
(11.39)
To simplify the problem, we want to eliminate
from the objective function so that only
is left to determine. We have
We can now eliminate
from the problem, which becomes
(⋅)
(11.40)
Here, since the objective has nothing to do with
we have dropped the conditions for it.
Without the IC condition in (11.40), the optimal solution
∗
must satisfy Euler equation:
implying
∗
Since this solution is indeed decreasing, the IC condition in (11.40) is satisfied. Hence, it is the
solution of problem (11.40).
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We now go back to solve for
∗
By the
condition in (11.39), we have
∗
implying
∗
∗
6.3. Labor Market
Consider the Spence model in the previous chapter. There is a single firm, which is now a
monoposony in the labor market. Instead of two possible types, we assume that the workers
have a continuum of possible types
is
and the density function is
tion level
where
with
The distribution function
for all
Each worker obtains an educa-
with utility function
where
represents the agent’s cost for education. Assume that
Let the revenue function be
for type
is differentiable with
is the wage and
is differentiable and
The firm’s payoff is
and
118
that a high-ability worker is more productive, and an increasing
where
A decreasing
means
means that education
has a positive effect on productivity. The firm does not observe a worker’s type. It uses an
allocation scheme to
to induce each worker to report his type truthfully.
Asymmetric Information
Assume that
linear in
is known only to the worker himself. Since each worker’s utility function is
we can use Proposition 11.3. Let
if the worker’s type is
which is the utility value
and he tells the truth. Suppose that the reservation utility value is
By
Proposition 11.3, the firm’s problem is
(⋅),
(⋅)
(11.41)
Here,
is the
in Proposition 11.3. To simplify the problem, we want to eliminate
from the objective function so that we need to find one function
is decreasing. Hence, the
118
Here,
corresponds to the
condition is equivalent to
in Spence’s model and
only. By the
condition,
We have
A bad type has a big
and small
productivity
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We can now eliminate
( )
from problem (11.41). Let
( )
Then, the problem
becomes
(⋅),
( )
(11.42)
for any solution. Thus, problem (11.42) can be fur-
It is obvious that we must have
ther written as
(11.43)
(⋅)
Without condition
the optimal solution
∗
∗
must satisfy the Euler equation:
∗
(11.44)
This implies119
∗
which holds if
∗
∗
∗
∗
is increasing. By this condition,
condition
is satisfied and
∗
∗
is decreasing in
implying that
is indeed the solution of problem (11.43).
∗
To find the optimal allocation scheme
∗
we further find
from the
condition:
∗
which then gives us
∗
∗
∗
∗
Complete Information
When a worker’s type is publicly observable, the IC conditions are no longer necessary.
Then, the firm’s problem (11.41) becomes120
119
Here,
∗
can be a strictly decreasing function, by which the company can recover the applicants’ types
completely. However, this does not mean that the solution is the complete-information solution. This only means
that the solution is a separating equilibrium. The complete-information is
∗∗
which is generally more efficient.
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(⋅),
(⋅)
Obviously, we must have
∗∗
for all
Then, the Euler equation for the optimal solution
is
∗∗
∗∗
Since
∗∗
maximizes the total social surplus
it is
ex-post efficient. This is not surprising since a solution under complete information is known
to be ex-ante efficient and, by Proposition 11.6, ex-ante efficiency implies ex-post efficiency.
and (11.44) implies
Since
∗
∗
∗
we have
∗
∗∗
∗
∗∗
In other words, under asymmetric information, only the best type has an ex-post efficient
education level, while all other types have their education level distorted downward.
6.4. Optimal Auction
Consider the auction in Example 11.4 again. There is a seller selling an indivisible object
to
buyers
Each buyer has a linear utility function of the form
where
or
A feasible allocation scheme is
where
∈ℕ
Denote
where
and
buyer s type is independently drawn from
and
for all
Assume that
with density
(
(
)
)
Each
and distribution
is increasing in
for all and
,…,
The expected revenue is
∈ℕ
120
∈ℕ
∈ℕ
We may interpret the model as having many applicants who distribute alone
ty function
with population densi-
Hence, the objective function is the per capita profit.
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Facing the interim IR conditions
for all
and
the seller wishes to choose
an IC allocation scheme that maximizes his expected revenue. Thus, by Proposition 11.3, the
seller’s problem is
{
(⋅),
(⋅)}
∈ℕ
∈ℕ
(11.45)
∈ℕ
By
the
condition is equivalent to
We have
Then, the expected revenue in (11.45) becomes
∈ℕ
Since
we have
∈ℕ
∈ℕ
∈ℕ
Hence, the seller’s problem is to choose functions
expected revenue subject to
the solution must have
ing functions
and
for all
and values
to maximize this
conditions for all
It is evident that
Hence, the seller’s problem is reduced to choos-
for all to maximize
(11.46)
∈ℕ
subject to
and
conditions. We ignore
for the time being. Then, by the Hamilton
approach, an inspection of (11.46) indicates that the solution
the
condition is, for all
∗
of the problem without
and
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∗
(11.47)
condition. Thus,
This solution indeed satisfies the
problem
∗
is the solution of the seller’s
(11.45).121
, the optimal transfer scheme is
By the definition of
∗
Here, the term
∗
∗
∗
∗
indicates the winner’s benefit
(11.48)
∗
but the term
indicates
that the payment is lower than the benefit.
Discussion:
1. The agent who has the largest value of
does not necessarily have the largest valuation
Thus, the optimal solution need not be ex-post efficient.
2. In the case of symmetric bidders in which
and
for all
where
the optimal auction always gives the object to the bidder
is large enough so that
with the highest valuation. This means that the solution is ex-post efficient in this case.
Further, the payment scheme is based on the second price. To see this, if
among
and
is the highest
is the second highest, (11.48) implies
∗
That is, the winner pays
which is the second highest bid in the direct mechanism.
Hence, the second-price sealed-bid auction is optimal. Using the revenue equivalence theorem, the first-price sealed-bid auction is also optimal.
6.5. A Buyer-Seller Model with Quasi-Linear Utility122
Consider a model with one buyer and one seller who have utility functions
respectively, where
is private information of the seller and
and
and
are commonly
observable variables based on which a contract can be specified. The contract
is pro-
be the buyer’s subjective distribution function for
where
posed by the buyer. Let
Using (11.15)–(11.16), the buyer’s ex-ante optimization problem is
121
When two individuals have the same largest
is increasing,
since
122
∗
give the item arbitrarily to one of them. Furthermore,
is increasing in
This example is from Guesnerie-Laffont (1984). See also Laffont (1989, p.159).
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(⋅), (⋅)
(11.49)
Depending on the interpretation of the problem, we may alternatively consider an ex-post IR
condition:
We may interpret the ex-ante IR condition as a situation in which the seller first agrees to
participate before he knows his own type and the trade
with a report type
occurs
later when the seller knows his own type. In contrast, the ex-post IR condition ensures that
every agent’s IR condition is satisfied even when every agent knows his own type.
Consider quasi-linear utility functions of the form:
We can interpret
as the price of the good and the buyer purchases
for a price
Then, the
buyer’s problem becomes
(⋅), (⋅)
(11.50)
The IR condition must be binding, by which the objective function becomes
By assuming
123
the
in the objective function, the
condition is satisfied if
and
Since
is no longer involved
are no longer needed. Therefore, the optimization
problem becomes
(⋅)
(11.51)
The Lagrangian function of (11.51) is
123
Laffont (1989, p.155, expression (7)) has this assumption. This assumption is actually suitable only if
is a
public good.
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where
and
Then, the Hamiltonian function is
The conditions for an optimal solution
∗
are
∗
∗
∗
(11.52)
∗
On the other hand, under complete information, the problem is
(⋅), (⋅)
Since the IR condition must be binding, the problem becomes
(11.53)
(⋅)
Let
∗∗
be the solution of problem (11.53). By the Hamiltonian method,
∗∗
is a solution
of the following equation:
∗∗
and
By the concavity of
in
∗∗
(11.54)
the solution is unique. Thus, any solution satisfying (11.54)
must be the optimal solution of problem (11.53).
