MICROECONOMICS
Solutions Manual
Frank A. Cowell
STICERD and Department of Economics
London School of Economics
Revised March 2007
ii
Contents
Contents
iii
1 Introduction
1
2 The Firm
3
3 The Firm and the Market
31
4 The Consumer
41
5 The Consumer and the Market
63
6 A Simple Economy
81
7 General Equilibrium
93
8 Uncertainty and Risk
115
9 Welfare
137
10 Strategic Behaviour
153
11 Information
195
12 Design
231
13 Government and the Individual
265
iii
CONTENTS
CONTENTS
iv
Chapter 1
Introduction
This manual contains outline answers to all 132 of the end-of-chapter exercises in
Microeconomics: Principles and Analysis by Frank Cowell (Oxford University
Press 2006). For convenience the (slightly edited) versions of the questions are
reproduced here as well. Almost every question begins on a new page so that
instructors can print o¤ individual problems and outline answers for classroom
use.
Some of the exercises are based on key contributions to the literature. For
the bibliographic references consult the original question in the printed text .
1
Microeconomics
c Frank Cowell 2006
CHAPTER 1. INTRODUCTION
2
Chapter 2
The Firm
Exercise 2.1 Suppose that a unit of output q can be produced by any of the
following combinations of inputs
0:2
0:5
z1 =
0:3
0:2
; z2 =
; z3 =
0:5
0:1
1. Construct the isoquant for q = 1.
2. Assuming constant returns to scale, construct the isoquant for q = 2.
3. If the technique z4 = [0:25; 0:5] were also available would it be included in
the isoquant for q = 1?
Outline Answer
z2
0.5
z1
• • z4
z2
0.2
•
0.1
0
0.2 0.3
z3
•
q=1
z1
0.5
Figure 2.1: Isoquant –simple case
3
Microeconomics
CHAPTER 2. THE FIRM
z2
z1
• • z4
0.5
z2
•
0.2
z3
•
0.1
0
0.2 0.3
q=1
z1
0.5
Figure 2.2: Isoquant –alternative case
z2
0.5
z1
•
q=2
z2
0.2
•
0
z3
•
0.1
0.2 0.3
q=1
z1
0.5
Figure 2.3: Isoquants under CRTS
c Frank Cowell 2006
4
Microeconomics
1. See Figure 2.1 for the simplest case. However, if other basic techniques
are also available then an isoquant such as that in Figure 2.2 is consistent
with the data in the question.
2. See Figure 2.3. Draw the rays through the origin that pass through each
of the corners of the isoquant for q = 1. Each corner of the isoquant for
q = 2. lies twice as far out along the ray as the corner for the case q = 1.
3. Clearly z4 should not be included in the isoquant since z4 requires strictly
more of either input to produce one unit of output than does z2 so that it
cannot be e¢ cient. This is true whatever the exact shape of the isoquant
in –see Figures 2.1 and 2.2
c Frank Cowell 2006
5
Microeconomics
CHAPTER 2. THE FIRM
Exercise 2.2 A …rm uses two inputs in the production of a single good. The
input requirements per unit of output for a number of alternative techniques are
given by the following table:
Process
Input 1
Input 2
1
9
4
2
15
2
3
7
6
4
1
10
5
3
9
6
4
7
The …rm has exactly 140 units of input 1 and 410 units of input 2 at its disposal.
1. Discuss the concepts of technological and economic e¢ ciency with reference to this example.
2. Describe the optimal production plan for the …rm.
3. Would the …rm prefer 10 extra units of input 1 or 20 extra units of input
2?
Outline Answer
1. As illustrated in …gure 2.4 only processes 1,2,4 and 6 are technically e¢ cient.
2. Given the resource constraint (see shaded area), the economically e¢ cient
input combination is a mixture of processes 4 and 6.
z2
4
•5
Economically Efficient
Point
Attainable
Set
•3
6
.
1
2
z1
0
Figure 2.4: Economically e¢ cient point
3. Note that in the neighbourhood of this e¢ cient point MRTS=1. So, as
illustrated in the enlarged diagram in Figure 2.5, 20 extra units of input
2 clearly enable more output to be produced than 10 extra units of input
1.
c Frank Cowell 2006
6
Microeconomics
20
Original
Isoquant
10
Figure 2.5: E¤ect of increase in input
c Frank Cowell 2006
7
Microeconomics
CHAPTER 2. THE FIRM
Exercise 2.3 Consider the following structure of the cost function: C(w; 0) =
0, Cq (w; q) = int(q) where int(x) is the smallest integer greater than or equal to
x. Sketch total, average and marginal cost curves.
Outline Answer
From the question the cost function is given by
C(w; q) = kq
k + 1; k
1<q
k; k = 1; 2; 3:::
so that average cost is
k+
1
k
q
;k
1<q
k; k = 1; 2; 3:::
–see Figure 2.6.
C(w,q)
Cq(w,q)
C(w,q)/q
0
q
1
2
3
Figure 2.6: Stepwise marginal cost
c Frank Cowell 2006
8
Microeconomics
Exercise 2.4 Suppose a …rm’s production function has the Cobb-Douglas form
q = z1 1 z2 2
where z1 and z2 are inputs, q is output and
1,
2
are positive parameters.
1. Draw the isoquants. Do they touch the axes?
2. What is the elasticity of substitution in this case?
3. Using the Lagrangean method …nd the cost-minimising values of the inputs
and the cost function.
4. Under what circumstances will the production function exhibit (a) decreasing (b) constant (c) increasing returns to scale? Explain this using …rst
the production function and then the cost function.
5. Find the conditional demand curve for input 1.
z2
z1
Figure 2.7: Isoquants: Cobb-Douglas
Outline Answer
1. The isoquants are illustrated in Figure 2.7. They do not touch the axes.
2. The elasticity of substitution is de…ned as
ij
@ log (zj =zi )
@ log j (z)= i (z)
:=
which, in the two input case, becomes
@ log
=
@ log
c Frank Cowell 2006
9
z1
z2
1 (z)
2 (z)
(2.1)
Microeconomics
CHAPTER 2. THE FIRM
In case 1 we have (z) = z1 1 z2 2 and so, by di¤erentiation, we …nd:
1 (z)
2 (z)
1
=
=
2
z1
z2
Taking logarithms we have
log
z1
z2
1
= log
1 (z)
log
2 (z)
2
or
1
u = log
v
2
where u := log (z1 =z2 ) and v := log ( 1 = 2 ). Di¤erentiating u with respect
to v we have
@u
= 1:
(2.2)
@v
So, using the de…nitions of u and v in equation (2.2) we have
@u
= 1:
@v
=
3. This is a Cobb-Douglas production function. This will yield a unique interior solution; the Lagrangean is:
L(z; ) = w1 z1 + w2 z2 + [q
z1 1 z2 2 ] ;
(2.3)
and the …rst-order conditions are:
@L(z; )
= w1
@z1
1 z1
1
1
@L(z; )
= w2
@z2
2 z1
1
z2 2
z2 2 = 0 ;
1
(2.4)
=0;
(2.5)
@L(z; )
= q z1 1 z2 2 = 0 :
(2.6)
@
Using these conditions and rearranging we can get an expression for minimized cost in terms of and q:
w1 z1 + w2 z2 =
1 z1
1
z2 2 +
2 z1
1
z2 2 = [
1
+
2]
q:
We can then eliminate :
q
1 z1
q
2 z2
w1
w2
=0
=0
which implies
z1 =
z2 =
1
w1
2
w2
q
q
:
(2.7)
Substituting the values of z1 and z2 back in the production function we
have
1
w1
c Frank Cowell 2006
1
q
2
w2
10
2
q
=q
Microeconomics
which implies
q= q
1
w1
w2
1
1
1+ 2
2
(2.8)
2
So, using (2.7) and (2.8), the corresponding cost function is
C(w; q)
= w1 z1 + w2 z2
=
[
1
+
2]
q
w1
1
2
w2
1
1
1+ 2
:
2
4. Using the production functions we have, for any t > 0:
(tz) = [tz1 ]
1
[tz2 ]
2
1+
=t
2
(z):
Therefore we have DRTS/CRTS/IRTS according as 1 + 2 S 1. If
we look at average cost as a function of q we …nd that AC is increasing/constant/decreasing in q according as 1 + 2 S 1.
5. Using (2.7) and (2.8) conditional demand functions are
1
H (w; q) = q
H 2 (w; q) = q
1 w2
2
1
1+ 2
1
1
1+ 2
2 w1
2 w1
1 w2
and are smooth with respect to input prices.
c Frank Cowell 2006
11
Microeconomics
CHAPTER 2. THE FIRM
Exercise 2.5 Suppose a …rm’s production function has the Leontief form
z1 z2
;
q = min
1
2
where the notation is the same as in Exercise 2.4.
1. Draw the isoquants.
2. For a given level of output identify the cost-minimising input combination(s) on the diagram.
3. Hence write down the cost function in this case. Why would the Lagrangean method of Exercise 2.4 be inappropriate here?
4. What is the conditional input demand curve for input 1?
5. Repeat parts 1-4 for each of the two production functions
q
q
=
=
1 z1
2
1 z1
+
+
2 z2
2
2 z2
Explain carefully how the solution to the cost-minimisation problem di¤ ers
in these two cases.
z2
•A
•B
z1
Figure 2.8: Isoquants: Leontief
Outline Answer
1. The Isoquants are illustrated in Figure 2.8 –the so-called Leontief case,
2. If all prices are positive, we have a unique cost-minimising solution at A:
to see this, draw any straight line with positive …nite slope through A
and take this as an isocost line; if we considered any other point B on the
isoquant through A then an isocost line through B (same slope as the one
through A) must lie above the one you have just drawn.
c Frank Cowell 2006
12
Microeconomics
z2
z1
Figure 2.9: Isoquants: linear
z2
z1
Figure 2.10: Isoquants: non-convex to origin
c Frank Cowell 2006
13
Microeconomics
CHAPTER 2. THE FIRM
3. The coordinates of the corner A are (
diately yields the minimised cost.
C(w; q) = w1
1 q;
1q
2 q)
+ w2
and, given w, this imme-
2 q:
The methods in Exercise 2.4 since the Lagrangean is not di¤erentiable at
the corner.
4. Conditional demand is constant if all prices are positive
H 1 (w; q) =
H 2 (w; q) =
1q
2 q:
5. Given the linear case
q=
1 z1
+
2 z2
Isoquants are as in Figure 2.9.
It is obvious that the solution will be either at the corner (q= 1 ; 0)
if w1 =w2 < 1 = 2 or at the corner (0; q= 2 ) if w1 =w2 > 1 = 2 , or
otherwise anywhere on the isoquant
This immediately shows us that minimised cost must be.
w1 w2
;
C(w; q) = q min
1
2
So conditional demand can be multivalued:
8
q
if
>
1
>
>
>
>
<
h
i
z1 2 0; q1
H 1 (w; q) =
if
>
>
>
>
>
:
0
if
H 2 (w; q) =
8
>
>
>
>
>
<
>
>
>
>
>
:
0
h
z2 2 0;
q
2
q
2
i
w1
w2
<
1
w1
w2
=
1
w1
w2
>
1
if
w1
w2
<
1
if
w1
w2
=
1
if
w1
w2
>
1
2
2
2
2
2
2
Case 3 is a test to see if you are awake: the isoquants are not convex
to the origin: an experiment with a straight-edge to simulate an
isocost line will show thatpit is almost like case 2 p
–the solution will
be either at
the
corner
(
q=
;
0)
if
w
=w
<
1
1
2
1 = 2 or at the
p
p
corner (0; q= 2 ) if w1 =w2 >
1 = 2 (but nowhere else). So the
cost function is :
r
p
q
C(w; q) = min w1
; w2 q= 2 :
1
c Frank Cowell 2006
14
Microeconomics
The conditional demand function is similar to, but slightly di¤erent
from, the previous case:
8
q
q
w1
1
>
if w
<
>
1
2
2
>
>
>
>
>
<
o
n
q
1
1
if w
=
z1 2 0; q1
H 1 (w; q) =
w2
2
>
>
>
>
>
q
>
>
1
1
:
0
if w
w2 >
2
H 2 (w; q) =
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
0
n
z2 2 0;
q
2
q
2
o
if
w1
w2
<
if
w1
w2
=
if
Note the discontinuity exactly at w1 =w2 =
c Frank Cowell 2006
15
w1
w2
p
>
q
q
q
1= 2
1
2
1
2
1
2
Microeconomics
CHAPTER 2. THE FIRM
Exercise 2.6 Assume the production function
h
(z) =
1 z1 +
i1
2 z2
where zi is the quantity of input i and i
0, 1<
1 are parameters.
This is an example of the CES (Constant Elasticity of Substitution) production
function.
1
1. Show that the elasticity of substitution is
1
.
2. Explain what happens to the form of the production function and the elasticity of substitution in each of the following three cases: ! 1, ! 0,
! 1.
3. Relate your answer to the answers to Exercises 2.4 and 2.5.
Outline Answer
1. Di¤erentiating the production function
(z) :=
h
1 z1
+
2 z2
i1
it is clear that the marginal product of input i is
i (z)
:=
Therefore the MRTS is
h
1 z1
1 (z)
2 (z)
+
=
2 z2
1
2
i1
1
i zi
1
(2.9)
1
z1
z2
(2.10)
which implies
log
z1
z2
=
1
1
log
1
1
2
log
1
1 (z)
:
2 (z)
Therefore
@ log
=
@ log
z1
z2
1 (z)
2 (z)
=
1
1
2. Clearly ! 1 yields = 0 ( (z) = min f 1 z1 ;
= 1 ( (z) = z1 1 z2 2 ), ! 1 yields = 1 ( (z) =
2 z2 g),
1 z1
+
! 0 yields
2 z2 ).
3. The case ! 1 corresponds to that in part 1 of Exercise 2.5; ! 0.
corresponds to that in Exercise 2.4; ! 1. corresponds to that in part 5
of Exercise 2.5 .
c Frank Cowell 2006
16
Microeconomics
Exercise 2.7 For the CES function in Exercise 2.6 …nd H 1 (w; q), the conditional demand for good 1, for the case where 6= 0; 1. Verify that it is decreasing
in w1 and homogeneous of degree 0 in (w1 ,w2 ).
Outline Answer
From the minimization of the following Lagrangean
L(z; ; w; q) :=
we obtain
1
2
m
X
wi zi + [q
(z)]
i=1
[z1 ]
1 1
q
= w1
(2.11)
[z2 ]
1 1
= w2
(2.12)
q
On rearranging:
1
q1
1
q1
w1
1
w2
2
=
[z1 ]
1
=
[z2 ]
1
Using the production function we get
1
q1
w1
1
1
1
q1
w2
1
+
2
2
1
=q
Rearranging we …nd
1
q1
1
1
1
=
1
1
+
[w1 ]
1
[w1 ]
1
2
[w2 ]
q1
1
Substituting this into (2.11) we get:
1
w1 =
1 [z1 ]
1
1
1
1
1
+
1
2
[w2 ]
1
q1
Rearranging this we have:
z1 =
"
1
+
1 w2
2 w1
2
1
#
1
q
Clearly z1 is decreasing in w1 if < 1. Furthermore, rescaling w1 and w2
by some positive constant will leave z1 unchanged.
c Frank Cowell 2006
17
Microeconomics
CHAPTER 2. THE FIRM
Exercise 2.8 For any homothetic production function show that the cost function must be expressible in the form
C (w; q) = a (w) b (q) :
z2
z1
0
Figure 2.11: Homotheticity: expansion path
Outline Answer
From the de…nition of homotheticity, the isoquants must look like Figure
2.11; interpreting the tangents as isocost lines it is clear from the …gure that
the expansion paths are rays through the origin. So, if H i (w; q) is the demand
for input i conditional on output q, the optimal input ratio
H i (w; q)
H j (w; q)
must be independent of q and so we must have
H i (w; q)
H j (w; q)
= j
i
0
H (w; q )
H (w; q 0 )
for any q; q 0 . For this to true it is clear that the ratio H i (w; q)=H i (w; q 0 ) must
be independent of w. Setting q 0 = 1 we therefore have
H 1 (w; q)
H 2 (w; q)
H m (w; q)
=
=
:::
=
= b(q)
H 1 (w; 1)
H 2 (w; 1)
H m (w; 1)
and so
H i (w; q) = b(q)H i (w; 1):
c Frank Cowell 2006
18
Microeconomics
Therefore the minimized cost is given by
C(w; q)
=
=
m
X
i=1
m
X
wi H i (w; q)
wi b(q)H i (w; 1)
i=1
= b(q)
m
X
wi H i (w; 1)
i=1
= a(w)b(q)
where a(w) =
Pm
i=1
c Frank Cowell 2006
wi H i (w; 1):
19
Microeconomics
CHAPTER 2. THE FIRM
Exercise 2.9 Consider the production function
q=
1 z1
1
+
2 z2
1
+
3 z3
1
1
1. Find the long-run cost function and sketch the long-run and short-run
marginal and average cost curves and comment on their form.
2. Suppose input 3 is …xed in the short run. Repeat the analysis for the
short-run case.
3. What is the elasticity of supply in the short and the long run?
Outline Answer
1. The production function is clearly homogeneous of degree 1 in all inputs
–i.e. in the long run we have constant returns to scale. But CRTS implies
constant average cost. So
LRMC = LRAC = constant
Their graphs will be an identical straight line.
z2
z1
Figure 2.12: Isoquants do not touch the axes
2. In the short run z3 = z3 so we can write the problem as the following
Lagrangean
h
i
1
1
1
1
^ ^ ) = w1 z1 + w2 z2 + ^ q
L(z;
; (2.13)
1 z1 + 2 z2 + 3 z3
or, using a transformation of the constraint to make the manipulation
easier, we can use the Lagrangean
L(z; ) = w1 z1 + w2 z2 +
where
1 z1
1
+
2 z2
1
k
(2.14)
is the Lagrange multiplier for the transformed constraint and
k := q
c Frank Cowell 2006
20
1
3 z3
1
:
(2.15)
Microeconomics
Note that the isoquant is
2
z2 =
k
1 z1
1:
From the Figure 2.12 it is clear that the isoquants do not touch the axes
and so we will have an interior solution. The …rst-order conditions are
wi
2
i zi
which imply
zi =
r
= 0; i = 1; 2
(2.16)
i
(2.17)
wi
; i = 1; 2
To …nd the conditional demand function we need to solve for . Using the
production function and equations (2.15), (2.17) we get
k=
2
X
1=2
j
j
from which we …nd
p
where
b :=
p
(2.18)
wj
j=1
=
1 w1
b
k
+
(2.19)
p
2 w2 :
Substituting from (2.19) into (2.17) we get minimised cost as
C~ (w; q; z3 )
=
2
X
wi zi + w3 z3
(2.20)
i=1
2
=
=
b
+ w3 z3
k
qb2
1 + w3 z3 :
1
3 z3 q
(2.21)
(2.22)
Marginal cost is
b2
1
3 z3
1
q
2
(2.23)
and average cost is
1
b2
w3 z3
:
1 +
q
3 z3 q
(2.24)
Let q be the value of q for which MC=AC in (2.23) and (2.24) – at the
minimum of AC in Figure 2.13 –and let p be the corresponding minimum
value of AC. Then, using p =MC in (2.23) for p p the short-run supply
8
if p <p
> 0
>
>
>
>
<
0 or q
if p =p
curve is given by q = S(w; p) =
>
>
>
h
i
>
>
: q = z3 1 pb
if p >p
p
3
c Frank Cowell 2006
21
Microeconomics
CHAPTER 2. THE FIRM
3. Di¤erentiating the last line in the previous formula we get
d ln q
p dq
1
1
=
= p
d ln p
q dp
2 p=b
1
>0
Note that the elasticity decreases with b. In the long run the supply curve
coincides with the MC,AC curves and so has in…nite elasticity.
marginal
cost
average
cost
q
Figure 2.13: Short-run marginal and average cost
c Frank Cowell 2006
22
Microeconomics
Exercise 2.10 A competitive …rm’s output q is determined by
q = z1 1 z2 2 :::zmm
where zi is its usage of input i and i > 0 is a parameter i = 1; 2; :::; m. Assume
that in the short run only k of the m inputs are variable.