∗
We now try to solve (11.52). Due to the condition
can either be increasing
or constant. Thus, to find a solution, we follow two steps:
•
If
∗
on an interval
and thus
•
If
∗
then by the Kuhn-Tucker condition we have
on
∗
or
By the Euler equation,
for some constant
by regions, by continuity of
know that
∗
on an interval
we must have
∗∗
,
for
and
∗
in near-
By the Euler equation, we
is continuous. Thus, by the Euler equation,
(11.55)
By continuity of
∗∗
we also have
∗∗
The conditions (11.55) and (11.56) determine
∗∗
(11.56)
and
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These two steps will give us a solution that can be verified to satisfy the conditions in (11.52).
An illustration of
∗
and
∗∗
is in the following figure.
Figure 11.7. The Optimal Contract
Remark. If the direct mechanism is so wonderful, why not we see it often in practice? The
reason is that the direct mechanism is convenient to use for theorists, but difficult to use for
practitioners. For example, suppose that a seller is to sell a piece of painting and there are
buyers with independently uniformly distributed types. By the direct mechanism, the seller
needs to know all untility functions
,
, set up the mechanism design problem
with IC and IR conditions, and solve the problem. In practice, the seller may not know the
utility functions; and even if she knows all utility functions, solving the mechanism design
problem is difficult. Instead, the seller can use the mechansm of the second-price sealed-bit
auction. With this auction, the seller only needs to announce this mechanism and waits for the
buyers to play out the game, and then receives the payment. As shown in Section 6.4 on optimal auction, this payment has the highest expected value that the seller can possibly receive.
That is, the seller’s expected incomes from the direct mechanism and the second-price sealedbit auction are identical. Therefore, the seller will simply use the second-price sealed-bit auction instead of the direct mechansim.
Notes
Good references for this chapter are Mas-Colell et al. (1995, Chapter 23) and Laffont
(1995).
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Chapter 12
Incentive Contracts
Besides asymmetric information, there is another type of information problem, called the
incentive problem. For example, in an employer-employee relationship, the employee’s applied effort may be observable by his employer but not verifiable to a court. If so, the effort
cannot be bounded by a contract. In this type of problem, information is symmetric: both the
employer and the employee have the same set of information. But the court cannot observe the
information. So the issue here is: how does the employer provide sufficient incentives in a
contract to motivate the employee?
1. The Standard Agency Model
In this section, we present the standard agency model developed by Mirrlees (1974) and
Holmström (1979). This section is mainly based on Holmström (1979). See also Laffont (1995,
p.180–198).124 The standard contract theory is about an output-sharing rule between a principal and an agent. The principal tries to hire the agent to work on a project. A problem is that
the agent’s effort may be observable to the principal but is not verifiable to a third party such
as a court and is thus non-contractable. Hence, the principal must provide incentives in a
contract for the agent to put in a sufficient effort voluntarily. The question is how to provide
sufficient incentives in a contract.
Let
tion of
be a random output from the firm. Given the agent’s effort
is
and the density function is
The effort
the distribution func-
is not verifiable or contract-
ible; more precisely, it is uncontractable ex ante since it is non-verifiable ex post. In fact, only
the output
is contractible (observable and verifiable ex post) in this model. The output is
random ex ante, but it becomes known and verifiable ex post. Thus a payment scheme
be based on the output ; the scheme pays
to the agent when output is
can
The set of
admissible contracts is
Here, condition
124
means limited liability to the agent. The agent’s utility function is
For more details of the standard principal-agent model, see Ross (1973), Stiglitz (1974), Harris and Raviv
(1979), and Shavell (1979).
Susheng Wang, HKUST
(12.1)
is the cost of effort which is private. The principal’s utility function is
where
In addition, assume that at least one of the parties is risk averse. This assumption is necessary
for the standard agency theory. We will discuss the special case when both the principal and
the agent are risk neutral in Section 5.
12.1. Verifiable Effort: The First Best
As a benchmark, we first consider the problem in the Arrow–Debreu world, in which effort
is verifiable. In this case, the principal specifies both an effort and a payment scheme in
a contract. That is, the principal can offer a contract of the form
to the agent. Let
be
the agent’s reservation utility, below which the agent will turn down the contract. The principal
is constrained by the participation constraint or individual rationality (IR) condition:
The principal’s problem is then
, (⋅)
∗∗
Its solution
∗∗
(12.3)
is said to be the first best and hence the above problem is called the first-
best problem. According to the first welfare theorem, it is Pareto optimal.
Given each
the support of the distribution function is the interval of
This interval will generally be dependent on
on which
The following assumption is im-
posed in the standard agency model.
Assumption 12.1. The support of the output distribution function is independent of
The Lagrangian for (12.3) is
where
is a Lagrange multiplier. The Hamiltonian for
Given Assumption 12.1, the FOCs for
and
can be defined as
from the Lagrangian and Hamiltonian are
respectively125
125 There
are no boundary conditions such as
and
for
for some given constants and
Thus, we will have two transversality conditions for the optimal contract. If we impose
and
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(12.5)
is
Equation (12.5) indicates that the marginal rate of substitution
constant across all possible states
implying the proportional sharing of risk in some sense.
We call it the first-best risk sharing. The marginal rate of substitution (MRS) is constant because the Lagrangian is a linear social welfare function, where the weight
is endogenous.
This result is the same as that in the Arrow-Debreu world, in which the MRS is equal to the
price ratio and thus constant in equilibrium.
Equation (12.4) means that the marginal social utility of
of
By (12.5), we have
equals the marginal social cost
implying a binding IR condition at the optimum.
By differentiating (12.5) with respect to
the principal is not risk neutral, we have
we have
Hence, if
implying that both the agent’s income
and the principal’s income increase strictly with
If the principal is risk neutral, the contract will be a constant, i.e., the principal absorbs all
the risk. Similarly, if the agent is risk neutral, the agent takes all the risk. This risk sharing
arrangement of a risk-neutral party taking all the risk makes sense. When a risk-averse party
faces risk, it demands a risk premium (an extra payment to compensate for bearing the risk),
while a risk-neutral party cares about expected income only. Hence, the risk-neutral principal
can do better by taking all the risk.
12.2. Nonverifiable Effort: The Second Best
Suppose now that
is not verifiable. Since the principal cannot impose an effort level, she
has to offer incentives in a contract to induce the agent to work hard. Given an income-sharing
rule
the agent is free to choose an effort and his problem is:
This is the incentive compatibility (IC) condition. The principal still decides the incomesharing rule
and effort
But whereas before, the principal has only to ensure that the
agent accepts the contract (the IR condition) and the effort can be imposed upon the agent,
now the principal has to provide not only an incentive for the agent to accept the contract but
also an incentive to accept the effort (the IC condition). Hence the principal’s problem is
then the conditions are
However, since
doesn’t contain
these conditions are all satisfied. Note also that by assumption the optimal
solution must be a continuous function if the admissible set is assumed to contain continuous contracts only.
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, (⋅)
The FOC of the IC condition (12.6) is
(12.7)
The second-order condition (SOC) for the IC condition (12.6) is
The IC condition (12.6) is too difficult to handle in an optimization problem. As an alternative,
the so-called first-order approach (FOA) is to substitute the original IC condition (12.6) by its
FOC (12.7) in the principal’s problem. Rogerson (1985) and Jewitt (1988) find conditions
under which the FOA is valid (i.e., by which the solution of the agency problem is indeed the
right one). Rogerson (1985) limits distributions to have finite states. Under the FOA, we call
(12.7) the IC condition. Thus, the principal’s problem becomes
, (⋅)
(12.8)
Its solution
∗
∗
is said to be the second best and hence the above problem is called the
second-best problem. Due to the additional constraint, this solution is inferior to the first best.
The IC and IR conditions provide different functions. An IR condition induces the agent
to participate, but the agent may not work hard or invest an effort up to the principal’s desired
amount. The principal has two ways to ensure the agent invests the desired amount of effort.
One way is to write it in the contract if this amount is verifiable. The other way is to provide
incentives in the contract for the agent to invest the desired amount voluntarily.
The Lagrangian for (12.8) is
where
and
The FOCs for
are Lagrange multipliers, and
and
The Hamiltonian for
is
from the Lagrangian and Hamiltonian are respectively
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(12.9)
(12.10)
The second equation is called the Euler equation, which determines the optimal contract. Here,
we can see that if both parties are risk neutral, (12.10) cannot hold. This is the reason that one
of the parties must be risk averse.
Notice that the likelihood function is
i.e.,
if
( ; )
( ; )
( ; )
( ; )
is the increase in
is increasing in
( ; )
(MLRP).
for a unit increase in
measures a kind of correlation between
( ; )
And,
and
are complementary
The latter condition is called the monotone likelihood ratio property
means that a higher value of
chance. Thus,
and
and
A higher value of
is more likely to come from a higher value of
rather than by
is the marginal increase in the principal’s belief that the effort is
from
the observation of
generates an improvement in
We further assume that an increase in
in the sense of
first-order stochastic dominance (FOSD), as stated in the following assumption.
Assumption 12.2. (FOSD).
for all
with a strict inequality on a set of
that has a non-zero probability measure.
Theorem 12.1 (Holmström 1979, Shavell 1979). Under Assumption 12.1 and Assumption
12.2, FOA and
we have
for the optimal solution of (12.8).
Let
Proof. Suppose, on the contrary, that
define
and for the
in (12.10),
by
(12.11)
over
The advantage of
∗
satisfying
Since
have
and
Then by (12.10), for
we have
we must have
Similarly, for
satisfying
∗
we
Thus,
∗
Let
∗
is that we know
be the support of
where
and
∗
are independent of
∗
Then
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∗
∗
∗
∗
Since
and
implies that
for all
∗
we must have
Also, by Assumption 12.2,
∗
(12.9)
on a non-zero-probability-
measure set. Therefore,
∗
However, if FOA is valid, the SOC for (12.6) must hold, which means
∗
∗
(12.12)
Then (12.9) cannot hold. This is a contradiction. We thus must have
Theorem 12.1 indicates that the risk sharing is no longer the first best. By deviating from
the first-best risk sharing, the principal can provide incentives in the contract. To understand
the result intuitively, assume that the principal is risk neutral. Then, the first-best risk sharing,
as indicated by (12.5), requires full insurance (no risk) for the agent, i.e.,
is constant.
However, if the agent’s action is not verifiable, under a fixed contract, the agent will choose
zero effort. Thus, for the agent to have the incentive to work, the contract with uncontractible
effort must relate a payment to output. This means that the agent must share some risk, implying inefficient risk sharing in the case of nonverifiable effort.
Let us compare the first-best and second-best contracts. As shown in Figure 12.1, an increase in
Let
∗∗
causes the density curve to shift to the right. This implies that
be the first-best contract determined by (12.5) and
∗
be the second-best contract de-
termined by (12.10). Since the left-hand side of (12.10) is increasing in
∗
∗∗
∗
we have
∗∗
Thus, the two contracts should have the relationship shown Figure 1.2, which means that, in
order to motivate the agent to work hard, the principal must use a steeper contract curve than
the first-best contract to provide incentives. The second-best contract rewards high output and
punishes low output; at the same time, the second-best contract must also provide a proper
risk-sharing scheme.
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s
s*
s**
f ( x, a ')
f ( x, a )
x
x
Figure 12.1. A Shift in Density Function
Figure 12.2. First- vs Second-Best Contracts
Example 12.1.126 The principal’s problem is
∈ , ∈
(12.13)
where
is the effort space, and
is the contract space that consists of continuous con-
tracts satisfying the limited liability condition:
The density function
mean
Let
states that the output follows the exponential distribution with
and variance
We find the second-best solution:
∗
∗
and the first-best solution:
∗∗
∗∗
Example 12.2. Consider
We have
Hence,
126 This
example is from Holmström (1979, p.79), but his solution seems to be erroneous.
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13. Two-State Agency Models
Two-state models are popular. For example, insurance models typically have only two
states. With only two possible states, the agency problem becomes simpler. A good reference
is the work of Mas-Colell et al. (1995, Chapter 14).
Again, there is a principal who tries to hire an agent. The output can either be a high output
or a low output
utility is
The probability of achieving
is
The agent’s reservation
The private cost of effort is
13.1. Verifiable Effort
Suppose that the principal pays
for
and
for
The IR condition is
(12.14)
Hence, if effort is verifiable, the principal’s problem is
,
,
The Lagrange function for (12.15) is
where
is a Lagrange multiplier and
The FOCs for
are respectively
which imply
We immediately have
That is, the optimal salary
and
∗∗
is independent of output and is
∗∗
The institu-
tion is that, since the agent is risk averse and the principal is risk neutral, the principal takes
all the risk. This is also indicated by the standard agency theory, as implied by (12.10). Since
the IR condition (12.14) must be binding, implying
∗∗
∗∗
(12.17)
Equation (12.16) then becomes
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∗∗
∗∗
implying
∗∗
∗∗
(12.18)
∗∗
Equations (12.17) and (12.18) determine the first-best solution
∗∗
∗∗
13.2. Nonverifiable Effort
With uncontractible effort, again suppose that the principal pays
for
and
for
Given the wage contract, the agent considers his own problem:
which implies the FOC:
The principal’s problem is to induce sufficient effort by considering the agent’s IC and IR
conditions, i.e.,
,
,
(12.19)
The Lagrange function for (12.19) is
where
is a Lagrange multiplier and
The FOCs for
are respectively
implying
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The third equation implies
resulting in a binding IR condition. By Theorem 12.1,
These three equations together with the IC and IR conditions determine
By the IC condition, we have
∗
∗
∗
∗
∗
i.e., the agent shares some risks.
No matter whether effort is verifiable or not, the agent’s welfare is
in both solutions. But
the principal yields a higher profit from problem (12.15) than problem (12.19) since the former
has fewer conditions. In other words, the solution with contractible effort Pareto-dominates
the solution with uncontractible effort, since the agent is indifferent but the principal is better
off with contractible effort.
13.3. Example: Insurance
Consider a monopolistic insurance company with a single type of consumer. A consumer
can influence his own probability
private cost
of accident by putting in some preventive effort
which is increasing and convex. Assume that
with
is a decreasing function,
indicating that a higher effort reduces the chance of an accident occurring. The company offers
premium
for compensation
Laffont (1995, 125–128) and Salanie (1999, 134–135) provide
good references for this example.
Insurance under Complete Information: The First Best
If
is verifiable, the company can offer a contract of the form
and its problem is
, ,
The Lagrangian is
where
is a Lagrange multiplier and
The FOCs are
implying
Then, if
we have
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implying full insurance:
That is, the company will offer full insurance under complete
information. This is implied by the standard agency theory, in particular, by (12.5). The intuition is clear. Since the company is risk neutral and the consumer is risk averse, to avoid having
to compensate the consumer for taking risk, the company takes all the risk. There is no incentive problem here since the company can force the consumer to take action
∗∗
Then,
The IR condition must be binding, which implies
and
This equation determines the optimal
∗∗
under complete information.
Insurance under Incomplete Information: The Second Best
If effort
is not verifiable so that the company cannot impose it in accident prevention, it
can only offer incentives in a contract
contract, the consumer’s preferred effort
to induce a desirable
After accepting the
is determined by:
The consumer’s marginal utility in accident prevention is:
If
we will have
for any
implying that the consumer’s preferred effort is
That is, with full insurance, the consumer would have no incentive to make any effort to prevent accidents from occurring. Obviously, the company cannot offer full insurance in this case.
Hence, to provide incentives for accident prevention, the company ensures that the
contract must be the consumer’s preferred
in a
To do this, the company includes the FOC of
(12.20) in its optimization problem [the SOC of (12.20) is automatically satisfied]. Hence, the
company’s problem is
, ,
The IR condition must be binding. Thus the two constraints become
Let
Then
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Then the company’s problem becomes
This problem determines the optimal
∗
under incomplete information.