1. Find the long-run average and marginal cost functions for this …rm. Under
what conditions will marginal cost rise with output?
2. Find the short-run marginal cost function.
3. Find the …rm’s short-run elasticity of supply. What would happen to this
elasticity if k were reduced?
Outline Answer
Write the production function in the equivalent form:
log q =
m
X
i
log zi
(2.25)
i=1
The isoquant for the case m = 2 would take the form
z2 = qz1
1
1
2
(2.26)
which does not touch the axis for …nite (z1 ; z2 ).
1. The cost-minimisation problem can be represented as minimising the Lagrangean
#
"
m
m
X
X
(2.27)
wi zi +
log q
i log zi
i=1
i=1
where wi is the given price of input i, and is the Lagrange multiplier
for the modi…ed production constraint. Given that the isoquant does not
touch the axis we must have an interior solution: …rst-order conditions are
wi
i zi
1
= 0; i = 1; 2; ::; m
(2.28)
i
; i = 1; 2; ::; m
(2.29)
which imply
zi =
wi
Now solve for . Using (2.25) and (2.29) we get
zi i =
q=
m
Y
i
wi
zi i =
i=1
c Frank Cowell 2006
23
i
; i = 1; 2; ::; m
A
m
Y
i=1
wi
i
(2.30)
(2.31)
Microeconomics
CHAPTER 2. THE FIRM
where :=
we …nd
Pm
j=1
Qm
and A := [ i=1
j
i
i
1=
]
are constants, from which
1=
q
i=1 wi
= A Qm
i
1=
= A [qw1 1 w2 2 :::wmm ]
:
(2.32)
Substituting from (2.32) into (2.29) we get the conditional demand function:
i
1=
H i (w; q) = zi =
A [qw1 1 w2 2 :::wmm ]
(2.33)
wi
and minimised cost is
C (w; q)
m
X
=
1=
wi zi = A [qw1 1 w2 2 :::wmm ]
(2.34)
i=1
Bq 1=
=
(2.35)
1=
where B := A [w1 1 w2 2 :::wmm ] . It is clear from (2.35) that cost is
increasing in q and increasing in wi if i > 0 (it is always nondecreasing
in wi ). Di¤erentiating (2.35) with respect to q marginal cost is
Cq (w; q) = Bq
1
(2.36)
Clearly marginal cost falls/stays constant/rises with q as
T 1.
2. In the short run inputs 1; :::; k (k m) remain variable and the remaining
inputs are …xed. In the short-run the production function can be written
as
k
X
log q =
(2.37)
i log zi + log k
i=1
where
k
m
X
:= exp
i
i=k+1
log zi
!
(2.38)
and zi is the arbitrary value at which input i is …xed; note that B is
…xed in the short run. The general form of the Lagrangean (2.27) remains
unchanged, but with q replaced by q= k and m replaced by k. So the
…rst-order conditions and their corollaries (2.28)-(2.32) are essentially as
before, but and A are replaced by
k
:=
k
X
(2.39)
j
j=1
and Ak :=
hQ
k
i=1
i
i
i
1=
k
. Hence short-run conditional demand is
~ i (w; q; zk+1 ; :::; zm ) =
H
c Frank Cowell 2006
i
wi
Ak
24
q
k
1=
w1 1 w2 2 :::wk k
k
(2.40)
Microeconomics
and minimised cost in the short run is
C~ (w; q; zk+1 ; :::; zm )
=
k
X
wi zi + ck
i=1
=
q
k Ak
=
k
k Bk q
where
m
X
ck :=
1=
1=
w1 1 w2 2 :::wk k
k
+ ck
k
+ ck (2.41)
(2.42)
wi zi
(2.43)
i=k+1
1=
is the …xed-cost component in the short run and Bk := Ak [w1 1 w2 2 :::wk k = k ]
Di¤erentiating (2.42) we …nd that short-run marginal cost is
C~q (w; q; zk+1 ; :::; zm ) = Bk q
1
k
k
3. Using the “Marginal cost=price” condition we …nd
1
Bk q
k
k
=p
(2.44)
where p is the price of output so that, rearranging (2.44) the supply function is
k
p 1 k
q = S (w; p; zk+1 ; :::; zm ) =
(2.45)
Bk
wherever MC AC. The elasticity of (2.45) is given by
@ log S (w; p; zk+1 ; :::; zm )
=
@ log p
1
It is clear from (2.39) that k
k 1
elasticity in (2.46) must fall as k falls.
c Frank Cowell 2006
25
k 2 :::
k
>0
(2.46)
k
and so the positive supply
k
.
Microeconomics
CHAPTER 2. THE FIRM
Exercise 2.11 A …rm produces goods 1 and 2 using goods 3,...,5 as inputs. The
production of one unit of good i (i = 1; 2) requires at least aij units of good j, (
j = 3; 4; 5).
1. Assuming constant returns to scale, how much of resource j will be needed
to produce q1 units of commodity 1?
2. For given values of q3 ; q4 ; q5 sketch the set of technologically feasible outputs of goods 1 and 2.
Outline Answer
1. To produce q1 units of commodity 1 a1j q1 units of resource j will be
needed.
q1 a1i + q2 a2i Ri :
2. The feasibility constraint for resource j is therefore going to be
q1 a1j + q2 a2j
Rj :
Taking into account all three resources, the feasible set is given as in Figure
2.14
q1
points
points satisfying
satisfying
qq11aa1313 +
+ qq22aa2323 ≤≤ RR33
points
points satisfying
satisfying
qq11aa1414 +
+ qq22aa2424 ≤≤ RR44
Feasible
Set
points
points satisfying
satisfying
qq11aa1515 +
+ qq22aa2525 ≤≤ RR55
q2
Figure 2.14: Feasible set
Exercise 2.12 [see Exercise 2.4]
c Frank Cowell 2006
26
Microeconomics
Exercise 2.13 An agricultural producer raises sheep to produce wool (good 1)
and meat (good 2). There is a choice of four breeds (A, B, C, D) that can be
used to stock the farm; each breed can be considered as a separate input to the
production process. The yield of wool and of meat per 1000 sheep (in arbitrary
units) for each breed is given in Table 2.1.
wool
meat
A
20
70
B
65
50
C
85
20
D
90
10
Table 2.1: Yield per 1000 sheep for breeds A,...,D
1. On a diagram show the production possibilities if the producer stocks exactly 1000 sheep using just one breed from the set {A,B,C,D} .
2. Using this diagram show the production possibilities if the producer’s 1000
sheep are a mixture of breeds A and B. Do the same for a mixture of
breeds B and C; and again for a mixture of breeds C and D. Hence draw
the (wool, meat) transformation curve for 1000 sheep. What would be the
transformation curve for 2000 sheep?
3. What is the MRT of meat into wool if a combination of breeds A and B are
used? What is the MRT if a combination of breeds B and C are used?And
if breeds C and D are used?
4. Why will the producer not …nd it necessary to use more than two breeds?
5. A new breed E becomes available that has a (wool, meat) yield per 1000
sheep of (50,50). Explain why the producer would never be interested in
stocking breed E if breeds A,...,D are still available and why the transformation curve remains una¤ ected.
6. Another new breed F becomes available that has a (wool, meat) yield per
1000 sheep of (50,50). Explain how this will alter the transformation
curve.
Outline Answer
1. See Figure 2.15.
2. See Figure 2.15.
3. The MRT if A and B are used is
is going to be
20
85
50
=
65
3
:
2
70
20
50
=
65
4
. If B and C are used it
9
4. In general for m inputs and n outputs if m > n then m
redundant.
n inputs are
5. As we can observe in Figure 2.15, by using breed E the producer cannot
move the frontier (the transformation curve) outwards.
c Frank Cowell 2006
27
Microeconomics
CHAPTER 2. THE FIRM
80
meat
60
40
20
0
0
20
40
60
80
100
wool
Figure 2.15: The wool and meat tradeo¤
6. As we can observe in Figure 2.16 now the technological frontier has moved
outwards: one of the former techniques is no longer on the frontier.
80
meat
60
40
20
0
0
20
40
60
80
wool
Figure 2.16: E¤ect of a new breed
c Frank Cowell 2006
28
100
Microeconomics
Exercise 2.14 A …rm produces goods 1 and 2 uses labour (good 3) as input
subject to the production constraint
2
2
[q1 ] + [q2 ] + Aq3
0
where qi is net output of good i and A is a positive constant. Draw the transformation curve for goods 1 and 2. What would happen to this transformation
curve if the constant A had a larger value?
Outline Answer
1. From the production function it is clear that, for any given value q3 ,
the transformation curve is the boundary of the the set of points (q1 ; q2 )
satisfying
2
2
[q1 ] + [q2 ]
q1 ; q 2
Aq3
0
where the right-hand side of the …rst expression is positive because q3 is
negative. This is therefore going to be a quarter circle as in Figure 2.17.
2. See Figure 2.17.
q2
Large A
Small A
q1
Figure 2.17: Transformation curves
c Frank Cowell 2006
29
Microeconomics
c Frank Cowell 2006
CHAPTER 2. THE FIRM
30
Chapter 3
The Firm and the Market
Exercise 3.1 (The phenomenon of “natural monopoly”) Consider an industry
in which all the potential member …rms have the same cost function C. Suppose
it is true that for some level of output q and for any nonnegative outputs q; q 0 of
two such …rms such that q + q 0 q the cost function satis…es the “subadditivity”
property
C (w; q + q 0 ) < C (w; q) + C (w; q 0 ) .
1. Show that this implies that for all integers N > 1
q
C (w; q) < N C w;
, for 0 q
N
q
2. What are the implications for the shape of average and marginal cost
curves?
3. May one conclude that a monopoly must be more e¢ cient in producing
this good?
Outline Answer
1. If q 0 = q then
C (w; 2q) < 2C (w; q) .
Hence
C (w; q) + C (w; 2q) < 3C (w; q) .
0
and so, putting q = 2q we have
C (w; 3q) < C (w; q) + C (w; 2q) < 3C (w; q) .
The result then follows by iteration.
2. If there are economies of scale the average cost of production is decreasing
and marginal cost will always be below it. Nevertheless “subadditivity”
does not imply economies of scale and therefore we can also observe a
standard U-shaped average cost curve.
3. It is cheaper to produce in a single plant rather than using two identical
plants. But a monopoly may distort prices.
31
Microeconomics
CHAPTER 3. THE FIRM AND THE MARKET
Exercise 3.2 In a particular industry there are n pro…t-maximising …rms each
producing a single good. The costs for …rm i are
C0 + cqi
where C0 and c are parameters and qi is the output of …rm i. The goods are
not regarded as being exactly identical by the consumers and the inverse demand
function for …rm i is given by
Aq
pi = Pn i
1
j=1 qj
where
measures the degree of substitutability of the …rms’products, 0 <
1.
1. Assuming that each …rm takes the output of all the other …rms as given,
write down the …rst-order conditions yielding …rm 1’s output conditional
on the outputs q2 ; :::; qn . Hence, using the symmetry of the equilibrium,
show that in equilibrium the optimal output for any …rm is
qi =
A [n 1]
n2 c
and that the elasticity of demand for …rm i is
n
n +
n
2. Consider the case = 1. What phenomenon does this represent? Show
thatqthe equilibrium number of …rms in the industry is less than or equal
to CA0 .
Outline Answer
1. We begin by computing the equilibrium for a typical …rm i. Pro…ts for
the …rm are
Aqi
C0 cqi
(3.1)
i =
K
where
n
X
K :=
qj
j=1
The …rst-order condition for maximising (3.1) with respect to qi (taking
all the other qj as given) is
A qi
@ i
=
@qi
K
1
A qi2
K2
1
c=0
(3.2)
If all …rms are identical, then in equilibrium all …rms must produce the
same amount and so
K = nqi
(3.3)
c Frank Cowell 2006
32
Microeconomics
Substituting (3.3) in (3.2) we get
A
n
A
n2
cqi = 0
(3.4)
from which the result follows immediately. To …nd the elasticity of demand
for …rm i take logs of the inverse demand curve (in the question) and
di¤erentiate with respect to qi
qi @pi
=1
pi @qi
+
qi
K
(3.5)
To …nd the elasticity in the neighbourhood of the equilibrium substitute
(3.3) in (3.5) and take the reciprocal.
2. The case = 1 represents a situation where the goods are perfect substitutes. We then …nd that …rm i’s pro…ts are
i
=
=
=
Aqi
C0 cqi
K
A
A[n 1]
C0
n
n2
A
C0
n2
(3.6)
(3.7)
Requiring that the right-hand side of (3.7) be non-negative implies
r
A
n
(3.8)
C0
c Frank Cowell 2006
33
Microeconomics
CHAPTER 3. THE FIRM AND THE MARKET
Exercise 3.3 A …rm has the cost function
1
F0 + aqi2
2
where qi is the output of a single homogenous good and F0 and a are positive
numbers.
1. Find the …rm’s supply relationship between output and price p;
p explain
carefully what happens at the minimum-average-cost point p := 2aF0 .
2. In a market of a thousand consumers the demand curve for the commodity
is given by
p = A bq
where q is total quantity demanded and A and b are positive parameters.
If the market is served by a single price-taking …rm with the cost structure
in part 1 explain why there is a unique equilibrium if b a A=p 1 and
no equilibrium otherwise.
3. Now assume that there is a large number N of …rms, each with the above
cost function: …nd the relationship between average supply by the N …rms
and price and compare the answer with that of part 1. What happens as
N ! 1?
4. Assume that the size of the market is also increased by a factor N but that
the demand per thousand consumers remains as in part 2 above. Show
that as N gets large there will be a determinate market equilibrium price
and output level.
Outline Answer
1. Given the cost function
1
F0 + aqi2
2
marginal cost is aqi and average cost is F0 =qi + 12 aqi . Marginal cost intersects average cost where
1
aqi = F0 =qi + aqi
2
i.e. where output is
q :=
and marginal cost is
p :=
p
2F0 =a
p
2aF0
(3.9)
(3.10)
For p > p the supply curve is identical to the marginal cost curve qi = p=a;
for p < p the …rm supplies 0 to the market; at p = p the …rm supplies
either 0 or q. There is no price which will induce a supply in the interior
of the interval 0; q . Summarising, …rm i’s optimal output is given by
8
p=a; if p > p
>
>
>
>
<
q 2 f0; qg if p = p
qi = S(p) :=
(3.11)
>
>
>
>
:
0; if p < p
c Frank Cowell 2006
34
Microeconomics
2. The equilibrium, if it exists, is found where supply=demand at a given
price. This would imply
p
a
=
p
=
A
p
b
aA
a+b
which would, in turn, imply an equilibrium quantity
q=
A
but it can only be valid if a+b
i
h
1
b.
equivalent to a A
p
A
a+b
q. Noting that q = p=a this condition is
3. If there are N such …rms, each …rm responds to price as in (3.11), and so
PN
the average output q := N1 i=1 qi is given by
8
p=a; if p > p
>
>
>
>
<
q 2 J(q) if p = p
q=
(3.12)
>
>
>
>
:
0; if p < p
where J(q) := f Ni q : i = 0; 1; :::; N g. As N ! 1 the set J(q) becomes
dense in [0; q], and so we have the average supply relationship:
8
p=a; if p > p
>
>
>
>
<
q 2 [0; q] if p = p
q=
>
>
>
>
:
0; if p < p
(3.13)
4. Given that in the limit the average supply curve is continuous and of the
piecewise linear form (3.13), and that the demand curve is a downwardsloping straight line, there must be a unique market equilibrium. The
equip
p
A p
A
2aF0
which, using (3.10) is
.
2aF0 ;
librium will be found at p; b
b
Using (3.9) this can be written p; q where
:=
In the equilibrium a proportion
…rms produce 0.
c Frank Cowell 2006
A p
bp=a
of the …rms produce q and 1
35
of the
Microeconomics
CHAPTER 3. THE FIRM AND THE MARKET
Exercise 3.4 A …rm has a …xed cost F0 and marginal costs
c = a + bq
where q is output.
1. If the …rm were a price-taker, what is the lowest price at which it would be
prepared to produce a positive amount of output? If the competitive price
were above this level, …nd the amount of output q that the …rm would
produce.
2. If the …rm is actually a monopolist and the inverse demand function is
1
Bq
2
p=A
(where A > a and B > 0) …nd the expression for the …rm’s marginal
revenue in terms of output. Illustrate the optimum in a diagram and show
that the …rm will produce
A a
q :=
b+B
What is the price charged p and the marginal cost c at this output
level? Compare q and q :
3. The government decides to regulate the monopoly. The regulator has the
power to control the price by setting a ceiling pmax . Plot the average and
marginal revenue curves that would then face the monopolist. Use these
to show:
(a) If pmax > p
and p
(b) If pmax < c
the …rm’s output and price remain unchanged at q
the …rm’s output will fall below q .
(c) Otherwise output will rise above q .
Outline Answer
1. Total costs are
1
F0 + aq + bq 2
2
So average costs are
1
F0
+ a + bq
q
2
which are a minimum at
F0
b
(3.14)
p
2bF0 + a
(3.15)
q=
where average costs are
r
2
Marginal and average costs are illustrated in Figure 3.1: notice that MC
is linear and that AC has the typical U-shape if F0 > 0. For a price above
the level (3.15) the …rst-order condition for maximum pro…ts is given by
p = a + bq
c Frank Cowell 2006
36
Microeconomics
a+bq
marginal
cost
F/q+a+0.5bq
P
average cost
q* = P—−—a
b
q
—
q
Figure 3.1: Perfect competition
from which we …nd
q :=
p
a
b
–see …gure 3.1.
2. If the …rm is a monopolist marginal revenue is
@
Aq
@q
1 2
Bq = A
2
Bq
Hence the …rst-order condition for the monopolist is
A
from which the solution q
c
p
Bq = a + bq
(3.16)
follows. Substituting for q
=A
=A
Bq
1
Bq
2
=
=c
Ab + Ba
B+b
1 A a
+ B
2 b+B
we also get
(3.17)
(3.18)
–see …gure 3.2.
3. Consider how the introduction of a price ceiling will a¤ect average revenue.
Clearly we now have
AR(q) =
pmax if q q0
A 12 Bq if q
q0
(3.19)
where q0 := 2 [A pmax ] =B: average revenue is a continuous function of
q but has a kink at q0 . From this we may derive marginal revenue which
is
pmax if q < q0
MR(q) =
(3.20)
A Bq if q > q0
c Frank Cowell 2006
37
Microeconomics
CHAPTER 3. THE FIRM AND THE MARKET
p
marginal
cost
p**
a+bq
F/q+a+0.5bq
average cost
c**
average
revenue
marginal
revenue
A − Bq
A − 0.5Bq
q**
q
Figure 3.2: Unregulated monopoly
– notice that there is a discontinuity exactly at q0 . The modi…ed curves
(3.19) and (3.20) are shown in Figure 3.3: notice that they coincide in
the ‡at section to the left of q0 . Clearly the outcome depends crucially
on whether MC intersects (modi…ed) MR (a) to the left of q0 , (b) to the
right of q0 , (c) in the discontinuity exactly at q0 . Case (c) is illustrated,
and it is clear that output will have risen from q to q0 . The other cases
can easily be found by appropriately shifting the curves on Figure 3.3 .
p
marginal
cost
p**
pmax
c**
average
revenue
q**
marginal
revenue
q0
q
Figure 3.3: Regulated Monopoly
c Frank Cowell 2006
38
Microeconomics
Exercise 3.5 A monopolist has the cost function
1 2
C(q) = 100 + 6q + [q]
2
1. If the demand function is given by
1
q = 24
p
4
calculate the output-price combination which maximises pro…ts.
2. Assume that it becomes possible to sell in a separate second market with
demand determined by
3
p:
q = 84
4
Calculate the prices which will be set in the two markets and the change
in total output and pro…ts from case 1.
3. Now suppose that the …rm still has access to both markets, but is prevented
from discriminating between them. What will be the result?
Outline Answer
1. Maximizing the simple monopolist’s pro…ts
0
= (96
4q)q
100 + 6q +
q2
2
with respect to q yields optimum output of q0 =10. Hence p0 = 56 and
0 = 350:
2. Now let the monopolist sell q1 in market 1 for price p1 and q2 in market
2 for price p2 :The new problem is to choose q1 ; q2 so as to maximise the
function
4
(q1 + q2 )2
4q1 )q1 + (112
q2 )q2
100 + 6q1 + 6q2 +
:
12 = (96
3
2
First-order conditions yield
9q1 + q2
11
q1 + q2
3
=
90
=
106:
Solving we …nd q1 = 7; q2 = 27 and hence p1 = 68; p2 = 76 and
1646.
12
=
3. If we abandon discrimination, a uniform price pb must be charged. If
pb > 112 nothing is sold to either market. If 112 > pb > 96 only market
2 is served. If 96 > pb both market are served and the demand curve
is qb = 108 pb. Clearly this is the relevant region. Maximising simple
monopoly pro…ts we …nd qb = 34; pb = 74 and b = 1634.
Hence the total output is identical to that under discrimination, p1 < pb <
p2 and 12 > b : These results are quite general.
c Frank Cowell 2006
39
Microeconomics
c Frank Cowell 2006
CHAPTER 3. THE FIRM AND THE MARKET
40
Chapter 4
The Consumer
Exercise 4.1 You observe a consumer in two situations: with an income of
$100 he buys 5 units of good 1 at a price of $10 per unit and 10 units of good 2
at a price of $5 per unit. With an income of $175 he buys 3 units of good 1 at
a price of $15 per unit and 13 units of good 2 at a price of $10 per unit. Do the
actions of this consumer conform to the basic axioms of consumer behaviour?
x2
x′′ = (3,13)
x′ = (5,10)
p1 / p 2 = 2
p1 / p2 = 1.5
x1
Figure 4.1: WARP violated
Outline Answer
At the original price ratio p1 =p2 = 2 the choice is x0 = (5; 10); but at those
prices the and with that budget the consumer could have a¤orded x00 = (3; 13):
x0 is revealed-preferred to x00 . But at the new price ratio p1 =p2 = 1:5 x00 is
chosen, although x0 is still a¤ordable: x00 is revealed-preferred to x0 . This
violates WARP –see Figure 4.1.
41
Microeconomics
CHAPTER 4. THE CONSUMER
Exercise 4.2 Draw the indi¤ erence curves for the following four types of preferences:
Type
Type
Type
Type
A
B
C
D
:
log x1 + [1
:
x1 + x2
2
2
:
[x1 ] + [x2 ]
: min f x1 ; x2 g
] log x2
where x1 ; x2 denote respectively consumption of goods 1 and 2 and ; ; ; are
strictly positive parameters with < 1. What is the consumer’s cost function
in each case?
x2
x2
A
B
x1
x1
x2
x2
D
C
x1
x1
Figure 4.2: Indi¤erence curves: four cases
Use the fact that expenditure minimisation for the household and costminimisation for the …rm are essentially the same problem. The indi¤erence
curves in Figure 4.2 are identical to the isoquants depicted in Exercises 2.4, 2.5.
So, substituting the notation in Exercise 2.4 and 2.5we get:
Case A: C(p; ) = C(p; ) = e
p1
Case B: C(p; ) =
min(p1 = ; p2 )
p
p
Case C: C(p; ) =
min(p1 = ; p2 )
Case D: C(p; ) =
c Frank Cowell 2006
p1
+ p2
.
42
h
p2
1
i1
.
Microeconomics
Exercise 4.3 Suppose a person has the Cobb-Douglas utility function
n
X
ai log(xi )
i=1
where xi is the quantity
Pnconsumed of good i, and a1 ; :::; an are non-negative
parameters such that
j=1 aj = 1. If he has a given income y, and faces
prices p1 ; :::; pn , …nd the ordinary demand functions. What is special about the
expenditure on each commodity under this set of preferences?
Outline Answer
The relevant Lagrangean is
n
X
i
log xi +
i=1
"
y
n
X
pi xi
i=1
#
(4.1)
The …rst-order conditions yield:
xi
i
; i = 1; 2; :::; n:
pi
n
X
=
pi xi
=
y
(4.2)
(4.3)
i=1
From the n + 1 equations (4.2,4.3) we get at the optimum: y =
1= . So the demand functions are
xi =
iy
; i = 1; 2; :::; n:
pi
Pn
i=1
i=
=
(4.4)
and expenditure on each commodity i is
ei := pi xi =
–a constant proportion of income.
c Frank Cowell 2006
43
i y;
(4.5)
Microeconomics
CHAPTER 4. THE CONSUMER
Exercise 4.4 The elasticity of demand for domestic heating oil is 0:5, and
for gasoline is 1:5. The price of both sorts of fuel is 60c/ per litre: included
in this price is an excise tax of 48c/ per litre. The government wants to reduce
energy consumption in the economy and to increase its tax revenue. Can it do
this (a) by taxing domestic heating oil? (b) by taxing gasoline?
Outline Answer
Let p be the untaxed price, and
the excise tax. Government revenue
is T = x, and the purchase price is p + . Clearly an increase in would
reduce consumption, and =[ + p] = 0:8: The e¤ect on tax revenue is given by
@T =@ = x + @x=@ = x[1 + 0:8"]. If (a) " = 0:5 then this is positive. If (b)
" = 1:5 then it is negative.
Exercise 4.5 De…ne the uncompensated and compensated price elasticities as
"ij :=
pj @H i (p; )
pj @Di (p;y)
; "ij :=
xi
@pj
xi
@pj
and the income elasticity
"iy :=
y @Di (p;y)
:
xi
@y
Show how the Slutsky equation can be expressed in terms of these elasticities
and the expenditure share of each commodity in the total budget.
Use the fact that each demand function Di is homogeneous of degree zero
in all prices and income. Then, using the standard lemma for homogenous
functions, we have for each i = 1; :::; n :
n
X
pj
j=1
which implies
@Di (p; y)
@Di (p; y)
+y
@pj
@y
n
X
=
0 Di (p; y)
=
0
"ij + "iy = 0:
j=1
Moreover we can rewrite the Slutsky equation as
"ij = "ij
where vj =
vj "iy
pj xj
is the expenditure share of commodity j.
y
c Frank Cowell 2006
44
Microeconomics
Exercise 4.6 You are planning a study of consumer demand. You have a data
set which gives the expenditure of individual consumers on each of n goods. It
is suggested to you that an appropriate model for consumer expenditure is the
Linear Expenditure System:
2
3
n
X
ei = i pi + i 4y
pj j 5
j=1
where pi is the price of good i, ei is the consumer’s expenditure on good i, y
is the consumer’
Pn s income, and 1 ; :::; n , 1 ; :::; n are non-negative parameters
such that j=1 j = 1.
1. Find the e¤ ect on xi , the demand for good i, of a change in the consumer’s
income and of an (uncompensated) change in any price pj .
2. Find the substitution e¤ ect of a change in price pj on the demand for good
i.
3. Explain how you could check that this demand system is consistent with
utility-maximisation and suggest the type of utility function which would
yield the demand functions implied by the above formula for consumer
expenditure. [Hint: compare this with Exercise 4.3]
Outline Answer
1. We have
xi =
+
i
i
pi
2
n
X
4y
j=1
3
pj j 5
(4.6)
Notice that ( 1 ; :::; n ) play
Pnthe role of “subsistence minima”of the n commodities, and so y0 := j=1 pj j can be considered as the subsistence
minimum expenditure, and the remaining budget y y0 as “discretionary
expenditure”; i is then the proportion of discretionary expenditure spent
on discretionary purchases of commodity i: pi [xi
y0 ]. Comi ] = [y
pare this with (4.5). From (4.6) we have:
@xi
@y
@xi
@pj
@xi
@pi
=
=
=
i
(4.7)
pi
i j
pi
"
if j 6= i
i
pi
i
+
y
(4.8)
Pn
j=1
pi
pj
j
#
(4.9)
2. Apply Slutsky equation using (4.7) and (4.8) to establish
dxi
dpj
c Frank Cowell 2006
=
=con tant
45
i
xj
pi
j
, if j 6= i
(4.10)
Microeconomics
CHAPTER 4. THE CONSUMER
3. Check that demand function (4.6) is homogeneous of degree 0 in prices
and income, and that the sum of the right-hand side of the equation in the
question adds up to total income. Check that cross-substitution e¤ects are
symmetric, and that own-price substitution e¤ects are negative. Using the
analogy with part (b) we can see that the demand system is similar, but
with the commodity origin shifted from 0 to the point ( 1 ; :::; n );so we
expect the indi¤erence curves from which the demand system was derived
will look like Cobb-Douglas contours with the origin shifted to the point
( 1 ; :::; n ). The utility function will then be
n
X
i
log(xi
i=1
c Frank Cowell 2006
46
i)
:
(4.11)
Microeconomics
Exercise 4.7 Suppose a consumer has a two-period utility function of the form
labelled type A in Exercise 4.2. where xi is the amount of consumption in period
i. The consumer’s resources consist just of inherited assets A in period 1, which
is partly spent on consumption in period 1 and the remainder invested in an
asset paying a rate of interest r.
1. Interpret the parameter
in this case.
2. Obtain the optimal allocation of (x1 ; x2 )
3. Explain how consumption varies with A, r and
.
4. Comment on your results and examine the “income” and “substitution”
e¤ ects of the interest rate on consumption.
Outline Answer
1. The parameter captures the consumer’s “impatience”: the higher is
the more steeply sloped will be the indi¤erence curves in Figure 4.3. Note
1
is the price of consumption in period 2 relative to the price of
that 1+r
consumption in period 1; so the lifetime budget constraint, expressed in
terms of period-1 prices, is:
x2
1+r
A
] log x2 +
A
x1 +
(4.12)
and so the Lagrangean is:
log x1 + [1
x1
x2
1+r
(4.13)
2. We can be sure an interior maximum will exist (examine the indi¤erence
curve in Figure 4.3). First-order conditions are
x1
1
x2
x
x1 + 2
1+r
From these we …nd
period is:
=
1
A
x1
x2
=
=
1
1+r
= A
and therefore optimum consumption in each
=
=
A
[1 + r] [1
]A
(4.14)
(4.15)
So we can see that the smaller is (the lower is the level of impatience), or
the larger is r (the rate of interest), the more consumption will be “tilted”
toward period 2.
c Frank Cowell 2006
47
Microeconomics
CHAPTER 4. THE CONSUMER
x2
x*
1+r
x1
A
Figure 4.3: Equilibrium in 2-period case
3. The e¤ect of an increase in assets is:
@x1
@A
@x2
@A
=
(4.16)
=
[1 + r] [1
]
(4.17)
leaving the proportion spent on consumption in each period unaltered.
The e¤ect of an increase in the interest rate is:
@x1
@r
@x2
@r
=
0
=
[1
(4.18)
]A
(4.19)
4. To …nd the substitution e¤ect we need to use the Slutsky equation. In a
conventional 2-commodity model this would be given by
@x1
dx1
=
@p2
dp2
x2
=constant
@x1
@y
(4.20)
Taking 1=[1 + r] as the “price” p2 of consumption in period 2, with
A =lifetime budget y and price of period-1 consumption de…ned as 1.
Noting that in this case dp2 = 1=[1 + r]2 dr we can rewrite (4.20) as
@x1
dx1
=
@r
dr
+
=constant
x2 @x1
[1 + r]2 @A
(4.21)
Rearranging this, the substitution e¤ect for good 1 of an increase in r may
c Frank Cowell 2006
48
Microeconomics
then be found (using 4.16 and 4.18) as:
dx1
dr
=
=constant
=
c Frank Cowell 2006
49
@x1
@r
x2 @x1
[1 + r]2 @A
x2
[1 + r]2
<0
(4.22)
Microeconomics
CHAPTER 4. THE CONSUMER
Exercise 4.8 Suppose a consumer is rationed in his consumption of commodity
1, so that his consumption is constrained thus x1
a. Discuss the properties
of the demand functions for commodities 2; :::; n of a consumer for whom the
rationing constraint is binding.
Outline Answer
Use the standard analysis on the short-run for the …rm (see Chapter 2) to
get insight on the economics of the consumer under rationing. In the case of the
…rm has to cope with the side-constraint z3 = z3 in the short run; the consumer
has to cope with the rationing constraint x1 a: if the constraint is slack then
it is irrelevant (the consumer does not use all his ration); if it is binding, then
the problem is just like that of the …rm. The solution is at x0 in Figure 4.4
x2
x′
x1
a
Figure 4.4: Ration
c Frank Cowell 2006
50
Microeconomics
Exercise 4.9 A person has preferences represented by the utility function
U (x) =
n
X
log xi
i=1
where xi is the quantity consumed of good i and n > 3.
1. Assuming that the person has a …xed money income y and can buy commodity i at price pi …nd the ordinary and compensated demand elasticities
for good 1 with respect to pj , j = 1; :::; n.
2. Suppose the consumer is legally precommitted to buying an amount An
of commodity n where pn An < y. Assuming that there are no additional
constraints on the choices of the other goods …nd the ordinary and compensated elasticities for good 1 with respect to pj , j = 1; :::n. Compare
your answer to part 1.
3. Suppose the consumer is now legally precommitted to buying
Pn an amount Ak
of commodity k, k = n r; :::; n where 0 < r < n 2 and k=n r pk Ak < y.
Use the above argument to explain what will happen to the elasticity of good
1 with respect to pj as r increases. Comment on the result.
Outline Answer
1. For the speci…ed utility function it is clear that the indi¤erence curves do
not touch the axes for any …nite xi , so we cannot have a corner solution.
The budget constraint is
n
X
pi xi y:
i=1
The problem of maximising utility subject to the budget constraint is
equivalent to maximising the Lagrangean
#
"
n
n
X
X
log xi +
y
pi xi :
i=1
i=1
The FOC are
1
xi
pi = 0; i = 1; :::; n
(4.23)
and the (binding) budget constraint. From (4.23) we get
n
n
X
pi xi = 0:
(4.24)
and so, using the budget constraint, we …nd
value of into (4.23) we …nd:
= n=y. Substituting the
i=1
(a) The ordinary demand function for good i is
xi =
c Frank Cowell 2006
51
y
npi
(4.25)
Microeconomics
CHAPTER 4. THE CONSUMER
The
Pn indirect utility function V is given by
i=1 log xi . So, from (4.25) we have:
= V (p; y) = U (x ) =
yn
nn p1 p2 p3 :::pn
= log
(4.26)
Inverting the relation (4.26) the cost function C is given by
1
1
y = C(p; ) = [nn p1 p2 p3 :::pn e ] n = n [p1 p2 p3 :::pn e ] n
(4.27)
Di¤erentiating (4.27) the compensated demand for good 1 is
1
n
1
(4.28)
x1 = p1 n [p2 p3 p4 :::pn e ] n
(b) From (4.25) we have the elasticities
@ log x1
@ log p1
y=const
@ log x1
@ log pj
y=const
=
1;
=
0; j = 2; :::; n:
(c) From (4.28) we have the compensated elasticities
@ log x1
@ log p1
@ log x1
@ log pj
=
1
n
n
=const
=
=const
< 0;
1
> 0; j = 2; :::; n
n
2. The problem now is equivalent to maximising
n
X1
log xi + log An
i=1
subject to
n
X1
pi xi
y0 ;
i=1
where y 0 := y pn An . Reusing the method above, the ordinary and
compensated demand functions are, respectively,
x1 =
y0
y pn An
=
[n 1] p1
[n 1] p1
2
x1 = p1n
n
1
[p2 p3 p4 :::pn
1e
]n
(4.29)
1
(4.30)
1
(a) So now, from (4.29) we have
c Frank Cowell 2006
@ log x1
@ log p1
y=const
@ log x1
@ log pj
y=const
=
=
52
1;
0; j = 2; :::; n
1:
Microeconomics
(as before) but
@ log x1
@ log pn
pn An
<0
y0
=
y=const
The reason for this last result is that if the person is forced to buy the
…xed amount An then changing pn is equivalent to a simple income
e¤ect (think about what happens to y 0 ).
(b) From (4.28) we have
@ log x1
@ log p1
@ log x1
@ log pj
=const
@ log x1
@ log pn
=const
=
=const
=
=
2
n
n
< 0;
1
1
n
1
; j = 2; :::; n
1:
0:
3. The problem is just a generalisation of part 2. The person is maximising
m
X
log xi + log A0
i=1
Pm
Qn
subject toP i=1 pi xi
y 0 , where m := n r 1, A0 := k=n r Ak and
n
0
y := y
k=n r pk Ak . Ordinary and compensated demands are
Pn
y
y0
k=n r pk Ak
=
(4.31)
x1 =
mp1
mp1
1
m
1
x1 = p1 m [p2 p3 p4 :::pm e ] m
(4.32)
and so we have .
@ log x1
@ log p1
@ log x1
@ log pj
=const
@ log x1
@ log pk
=const
=
=const
1
m
< 0;
m
(4.33)
=
1
> 0; j = 2; :::; m:
m
(4.34)
=
0; k = n
(4.35)
r; :::; n:
Given that m = n r 1 it is clear that as r increases the elasticity (4.33)
decreases in absolute value and (4.34) increases. We also have
@ log x1
@ log pk
=
y=const
pk Ak
< 0; k = n
y0
r; :::; n
The model can be used to illustrate in part the comparative statics of someone who is subject to a quota ration xi
Ai where the rationing constraint
is assumed to be binding in the case of goods n r to n. However, it is not
rich enough to allow us to determine which commodities are consumed at a
conventional equilibrium with MRS =price ratio, like (4.29), and which will be
constrained by the ration. Parts 2 and 3 show clearly how the compensated
demand becomes “steeper” (less elastic with respect to its own price) the more
external constraints are imposed –as in the “short-run” problem of the …rm.
c Frank Cowell 2006
53
Microeconomics
CHAPTER 4. THE CONSUMER
Exercise 4.10 Show that if the utility function is homothetic, then ICV = IEV
Outline Answer
Let x0 be optimal for p0 at
0
and x1 be optimal for p1 at
0
:
x2
αx1
x1
αx0
υ1
x0
υ0
x1
0
Figure 4.5: Homothetic preferences
Because of homotheticity, x0 must be optimal for p0 at
optimal for p1 at 1 : see Figure 4.5.
Hence
X
C(p0 ; 0 ) =
p0i x0i ;
X
C(p0 ; 1 ) =
p0i x0i ;
X
C(p1 ; 0 ) =
p1i x1i ;
X
C(p1 ; 1 ) =
p1i x1i
So in this special case we have
and the result follows.