14. Linear Contracts under Risk Neutrality
Linear contracts are the simplest form of contracts and they are very popular in applications. They offer a simple incentive mechanism. Examples of linear contracts are many: contractual joint ventures, equity joint ventures, crop-sharing contracts, and fixed-price contracts,
etc. Kim and Wang (1998), and Wang and Zhu (2005) provide some of the best references.
A key theoretical question is: under what conditions is a linear contract optimal? Here, an
optimal linear contract is (weakly) better than any possible admissible contract, not just better
than any linear contract. When both parties in a contractual relationship are risk neutral, we
say there is double risk neutrality; similarly, when both parties in a contractual relationship
have moral hazard, we say there is double moral hazard. This section shows that, with moral
hazard and uncertainty, a linear contract can be optimal only under double risk neutrality, and
it can be the first best only under single moral hazard.
From (12.10), we know that the optimal contract is generally nonlinear if one of the parties is risk averse. For example, if the principal is risk neutral with utility function
all
for
then (12.10) implies
∗
For some popular output distributions,
follows the normal distribution
is linear in
then
For example, if output
If that’s the case,
the optimal contract is nonlinear for a risk-averse agent. Hence, to find optimal linear contracts, we assume double risk neutrality in this section and specify utility functions
and
for all
for the agent and principal, respectively.
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14.1. Single Moral Hazard
Let
be the agent’s effort and
be the density function of output
Let
be the
expected revenue
The First Best
With verifiable effort, the principal offers a contract of the form
Hence, the prin-
cipal’s problem is
∗∗
∈ , ∈
∗
If the IR condition is not binding in equilibrium, given an optimal contract
ciapal can offer contract
∗
instead for some
the prin-
to satisfy the IR condition. This
∗
contract raises the profit’s profit, which contradicts the fact that contract
is optimal.
Hence, the IR condition must be binding at the optimum. With a binding IR condition, the
problem becomes a problem of social welfare maximization:
∗∗
∈ , ∈
Since the principal’s objective function is independent of the contract
be solved in two steps. First, by maximizing
this problem can
the first-best effort
∗∗
is deter-
mined by
∗∗
∗∗
Second, given the optimal effort, an optimal contract will have to satisfy the IR condition only.
To find an optimal contract, consider a fixed contract
binding IR condition,
∗∗
∗∗
where
is a constant. By the
is such a contract. This is a first-best contract since it
supports the first-best effort.
The Second Best
With unverifiable effort, the principal will still offer a contract of the form
principal has to provide incentives for the agent to accept this
But the
For this purpose, an IC is
introduced. Hence, the principal’s problem is
∗
∈ , ∈
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If the IR condition is not binding in equilibrium, given an optimal contract
ciapal can offer contract
∗
instead for some
∗
the prin-
to satisfy the IR condition. This
contract also satisfies the IC condition. It raises the profit’s profit, which contradicts the fact
∗
that contract
is optimal. Hence, the IR condition must be binding at the optimum. With
a binding IR condition, the problem becomes a problem of social welfare maximization:
∗
∈ , ∈
Since the principal’s objective function is independent of
two steps. First, by maximizing
the second-best effort
∗
implying
∗
∗∗
this problem can be solved in
∗
is determined by
∗
Second, given the optimal effort, an optimal contract will have to satisfy the
IC and IR conditions only. We can find many contracts that can satisfy these conditions for a
given
∗
For a linear contract
∗
the IC and IR conditions become
∗
∗
∗
which imply
∗
∗
This
∗
∗
∗
is a second-best contract since it supports the second-best effort.
Proposition 12.1. With single moral hazard and double risk neutrality, the linear contract
∗
∗
is optimal, implying the first-best effort
∗∗
Interestingly, this solution can be implemented by a change of ownership: the principal
can simply sell the firm to the agent for payment
∗
When the agent is the owner, the incen-
tive problem disappears. This solution leads to an important idea: incentive problems may be
solved through an organizational approach. Coase (1960) contributes precisely by proposing
this idea. However, in this chapter, as we mentioned at the beginning, all contracts are treated
as complete contracts. With complete contracts, an ownership transfer is not allowed. Refer to
Wang (2012) on incomplete contracts. When we use incomplete contracts, ownership transfers are allowed.
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14.2. Double Moral Hazard
We have so far allowed the agent to invest only. In this subsection, we allow both parties
in a joint venture to invest. In addition, we will use a bargaining model setting instead of the
principal-agent setting.
Consider two agents,
respectively from
and
and
and
engaged in a joint project. Efforts (investments)
are private information. Let
be the private costs of effort. Let
the ex post revenue depending on the state
be the effort space. Let
be the joint effort, and
and joint effort
and
Let
be
follow a density function
ex ante. Thus, the expected revenue is
Assumption 12.3.
is strictly increasing in
Assumption 12.4.
and
Assumption 12.5.
and
are convex and strictly increasing.
is concave and strictly increasing in
Given a contract
that specifies payments
and
and
to agents 1 and 2,
respectively, the two agents play a Nash game to determine their efforts. In other words, given
agent 1 chooses his effort
by maximizing his own expected utility:
∈
which implies the FOC and SOC:
Similarly, given
agent 2 chooses his effort
by maximizing his own utility.
Assume that contracting negotiation leads to social welfare maximum. This assumption is
standard in the literature and is imposed on any negotiation outcome. Then the problem of
maximizing social welfare can be written as
∈ ,
,
∈
(12.22)
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where the last constraint is the resource constraint (RC). Since we allow a fixed transfer in the
contract, IR conditions are unnecessary.
The first-best problem is
,
∈
where the contract needs to satisfy RC only.
Proposition 12.2. Under Assumptions 1.7–1.9, with double moral hazard and double risk
neutrality, there exists a linear output-sharing rule
∗
where
∗
(
∗
(
∗
)
,
∗
)
that induces the second-best efforts
,
In addition,
Proof. Conditions
∗
∗
∗
determined by
∈
and the first-best outcome is not obtainable.
and
imply127
(12.23)
So the problem is equivalent to
∈ ,
,
∈
(12.24)
This problem can be solved in two steps. First, we find a solution
∗
∗
from the following
problem:
,
∈
(12.25)
This problem is not related to a contract. Second, given
that satisfies
and
sharing contract of the form
∗
∗
we look for a contract
There are many such contracts. Consider a simple
for
with
∗
∗
127 Also:
(
,
)
(
,
)
∗
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It is easy to verify that this contract satisfies
. For RC, by (12.23), we have
Thus,
Conditions
and
have now been verified. Since
Further, for contract
for both
and
we must have
we have
∗
Hence, by Assumption 12.4 and Assumption 12.5, condition
can also be verified similarly. Finally, since the first-best solution
∗∗
∗∗
by condition (12.23), the solution
∗
∗∗
∗
∗
is satisfied. Condition
∗∗
∗∗
satisfies
∗∗
of problem (12.25) cannot be the first best. The
proposition is thus proven.
The model in (12.22) is a model of two equal partners. An alternative setup is a principalagent model. If one of the agents, say
add an
is the principal and the other is the agent, we need to
condition of the form
for the agent into problem (12.22). The principal’s problem becomes
∈ ,
,
∈
(12.26)
The IR condition in (12.26) must be binding for the optimal solution. Hence, the problem
becomes
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∈ ,
,
∈
(12.27)
Comparing (12.27) with (12.22), problem (12.27) has an extra condition. In this case, Proposition 12.2 still holds except that the optimal contract is a linear contract with the form
∗
∗
where
∗
∗
is determined by the IR condition.
In summary, an optimal linear contract exists under double risk neutrality; it is the first
best under single moral hazard and it is the second best under double moral hazard.
Example 12.3. Consider the following parametric case:
where
is a random variable with
∗
and
The second-best solution is
∗
∗
The first-best solution is
∗∗
∗∗
There is no problem of risk sharing in a model under double risk neutrality since both
parties care about expected incomes only. Hence, under single moral hazard, when a single
mechanism (a contract) is sufficient to handle the incentive problem, the optimal solution
achieves the first best. However, under double moral hazard, when a single mechanism is not
sufficient to handle the two incentive problems, the optimal solution cannot achieve the first
best.
For the optimal linear solution in Proposition 12.2, the value of
∗
reflects the relative im-
portance of player in the project. Example 12.3 indeed shows this, where individual ’s marginal contribution to the project is represented by the parameter
and the output share
∗
indeed reflects his importance.
Bhattacharyya and Lafontaine (1995) are the first to provide such a result as a special case
of Proposition 12.2. Their result is for a special output process of the form
special distribution function
function
where
with a
is a random shock with distribution
Kim & Wang (1998) are the first to provide this general theory on optimal linear
contracts, with Wang & Zhu (2005) providing the proof for it.
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The optimality of linear contracts in this section is based on risk neutrality. Without double risk neutrality, linear contracts are generally not optimal. However, linear contracts are
very popular in reality and most parties involved are likely to be risk averse. This is puzzling
within the framework of complete contracts. Our theory of incomplete contracts in Chapter 2
will provide a resolution to this puzzle. Linear contracts must be optimal incomplete contracts.
Contract theory is based on either a principal-agent model or a bargaining model. The
bargaining model is more general than the principal-agent model. In a bargaining model, both
parties have bargaining powers and they bargain to reach a contractual agreement; in a principal-agent model, the principal has 100% bargaining power ex ante and she offers a take-itor-leave-it contract, which often causes the agent to have zero surplus (with a binding IR
condition). However, the principal-agent model is generally equivalent to the bargaining
model: they lead to the same solution if the IR condition is binding. For example, we have
shown the equivalence of problems (12.22) and (12.26).
In the principal-agent model, the objective of the problem is to maximize the principal’s
profits; in the bargaining model, the objective is to maximize social welfare. See, for example,
the theory of the Nash bargaining solution (Myerson, 1991, Chapter 8; Osborne & Rubinstein,
1994, Chapters 7 and 15), in which the players maximize social welfare. In the bargaining
model, the principal’s profit maximizing behavior is reflected in the principal’s IC condition.
For example, in the bargaining model (12.22), the objective is to maximize social welfare; in
the principal-agent model (12.26), the objective is to maximize the principal’s expected profit.
A contractual solution implicitly allows a upfront transfer (in money, payoff or benefit).
For example, if
∗
∗
is an optimal contract, the actual contract can be
, where
is a upfront transfer. This transfer will depend on the bargaining powers of the two parties.
The bargaining model may or may not explicitly take this
into account, but the principal-
agent model does not. In the bargaining model, we may or may not have explicit IR conditions
for both the principal and the agent; in the principal-agent model, we will have an explicit IR
condition for the agent. In a principal-agent model, since the agent has no bargaining power,
an IR is needed to protect the agant’s interest; in a bargaining model, since both parties have
bargaining powers, no IR condition is necessary. In sum, in the principal-agent model, a upfront transfer and the principal’s IR condition are implicit; but in the bargaining model, these
two items may or may not be explicit. In model (12.22), we do not explicitly mention IR condi∗
tions. This means that the actual income scheme should be
defined in Proposition 12.2 and
∗
∗
∗
and
∗
∗
∗
where
∗
is
satisfy the IR conditions of the two parties and
There are an infinite number of such pairs
∗
∗
; which pair will be in the
actual solution depends on the bargaining powers of the two parties.
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15. A Conditional Fixed Contract
We now modify the standard agency model from yet another angle in an attempt to find a
simple optimal contract. We have so far found a simple optimal contract, a linear contract, in
a model under uncertainty only when there is no risk aversion. However, risk aversion is an
important parameter in many applications. In this section, we find two conditions under
which the agency model with risk aversion has a simple optimal contract known as a conditional fixed contract. This contract is very convenient for applied agency problems.
Real-world contracts between principals and agents are typically very simple and often
have a bonus structure that specifies wage increases for certain minimum levels of performance. How can contracts be so simple in reality? We provide an answer by establishing the
optimality of a conditional fixed contract for the standard agency model under risk aversion.
15.1. The Model
We now abandon Assumption 12.1, and instead assume that the support of output is
where the boundaries of the support
pendent on effort
We allow the special cases with
and those special cases with
and/or
and
are generally de-
and/or
being independent of
for any
We will also abandon
the FOA.
the agent’s expected utility is
Given a utility function
( )
( )
Hence, the principal’s problem is
( )
∈ , ∈
( )
( )
(12.28)
( )
Here, we assume that
If
we can replace
by
and the model
remains the same.
This model is the same as the standard agency model except that we allow the boundaries
of the domain to be dependent on effort. It turns out that this dependence is important for our
alternative solution to the standard agency problem.
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Assumption 12.6. Utility function
is concave, onto and strictly increasing.
( )
( )
Assumption 12.7. The expected revenue
Assumption 12.8 (FOSD).
for any
is increasing and concave
and
These three assumptions are natural requirements. In particular, , being onto and strictly increasing, ensures that
is well defined, and Assumption 12.8 is required for any
sensible contract theory.
15.2. The Optimal Contract
The optimal conditional fixed contract is stated in the following proposition.
Proposition 12.3. Let
∗
be the solution of the following equation:
∗
∗
(12.29)
and suppose that the following two conditions are satisfied:
∗
∗
∗
∗
(12.30)
∗
Then under Assumptions 12.6–12.8, the optimal effort is
(12.31)
∗
and the optimal contract is
∗
∗
∗
Furthermore,
∗
∗
∗
is the first-best solution.
This optimal contract is called a conditional fixed contract since it makes a fixed payment
conditional on the performance to be above a cutoff point.
As shown in Figure 12.3, this solution looks distinctly different from the solution of the
standard agency problem. It, in fact, looks puzzling at first glance because of its simplicity.
However, the intuition for the solution turns out to be quite simple. To induce the optimal
effort
∗
the FOA suggests that the principal should associate each output level with a pay-
ment. This alternative approach suggests that the principal can induce
∗
by simply offering a
bonus at a minimum level of performance. For the latter strategy to work, the dependence of
the distribution function on effort at the left boundary of its domain is crucial. Condition
(12.31) is precisely for establishing this dependence which is indeed crucial for the solution.
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s* ( x)
s
A( a * )
x
B(a* )
Figure 12.3. The Optimal Conditional Fixed Contract
We can immediately see some interesting features of the solution. First, the solution
achieves the first best. This is impressive considering the simplicity of the contract.
Second, it is a closed-form solution with a clear promotion component. In fact, the pay at
the minimum level of performance is a jump, i.e., it could be in the form of a promotion, bonus, or a change of nature of the contract. This solution looks very much like a typical employment contract, in which a fixed wage rate is given based on a certain level of education and
working experience plus a potential bonus or promotion based on a minimum level of performance.
Third, the solution has an interesting form with simple and intuitive expressions deter∗
mining the optimal contract and effort. Based on the fact that
∗
(12.30) means that, for a percentage increase in
∗
∗
for
the percentage increase in pay
∗
is
greater than the percentage increase in cost. Condition (12.31) means that, for an increase in
∗
the increase in the probability of an output being larger than
∗
is greater than the
percentage increase in cost.
We make some technical remarks about the conditions in Proposition 12.3. First, since
both
in
and
are convex,
Also, since
is decreasing in
is convex in
∗
implying that
[ ( )]
is increasing
from (12.29) must be unique.
Second, condition (12.30) means that the bonus elasticity of effort is larger than the cost
elasticity of effort. Since
is convex,
is more convex than
that condition (12.30) can be easily satisfied for any
where
is. This indicates
For example, for
is the relative risk aversion and
and
(12.30) is satisfied for any
We consider two examples: one uses the exponential distribution and the other uses the
uniform distribution.