P 1 1
p x
ICV = P 0i 0i
pi xi
P 1 1
p x
IEV = P i0 i0
pi xi
c Frank Cowell 2006
54
1
and
x1 be
Microeconomics
Exercise 4.11 Suppose an individual has Cobb-Douglas preferences given by
those in Exercise 4.3.
1. Write down the consumer’s cost function and demand functions.
2. The republic of San Serrife is about to join the European Union. As a
consequence the price of milk will rise to eight times its pre-entry value. but
the price of wine will fall by …fty per cent. Use the compensating variation
to evaluate the impact on consumers’ welfare of these price changes.
3. San Serrife economists have estimated consumer demand in the republic
and have concluded that it is closely approximated by the demand system
derived in part 1. They further estimate that the people of San Serrife
spend more than three times as much on wine as on milk. They conclude
that entry to the European Union is in the interests of San Serrife. Are
they right?
Outline Answer
1. Using the results from previous exercises we immediately get
1
p1
C(p; ) =
2
p2
1
:::
pn
2
n
:
n
This is su¢ cient. However, it may be useful to see the proof from …rst
principles. The relevant Lagrangean is
#
"
n
n
X
X
(4.36)
pi xi +
i log xi
i=1
i=1
The …rst-order conditions are:
i
xi =
; i = 1; 2; :::; n:
pi
=
n
X
i
(4.37)
log xi
(4.38)
i=1
From the n equations (4.37) we get at the optimum:
Pn
n
X
pi xi
= Pi=1
=
pi xi = y
n
i
i=1
where y is the budget, or minimised cost and
we get, using (4.37):
=
n
X
i
log
i
+ log
i=1
Using (4.39) and writing
n
X
i=1
Pn
i=1
i
log
i
=
i
Pn
55
i
i=1
n
X
i
= 1. From (4.38)
log pi
(4.40)
i=1
log A, equation (4.40) gives:
y = Ae p1 1 p2 2 :::pnn = C(p; ):
c Frank Cowell 2006
(4.39)
i=1
(4.41)
Microeconomics
CHAPTER 4. THE CONSUMER
This is the required cost function. The demand functions are known from
Exercise 4.2 or are obtained immediately from (4.37) and (4.39):
xi =
iy
; i = 1; 2; :::; n:
pi
(4.42)
2. Let p denote the original price vector, p
^ the price vector after entry.
Observe that p^milk = 8pmilk ; p^wine = 12 pwine . So, using (4.41):
C(^
p; ) = Ae p^1 1 p^2 2 :::^
pnn = Ae p1 1 p2 2 :::pnn b = bC(p; ):
(4.43)
where
1
(4.44)
[ ] wine = 23 milk wine
2
Using the de…nition in the notes the compensating variation is therefore
b := [8]
milk
CV(p ! p
^ ) := C(p; )
C(^
p; ) = [1
b] C(p; )
(4.45)
Clearly, the consumer will bene…t from p ! p
^ if CV(p ! p
^ ) > 0: the cost
of living – interpreted as the cost of hitting the original level of utility –
goes down. This condition is satis…ed if, and only if, b < 1.
3. Notice that, from (4.42), i = pi xi =y – the budget share of commodity
i. So, since we are told that wine > 3 milk it is clear that b < 1. The
economists are right!
c Frank Cowell 2006
56
Microeconomics
Exercise 4.12 In a two-commodity world a consumer’s preferences are represented by the utility function
1
U (x1 ; x2 ) = x12 + x2
where (x1 ; x2 ) represent the quantities consumed of the two goods and
non-negative parameter.
is a
1. If the consumer’s income y is …xed in money terms …nd the demand
functions for both goods, the cost (expenditure) function and the indirect
utility function.
2. Show that, if both commodities are consumed in positive amounts, the
compensating variation for a change in the price of good 1 p1 ! p01 is
given by
2 2
1
p2 1
:
0
4
p1
p1
3. In this case, why is the compensating variation equal to the equivalent
variation and consumer’s surplus?
Outline Answer
x2
x1
Figure 4.6: Possible corner solution
First sketch the utility function. Note that the indi¤erence curves touch
the axes – it is possible that one or other commodity is not consumed at the
optimum – See Figure 4.6. In this case it is easiest to substitute directly from
the budget constraint (binding at the optimum)
p1 x1 + p2 x2 = y
into the utility function. The consumer will then choose x1 to maximise
1
x12 +
c Frank Cowell 2006
y
57
p1 x1
p2
Microeconomics
CHAPTER 4. THE CONSUMER
The FOC is
2
p1
=0
p2
1
2
[x1 ]
which suggests that demands are
x1
x2
2
1
=
D (p1 ; p2 ; y)
D2 (p1 ; p2 ; y)
=4
h
p2
2p1
i2
2
y
p2
p2
4 p1
3
5
But this neglects the possibility that we may be at a corner. Note that a strictly
positive amount of good 2 requires
2
p1 > p1 :=
p22
4 y
So demand functions are given by
x1 = D1 (p1 ; p2 ; y) =
x2 = D2 (p1 ; p2 ; y) =
Also, if p1 > p1 , maximised utility is
V (p1 ; p2 ; y)
Otherwise V (p1 ; p2 ; y) =
implies:
q
y
p1 :
8 h
i2
>
< 2pp12
>
:
8
<
y
p1
2 p
2
4 p1
y
p2
:
0
= U (x1 ; x2 )
2
p2
y
=
+
2p1
p2
2
p2
y
=
+
4p1
p2
if p1 > p1
(4.46)
otherwise
if p1 > p1
(4.47)
otherwise
2
p2
4p1
(4.48)
Also note ( for the case p1 > p1 ) that (4.48)
2
V1 (p1 ; p2 ; y)
=
V2 (p1 ; p2 ; y)
=
Vy (p1 ; p2 ; y)
=
p2
x1
=
<0
4p21
p2
2
y
x2
=
<0
4p1
p22
p2
1
>0
p2
so that we immediately see that Roy’s identity holds. To …nd the cost function
write = V (p; y) and solve for y in terms of p and . This gives
2
=
C(p; )
c Frank Cowell 2006
=
p2
C(p; )
+
4p1
p2
2 2
p2
p2
4p1
58
(4.49)
Microeconomics
for the case p1 > p1 and
C(p; ) = p1
otherwise.
h i2
2 Using the de…nition of the compensating variation
2
V (p1 ; p2 ; y)
V (p01 ; p2 ; y
CV)
CV(p ! p0 ) =
=
=
2 2
p2
4
y
p2
+
4p1
p2
2
y CV
p2
+
4p01
p2
1
p01
1
p1
(4.50)
As an alternative method use the consumer’s cost function (4.49). Clearly
we have:
2 2
2 2
p2
p2
C(p; ) C(p0 ; ) =
0
4p1
4p1
3 In this case the income e¤ect on commodity 1 is zero if p1 > p1 – see
equation (4.46). In the special case of zero income e¤ects CV=EV=CS
c Frank Cowell 2006
59
Microeconomics
CHAPTER 4. THE CONSUMER
Exercise 4.13 Take the model of Exercise 4.12. Commodity 1 is produced by
a monopolist with …xed cost C0 and constant marginal cost of production c.
Assume that the price of commodity 2 is …xed at 1 and that c > 2 =4y.
1. Is the …rm a “natural monopoly”?
2. If there are N identical consumers in the market …nd the monopolist’s
demand curve and hence the monopolist’s equilibrium output and price p1 .
3. Use the solution to Exercise 4.12 to show the aggregate loss of welfare
L(p1 ) of all consumers’ having to accept a price p1 > c rather than being
able to buy good 1 at marginal cost c. Evaluate this loss at the monopolist’s
equilibrium price.
4. The government decides to regulate the monopoly. Suppose the government
pays the monopolist a performance bonus B conditional on the price it
charges where
B = K L(p1 )
and K is a constant. Express this bonus in terms of output. Find the
monopolist’s new optimum output and price p1 . Brie‡y comment on the
solution.
Outline Answer
1. It is easy to see that the cost function is subadditive and therefore the
…rm is a natural monopoly.
2. Because consumers are identical we can just multiply the demand of one
consumer by N to get the market aggregates. We use this throughout the
answer.
If p2 is normalized to 1 then, given that there are N identical consumers
the market demand curve is given by
2
q = N x1 = N
2p1
which on rearranging gives
p1 =
2
s
N
q
(4.51)
p
This gives the average revenue curve. So total revenue is 2 N q. Given
the structure of costs speci…ed in the question the monopolist’s pro…ts are
p
N q C0 cq
(4.52)
p1 q [C0 + cq] =
2
Di¤erentiating (4.52) we …nd the FOC characterising the monopolist’s
optimum as
s
N
c=
(4.53)
4 q
c Frank Cowell 2006
60
Microeconomics
where the expression on the right-hand side of (4.53) is marginal revenue.
Using (4.53) we …nd that the monopolist’s equilibrium output is given by
h i2
q =N
4c
Using (4.51) the price charged will be
s
p1 =
2
N
2
N [ =4c]
in other words
p1 = 2c
(4.54)
Maximised pro…ts are
p1 q
cq
C0
=
2cN
2
N
16c
=
h
4c
i2
cN
h
4c
i2
C0
C0
3. Using (4.50) and multiplying by N , the (absolute) loss of welfare of each
consumer of having to buy at price p1 rather than at marginal cost c is
given by
N 2 1
1
L(p1 ) = N CV =
(4.55)
4
c p1
Using (4.54) to evaluate (4.55)
L(p1 )
1
N 2 1
4
c 2c
2
N
>0
8c
=
=
4. Pro…ts, including the performance bonus, are now
p
p1 q C0 cq + B =
N q C0 cq + B
2
(4.56)
where, from the de…nition in the question and (4.55)
B
= K
= K
L(p1 )
N 2 1
4
c
1
p1
(4.57)
Given (4.51) we …nd that (4.57) can be written as
r
2
N 2
q
1
B(q) =
+K
4
N
c
2
p
N
=
Nq
+K
2
4c
(4.58)
and so, substituting from (4.58) into (4.56) we get
p1 q
C0
c Frank Cowell 2006
cq + B =
p
Nq
61
2
cq + K
C0
N
4c
(4.59)
Microeconomics
CHAPTER 4. THE CONSUMER
The FOC for maximising (4.59)
2
s
N
=c
q
and so
q
p1
= N
h
2c
= c < p1
i2
>q
Of course this is just the solution “price equal to marginal cost.”The bonus
scheme has made the monopolist simulate the outcome of a competitive
industry.
c Frank Cowell 2006
62
Chapter 5
The Consumer and the
Market
Exercise 5.1 A peasant consumer has the utility function
a log (x1 ) + [1
a] log (x2
k)
where good 1 is rice and good 2 is a “basket” of all other commodities, and
0 < a < 1, k 0.
1. Brie‡y interpret the parameters a and k.
2. Assume that the peasant is endowed with …xed amounts (R1 ; R2 ) of the two
goods and that market prices for the two goods are known. Under what
circumstances will the peasant wish to supply rice to the market? Will the
supply of rice increase with the price of rice?
3. What would be the e¤ ect of imposing a quota ration on the consumption
of good 2?
Outline Answer
1. k is the minimum amount of “other goods”required for subsistence. Consider the amount of income that is in excess of this minimum (i.e. the
amount y k): the parameter a is the proportion of this that will be
spent on cereal consumption.
2. The person’s budget constraint is
px1 + x2
y
(5.1)
where
y := pR1 + R2 ;
(5.2)
and p is the price of rice. The Lagrangean for this problem is
log (x1 ) + [1
] log (x2
63
k) + [y
px1
x2 ]
Microeconomics
CHAPTER 5. THE CONSUMER AND THE MARKET
and so the FOCs for an interior optimum are:
p
x1
1
x2
k
=
0
=
0
which , on rearranging, give
px1
=
k
=
x2
1
:
Using the budget constraint we then have y
FOC, the demand for rice will be given by
x1
=
=
p
[y
k = 1= and so, using the
k]
R1 +
p
[R2
k]
(5.3)
The person will want to supply to the market if (5.3) is less than R1 . The
supply of rice is
S(p) := R1 x1
(5.4)
so that
S(p) = R1 [1
]
p
[R2
k]
(5.5)
Supply of rice increases/decreases with its price according as R2 ? k.
3. Let the quota ration on good 2 be c. If the peasant’s optimal consumption
of good 2 (found from 5.1-5.3) is less than or equal to c then obviously
nothing changes: (5.5) is still the relevant relationship. Otherwise we have
x2 = c and utility maximisation and conditions (5.1,5.2) imply
x1 = R1 +
R2
c
p
(5.6)
Supply of rice to the market must be less than without the ration and is
given by c pR2 ; if supply is positive, it must be decreasing in the price of
rice.
c Frank Cowell 2006
64
Microeconomics
Exercise 5.2 Take the model of Exercise 5.1. Suppose that it is possible for
the peasant to invest in rice production. Sacri…cing an amount z of commodity
2 would yield additional rice output of
b 1
e
z
where b > 0 is a productivity parameter.
1. What is the investment that will maximise the peasant’s income?
2. Assuming that investment is chosen so as to maximise income …nd the
peasant’s supply of rice to the market.
3. Explain how investment in rice production and the supply of rice to the
market is a¤ ected by b and the price of rice. What happens if the price of
rice falls below 1=b?
Outline Answer
1. By de…nition net income is
y = pR1 + R2 + bp 1
e
z
z
The FOC for an interior maximum is
@y
= bpe
@z
z
1=0
which has as a solution
z = log(bp)
Clearly this only makes sense if bp
optimal investment is
1: otherwise investment is zero. So,
z (p; b) := max (log(bp); 0)
So maximised income is:
h
y (p; b) := pR1 + R2 + bp 1
e
z (p;b)
i
z (p; b)
2. In this model the total output of rice is now
h
i
R1 + b 1 e z (p;b)
the consumption of rice is
x1 =
y (p; b)
p
k
and so the supply of rice is given by
h
S (p; b) = R1 + b 1
c Frank Cowell 2006
e
65
z (p;b)
i
y (p; b)
p
k
:
Microeconomics
CHAPTER 5. THE CONSUMER AND THE MARKET
3. Evaluating y for p
1=b we have
pR1 + R2 + bp
1
log (bp)
and so
S (p; b) = R1 + b 1
1
bp
pR1 + R2 + bp
1
log (bp)
p
k
:
which simpli…es to
S (p; b) = [1
] R1 + b
1
p
R2
log (bp)
p
k
:
(5.7)
Di¤erentiating (5.7):
@S (p; b)
= [1
@p
]
R2
1
+
2
p
@S (p; b)
= [1
@b
]+
k
log (bp)
p2
bp
:
Clearly supply increases with b. For small p supply increases with p; this
may be reversed for large p.
c Frank Cowell 2006
66
Microeconomics
Exercise 5.3 Consider a household with a two-period utility function of the
form
log x1 + [1
] log x2
where xi is the amount of consumption in period i. . Suppose that the individual receives an exogenously given income stream (y1 ; y2 ) over the two periods,
assuming that the individual faces a perfect market for borrowing and lending at
a uniform rate r.
1. Examine the e¤ ects of varying y1 ,y2 and r on the optimal consumption
pattern.
2. Will …rst-period savings rise or fall with the rate of interest?
3. How would your answer be a¤ ected by a total ban on borrowing?
Outline Answer
1. If the individual can lend or borrow as much as he wishes at the going
interest rate the given income stream is equivalent to an asset A in period
1 where
y2
(5.8)
A = y1 +
1+r
x2
x*
y
1+r
x1
S
Figure 5.1: Equilibrium of a saver
2. The time-speci…c income endowments are the individual resources and
relative prices are as in Exercise 4.7 . So we immediately …nd the optimal
consumption. Equilibrium for persons with two di¤erent income streams
c Frank Cowell 2006
67
Microeconomics
CHAPTER 5. THE CONSUMER AND THE MARKET
x2
y
x*
1+r
x1
B
Figure 5.2: Equilibrium of a borrower
x2
x′ = y
x1
Figure 5.3: Income-constrained equilibrium
c Frank Cowell 2006
68
Microeconomics
–a saver and a borrower –is illustrated in Figures 5.1 and 5.2 respectively.
Savings in the …rst period must be:
S := y1
x1 = y1
A = [1
] y1
y2
1+r
(5.9)
Clearly this increases with r.
3. If there is a ban on borrowing and y1 is low then the consumer may be
constrained such that x1 = y1 , and variations in r may leave the solution
unchanged –see point x0 in Figure 5.3.
c Frank Cowell 2006
69
Microeconomics
CHAPTER 5. THE CONSUMER AND THE MARKET
Exercise 5.4 A consumer lives for two periods and has the utility function
log (x1
k) + [1
] log (x2
k)
where xt is consumption in period t, and , k are parameters such that 0 < < 1
and k 0. The consumer is endowed with an exogenous income stream (y1 ; y2 )
and he can lend on the capital market at a …xed interest rate r, but is not allowed
to borrow.
1. Interpret the parameters of the utility function.
2. Assume that y1
y where
y := k +
y2 k
1+r
1
Find the individual’s optimal consumption in each period.
3. If y1
y what is the impact on period 1 consumption of
(a) an increase in the interest rate?
(b) an increase in y1 ?
(c) an increase in y2 ?
4. How would the answer to parts 2 and 3 change if y1 < y ?
Outline Answer
1. The number k represents a subsistence minimum consumption; is the
share of period-1 non-subsistence consumption in non-subsistence lifetime
income.
2. The budget constraint is
x1 +
x2
1+r
y
(5.10)
y := y1 +
y2
:
1+r
(5.11)
] log (x2
k) +
where
The Lagrangean is
log (x1
k) + [1
y
and so the FOCs for an interior optimum are:
x1
1
x2
c Frank Cowell 2006
k
k
1
1+r
70
=
0
=
0
x1
1
x2
1+r
Microeconomics
which , on rearranging give
x1
k
=
x2 k
1+r
=
1
:
Using the budget constraint we then have y k k=[1 + r]1 = 1= and
so, using the FOC, the demand for period-1 consumption will be given by
x1 = k +
y
k
k
= [1
1+r
]k +
y1 +
y2 k
1+r
(5.12)
The person will save a positive amount if (5.12) is less than y1 . Saving is
given by:
y2 k
(5.13)
S(r; y1 ; y2 ) := [1
] [y1 k]
1+r
which is positive if
y1 > y := k +
y2 k
1+r
1
3. Di¤erentiating (5.12) we have
x1 = k +
y
k
k
= [1
1+r
@x1
@y1
@x1
@y2
@x1
@r
=
=
]k +
y1 +
>0
1+r
=
y2 k
1+r
(5.14)
(5.15)
>0
2
[1 + r]
(5.16)
[y2
k]
(5.17)
Period-1 consumption decreases/increases with the interest rate according
as y2 ? k.
4. If y1 < y the individual is constrained in period 1 by current income.x1 =
y1 . Clearly
@x1
@x
@x
= 1; 1 = 0; 1 = 0:
@y1
@y2
@r
c Frank Cowell 2006
71
Microeconomics
CHAPTER 5. THE CONSUMER AND THE MARKET
Exercise 5.5 In an n-commodity economy a consumer lives for T periods and
has preferences that are represented by a di¤ erentiable utility function.
1. Show that Leontief ’s Theorem implies that if the marginal rate of substitution between commodity i and commodity j in period t is independent of
consumption in any other period then the person’s preferences can always
be represented by the additive form
T
X
v (xt ; t)
(5.18)
t=1
where xt is the vector of consumption goods at time t and the within-period
utility function v ( ; t) may be di¤ erent in each period.