Example 12.4. Suppose
( )
where
is an arbitrary increasing function,
is the relative risk aversion, and
These functions include those in Example 12.1 as a special case. The key difference
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between the two examples is that we allow the boundary
to be dependent on effort. This
dependence is crucial for our solution. We have
Equation (12.29) becomes
∗
∗
Condition (12.30) is satisfied for any
(12.32)
Condition (12.31) becomes
∗
for some
Let
(12.33)
Then (12.33) is satisfied for any
and for
and (12.32)
implies
∗
Example 12.5. Suppose
where
and
is an arbitrary increasing function,
is the relative risk aversion,
We have
Equation (12.29) becomes
∗
Condition (12.30) is satisfied for any
∗
Condition (12.31) becomes
∗
which is satisfied when
(12.34)
∗
is sufficiently small. Again, if we let
for some
then
(12.34) implies
∗
16. A Suboptimal Linear Contract
In many published papers, researchers often use a limited set of admissible contracts for
the standard agency model and find an optimal contract from this admissible set. In particular,
researchers often limit admissible contracts to linear contracts and find an optimal linear
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contract from this set. This optimal contract is obviously inferior to the optimal contract in the
standard agency model and hence we call it a suboptimal linear contract or a third-best linear
contract.
Specifically, for the standard agency model in Section 1, let the set of admissible contracts
be
The principal’s problem can be written as
, , ∈ℝ
(12.35)
The Lagrangian for (12.35) is
where
and
are Lagrange multipliers and
Under Assumption 12.1, the FOCs for prob-
lem (12.35) are
From these three equations, we can try to solve for
the suboptimal linear contract is
∗
∗
∗
∗
∗
This solution is suboptimal and
∗
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Appendix A:
General Optimization
In this and next appendices, we will simply list some useful mathematical results without
proofs and without much explanation.
1. Gateaux Differentiation
A popular definition of differentiation for a multi-variable vector-valued function
is the so-called Gateaux differentiation. Given
It turns out that
any
is linear in
We call this
or
For function
(
)
and
×
That is, there is a matrix
the Gateaux derivative of
w.r.t.
at
define
such that
for
denoted by
That is,
×
we can again define its Gateaux derivative as above, called it the
second-order Gateaux derivative, denoted as
if the partial derivatives of
or
We find that, for
exist, under mild conditions, we have
×
This matrix is called the Jacobian matrix in economics.
For
have
if the second-order partial derivatives of
exist, under mild conditions, we
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×
In this case,
is called the gradient, and
is called the Hessian matrix in economics.
Theorem A.1. (Composite Mapping Theorem). Given two open sets
two mappings
and
differentiable at
then the mapping
where
is also differentiable at
and
and
and
respectively,
and
is a matrix product.
Proposition A.1.
( ⋅ )
• For any vector
• For any matrix
(
• For any matrix
(
)
)
2. Positive Definite Matrix
Convexity is directly related to minimization, and concavity is directly related to maximization. The concavity and convexity of functions are directly related to the positive and negative definiteness of matrices.
We deal with symmetric matrices only when we talk about definiteness. The notation
×
or
×
indicates that
is an
matrix. Our matrices contain real en-
tries only (with real coefficients). For a symmetric matrix
×
This definition has to do with the Taylor expansion for a function
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where
is some point between
At optimum,
and
is the gradient, and
Hence, the sign of
determines whether or not
is the Hessian matrix.
or the definiteness of
is optimal.
Given a matrix
for
×
with
define a
minor as
,⋯,
In particular, define the principal minors as
{ }
{ , ,…, }
That is,
×
Theorem A.2. For a symmetric matrix
(a)
{ , }
for all
(b)
for all
(c)
{ ,⋯,
(d)
}
for all permutations
{ ,⋯,
}
for all permutations
and all
and all
Proposition A.2.
• If
then
•
•
• If
• If
is a tall full-rank matrix, then
and
and
then
•
iff all its eigenvalues are positive.
•
iff all its eigenvalues are strictly positive.
3. Concavity
Given vectors
a convex combination of the vectors is
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Given any two points
the set of all convex combinations of
and
is the interval
connecting the two points:
We can similarly define
and
interval of any two points in
A set
is called a convex set if the
is contained in
Proposition A.3. (Properties of Convex Sets).
1. Any intersection of convex sets is also convex.
2. The Cartesian product of convex sets is also convex.
A function
is concave if its domain
If the inequality holds strictly, we say that
if
is convex and
is strictly concave. A function
is (strictly) convex
is (strictly) concave. See Figure 4.
Figure 4. A Concave Curve
Theorem A.3. (Properties of Concave Functions). Let the domain
1. A function
is concave iff
be a convex set.
is convex.
2. Concave functions are continuous in the interior of their domain.
3. A function
is concave iff
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for all
and all convex combinations
twice differentiable
Theorem A.4. Given convex
1.
is convex
is strictly convex.
2.
3.
is concave
4.
is strictly concave.
twice differentiable
Corollary A.1. Given convex
and let
{ ,⋯,
}
be a
principal minors of
1.
is convex
2.
is concave
3.
{ ,⋯,
for any
}
{ ,⋯,
for any
}
for any
is strictly convex.
for any
4.
Example A.1. For function
is strictly concave.
defined by
we have
Example A.2. For Cobb-Douglas function
defined by
we have
Proposition A.4. Let
(1)
is concave
(2)
is strictly concave
be differentiable. Then,
for all
for all
This proposition is illustrated in Figure 5.
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f
.
x
y
Figure 5. Characterization of Concave Functions
4. Quasi-Concavity
Given a convex set
we say that
is quasi-concave if
is strictly quasi-concave if
See Figure 6.
is (strictly) quasi-convex if
is (strictly) quasi-concave.
x2
x2
x1
x1
Figure 6. Strict and Nonstrict Quasi-Concave Curves
Theorem A.5. Given convex set
•
is quasi-concave iff the upper contour set
•
is quasi-convex iff the lower contour set
The border of a contour set
when
is convex for all
is convex for all
is called a level set. In economics,
is a utility function, a level set is called an indifference curve; when
function, it is called an isoquant; and when
is a production
is a cost function, it is called an isocost curve.
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Theorem A.6.
(a) Concave functions are quasi-concave; convex functions are quasi-convex.
(b) Strictly concave functions are strictly quasi-concave. Strictly convex functions are strictly
quasi-convex.
(c) If
monotonic functions are both quasi-concave and quasi-convex.
Quasi-concavity however does not necessarily imply concavity; see Figure 7 for a quasiconcave curve.
f
x
Figure 7. A Quasi-Concave Curve
In summary,
Given function
denote
and the bordered Hessian matrix of
as
It is actually the Hessian matrix of the Lagrangian function at an optimal point of
Theorem A.7. Given convex set
cipal minors
and the prin-
of
1. For
is quasi-convex
2. For
is quasi-concave
3. For
twice differentiable function
or
is strictly quasi-convex.
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4. For
or
is strictly quasi-concave.
defined by
Example A.3. Consider function
we have
Example A.4. Consider Cobb-Douglas function
defined by
we have
×
Theorem A.8. Let
×
be symmetric,
be the principal minors of
have full rank,
and
Then,
1.
for
satisfying
2.
for
satisfying
5. Unconstrained Optimization
For a function
ue
if
we call
or
the stationary value. Given a stationary point
a stationary point, and its valif
is a scalar, there are three possible
situations:
1. If
changes its sign from positive to negative when
passes
then
is called a local
passes
then
is called a local
maximum point (or relative maxima).
2. If
changes its sign from negative to positive when
minimum point (or relative minima).
3. If
does not change its sign when
passes
then
or
is called a reflection
point.
Theorem A.9. (Weierstrass). Let
be continuous and
be compact. Then, the
following unconstrained optimization problem has at least one solution:
∈
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It is a unconstrained problem since it has no explicit equality or inequality constraints defined by some functions (functional constraints).
Theorem A.10. Let
∗
(a) If
is an interior solution of
(A.1)
∈
then (FOC)
∗
(b) If
∗
∗
∗
and (SONC)
∗
and (SOSC)
is the maximum point of
then there is a neighborhood
∗
on
of
∗
such that
(Local maximum).
∗
then any point
∗
satisfying
∗
(c) If
is concave on
is a maximum point.
(d) If
is strictly quasi-concave, a unique local maximum over a convex set
is the unique
global maximum.
6. Constrained Optimization
When an optimization problem has explicit functional constraints, we have constrained
optimization.
Theorem A.11. (Lagrange). For
and
consider problem
(A.2)
Define the Lagrangian function
• If
∗
solves and if
∗
has full rank, then there exists
(Lagrangian multiplier) such
that
∗
and
∗
∗
• If the FOC is satisfied,
∗
and
∗
then
∗
∗
is a unique local maximum.
Explain FOC: By observing Figure 8, since the curve
are tangent at
there exists a
such that
∗
their gradients at
∗
∗
∗
and the indifference curve
must be linearly dependent. In other words,
which is the FOC.
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x2
.
′(
∗
′(
∗
)
)
=
=0
∗
∗
x1
=0
Figure 8. The FOC for the Lagrangian Function
Explain SOC: The tangent line is
∗
∗
∗
∗
By inspecting Figure 8, we know that
(A.2) is equivalent to
or
∗
∈
where
∗
∗
Let
We know that the optimal solution is
vector space, the SOC is that
is concave at
∗
∗
for
Since
is a
which is precisely the SOC in Theorem
A.11.. By this explanation, we know that the strict version of this SOC should be locally sufficient. Therefore, the SOC is the concavity condition of
defined by
on subspace
and the subspace
is
∗
How do we verify the SOC? Let
Let
be the principal minors of
∗
By Theorem A.7., the SOC is guaranteed if
for
Theorem A.12. (Kuhn-Tucker). For differentiable
and
consider prob-
lem
Let
If
∗
is a solution, then there exists
such that
∗
(Kuhn-Tucker Condition) and
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∗
The Lagrange and Kuhn-Tucker theorems are very useful in economics. The following result is useful when global maximization is required.
Theorem A.13. (Global Maximum). Let
Let
∗
and
be quasi-concave, where
satisfy the Kuhn-Tucker condition and the FOC for (A.3). Then,
∗
is a
global maximum point if
∗
(1)
(2)
and
is locally twice continuously differentiable, or
is concave.
7. The Envelope Theorem
We often deal with maximum and minimum value functions in economics. The following
result on differentiation of these types of functions is very useful.
is differentiable,
Theorem A.14. (Envelope). Suppose
∗
and
is an interior solution of
∈
Then,
∗(
)
The key advantage of the envelope theorem is that you can find the derivative of function
without actually solving for
first. Given the knowledge of
∗
we can use function
to find
Notice that the optimization problem in the envelope theorem is a unconstrained problem.
This means that a constrained optimization problem should first be translated into a unconstrained optimization problem using a Lagrangian function before we can apply the envelope
theorem.
8. Homogeneous Functions
Besides concave and convex functions, homogeneous functions are of special interest to
economists. A function
is homogeneous of degree
if
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In particular, when
say that
we also say that
is linearly homogeneous; and when
we also
is zero homogeneous.
Theorem A.15. (Euler Law). If
Theorem A.16. If
is linearly homogeneous, then
is homogeneous of degree
then
is homogeneous of degree
Notes
Most results in this chapter can be found in Chiang (1984), Takayama (1993), Varian
(1992), and Mas-Colell et al. (1995).
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Appendix B:
Dynamic Optimization
This appendix presents theories on two types of dynamic models: discrete-time stochastic
models and continuous-time deterministic models.
1. Discrete-Time Stochastic Model
In this section, time is discrete, meaning that it takes values of
time may be
The current
and you need to make a plan now for future actions. Many things in the
future are unknown at present. These unknown things can usually be represented by random
variables and you are supposed to know the conditional distribution functions of these random variables. Hence, your plan will be dependent on these distribution functions as well as
the realized values of the random variables when the time comes.
1.1.
The Markov Process
What is a Markov process? Suppose that you are interested in a variable or a vector
value depends on time. We denote its value at time as
not know
you know
Suppose that you know the distribution function of
knowledge of
where
At time
Its
but you do
conditional on your
i.e.,
is the distribution function of
at time
you have the knowledge of the distribution function of
In general, at
you know
and
conditional on your knowledge of
that is,
In this case, we call
x0
x1
xt
0
1
t
t
a random process. In general, the distribution function
dependent. If the distribution function is time-independent, i.e.,
is time-
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we call
a stationary process. In addition, if the distribution function’s dependence on
past history has a fixed length of time, i.e., there exists an integer
then we call
such that128
an nth-order Markov process. For example, a first-order Markov process
is defined by
1.2. The Bellman Equation
Let
and
,
where
Consider the problem:
,⋯
(B.1)
is a random vector not known until period
expectation operator conditional on period information
is injected by the random process
is the mathematical
In this problem, the uncertainty
from the motion equation
Let
,
,⋯
(B.2)
We have
,
,⋯
,
,⋯
Thus, the solution of (B.1) must be the solution of the following so-called Bellman equation:
Conversely, the solution of the Bellman equation (B.3) is generally the solution of (B.1) if the
following so-called transversality condition is satisfied:
128 Here,
we implicitly assume
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∗
(B.4)
→
The typical way of solving problem (B.3) is to use the FOC and the envelope theorem:
,
,
,
If
(B.6)
,
∗
is known, (B.5) gives us the policy function:
To determine
we need (B.6).
These two equations together determine the solution.
Theorem B.1. If
and
Markov process, then
are time invariant,
and
and
is a first-order
defined in (B.2) is also time invariant.
from (B.5) and (B.6) will be time invariant:
By Theorem B.1, a solution
∗
and
1.3. The Lagrange Method
We can alternatively use the Lagrange theorem in Appendix A to solve problem (B.1). The
Lagrangian function for (B.1) is
The problem becomes
,
,
,⋯
,⋯
The FOCs are
,
,
,
,
For
,
,
,
(B.8)
,
By the Envelope theorem,
,
By (B.8) and (B.9), we have
,
Substituting this into (B.7) and (B.8) yields
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,
,
,
,
These two equations are the same as (B.5) and (B.6).
2. The Continuous-Time Deterministic Model
2.1. The General Model
Theorem B.2. Suppose that
is continuous w.r.t. its first argument,
continuously differentiable w.r.t. its second and third arguments. Let the set of admissible
controls be:
∗
Then, the solution
of
(B.10)
∈
must satisfy the Euler equation:
∗
Here,
∗
∗
∗
(B.11)
can be either finite or infinite, and the terminal value
If the terminal value
is free, the transversality condition is
∗
∗
(B.12)
is free, the transversality condition is
If the initial value
∗
If the terminal condition is
∗
Conversely, if
can be either fixed or free.
∗
(B.13)
the transversality conditions are
∗
∗
∗
is concave in
then any
∗
∗
(B.14)
satisfying the Euler equation (B.11)
and the initial and terminal conditions is a solution of (B.10).
in
The concavity condition of
is too strong and it often fails. Without it, a
second-order condition is needed. A necessary condition is
∗
∗
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This is the so-called Legendre condition. However, this condition is far from being sufficient.
In fact, a strict inequality of this condition is not sufficient even for a locally optimal path.
Since the Euler equation is a second-order differential equation for
two boundary con-
ditions are needed to pin down the two arbitrary constants in the general solution. For problem (B.10), the two boundary conditions for the Euler equation are
and
when one of the boundary condition is missing, a transversality condition, such as those in
(B.12)–(B.14), is needed to replace it.
2.2. Special Models
Problem (B.10) is a unconstrained problem. Economics problems often have constraints,
such as consumer budget constraints, government budget constraints, and resource constraints, etc. We now present a few cases of constrained problems, which are special cases of
Theorem B..
Theorem B.3. (Special Model I). Let
For problem
define the Hamiltonian as
Under minor differentiability conditions, if
such that
∗
∗
is a solution, then there exists a function
is a solution of
(B 15)
(B 16)
with transversality conditions:
→
When the terminal value
is free, the transversality condition is
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Theorem B.4. (Special Model II). Let
For problem
(
)
define the Hamiltonian as
Under minor differentiability conditions, if
such that
∗
∗
is a solution, then there exists a function
is a solution of
with transversality condition
→
Notes
A good reference for Section 1 is Stokey–Lucas (1989, p.239–259); see also Sargent (1987).