2. Show that if the MRSij in period t equals the MRSij in period t0 whenever
xt = xt0 then within-period utility in (5.18) can be written as
v (x; t) = u (x) ' (t)
(5.19)
3. Suppose that v in (5.19) has the property that, whenever the consumption
vector is the same in all periods, the MRS of good i in period t for good
i in period t + 1 is independent of t. Show that within-period utility can
then be written
v (x; t) = t u (x)
(5.20)
Outline Answer
1. Consider the utility function
U (x1 ; x2 ; :::; xT )
If MRS between two goods in period t < T is independent of consumption
in period T then, by Leontief ’s Theorem there exist functions and
such that
U (x1 ; x2 ; :::; xT ) =
( (x1 ; x2 ; :::; xT
1 ) ; xT )
(5.21)
Given that is a continuous monotonic function and that a utility function is de…ned up to an arbitrary monotonic transformation we can rewrite
the right-hand side of (5.21 as
(x1 ; x2 ; :::; xT
1)
+ f (xT )
for some function f . Repeating this argument we get for T
we get
U (x1 ; x2 ; :::; xT ) = +v (x1 ; 1) + v (x2 ; 2) + ::: + v (xT
1; T
1, T
1) + v (xT ; T )
where v ( ; T ) is just the function f in a more convenient notation.
c Frank Cowell 2006
72
2, ...
Microeconomics
2. Without loss of generality let t0 = 1. If the consumption bundle in period
t is given as x then the condition on the MRSij implies
vi (x; 1)
vi (x; t)
=
vj (x; t)
vj (x; 1)
which means
vi (x; t)
vj (x; t)
=
vi (x; 1)
vj (x; 1)
But this means that each side of the equation is independent of x i and j;
we may write it as ' (t). Therefore we may write
vi (x; t) = ui (x) ' (t)
where u ( ) is some function independent of t and ui ( ) is the partial derivative with respect to the ith component. Integrating yields the result.
3. The required MRS is given by
' (t + 1)
ui (x) ' (t + 1)
=
:
ui (x) ' (t)
' (t)
If the right-hand side of this is independent of t then call this ratio . We
have
' (2)
' (3)
' (t)
de…ning ' (1) arbitrarily as
c Frank Cowell 2006
=
=
' (1)
2
' (1)
:::
= t 1 ' (1)
gives the result.
73
Microeconomics
CHAPTER 5. THE CONSUMER AND THE MARKET
Exercise 5.6 Suppose an individual lives for T periods and has a utility function of the form (5.18) where within-period utility is given by (5.20), consumption in period t is given by the scalar xt and the function u has the isoelastic
form
1
u (x) =
x1
(5.22)
1
where is a non-negative parameter.
1. Suggest an appropriate limiting form for u as
! 1.
2. The individual has a known exogenous income stream (y1 ; y2 ; :::; yT ) and
the interest rate in period t is rt . Find the …rst-order condition for maximising utility.
3. Using the approximation log (1 + z) ' z show that the optimum consumption path is approximated by
log (xt ) = k + rt :
where k and are
4. How is
constants.
related to the parameter ?
Outline Answer
1. Note that, taking x1
1 as a function of , we have
@
@
x1
x1
1 =
log x
(5.23)
So, by l’Hôpital’s rule, we know that, for any x > 0:
lim
x1
1
1
!1
=
lim
=
log x
!1
x1
log x
1
This gives the appropriate limiting form.
2. The price of one unit of consumption in period t expressed in terms of
values at t 1 is 1= [1 + rt ] for t = 2; 3; :::. So the price of period t’s
consumption expressed in units of consumption in period 1 is
pt :=
t
Y
1
; t = 2; 3; :::
1
+
ri
i=2
we may arbitrarily de…ne p1 := 1. Note that
pt
pt
:=
1
1
; t = 2; 3; :::
1 + ri
So the relevant budget constraint is
T
X
pt xt
t=1
c Frank Cowell 2006
74
A
(5.24)
Microeconomics
where
A :=
T
X
pt y t
(5.25)
t=1
is the present value of lifetime income. The problem is then in the familiar form of consumer optimisation with many goods and is a simple
generalisation of the problem in Exercise 5.3. The Lagrangean is
"
#
T
T
X
1 X t 1
xt +
A
pt xt
(5.26)
1 t=1
t=1
The FOC for a maximum is
t
xt
pt = 0
3. Rearranging the FOC we have
t log
log xt = log
+ log pt
(5.27)
and for the previous period we have
[t
1] log
log xt
1
= log
+ log pt
Subtracting and using the di¤erence operator we get:
log xt
=
=
log
log pt
log + log (1 + rt )
On rearranging we get the result.
4. From the above we have
log (xt ) = k + rt :
where k := log = and
c Frank Cowell 2006
:= 1= .
75
1
(5.28)
Microeconomics
CHAPTER 5. THE CONSUMER AND THE MARKET
Exercise 5.7 Suppose a person is endowed with a given amount of non-wage
income y and an ability to earn labour income which is re‡ected in his or her
market wage w. He or she chooses `, the proportion of available time worked,
1
in order to maximise the utility function x [1 `]
where x is total money
income –the sum of non-wage income and income from work. Find the optimal
labour supply as a function of y, w, and . Under what circumstances will the
person choose not to work?
Outline Answer
The consumer’s problem is to maximise utility subject to
x = w` + y
`
0
We have to be careful of the non-negativity constraint on ` here. Substituting
the budget constraint into the utility function the problem becomes:
1
max [[w` + y] [1
`]
` 0
(5.29)
Take log of the maximand in (5.29) and di¤erentiate with respect to ` to get:
1
1
w
w` + y
(5.30)
`
which is zero if
0
=
w` + w
` =
[1
y
] :
w
[1
]y
But for this to make sense as a solution we must have `
w
y
1
:
In other words optimal labour supply is given by
8
[1
] wy , if w y 1
<
` =
:
0,
otherwise
c Frank Cowell 2006
76
0 which requires
(5.31)
::
(5.32)
Microeconomics
Exercise 5.8 A household consists of two individuals who are both potential
workers and who pool their budgets. The preferences are represented by a single
utility function U (x0 ; x1 ; x2 ) where x1 is the amount of leisure enjoyed by person
1, x2 is the amount of leisure enjoyed by person 2, and x0 is the amount of the
single, composite consumption good enjoyed by the household. The two members
of the household have, respectively (T1 ; T2 ) hours which can either be enjoyed as
leisure or spent in paid work. The hourly wage rates for the two individuals are
w1 , w2 respectively, they jointly have non-wage income of y, and the price of
the composite consumption good is unity.
1. Write down the budget constraint for the household.
2. If the utility function U takes the form
U (x0 ; x1 ; x2 ) =
2
X
i
log(xi
i)
:
(5.33)
i=0
where i and i are parameters such that i 0 and i > 0, 0 + 1 + 2 =
1, interpret these parameters. Solve the household’s optimisation problem
and show that the demand for the consumption good is:
x0 =
+
0
0
[[y + w1 T1 + w2 T2 ]
[
0
+ w1
1
+ w2
2 ]]
3. Write down the labour supply function for the two individuals.
4. What is the response of an individual’s labour supply to an increase in
(a) his/her own wage,
(b) the other person’s wage, and
(c) the non-wage income?
Outline Answer
The time endowments of the two individuals along with the non-wage income
correspond to the resources Ri . Given that the price of good 0 is normalised
to1 the budget constraint is therefore
x0 + w1 x1 + w2 x2
y + w1 T1 + w2 T2
(5.34)
Because utility function is of same form as in exercise4.6, solution is essentially
as for model in 4.6 –see equation. Demand for consumption of commodity i is
xi =
with w0
i
+
i
wi
[[y + w1 T1 + w2 T2 ]
[
0
+ w1
1
+ w2
2 ]]
(5.35)
1. So supply of labour by person 1 is:
`1 := T1
x1 = [T1
1 ] [1
1]
1
w1
[[y + w2 T2 ]
[
0
+ w2
2 ]]
(5.36)
Therefore we have:
@`1
= 12 [[y + w2 T2 ]
@w1
w1
c Frank Cowell 2006
77
[
0
+ w2
2 ]]
(5.37)
Microeconomics
CHAPTER 5. THE CONSUMER AND THE MARKET
which may be positive or negative, depending on non-wage income, minimum
consumption needs and the wage of the other person. Also
@`1
@w2
@`1
@y
=
=
1
w1
1
w1
[T2
<0
2]
<0
(5.38)
(5.39)
The other person’s wage acts on person 1’s labour supply in a manner that is
very similar to non-wage income.
c Frank Cowell 2006
78
Microeconomics
Exercise 5.9 Let the demand by household 1 for good 1 be given by
9
8
y
if p1 > a =
<
4p1
y
if p1 < a
;
x11 =
;
: y 2p1 y
or
if
p
=
a
i
4a
2a
where a > 0.
1. Draw this demand curve and sketch an indi¤ erence map that would yield
it.
2. Let household 2 have identical income y: write down the average demand
of households 1 and 2 for good 1 and show that at p1 = a there are now
three possible values of 21 [x11 + x21 ].
3. Extend the argument to nh identical consumers. Show that nh ! 1 the
possible values of the consumption of good 1 per household becomes the
y
y
; 2a
entire segment 4a
.
Outline Answer
1. From the question the demand function for 1 consumer looks like that
in the …gure 5.4 below: there is a discontinuity between the two points
y
y
4a ; a and 2a ; a . By analogy with the analysis of the …rm it is clear
that the indi¤erence curve must look like that depicted in the following
Figure 5.5.
p1
a
x1
y/4a
y/2a
Figure 5.4: Demand by one-household
2. If household 2 has identical income y then average demand of households
1 and 2 for good 1 must be identical to that of household 1 for p1 < a and
p1 > a. At p1 = a there are the following possibilities:
(a)
Both households at
c Frank Cowell 2006
y
4a :
so 12 x11 + x12 =
79
y
4a .
Microeconomics
CHAPTER 5. THE CONSUMER AND THE MARKET
x2
x1
Figure 5.5: Non-quasiconcave U
y
1
1
1
2a : so 2 x1 + x2
y
y
4a , the other at 2a :
Both households at
=
One household at
so
y
2a .
1
1
2 x1
+ x12 =
3y
4a .
3. If there are nh identical consumers and p1 = a then if h of them are at
y
the remaining nh h of them are at 4a
so that average demand is
2
h
nh h y
nh + h y
+
=
nh
nh
4a
nh 4a
So at p1 = a the average demand for good 1 must belong to the set
1+
h y
: h = 0; 1; :::; nh
nh 4a
As nh ! 1 this set becomes dense in
c Frank Cowell 2006
80
y
y
4a ; 2a
.
y
2a
Chapter 6
A Simple Economy
Exercise 6.1 In an economy the activity of digging holes in the ground is carried out by self-employed labourers (single–person …rms). The production of
one standard-sized hole requires a minimum input of one unit of labour. No
self-employed labourer can produce more than one hole.
1. Draw the technology set Q for a single …rm.
2. Draw the technology set Q for two …rms.
3. Which of the basic production axioms are satis…ed by this simple technology?
Outline Answer
q2
q2
1
2
q1
q1
Figure 6.1: Convexi…cation of Production
1. See left-hand panel of Figure 6.1. Good 1 is labour, good 2 is holes.
2. See right-hand panel of Figure 6.1; this has been rescaled to be compatible
with the single-…rm case..
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Microeconomics
CHAPTER 6. A SIMPLE ECONOMY
3. Nonconvexities are present so that the divisibility assumption is violated.
Nevertheless the production can be (approximately) convexi…ed by increasing the number of …rms.
c Frank Cowell 2006
82
Microeconomics
Exercise 6.2 Consider the following four examples of technology sets Q:
q : q12 + q22 + q3 + q4 0; q1 ; q2 0; q3 ; q4 0
n
o
B :
q : q1 [ q2 ]
[ q3 ]
0; q1 0; q2 ; q3 0
A
:
1
log (q2 q3 ) 0; q1 0; q2 ; q3 0
2
n
q4
D :
q : q1 + q2 + max q3 ;
0; q1 ; q2 0; q3 ; q4
C
:
q : log q1
o
0
1. Check whether basic production axioms are satis…ed in each case.
2. Sketch their isoquants and write down the production functions.
3. In cases B and C express the production function in terms of the notation
used in chapter 2.
4. In cases A and D draw the transformation curve.
Outline Answer:
1. Axioms:
A:
B:
C:
D:
Additivity is not satis…ed
All axioms are satis…ed if
All axioms satis…ed.
All axioms are satis…ed if
=
=
< 1:
> 0:
2. Isoquants and production functions:
A: Isoquants are straight lines.
(q) = q12 + q22 + q3 + q4
B: If
=
=
< 1 isoquants will be similar to hyperbolas.
(q) = q1
[ q2 ]
C: Similar to B.
(q) = log q1
[ q3 ]
1
log (q2 q3 )
2
D: Isoquants are rectangular.
(q) = q1 + q2 + max q3 ;
3. Single-output production functions. Let z2 =
h
i1=
B: q1
[z2 ] + [z3 ]
p
z2 z3
C: q1
q2 and z3 =
4. Transformation curve (the boundary of the set Q).
A: A quarter circle.
D: A straight line.
c Frank Cowell 2006
83
q4
q3 . Then:
Microeconomics
CHAPTER 6. A SIMPLE ECONOMY
Exercise 6.3 Suppose two identical …rms each produce two outputs from a
single input. Each …rm has exactly 1 unit of input. Suppose that for …rm 1 the
amounts q11 , q22 it produces of the two outputs are given by
q11
q21
1
=
=
[1
1
]
where 1 is the proportion of the input that …rm 1 devotes to the production of
good 1 and and depend on the activity of …rm 2 thus
1+2 2
1 + 2[1
=
=
2
]:
where 2 is the proportion of the input that …rm 2 devotes to good 1. Likewise
for …rm 2:
q12
q22
0
0
=
=
0 2
0
[1
= 1+2 1
= 1 + 2[1
2
]
1
]:
1. Draw the production possibility set for …rm 1 if …rm 2 sets
…rm 2’s production possibility set if …rm 1 sets 1 = 12 .
2
=
1
2
and
2. Draw the combined production-possibility set.
Outline Answer:
1
1. The production possibility set for …rm 1 if …rm 2 sets 2 = are given by
2
the top left-hand panel of Figure 6.2 below. Firm 2’s position if …rm 1 sets
1
1
=
is given in the top right-hand panel. The combined production
2
possibility set is as shown in the lower panel.
1
; then by de…nition
=
= 2, and …rm 1’s production
2
1
1
1
boundary is given by all the points (2 ; 2[1
]) where 0
1. A
similar argument applies to …rm 2. Now consider the sum of their outputs.
1
2
This will be given by points such as 1 + 2 + 4 1 2 , 2
+ 4[1
1
2
2
1
2
][1
]. So, for example, if
=
= 0 we get output (0; 6); if
1
1
= 2 = 1 we get (6; 0) but if 1 = 2 = we get (2; 2).
2
2. Let
2
=
c Frank Cowell 2006
84
Microeconomics
q2`1
q2`2
Firm 1
Firm 2
q1`1
0
0
q2
Combined
Firms
Figure 6.2: Externalities and nonconvexity
c Frank Cowell 2006
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Microeconomics
CHAPTER 6. A SIMPLE ECONOMY
Exercise 6.4 Take the model of Exercise 2.14. Assuming that production is
organised to maximise pro…ts at given prices show that pro…t-maximising net
outputs of goods 1 and 2 are:
q1
=
q2
=
A
p1
2
A
p2
2
where pi is the price of good i expressed in terms of commodity 3, and that
maximised pro…ts are
2
2
[p1 ] + [p2 ]
:
=A
4
Outline Answer:
Using standard notation for net output pro…ts are given by
=
3
X
pi y i
(6.1)
i=1
If pro…ts are maximised then production must take place on the transformation
curve. Therefore, substituting we get:
2
To maximise
FOCs:
2
[y1 ] + [y2 ]
A
= p1 y 1 + p2 y 2
(6.2)
just maximise (6.2) with respect to y1 and y2 . This gives the
pi
2
yi = 0
A
y1
=
y2
=
(6.3)
i = 1; 2, and so
A
p1
2
A
p2
2
(6.4)
(6.5)
from which, using (6.2), pro…ts are
2
=A
c Frank Cowell 2006
2
[p1 ] + [p2 ]
:
4
86
(6.6)
Microeconomics
Exercise 6.5 Take the model of Exercise 5.3 but suppose that income is exogenously given at y1 for the …rst period only. Income in the second period can
be obtained by investing an amount z in the …rst period. Suppose y2 = (z),
where is a twice-di¤ erentiable function with positive …rst derivative and negative second derivative and (0) = 0, and assume that there is a perfect market
for lending and borrowing.
1. Write down the budget constraint.
2. Explain the rôle of the Decentralisation Theorem in this model
3. Find the household’s optimum and compare it with that of Exercise 5.3.
4. Suppose (z) were to be replaced by
a¤ ect the solution?
(z) where
> 1; how would this
Outline Answer:
1. The budget constraint is
x1 +
x2
1+r
A(z)
(6.7)
where the present value of lifetime income is
A(z) := y1
z+
(z)
1+r
(6.8)
x2
iy
*
ix′A= y ′
ix
*
1+r
x1
Figure 6.3: Lifetime income: self investment
2. Use the decentralisation result. Given that borrowing/lending is possible
at the rate of interest r the right thing to do is to maximise the value of
lifetime income (“pro…ts”) at point y in Figure 6.3 and then maximise
utility using this maximised lifetime income (point x ).
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Microeconomics
CHAPTER 6. A SIMPLE ECONOMY
y2
φ (z)
z
Figure 6.4: The production of human capital
3. Otherwise, without a bank loan one would be constrained to point x0 =
y0 := (y1 ; y2 ). Maximising lifetime income implies
@A(z)
=
@z
1+
z (z)
=0
1+r
(6.9)
for an interior solution. This …rst-order condition implies that the optimal
value z must satisfy
(6.10)
z (z ) = 1 + r;
the marginal rate of transformation of current income into future income
equals the price ratio.
1. It is obvious that if were to be replaced by
( > 1) the opportunity
set is “stretched” in the direction of the vertical axis in Figure 6.4. We
expect that maximised lifetime income will increase. So consumption will
increase in both periods. What happens to optimal investment? From the
…rst-order condition (6.9) we …nd
z (z
)=1+r
(6.11)
from which we could get the optimal investment z . If we increase then
the z obtained from (6.11) will change. To see by how much z changes
di¤erentiate (6.11) thus:
z (z
from which we have
c Frank Cowell 2006
) d +
dz
=
d
zz (z
z (z
) dz = 0
)
>0
(z
)
zz
88
(6.12)
Microeconomics
Exercise 6.6 Apply the model of Exercise 6.5 to an individual’s decision to
invest in education.
1. Assume the parameter represents talent. Will more talented people demand purchase more education?
2. How is the demand for schooling related to exogenous …rst-period income
y1 ?
Outline Answer:
The optimal purchase of education z increases with ability , but is independent of initial money income y1 .
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Microeconomics
CHAPTER 6. A SIMPLE ECONOMY
Exercise 6.7 Take the savings model of Exercise 5.4. Suppose now that by
investing in education in the …rst period the consumer can augment his future
income. Sacri…cing an amount z in period 1 would yield additional income in
period 2 of
1 e z
where
> 0 is a productivity parameter.
1. Explain how investment in education is a¤ ected by the interest rate. What
would happen if the interest rate were higher than
1?
2. How is the demand for borrowing a¤ ected by (i) an increase in the interest
rate r and (ii) an increase in the person’s productivity parameter ?
Outline Answer
1. By the separation theorem the right thing for the consumer to do is to
maximise net income and then choose optimal consumption. By de…nition
net income is
y2 + [1 e z ]
M = y1 z +
1+r
The FOC for a maximum is
e z
@M
=
@z
1+r
1=0
which has as a solution
z = z (r; ) := log
(6.13)
1+r
Clearly this only makes sense if
1+r
(6.14)
otherwise investment is zero. If (6.14) holds then z is decreasing in its
…rst argument and increasing in its second argument.
2. Assuming (6.14) holds, maximised income is:
M
= M (r; ) := y1
= y1 + log
log
1+r
y2 +
1+
1+r
1+r
+
y2 +
[1 e
1+r
z
]
(6.15)
Hence in this model optimal period-1 consumption is
x1 = k +
y1
k + log
1+r
1+
y2
k+
1+r
Borrowing is given by
B (r; ) := x1 + z (r; )
c Frank Cowell 2006
90
y1
:
(6.16)
Microeconomics
and so, on evaluating this from (6.13) and (6.16), we have
B (r; ) =
y2
k+
1+r
1 + [1
] log
1+r
[y1
k]
Dei¤erentiating we have
@B (r; )
@r
=
@B (r; )
@
=
y2
k+
2
[1 + r]
1
+
1+r
1
1+r
>0
If borrowing decreased with r in the no-education case – see equation
(5.17) in Exercise 5.4 – it must certainly decrease in this case too. Borrowing increases with .
c Frank Cowell 2006
91
Microeconomics
CHAPTER 6. A SIMPLE ECONOMY
c Frank Cowell 2006
92
Chapter 7
General Equilibrium
Exercise 7.1 Suppose there are 200 traders in a market all of whom behave
as price takers. Suppose there are three goods and the traders own initially the
following quantities:
100 of the traders own 10 units of good 1 each
50 of the traders own 5 units of good 2 each
50 of the traders own 20 units of good 3 each
All the traders have the utility function
1
1
1
U = x12 x24 x34
What are the equilibrium relative prices of the three goods? Which group of
traders has members who are best o¤ ?
Outline Answer:
For each group of traders the Lagrangean may be written
1
1
1
log xh1 + log xh2 + log xh3 +
2
4
4
h
[y h
p1 xh1
p2 xh2
p3 xh3 ]
where h = 1; 2; 3 and y 1 = 10p1 ; y 2 = 5p2 and y 3 = 20p3 : From the …rstorder conditions we …nd that for a trader of type h:
xh1
=
xh2
=
xh3
=
yh
2p1
yh
4p2
yh
4p3
Excess demand for good 1 and 2 are then:
93
Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
E1
E2
=
=
100x11 + 50x21 + 50x31
100x12 + 50x22 + 50x32
1000
250
Substituting in for xhi and y h and putting E1 = E2 = 0 we …nd
p2
p3
+ 500
p1
p1
750
p3
+ 250
4
p2
500 + 125
=
0
p1
p2
=
0
250
p3
p2
= 2 and
= 21 .
p1
p1
Using good 1 as numeraire we immediately see that y 1 = y 2 = y 3 = 10: All
are equally well o¤.
that implies
c Frank Cowell 2006
94
Microeconomics
Exercise 7.2 Consider an exchange economy with two goods and three persons.
Alf always demands equal quantities of the two goods. Bill’s expenditure on group
1 is always twice his expenditure on good 2. Charlie never uses good 2.
1. Describe the indi¤ erence maps of the three individuals and suggest utility
functions consistent with their behaviour.
2. If the original endowments are respectively (5; 0), (3; 6) and (0; 4), compute
the equilibrium price ratio. What would be the e¤ ect on equilibrium prices
and utility levels if
(a) 4 extra units of good 1 were given to Alf;
(b) 4 units of good 1 were given to Charlie?
Outline Answer:
1. Let
=
p1
so that values are measured in terms of good 2.
p2
(a) Alf’s (binding) budget constraint is
xa1 + xa2 = 5
Therefore, given the information in the question, the demand functions are
5
:
xa1 = xa2 =
+1
The utility function consistent with this behaviour is
U a (xa1 ; xa2 ) = min fxa1 ; xa2 g
–see Figure 7.1.
(b) Bill’s budget constraint is
xb1 + xb2 = 3 + 6
From the question we have
xb1 = 2xb2
Therefore:
xb1
=
xb2
=
2+
4
+ 2:
The utility function is Cobb-Douglas:
U b = 2 log xb1 + log xb2
–see Figure 7.2.
c Frank Cowell 2006
95
Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
x2
•
x1
(5,0)
Figure 7.1: Alf’s preferences and demand
x2
•
(3,6)
•
x1
Figure 7.2: Bill’s preferences and demand
c Frank Cowell 2006
96
Microeconomics
(c) Charlie’s budget constraint is
xa1 + xa2 = 4
Given the information in the question we have
xc1
xc2
= 4=
= 0
and utility is
U c = xc1
–see Figure 7.3.
x2
•
(4,0)
•
x1
Figure 7.3: Charlie’s preferences and demand
2. Excess demand for good 2 is :
E2 =
5
+ +2
+1
10:
Putting E2 = 0 yields = 2 or 4. Hence the equilibrium price ratio is
4. Utility levels are U a = 4, U b = log(54) and U c = 1:
(a) Excess demand is now
9
+ +2
+1
10
and the equilibrium price ratio is 2. Utility levels are U a = 6, U b =
log(64) and U c = 2:
(b) Excess demand for good 2 is :
E2 =
5
+ +2
+1
10:
Putting E2 = 0 yields = 2 or 4. Hence the equilibrium price ratio
is 4. Utility levels are U a = 4, U b = log(54) and U c = 4 + 1 = 5:
c Frank Cowell 2006
97
Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
Exercise 7.3 In a two-commodity economy assume a person has the endowment (0; 20).
1. Find the person’s demand function for the two goods if his preferences are
represented by each of the types A to D in Exercise 4.2. In each case
explain what the o¤ er curve must look like.
2. Assume that there are in fact two equal sized groups of people, each with
preferences of type A, where everyone in group 1 has the endowment (10; 0)
with = 21 and everyone in group 2 an endowment (0; 20) with = 34 . Use
the o¤ er curves to …nd the competitive equilibrium price and allocation.
Outline Answer:
1. The income of person h is 20.
(a) If he has preferences of type A then the Lagrangean is
log xh1 + [1
] log xh2 +
xh1
20
xh2
(7.1)
First order conditions for an interior maximum of (7.1) are
xh1
1
xh2
xh1
20
Solving these we …nd
=
1
20
xh2
=
0
=
0
=
0
and so the demands will be
20
xh =
20 [1
]
(7.2)
and the o¤er curve will simply be a horizontal straight line at xh2 =
20 [1
].
(b) If h has preferences of type B then demand will be
xh
xh
xh
= x0 , if >
2 [x0 ; x00 ], if =
= (20= ; 0), if <
where x0 := (0; 20), x00 := (20= ; 0), and their o¤er curve will consist
of the union of the line segment [x0 ; x00 ] and the line segment from
x00 to (1; 0).
(c) If group-2 persons have preferences of type C then their demands will
be
p
xh = x0 , if >
p
xh = x0 or x00 , if =
p
xh = (20= ; 0), if <
p
where x0 := (0; 20), x00 := (20= ; 0), and their o¤er curve will
consist of the union of the point x0 and the line segment from x00 to
(1; 0).
c Frank Cowell 2006
98
Microeconomics
x'
20
B
ce
ren
iffe
ind curve
A
20[1-α]
β
x1
x1
x''
x2
x2
x'
C
D
√γ
δ
x1
x1
x''
Figure 7.4: O¤er Curves for Four Cases
(d) If group-2 persons have preferences of type D then their demands will
be
#
"
xh =
20
+
20
+
and their o¤er curve is just the straight line xh2 = xh1 . These are
illustrated in Figure 7.4.
2. If a type-A person had an income of 10 units of commodity 1 then, by
analogy with part 1, demand would be
x1 =
10
10 [1
]
(7.3)
and the o¤er curve will simply be a vertical straight line at xh1 = 10 .
From (7.2) and (7.3) we have x11 = 10 21 = 5, x22 = 20 1 43 = 5.
Given that there are 10 units per person of commodity 1 and 20 units per
person of commodity 2 the materials balance condition then means that
c Frank Cowell 2006
99
Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
the equilibrium allocation must be
Solving for
must be 3.
x1
=
5
15
x2
=
5
5
from (7.2) and (7.3) we …nd that the equilibrium price ratio
c Frank Cowell 2006
100
Microeconomics
Exercise 7.4 The agents in a two-commodity exchange economy have utility
functions
U a (xa ) = log(xa1 ) + 2 log(xa2 )
U b (xb ) = 2 log(xb1 ) + log(xb2 )
where xhi is the consumption by agent h of good i, h = a; b; i = 1; 2. The property
distribution is given by the endowments Ra = (9; 3) and Rb = (12; 6).
1. Obtain the excess demand function for each good and verify that Walras’
Law is true.
2. Find the equilibrium price ratio.
3. What is the equilibrium allocation?
4. Given that total resources available remain …xed at R := Ra + Rb =
(21; 9), derive the contract curve.
Outline Answer:
1. To get the demand functions for each person we need to …nd the utilitymaximising solution. The Lagrangean for person a is
La (xa ;
a
) := log (xa1 ) + 2 log (xa2 ) +
a
[9p1 + 3p2
p1 xa1
p2 xa2 ]
First-order conditions are
1
x1
2
x2
a
p1 = 0
a
p2 = 0
9p1 + 3p2 p1 x1a
p2 x2a = 0
9
>
=
>
;
De…ne := p1 =p2 and normalise p2 arbitrarily at 1. Then, rearranging
the FOC we get
9
1
x1
=
a =
2
(7.4)
a = x2
;
9 + 3 = x1a + x2a
Subtracting the …rst two equations from the third in (7.4) we can see that
1
a
= 1+3
. Substituting back for the Lagrange multiplier a into the
…rst two parts of (7.4) we see that the …rst-order conditions imply:
x1a
x2a
=
3+ 1
6 +2
(7.5)
Using exactly the same method for person b we would …nd
x1b
x2b
=
8+ 4
4 +2
(7.6)
Using the de…nition we can then …nd the excess demand functions by
evaluating:
Ei := xi a + xi b Ria Rib
(7.7)
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Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
Doing this we get
5
E1
E2
=
10
10
5
(7.8)
Now construct the weighted sum of excess demands. It is obvious that
E1 + E2 = 0
(7.9)
thus con…rming Walras’Law. In equilibrium the materials’balance condition must hold and so excess demand for each good must be zero, unless
the corresponding equilibrium price is zero (markets clear).
2. Solving for E1 = 0 in (7.8) we …nd
prices.
3. The allocation is
x1b
x2b
=
1
2
for the (normalised) equilibrium
16
4
=
4. The contract curve is traced out by the MRS condition
MRSa12 = MRSb12
(7.10)
and the materials balance condition
E=0
(7.11)
From (7.11) we have
xb1
xb2
=
21 xa1
9 xa2
(7.12)
Applying (7.10) we then get
21
2xa1
=
xa2
2 [9
xa1
xa2 ]
(7.13)
which implies that the equation of the contract curve is:
xa2 =
c Frank Cowell 2006
102
12xa1
:
xa1 + 7
(7.14)
Microeconomics
Exercise 7.5 Which of the following sets of functions are legitimate excess demand functions?
9
E1 (p) = p2 + p101 =
E2 (p) = p1
(7.15)
;
E3 (p) = p103
9
3
>
E1 (p) = p2p+p
=
1
3
E2 (p) = p1p+p
(7.16)
2
>
2 ;
E3 (p) = p1p+p
3
p3 9
E1 (p) = p1 =
E2 (p) = pp32
(7.17)
;
E3 (p) = 2
Outline Answer:
The …rst system is not homogeneous of degree zero in prices. The second
violates Walras’Law. The third one is both homogeneous and satis…es Walras’
Law.
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Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
Exercise 7.6 In a two-commodity economy let be the price of commodity 1
in terms of commodity 2. Suppose the excess demand function for commodity 1
is given by
1 4 + 5 2 2 3:
How many equilibria are there? Are they stable or unstable? How might your
answer be a¤ ected if there were an increase in the stock of commodity 1 in the
economy?
Outline Answer:
The excess demand for commodity 1 at relative price
E( ) := 1
4 +5
2
2
3
]2 [1
= [1
can be written
2 ]:
So that
dE( )=d =
4 + 10
6
2
–see Figure 7.5. From this we see that there are two equilibria as follows:
1.
= 0:5. Here dE( )=d < 0 and so it is clear that the equilibrium is
locally stable
= 1. Here dE( )=d = 0. But the graph of the function reveals
that it is locally stable from “above” (where > 1) and unstable
from “below” (where < 1).
If there were an increase in the stock of commodity 1 the excess demand
function would be shifted to the left in Figure 7.5 –then there is only one,
stable equilibrium.
ρ
1.5
1
0.5
-1
-0.5
•
•
°
0
E
0.5
Figure 7.5: Excess demand
c Frank Cowell 2006
104
1
Microeconomics
Exercise 7.7 Consider the following four types of preferences:
Type
Type
Type
Type
A
B
C
D
:
log x1 + [1
:
x1 + x2
2
2
:
[x1 ] + [x2 ]
: min f x1 ; x2 g
] log x2
where x1 ; x2 denote respectively consumption of goods 1 and 2 and ; ; ; are
strictly positive parameters with < 1.
1. Draw the indi¤ erence curves for each type.
2. Assume that a person has an endowment of 10 units of commodity 1 and
zero of commodity 2. Show that, if his preferences are of type A, then his
demand for the two commodities can be represented as
x :=
x1
x2
=
10
10 [1
]
where is the price of good 1 in terms of good 2. What is the person’s
o¤ er curve in this case?
3. Assume now that a person has an endowment of 20 units of commodity
2 (and zero units of commodity 1) …nd the person’s demand for the two
goods if his preferences are represented by each of the types A to D. In
each case explain what the o¤ er curve must look like.
4. In a two-commodity economy there are two equal-sized groups of people.
People in group 1 own all of commodity 1 (10 units per person) and people
in group 2 own all of commodity 2 (20 units per person). If Group 1 has
preferences of type A with = 21 …nd the competitive equilibrium prices
and allocations in each of the following cases:
3
4
(a) Group 2 have preferences of type A with
=
(b) Group 2 have preferences of type B with
= 3.
(c) Group 2 have preferences of type D with
= 1.
5. What problem might arise if group 2 had preferences of type C? Compare
this case with case 4b
Outline Answer:
1. Indi¤erence curves have the shape shown in the …gure 7.6.
2. The income of a group-1 person is 10 . If group-1 persons have preferences
of type A then the Lagrangean is
log x11 + [1
c Frank Cowell 2006
] log x12 +
105
10
x11
x12
Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
z2
z2
(1)
(2)
z1
z1
z2
z2
(4)
(3)
z1
z1
Figure 7.6: Indi¤erence Curves
First order conditions for an interior maximum are
x11
1
x12
x11
10
Solving these we …nd
=
1
10
x12
=
0
=
0
=
0
and so the demands will be
x1 =
10
10 [1
]
and the o¤er curve will simply be a vertical straight line at x11 = 10 .
3. The income of a group-2 person is 20. So, if group-2 persons have preferences of type A, then their demands will be
20
x2 =
20 [1
]
and their o¤er curve will simply be a horizontal straight line at x22 =
20 [1
]. If group-2 persons have preferences of type B then their demands will be
x2
x2
x2
c Frank Cowell 2006
= x0 , if >
2 [x0 ; x00 ], if =
= (20= ; 0), if <
106
Microeconomics
where x0 := (0; 20), x00 := (20= ; 0), and their o¤er curve will consist of
the union of the line segment [x0 ; x00 ] and the line segment from x00 to
(1; 0). If group-2 persons have preferences of type C then their demands
will be
p
x2 = x0 , if >
p
x2 = x0 or x00 , if =
p
x2 = (20= ; 0), if <
p
where x0 := (0; 20), x00 := (20= ; 0), and their o¤er curve will consist
of the union of the point x0 and the line segment from x00 to (1; 0). If
group-2 persons have preferences of type D then their demands will be
"
#
20
+
20
+
x2 =
and their o¤er curve is just the straight line x22 = x21 .
4. In each case below we could work out the excess demand function, set
excess demand equal to zero, …nd the equilibrium price and then the equilibrium allocation. However, we can get to the result more quickly by
using an equivalent approach. Given that an equilibrium allocation must
lie at the intersection of the o¤er curves of the two parties the answer in
each case is immediate.
(a) From the above computations we have x11 = 10 21 = 5, x22 =
20 1 34 = 5. Given that there are 10 units per person of commodity 1 and 20 units per person of commodity 2 the materials balance
condition then means that the equilibrium allocation must be
Solving for
x1
=
5
15
x2
=
5
5
we …nd that the equilibrium price ratio must be 3.
x11
(b) We have
= 5, and so x21 = 5. Using the fact that the equilibrium
must lie on the group-2 o¤er curve we see that the solution must lie
on the straight line from (0; 20) to (20=3; 0) we …nd that x22 = 5 and,
from the materials balance condition x12 = 20 5 = 15 (as in the
previous case). By the same reasoning as in the previous case the
equilibrium price must be = 3.
(c) Once again we have x11 = 5, and so x21 = 5. Given that the group2 o¤er curve in this case is such that the person always consume’s
equal quantities of the two goods we must have x22 = 5 and so again
x12 = 20 5 = 15 (as in the previous cases). As before the equilibrium
price must be 3.
5. Note that the demand function and the o¤er curve for the group-2 people
is discontinuous. So, if there are relatively small numbers in each group
c Frank Cowell 2006
107
Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
there may be no equilibrium (the two o¤er curves do not intersect). In the
large numbers case we could appeal to a continuity argument and have
an equilibrium with proportion of group 2 at point x0 and the rest at x00 .
The equilibrium would then look very much like case 4b.
c Frank Cowell 2006
108
Microeconomics
Exercise 7.8 In a two-commodity exchange economy there are two equal-sized
groups of people. Those of type a have the utility function
1 a
[x ]
2 1
U a (xa ) =
1 a
[x ]
2 2
2
2
and a resource endowment of (R1 ; 0); those of type b have the utility function
U b xb = xb1 xb2
and a resource endowment of (0; R2 ).
1. How many equilibria does this system have?
2. Find the equilibrium price ratio if R1 = 5, R2 = 16.
Outline Answer:
For consumers of type a the relevant Lagrangean is
1 a
[x ]
2 1
2
1 a
[x ]
2 2
2
pxa1
+ [pR1
xa2 ]
where p is the price of good 1 in terms of good 2. The FOC for a maximum are
[xa1 ]
3
p
3
[xa2 ]
=
0
=
0
Rearranging and using the budget constraint we get
xa1
1=3
1=3
= p
1=3
=
h
i
pxa1 + xa2 = p2=3 + 1
xa2
So
1=3
xa2 =
1=3
= pR1
pR1
p2=3 + 1
=
For consumers of type b the Lagrangean is
log xb1 + log xb2 +
R2
pxb1
The FOC for a maximum are
1
xb1
xb2
p
1
=
0
=
0
Rearranging and using the budget constraint we get
c Frank Cowell 2006
xb1
=
xb2
=
109
1
p
1
xb2
Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
pxb1 + xb2 =
So
xb2 =
1
2
=
= R2
R2
2
The excess-demand function for good 2 is therefore
pR1
p2=3 + 1
R2
2
1. Excess demand is 0 where
p
where
p2=3
1=0
:= 2R1 =R2 . This is equivalent to requiring
p2=3 = p
1
The expression p2=3 is an increasing concave function through the origin.
It is clear that the straight line given by p
1 can cut this just once.
There is one equilibrium.