A good reference for Section 2 is Kamien–Schwartz (1991); see also Chiang (1992).
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[66] Stokey, N.L. and R.E. Lucas (1989): Recursive Methods in Economic Dynamics, Harvard
University Press.
[67] Takayama, T. (1993): Analytical Methods in Economics, University of Michigan Press.
[68] Varian, H.R. (1992): Microeconomic Analysis, 3rd edition, Norton.
[69] Wang, Susheng (2002): “Limit Contract in the Standard Agency Model,” Working Paper,
HKUST.
[70] Wang, Susheng (2003): “The Optimality of One-Step Contracts,” Working Paper, HKUST.
[71] Wang, Susheng (2004): “Explicit Solutions for Two-State Agency Problems,” Working
Paper, HKUST.
[72] Wang, S. (2008): Math in Economics, People’s University Publisher, Beijing, China.
[73] Wang, S. (2012a): Microeconomic Theory, 2nd ed., McGraw-Hill.
[74] Wang, S. (2012b): Organization Theory and Its Applications, Routledge Publisher, UK.
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[75] Wang, S.; Zhu, T. (2005): “Control Allocation, Revenue Sharing, and Joint Ownership,”
International Economic Review, 46 (3), 895–915.
[76] Wang, S. and L. Zhu (2001): “Variable Capacity Utilization and Countercyclical Pricing
under Demand Shocks,” Working Paper, HKUST.
[77] Wilson, C. (1977): “A Model of Insurance Markets with Incomplete Information,” Journal of Economic Theory, 16, 167–207.
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Index
absolute risk aversion .............................. 81
consumer surplus .................................... 156
admissible contracts ............................... 368
Consumption-Based Capital Asset Pricing
adverse selection .................................... 294
allocation .................................................. 95
allocation scheme ................................... 339
allocative efficiency ................................. 157
alternating-offer bargaining solution .... 280
Arrow-Debreu world ................................ 95
backward induction ................................ 213
Bayesian equilibrium (BE) ..................... 223
Bayesian Nash equilibrium (BNE) 254, 340
beliefs ...................................................... 219
Bertrand equilibrium ............................. 162
beta ..........................................................133
Black-Scholes formula ........................... 142
Capital Asset Pricing Model (CAPM) .....133
CDBE ...................................................... 246
certainty equivalent .................................. 72
cheap-talk game ..................................... 265
coalition .................................................. 282
Cobb-Douglas function ............................ 32
compensated demand function ............... 50
competitive firm ....................................... 25
competitive industry .............................. 144
complete market ................................ 95, 125
conditional demand function ................... 26
constrained Pareto optimal .....................331
Model ................................................... 136
contingent contracts economy ................ 122
contingent market equilibrium (CME) .. 125
continuation game ................................. 250
cooperative game .................................... 167
core ......................................................... 282
Cournot equilibrium ............................... 162
deadweight loss ....................................... 158
derivative security ................................... 141
desirable goods ......................................... 99
diffusion process ..................................... 137
direct mechanism .................................... 341
dominant strategy .......................... 169, 202
dominant strategy equilibrium (DSE) .... 169
dominant-strategy equilibrium (DSE) .. 203
double moral hazard .............................. 379
double risk neutrality ............................. 379
duality ........................................................ 51
duopoly .................................................... 162
economic cost ........................................... 24
economic rate of substitution .................. 27
economic rent ......................................... 180
economically efficient ............................... 14
EDBE ...................................................... 247
Edgeworth box ......................................... 95
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elasticity of scale ........................................19
incentive compatible (IC) ....................... 341
elasticity of substitution ........................... 23
incentive problem .................................. 368
equilibrium actions ................................ 222
income effect .............................................57
equilibrium path ..................................... 222
income elasticity ....................................... 86
Euler equation ........................................ 372
incomplete information games .............. 253
Euler law ................................................. 403
independent securities ............................ 125
European call option .............................. 142
indirect utility function ............................ 48
ex-ante efficient ...................................... 353
individual rationality (IR) condition ..... 369
expected utility ......................................... 74
integrability ............................................... 61
expected utility property .......................... 74
interim efficient ...................................... 353
expenditure function ................................ 50
isoquant ..................................................... 17
ex-post efficient .............................. 339, 353
Leontief production function ................... 32
extensive form ........................................ 194
linear contracts ....................................... 379
feasible allocation ..................................... 95
linearly homogeneous ............................ 403
first best .................................................. 369
local returns to scale ................................ 20
first-best risk sharing ............................. 370
lottery ....................................................... 73
first-order approach (FOA) ..................... 371
marginal cost product ............................. 179
first-order condition (FOC) ......................16
marginal rate of substitution ................. 370
first-order stochastic dominance (FOSD)90,
marginal rate of substitution (MRS) ....... 49
372
First-Price Sealed-Bid Auction .............. 348
marginal rate of transformation (MRT) .. 15,
18
full insurance ............................................ 84
marginal revenue product ...................... 179
general equilibrium .................................. 94
Marshallian demand function ................. 48
generalized backward induction ............. 215
mean-preserving spread .......................... 92
global returns to scale ...............................19
mixed strategy ........................................ 198
Hicksian demand function ....................... 50
monopolistically competitive industry ... 159
ideal solution ........................................... 271
monopoly ................................................. 151
imperfect information game ................... 195
monopsony ............................................. 184
implementable ........................................ 341
monotone likelihood ratio property (MLRP)
incentive compatibility (IC) condition .. 370
............................................................ 372
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Nash bargaining solution ....................... 275
relative risk aversion ................................ 82
Nash equilibrium (NE)........................... 199
relative risk premium ............................... 82
Nash strategy .................................. 162, 205
repeated game ......................................... 170
net output .................................................. 14
representative agent ............................... 130
no short sales .......................................... 134
Revelation Principle ................................ 341
noncooperative game .............................. 167
Revenue Equivalence Theorem .............. 351
normal form............................................ 196
risk averse ................................................. 80
off-equilibrium path ............................... 222
risk loving ................................................. 80
offer curve ................................................. 97
risk neutral ............................................... 80
oligopoly ................................................. 162
risk premium ............................................ 82
opportunity cost ....................................... 24
second best .............................................. 371
ordinary demand function ....................... 48
second-best problem ............................... 371
Pareto optimal (PO) ............................... 102
second-order condition (SOC) ................ 371
partial equilibrium ................................... 94
second-order stochastic dominance (SOSD)
perfect market .......................................... 95
perfect-information game ....................... 195
pooling equilibrium ........................ 263, 301
price-discriminating monopoly .............. 151
prisoners’ dilemma ................................. 169
producer surplus ..................................... 157
production frontier (PPF) ................... 14, 15
production function .................................. 17
production possibility set .................... 14, 15
pure exchange economy ........................... 95
pure-strategy NE .................................... 199
reaction function .................................... 164
reactive equilibrium ............................... 304
real subgame........................................... 214
Reduction of Compound Lottery (RCLA) 73
.............................................................. 90
Second-Price Sealed-Bid Auction .......... 349
security market equilibrium (SME) ....... 126
separating equilibrium ........................... 301
sequential equilibrium (SE) ................... 237
sequentially rational (SR) ...................... 220
Shapley value .......................................... 287
single-price monopoly ............................ 151
social welfare function ............................ 110
socially optimal ....................................... 110
stable equilibrium ................................... 163
Stackelberg equilibrium .......................... 162
standard agency model .......................... 368
standard Brownian motion ..................... 137
state price ................................................ 132
strategy ........................................... 196, 253
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subgame perfect Nash equilibrium (SPNE)
............................................................. 215
suboptimal linear contract ..................... 391
substitution effect ..................................... 57
substitution matrix ............................. 39, 56
support of the distribution function ...... 369
technologically efficient ............................ 14
third-best linear contract ....................... 391
transfer earnings .................................... 180
trembling-hand perfect (THP) .............. 205
trigger strategy ........................................ 170
truthfully implementable ........................ 341
Two-State Agency Model ....................... 375
utility function .......................................... 45
variance averse ........................................ 132
zero homogeneous ................................. 403
totally mixed strategy ............................. 205
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