2. If (R1 ; R2 ) = (10; 32) then
c Frank Cowell 2006
:= 20=32 = 5=8 and p = 8:
110
Microeconomics
Exercise 7.9 In a two-person, private-ownership economy persons a and b
each have utility functions of the form
V h p; y h = log y h
h
1
p1
1
log (p1 p2 )
2
h
2
p2
where h = a; b and h1 , h2 are parameters. Find the equilibrium price ratio as
a function of the property distribution [R].
Outline Answer:
Using Roy’s identity we have, for each h and each i :
Vih
:
Vyh
xhi =
Now we have
h h
=
i [y
h
= [y
p1
Vih
Vyh
p1
h
1
h
1
p2
p2 h2 ]
h 1
:
2]
1
1=2pi ;
Combining the two results we …nd for each h:
h
i
xhi =
De…ning
good 2:
i
a
i
+
E2 =
2
=
+
b
i
1 h h h
p1 R 1
2pi
h
1
i
h
+ p2 R2h
h
2
ii
and Ri = Ria + Rib we obtain the excess demand for
R2 +
1
[p1 [R1
2p2
1]
+ p2 [R2
2 ]]
Hence putting E2 = 0 we get the equilibrium price ratio thus:
p1
R2
=
p2
R1
c Frank Cowell 2006
111
2
1
:
Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
Exercise 7.10 In an economy there are large equal-sized groups of capitalists
and workers. Production is organised as in the model of Exercises 2.14 and 6.4.
Capitalists’ income consists solely of the pro…ts from the production process;
workers’ income comes solely from the sale of labour. Capitalists and workers
2
h
have the utility functions xc1 xc2 and xw
[R3 xw
1
3 ] respectively, where xi denotes the consumption of good i by a person of type h and R3 is the stock of
commodity 3.
1. If capitalists and workers act as price takers …nd the optimal demands for
the consumption goods by each group, and the optimal supply of labour
R3 xw
3.
2. Show that the excess demand functions for goods 1,2 can be written as
2p1
+
2p2
A
p1
2
1
2
2 [p1 ]
A
p2
2
where
is the expression
for pro…ts found in Exercise 6.4. Show that in
p
equilibrium p1 =p2 = 3 and hence show that the equilibrium price of good
1 (in terms of good 3) is given by
1=3
3
2A
p1 =
3. What is the ratio of the money incomes of workers and capitalists in equilibrium?
Outline Answer:
1. Given that the capitalist utility function is
xc1 xc2
it is immediate that in the optimum the capitalists spend an equal share
of their income on the two consumption goods and so
xci =
2pi
:
Worker utility is
xw
1
2
(7.18)
xw
3
(7.19)
xw
3]
[R3
and the budget constraint is
p1 xw
1
R3
Maximising (7.18) subject to (7.19) is equivalent to maximising
1
[R3
p1
c Frank Cowell 2006
xw
3]
112
[R3
2
xw
3] :
(7.20)
Microeconomics
The FOC is
1
2 [R3 xw
3]=0
p1
which gives optimal labour supply as:
xw
3 =
R3
(7.21)
1
2p1
(7.22)
and, from (7.19), the workers’optimal consumption of good 1 is
xw
1 =
1
2:
(7.23)
2 [p1 ]
2. The economy has no stock of good 1 or good 2; workers do not consume
good 2; so excess demand for the two goods is, respectively:
xc1 + xw
1
y1
=
xc2
y2
=
2p1
+
2p2
A
p1
2
1
2
2 [p1 ]
A
p2
2
(7.24)
(7.25)
To …nd the equilibrium set each of (7.24) and (7.25) equal to zero. This
gives
p1 + 1
3
(7.26)
2
(7.27)
= A [p1 ]
= A [p2 ]
Substituting in for pro…ts in (7.27) we have
2
2
[p1 ] + [p2 ]
2
= [p2 ]
4
and so
p
p1
= 3:
p2
and p2 we get
Substituting for
(7.28)
2
2
[p1 ] + 31 [p1 ]
3
+ 1 = A [p1 ]
4
and, on rearranging, this gives
p1 A
p1 =
3
2A
1=3
(7.29)
3. Pro…ts in equilibrium are
A
1+
4
1
3
2
[p1 ] =
A
A 3
2
[p1 ] =
3
3 2A
2=3
=
A
3
1=3
2
2=3
:
Given that the price of good 3 is normalised to 1, using (7.22)and (7.29)
total labour income in equilibrium is
1
c Frank Cowell 2006
1
1 3
=
2p1
2 2A
113
1=3
=
A
3
1=3
2
2=3
Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
So, workers and capitalists get the same money income in equilibrium!
Note that this is una¤ected by the value of A; increases in A could be
interpreted as technical progress and so the income distribution remains
unchanged by such progress.
c Frank Cowell 2006
114
Chapter 8
Uncertainty and Risk
Exercise 8.1 Suppose you have to pay $2 for a ticket to enter a competition.
The prize is $19 and the probability that you win is 13 . You have an expected
utility function with u(x) = log x and your current wealth is $10.
1. What is the certainty equivalent of this competition?
2. What is the risk premium?
3. Should you enter the competition?
Outline Answer:
1. Given the probabilities and payo¤s we have the following expected utility
if the person enters the lottery
Eu(x)
=
=
=
1
2
log (10 2 + 19) + log (10
3
3
1
2
log (27) + log (8)
3
3
log 3 + log 4 = log 12
2)
So the certainty equivalent is $12.
2. The expected wealth at the end of the period is
Ex =
=
1
2
[10 2 + 19] + [10
3
3
27 16
43
1
+
=
= 14
3
3
3
3
So the risk premium is $14 13
2]
$12 = $2 31 .
3. If the person does not enter the lottery he has just his initial wealth, $10.
So, in view of the answer to part (a) it makes sense to enter the lottery.
115
Microeconomics
CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.2 You are sending a package worth 10 000A
C. You estimate that
there is a 0.1 percent chance that the package will be lost or destroyed in transit. An insurance company o¤ ers you insurance against this eventuality for a
premium of 15A
C. If you are risk-neutral, should you buy insurance?
Outline Answer:
No you should not. Your expected loss is 10 euros whereas the premium is
15 euros.
c Frank Cowell 2006
116
Microeconomics
Exercise 8.3 Consider the following de…nition of risk aversion. Let P :=
f(x! ; ! ) : ! 2 g be a random prospect, where x! is the payo¤Pin state !
and ! is the (subjective) probability of state ! , and let Ex := !2 ! x! ,
the mean of the prospect, and let P := f( x! + [1
]Ex; ! ) : ! 2 g be a
“mixture” of the original prospect with the mean. De…ne an individual as risk
averse if he always prefers P to P for 0 < < 1.
1. Illustrate this concept in (xred ; xblue )-space and contrast it with the concept
of risk aversion used in the text
2. Show that this de…nition of risk aversion need not imply convex-to-theorigin indi¤ erence curves.
Outline Answer:
1. See Figure 8.1.
xBLUE
P
Pλ
P
xRED
Figure 8.1: Nonconvex indi¤erence curve
2. Let P be the prospect and P its mean. P can be any point in the line
joining them. The de…nition implies that moving along this line towards
P puts the person on a successively higher indi¤erence curves. In Figure
8.1 it is clear that this condition is consistent with there being indi¤erence
curves that violate the convex-to-the-origin property locally.
c Frank Cowell 2006
117
Microeconomics
CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.4 Suppose you are asked to choose between two lotteries. In one
case the choice is between P1 and P2 ;and in the other case the choice o¤ ered is
between P3 and P4 , as speci…ed below:
P1 :
$1; 000; 000 with probability 1
8
< $5; 000; 000 with probability 0.1
$1; 000; 000 with probability 0.89
P2 :
:
$0
with probability 0.01
P3 :
$5; 000; 000 with probability 0.1
$0
with probability 0.9
P4 :
$1; 000; 000
$0
with probability 0.11
with probability 0.89
It is often the case that people prefer P1 to P2 and then also prefer P3 to P4 .
Show that these preferences violate the independence axiom.
Outline Answer:
Let there be only three possible states of the world: red, blue and green,
with probabilities 0.01, 0.10, 0.89 respectively. Then the payo¤s in the four
prospects can be written
P1
P2
P3
P4
red
1
0
0
1
blue
1
5
5
1
green
1
1
0
0
where all the entries in the table are in millions of dollars. Note that P1 and P2
have the same payo¤ in the green state; P3 and P4 form a similar pair, except
that the payo¤ in the green state is 0. Axiom 8.2 states that if P1 is preferred
to P2 than any other similar pair of prospects (1; 1; z) and (0; 5; z) ought also
to be ranked in the same order, for arbitrary z: but this would imply that P4
is preferred to P3 , the opposite of the preferences as stated.
Note also that if the preferences had been such that P4 was preferred to P3
then the independence axiom would imply that P1 was preferred to P2 .
c Frank Cowell 2006
118
Microeconomics
Exercise 8.5 This is an example to illustrate disappointment. Suppose the
payo¤ s are as follows
x00 weekend for two in your favourite holiday location
x0 book of photographs of the same location
x
…sh-and-chip supper
Your preferences under certainty are x00 x0 x . Now consider the following
two prospects
8 00
with probability 0:99
< x
x0 with probability 0
P1 :
:
x with probability 0:01
8 00
with probability 0:99
< x
x0 with probability 0:01
P2 :
:
x with probability 0
Suppose a person expresses a preference for P1 over P2 . Brie‡y explain why
this might be the case in practice. Which of the three axioms State Irrelevance,
Independence, Revealed Likelihood, is violated by such preferences?
Outline Answer:
It is possible that, given the information that the …rst event (with payo¤ x00 )
has not happened you would then prefer x to x0 : photographs of your favourite
holiday spot may be too painful once you know that the holiday is not going to
happen. So you may prefer P1 over P2 .
These preferences violate the independence axiom. To see this, note that,
by the revealed likelihood axiom, since x0 is strictly preferred to x , it must be
the case that P20 is strictly preferred to P10 , where
P10 :
x0
x
with probability 0
with probability 1
P20 :
x0
x
with probability 0:01
with probability 0:99
But P10 and P20 can be written equivalently as
8
< x with probability
x0 with probability
P10 :
:
x with probability
8
< x with probability
x0 with probability
P20 :
:
x with probability
0:99
0
0:01
0:99
0:01
0
By the independence axiom if P20 is strictly preferred to P10 , then P2 must be
strictly preferred to P1 .
c Frank Cowell 2006
119
Microeconomics
CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.6 An example to illustrate regret. Let
P := f(x! ;
!)
:!2 g
P 0 := f(x0! ;
!)
:!2 g
be two prospects available to an individual. De…ne the expected regret if the
person chooses P rather than P 0 as
X
0
x! ; 0g
(8.1)
! max fx!
!2
Now consider the choices amongst prospects presented in Exercise 8.4. Show
that if a person is concerned to minimise expected regret as measured by (8.1),
then it is reasonable that the person select P2 when P1 is also available and then
also select P4 when P3 is available.
Outline Answer:
Denote the regret in (8.1) by r (P; P 0 ).
If I choose P2 when P1 is also available then the regret is
r (P2 ; P1 )
= 0:1 [0] + 0:89 [0] + :01 [1]
= 10; 000
Whereas, had I chosen P1 when P2 was available, then the regret would have
been
r (P1 ; P2 )
= 0:1 [4; 000; 000] + 0:89 [0] + :01 [0]
= 400; 000
If I choose P4 when P3 is also available then the regret is
r (P4 ; P3 )
= 0:1 [0] + 0:89 [0] + :01 [0]
= 0
Whereas, had I chosen P3 when P4 was available, then the regret would have
been
r (P3 ; P4 )
c Frank Cowell 2006
= 0:1 [0] + 0:89 [0] + :01 [5; 000; 000]
= 50; 000
120
Microeconomics
Exercise 8.7 An example of the Ellsberg paradox . There are two urns marked
Left and Right each of which contains 100 balls. You know that in Urn L
there exactly 49 white balls and the rest are black and that in Urn R there are
black and white balls, but in unknown proportions. Consider the following two
experiments:
1. One ball is to be drawn from each of L and R. The person must choose
between L and R before the draw is made. If the ball drawn from the chosen
urn is black there is a prize of $1000, otherwise nothing.
2. Again one ball is to be drawn from each of L and R; again the person must
choose between L and R before the draw. Now if the ball drawn from the
chosen urn is white there is a prize of $1000, otherwise nothing.
You observe a person choose Urn L in both experiments. Show that this
violates the Revealed Likelihood Axiom.
Outline Answer
The implication of the revealed likelihood axiom is that there exist subjective
probabilities ! . The result is proved by showing that it the stated behaviour
is inconsistent with the existence of subjective probabilities.
In this case the revealed likelihood axiom implies that for each urn there is
a given subjective probability of drawing a black ball L (left-hand urn) and R
(right-hand urn) such that preferences can be represented as
vblack (xblack ; xwhite ) + [1
] vwhite (xblack ; xwhite )
(8.2)
where = L or R and xblack and xwhite are the payo¤s if a black ball or a
white ball are drawn respectively.
Note that the representation (8.2) does not impose either the State Irrelevance Axiom (which would require that vblack ( ) and vwhite ( ) be the same
function) or the Independence axiom (which would require that vblack ( ) be a
function only of xblack etc.). Nor does it impose the common-sense requirement
that L = 0:49. All we need below is the very weak assumption that preferences
are not perverse:
vblack (1000; 0) > vwhite (1000; 0)
(8.3)
and
vblack (0; 1000) < vwhite (0; 1000)
(8.4)
Condition (8.3) simply says that if the $1000 prize is attached to a black ball
then the utility to be derived from having selected a black ball is higher than
selecting a white ball; condition (8.4) is the counterpart when the prize attaches
to the white ball..
Experiment 1 suggests that
L vblack
>
c Frank Cowell 2006
(1000; 0) + [1
R vblack (1000; 0) + [1
121
L ] vwhite
(1000; 0)
R ] vwhite (1000; 0)
(8.5)
Microeconomics
CHAPTER 8. UNCERTAINTY AND RISK
while experiment 2 suggests that
L vblack
>
(0; 1000) + [1
v
R black (0; 1000) + [1
L ] vwhite
(0; 1000)
]
v
R white (0; 1000)
(8.6)
We can see that (8.5) implies
L
[vblack (1000; 0)
vwhite (1000; 0)] >
from which we deduce that, given (8.3),
L
[vblack (0; 1000)
vwhite (0; 1000)] >
R
[vblack (1000; 0)
L
>
R
[vblack (0; 1000)
R.
vwhite (1000; 0)]
However (8.5) implies
vwhite (0; 1000)] :
So that, given (8.4), we would have L < R –a contradiction. Therefore the
revealed likelihood axiom must be violated.
c Frank Cowell 2006
122
Microeconomics
Exercise 8.8 An individual faces a prospect with a monetary payo¤ represented
by a random variable x that is distributed over the bounded interval of the real
line [a; a]. He has a utility function Eu(x) where
u(x) = a0 + a1 x
1
a2 x2
2
and a0 ; a1 ; a2 are all positive numbers.
1. Show that the individual’s utility function can also be written as '(Ex; var(x)).
Sketch the indi¤ erence curves in a diagram with Ex and var(x) on the
axes, and discuss the e¤ ect on the indi¤ erence map altering (i) the parameter a1 , (ii) the parameter a2 .
2. For the model to make sense, what value must a have? [Hint: examine
the …rst derivative of u.]
3. Show that both absolute and relative risk aversion increase with x.
Outline Answer:
1. Clearly
1
Eu(x) = a0 + a1 E(x) + a2 [(E(x))2
2
Marginal utility is a1
var(x)]:
a2 x:
2. For this to be non-negative we must have E(x)
a1 =a2 hence the indi¤erence curves are depicted with E(x) as good, var(x) as bad and
M RS = 2 [a1 =a2 E(x)] :
3.
ux (x) = a1 + a2 x
uxx (x) = a2
where and 0
c Frank Cowell 2006
x
(x)
=
(x)
=
%(x)
=
xmax :=
uxx (x)
=
ux (x)
1
xmax x
1
xmax =x 1
a1
a2 :
123
a2
a1 + a2 x
Microeconomics
CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.9 A person lives for 1 or 2 periods. If he lives for both periods he
has a utility function given by
U (x1 ; x2 ) = u (x1 ) + u (x2 )
(8.7)
where the parameter is the pure rate of time preference. The probability of
survival to period 2 is , and the person’s utility in period 2 if he does not
survive is 0.
1. Show that if the person’s preferences in the face of uncertainty are represented by the expected-utility functional form
X
(8.8)
! u (x! )
!2
then the person’s utility can be written as
u (x1 ) + 0 u (x2 ) :
What is the value of the parameter
0
(8.9)
?
2. What is the appropriate form of the utility function if the person could live
for an inde…nite number of periods, the rate of time preference is the same
for any adjacent pair of periods, and the probability of survival to the next
period given survival to the current period remains constant?
Outline Answer:
1. Consider the person’s lifetime utility with the consumption x1 and x2
in the two periods. If the person survives into the second period utility
is given by u (x1 ) + u (x2 ) otherwise it is just u (x1 ). Given that the
probability of the event “survive to second period” is expected lifetime
utility is
[u (x1 ) + u (x2 )] + [1
] u (x1 ) :
On rearranging we get
u (x1 ) +
in other words the form (8.9) with
u (x2 ) ;
0
=
(8.10)
.
2. Apply the argument to one more period. Now there is consumption
x1 ,x2 ; x3 in the three periods and the probability of surviving into period
t + 1 given that you have made it to period t is still . Consider the
situation of someone who survives to period 2. The person gets utility
u (x2 ) + u (x3 )
(8.11)
if he survives to period 3 and u (x2 ) otherwise. His expected utility for
the rest of his lifetime, contingent on having reached period 2 is therefore
[u (x2 ) + u (x3 )] + [1
= u (x2 ) + u (x3 )
c Frank Cowell 2006
124
] u (x2 )
(8.12)
Microeconomics
So now view the situation from the position of the beginning of the lifetime.
The person gets utility
u (x1 ) + [u (x2 ) +
u (x3 )]
(8.13)
if he makes it through to period 2, where the expression in square brackets
in (8.13) is just the rest-of-lifetime expected utility if you get to period 2,
taken from (8.12); of course if the person does not survive period 1 he gets
just u (x1 ). So, using the same reasoning as before, from the standpoint
of period 1 lifetime expected utility is now
[u (x1 ) + [u (x2 ) +
+ [1
] u (x1 ) :
u (x3 )]]
Rearranging this we have
u (x1 ) +
u (x2 ) +
2 2
u (x2 ) :
(8.14)
It is clear that the same argument could be applied to T > 2 periods and
that the resulting utility function would be of the form
u (x1 ) +
u (x2 ) +
2 2
u (x2 ) + ::: +
T T
u (x2 ) :
(8.15)
In other words we have the standard intertemporal utility function with
the pure rate of time preference replaced by the modi…ed rate of time
preference 0 := .
c Frank Cowell 2006
125
Microeconomics
CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.10 A person has an objective function Eu(y) where u is an increasing, strictly concave, twice-di¤ erentiable function, and y is the monetary value
of his …nal wealth after tax. He has an initial stock of assets K which he may
keep either in the form of bonds, where they earn a return at a stochastic rate
r, or in the form of cash where they earn a return of zero. Assume that Er > 0
and that Prfr < 0g > 0.
1. If he invests an amount in bonds (0 < < K) and is taxed at rate t
on his income, write down the expression for his disposable …nal wealth y,
assuming full loss o¤ set of the tax.
2. Find the …rst-order condition which determines his optimal bond portfolio
.
3. Examine the way in which a small increase in t will a¤ ect
.
4. What would be the e¤ ect of basing the tax on the person’s wealth rather
than income?
Outline Answer:
1. Suppose the person puts an amount
in bonds leaving the remaining
K
of assets in cash. Then, given that the rate of return on cash is
zero and on bonds is the stochastic variable r, income is
[K
]0 + r = r
If the tax rate is t then, given that full loss o¤set implies that losses and
gains are treated symmetrically, disposable income is
[1
t] r
and (disposable) …nal wealth is
x =
[K
]+
+ [1
[value of b onds]
[cash]
= K + [1
t] r
[incom e]
t] r:
(8.16)
Note that x is a stochastic variable and could be greater or less than initial
wealth K.
2. The individual’s optimisation problem is to choose
Using (8.16) the FOC for an interior solution is
E (ux (x) [1
to maximise Eu(x).
t] r) = 0;
which implies
E (ux (x)r) = 0:
(8.17)
Solving this determines
=
(t; K), the optimal bond purchases that
depends on the tax rate and initial wealth as well as the distribution of
returns and risk aversion.
c Frank Cowell 2006
126
Microeconomics
3. Take the FOC (8.17). Substituting for x from (8.16) and di¤erentiating
with respect to t we get
E uxx (x)
@
[1
@t
r+
E uxx (x)r2
t] r r
@
[1
@t
+
= 0;
t]
= 0:
so that
+
@
[1
@t
t]
@
@t
= 0
=
1
t
:
An increase in the tax rate increases the demand for bonds.
4. Final wealth is initial wealth plus income. If the tax is on wealth then
disposable …nal wealth is
x = [1
t] K + [1
t] r
(8.18)
instead of (8.16). Clearly the FOC (8.17) remains essentially unaltered
(the new tax just reduces total wealth). Di¤erentiating the FOC with x
de…ned by (8.18) we now …nd
E uxx (x)
K
r+
@
[1
@t
KE (uxx (x)r) + E uxx (x)r2
+
t] r r
= 0;
@
[1
@t
t]
= 0:
This implies
K
E (uxx (x)r)
@
+
[1
2
E (uxx (x)r )
@t
@
[1
@t
@
=
@t
1
t] =
t
+K
+
K
1
t] = 0:
E (uxx (x)r)
:
E (uxx (x)r2 )
E (uxx (x)r)
:
t E (uxx (x)r2 )
The …rst term on the right-hand side is positive; as for the second term,
the denominator is negative and the numerator is positive, given DARA.
So the impact of tax on bond-holding is now ambiguous.
c Frank Cowell 2006
127
Microeconomics
CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.11 An individual taxpayer has an income y that he should report to
the tax authority. Tax is payable at a constant proportionate rate t. The taxpayer
reports x where 0 x y and is aware that the tax authority audits some tax
returns. Assume that the probability that the taxpayer’s report is audited is
, that when an audit is carried out the true taxable income becomes public
knowledge and that, if x < y, the taxpayer must pay both the underpaid tax and
a surcharge of s times the underpaid tax.
1. If the taxpayer chooses x < y, show that disposable income c in the two
possible states-of-the-world is given by
cnoaudit
caudit
= y tx;
= [1 t st] y + stx:
2. Assume that the individual chooses x so as to maximise the utility function
[1
] u (cnoaudit ) + u (caudit ) :
where u is increasing and strictly concave.
(a) Write down the FOC for an interior maximum.
(b) Show that if 1
s > 0 then the individual will de…nitely underreport income.
3. If the optimal income report x satis…es 0 < x < y:
(a) Show that if the surcharge is raised then under-reported income will
decrease.
(b) If true income increases will under-reported income increase or decrease?
Outline Answer:
If the individual reports x then he pays tax tx –i.e. he underpays an amount
t [y x]. So
1. If the under-reporting remains undetected then
cnoaudit
= y
= y
tx
ty + t [y
x]
and if the audit takes place then
caudit
= y tx [1 + s] t [y
= [1 t st] y + stx
x]
2. The individual maximises
Eu(c) = [1
] u (y
tx) + u ([1
t
st] y + stx)
Di¤erentiating this we have
@Eu(c)
=
@x
t [1
] uc (y
tx) + st uc ([1
where uc ( ) denotes the …rst derivative of u.
c Frank Cowell 2006
128
t
st] y + stx)
Microeconomics
(a) If there is an interior maximum at x then the following FOC must
hold
[1
] uc (y tx ) = s uc ([1 t st] y + stx ) :
(b) If the person reports fully then
@Eu(c)
@x
=
t [1
=
[1
] uc (y
ty) + st uc ([1
t] y)
x=y
s ] tuc (y
ty)
Given that t and uc are positive it is clear that the above expression
is negative if 1
s > 0. Therefore the individual’s expected
utility would increase if he reduced x below y.
3. Di¤erentiating the FOC with respect to s and rearranging we get
=
@x
@x
s2 t ucc ([1 t st] y + stx )
@s
@s
st] y + stx ) + st [x
y] ucc ([1 t st] y + stx )
t [1
] ucc (y
uc ([1
t
tx )
(a) Therefore
uc (caudit ) + st [x
t
@x
=
@s
y] ucc (caudit )
(8.19)
where
:=
[1
] ucc (y
tx )
s2 ucc ([1
t
st] y + stx ) > 0
Given that uc > 0, x < y and ucc < 0 it is clear that the numerator
of (8.19) is positive.x increases with s so t [y x ] decreases.
(b) Di¤erentiating the FOC with respect to y we get
[1
] ucc (y
= s ucc ([1
t
tx ) 1
t
@x
@y
st] y + stx ) [1
t
st] + st
@x
@y
:
Therefore we have
@x
=
@y
s [1
t
t [[1
st] ucc (caudit ) [1
] ucc (cnoaudit )
] ucc (cnoaudit ) + s2 ucc (caudit )]
@ [y x ]
s [1 t st] ucc (caudit ) [1
] ucc (cnoaudit )
=1+
@y
[[1
] tucc (cnoaudit ) + s2 tucc (caudit )]
@ [y x ]
=
@y
1
t [1
t [[1
] ucc (cnoaudit )
succ (caudit )
] ucc (cnoaudit ) + s2 ucc (caudit )]
This is of ambiguous sign unless we assume DARA in which case it
is positive.
c Frank Cowell 2006
129
Microeconomics
CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.12 A risk-averse person has wealth y0 and faces a risk of loss
L < y0 with probability . An insurance company o¤ ers cover of the loss at
a premium > L. It is possible to take out partial cover on a pro-rata basis,
so that an amount tL of the loss can be covered at cost t where 0 < t < 1.
1. Explain why the person will not choose full insurance
2. Find the conditions that will determine t , the optimal value of t.
3. Show how t will change as y0 increases if all other parameters remain
unchanged.
Outline Answer:
1. Consider the person’s wealth after taking out (partial) insurance cover
using the two-state model (no loss;loss). If the person remained uninsured it would be (y0 ; y0 L); if he insures fully it is (y0
; y0
). So
if he insures a proportion t for the pro-rata premium wealth in the two
states will be
([1
t] y0 + t [y0
] ; [1
t] [y0
L] + t [y0
])
which becomes
(y0
t ; y0
t
[1
t] L)
So expected utility is given by
Eu = [1
] u (y0
t ) + u (y0
t
[1
t] L)
Therefore
@Eu
=
@t
[1
] uy (y0
t ) + [L
] uy (y0
t
[1
t] L)
Consider what happens in the neighbourhood of t = 1 (full insurance).
We get
@Eu
@t
=
[1
] uy (y0
) + [L
] uy (y0
)
t=1
=
[L
] uy (y0
)
We know that uy (y0
) > 0 (positive marginal utility of wealth) and, by
assumption, L < . Therefore this expression is strictly negative which
means that in the neighbourhood of full insurance (t = 1) the individual
could increase expected utility by cutting down on the insurance cover.
2. For an interior maximum we have
@Eu
=0
@t
which means that the optimal t is given as the solution to the equation
[1
] uy (y0
c Frank Cowell 2006
t
) + [L
130
] uy (y0
t
[1
t ] L) = 0
Microeconomics
3. Di¤erentiating the above equation with respect to y0 we get
[1
] uyy (y0
t
) 1
@t
+[L
@y0
] uyy (y0
t
[1
t ] L) 1
which gives
[1
] uyy (y0 t ) + [L
@t
=
@y0
[1
] 2 uyy (y0 t ) + [L
] uyy (y0
2
] uyy (y0
t
[1
t ] L)
t
[1
t ] L)
The denominator of this must be negative: uyy ( ) is everywhere negative
and the other terms are positive. The numerator is positive if DARA
holds: therefore an increase in wealth reduces the demand for insurance
coverage.
c Frank Cowell 2006
131
[
L]
@t
=0
@y0
Microeconomics
CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.13 Consider a competitive, price-taking …rm that confronts one of
the following two situations:
“uncertainty”: price p is a random variable with expectation p.
“certainty”: price is …xed at p.
It has a cost function C(q) where q is output and it seeks to maximise the
expected utility of pro…t.
1. Suppose that the …rm must choose the level of output before the particular
realisation of p is announced. Set up the …rm’s optimisation problem and
derive the …rst- and second-order conditions for a maximum. Show that,
if the …rm is risk averse, then increasing marginal cost is not a necessary
condition for a maximum, and that it strictly prefers “certainty” to “uncertainty”. Show that if the …rm is risk neutral then the …rm is indi¤ erent
as between “certainty” and “uncertainty”.
2. Now suppose that the …rm can select q after the realisation of p is announced, and that marginal cost is strictly increasing. Using the …rm’s
competitive supply function write down pro…t as a function of p and show
that this pro…t function is convex. Hence show that a risk-neutral …rm
would strictly prefer “uncertainty” to “certainty”.
Outline Answer:
1. Pro…t is given by
:= pq
C(q)
where p is a random variable. Maximising expected utility of pro…t Eu( )
by choice of q requires the FOC
E(u ( )p)
E(u ( ))Cq = 0
where u ( ) is the …rst derivative of u( ). This will represent a maximum
if
d2 Eu
< 0:
dq 2
We …nd that this implies
E(u
[p
Cq ]2 )
E(u
)Cqq < 0:
Notice that since the …rst term is negative for a risk-averse …rm then the
condition can be satis…ed not only if Cqq > 0 but also if Cqq < 0 and jCqq j
is not too large. Now consider transforming p to pb thus: pb = (1
)p + p
then pb has the same mean as p but is less dispersed. Maximised utility for
the random variable pb is
Eu([(1
)p + p]q
C(q ))
where q is the output satisfying the …rst-order conditions for a maximum.
Di¤erentiate this expected utility with respect to
@Eu
= [E(u [p
@
c Frank Cowell 2006
p])]q + [E(u [b
p
132
Cq ])]
@q
@
Microeconomics
where the last term vanishes because of the …rst order condition. So
@Eu
has the sign of E(u [p p]): But this must be positive if u is
@
decreasing with and will be zero if u is constant. Hence the …rm strictly
prefers certainty if it is risk averse and is indi¤erent between certainty and
uncertainty if it is risk neutral.
2. For any known realization p we may write q = S(p) where S is the competitive …rm supply curve. Pro…ts as a function of P may thus be written:
(p) = pS(p)
C(S(p))
which implies
d (p)
= [p
dp
Cq ]Sp (p) + S(p) = S(p)
(8.20)
where Sp (p) is the slope of the supply curve at p, a positive number.
Therefore, di¤erentiating (8.20) we have
d2 (p)
= Sp (p) > 0:
dp2
Hence ( ) is increasing and convex. So it is immediate that E (p) >
c Frank Cowell 2006
133
(p):
Microeconomics
CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.14 Every year Alf sells apples from his orchard. Although the market price of apples remains constant (and equal to 1), the output of Alf ’s orchard
is variable yielding an amount R1 ; R2 in good and poor years respectively; the
probability of good and poor years is known to be 1
and respectively. A
buyer, Bill, o¤ ers Alf a contract for his apple crop which stipulates a down payment (irrespective of whether the year is good or poor) and a bonus if the year
turns out to be good.
1. Assuming Alf is risk averse, use an Edgeworth box diagram to sketch the
set of such contracts which he would be prepared to accept. Assuming that
Bill is also risk averse, sketch his indi¤ erence curves in the same diagram.
2. Assuming that Bill knows the shape of Alf ’s acceptance set, illustrate the
optimum contract on the diagram. Write down the …rst-order conditions
for this in terms of Alf ’s and Bill’s utility functions.
b
xRED
0
b
a
xBLUE
R2
•D
b
xBLUE
a
0
R1
a
xRED
Figure 8.2: Acceptable contracts
Outline Answer:
1. In Figure 8.2 the contours represent Alf’s indi¤erence curves: note that
they are convex to the point 0a (risk aversion) and that they have the same
slope [1
] = where they cross the 45 ray through 0a (consequence of
von-Neumann utility function). Point D represents the initial endowment;
Alf’s endowment is (R1 ; R2 ). Alf’s indi¤erence curve through point D
represents the boundary of the set of consumptions that Alf would regard
as being at least as good as the initial endowment: the shaded area is his
acceptance set. The buyer (Bill) has an endowment K that is independent
of the state of the world –see Figure 8.3. Note that the indi¤erence curves
for Bill also have the slope [1
] = where they cross the 45 ray through
0b :
2. Point E in Figure ?? represents the optimum contract (from Bill’s point
of view) since it is a point of common tangency of two indi¤erence curves.
c Frank Cowell 2006
134
Microeconomics
b
K
xRED
0
b
a
xBLUE
K
•D
b
xBLUE
a
0
a
xRED
Figure 8.3: Buyer’s situation
b
xRED
0
b
a
xBLUE
E
R2
••
D
b
xBLUE
a
0
R1
Figure 8.4: Optimal contract
At E we have that
c Frank Cowell 2006
ua0 (xa1 )
ub0 (xb )
= b0 1b :
a
a0
u (x2 )
u (x2 )
135
a
xRED
Microeconomics
CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.15 In exercise 8.14, what would be the e¤ ect on the contract if (i)
Bill were risk neutral; (ii) Alf risk neutral?
Outline Answer:
In case (i) Bill’s indi¤erence curves become lines with slope [1
] = and
the optimum is at E in Figure 8.5. In case (ii) Alf’s indi¤erence curves become
lines with slope [1
] = and the optimum is at the endowment point D.
b
xRED
0
b
a
xBLUE
E
•
R2
•D
b
xBLUE
a
0
R1
Figure 8.5: Optimal contract: risk-neutral buyer
c Frank Cowell 2006
136
Chapter 9
Welfare
Exercise 9.1 In a two-commodity exchange economy there are two large equalsized groups of traders. Each trader in group a has an endowment of 300 units
of commodity 1; each person in group b has an endowment of 200 units of
commodity 2. Each a-type person has preferences given by the utility function
U a (xa ) = xa1 xa2
and each b-type person’s utility can be written as
U b (xb ) =
xb1 xb2
xa1
where xhi means the consumption of good i by an h-type person.
1. Find the competitive equilibrium allocation
2. Explain why the competitive equilibrium is ine¢ cient.
3. Suggest a means whereby a benevolent government could achieved an e¢ cient allocation.
Outline Answer:
1. Notice that the term xa1 that appears in the b-type’s utility function is
a negative externality: the more that a-people consume of commodity 1,
the lower is every b-person’s utility. This is the reason that the competitive equilibrium will be ine¢ cient. To derive the competitive equilibrium,
notice that the term xa1 is virtually irrelevant to the b-people’s behaviour
(they cannot do anything about it). Both a-people and b-people thus have
Cobb-Douglas utility functions, and we know that their demands will be
given by:
1 yh
xi h =
; h = a; b; i = 1; 2:
2 pi
Incomes are y a = 300p1 , y b = 200p2 . Using this information we can see
that total demand for commodity 1 is
N x1a + x1b = N
137
1
1 200
300 +
2
2
Microeconomics
CHAPTER 9. WELFARE
where N is the large unknown number of traders in each group and :=
p1 =p2 (Notice that only the price ratio matters in the solution). Clearly
there are 300N units of commodity 1 available, so the excess demand
function for commodity 1 is:
E1 = N 150 +
100
300
By Walras’Law we know that if E1 = 0 then E2 = 0 also. Clearly E1 = 0
when
= 2=3; this is the equilibrium price ratio. Using the demand
functions we …nd
x1a
x2a
=
x1b
x2b
=
1
2
1
2
1
2
1
2
300
300
=
150
100
(9.1)
200=
200
=
150
100
(9.2)
This is the competitive equilibrium allocation.
2. To verify that this allocation is ine¢ cient consider the following. Since
there is a negative externality, it is likely that in the competitive equilibrium the a-people are consuming too much of commodity 1. So let us
change the allocation in such a way that the a-people consume less of commodity 1 ( xa1 < 0) but where the a-people’s utility remains unchanged:
this means that their consumption of good 2 must be increased, by an
amount
xa2 =
xa1 > 0 (remember that in equilibrium equals the
marginal rate of substitution). Now since there is a …xed total amount of
each commodity, the b-people’s consumptions must move in exactly the
opposite direction; so xb1 =
xa1 > 0 and
xb2 =
xa2 < 0. The
e¤ect on their utility can be computed thus:
log(U b )
=
=
=
1
xb1
1
xb2
xb2
1 2
1
+
150 100 3
1
xa1 > 0
150
xb1 +
1
xa1
xa1
1
xa1
150
So, as we expected, it is possible to move away from the competitive
equilibrium in such a way that some people’s utility is increased, and
no-one else’s utility decreases.
3. We might think that for e¢ ciency the relative price of commodity 1 should
just be increased to the a-people, relative to that facing the b-people. But
this will not work since their income is determined by p1 , and, as the
demand function reveals, their resulting consumption of commodity 1 is
independent of price. A rationing scheme may (in this case) be simpler.
c Frank Cowell 2006
138
Microeconomics
Exercise 9.2 Consider a constitution based on a system of rank-order voting
whereby the worst alternative gets 1 point, the next worst, 2, ... and so on, and
the state that is awarded the most points by the citizens is the one selected. Alf ’s
ranking of social states changes during the week. Bill’s stays the same:
Monday:
Alf Bill
Tuesday:
Alf Bill
0
0
00
0
00
00
0
00
What is the social ordering on Monday? What is it on Tuesday? How does
this constitution violate the IIA Axiom?
Outline Answer:
Use “ ” to denote strict preference and “~” to denote indi¤erence. The
voting points scored for the three states ; 0 ; 00 are as follows; Monday: [5,5,2],
00
0
00
Tuesday: [5,4,3]. So the social ordering changes from ~ 0
to
.
0
Notice that the rankings by Alf and Bill of and remain unchanged; yet their
reranking of 00 with reference to other alternatives changes the social ordering
of and 0 from indi¤erence to strict preference.
c Frank Cowell 2006
139
Microeconomics
CHAPTER 9. WELFARE
Exercise 9.3 Consider an economy that consists of just three individuals, fa; b; cg
and four possible social states of the world. Each state-of-the-world is characterised by a monetary payo¤ y h thus:
0
00
000
a
3
1
5
2
b
3
4
1
6
c
3
4
3
1
Suppose that person h has a utility function U h = log(y h ).
1. Show that if individuals know the payo¤ s that will accrue to them under
each state-of-the-world, then majority voting will produce a cyclic decision
rule.
2. Show that the above conditions can rank unequal states over perfect equality.
3. Show that if people did not know which one of the identities fa; b; cg they
were to have before they vote, if they regard any one of these three identities
as equally likely and if they are concerned to maximise expected utility, then
majority voting will rank the states strictly in the order of the distribution
of the payo¤ s.
4. A group of identical schoolchildren are to be endowed at lunch time with
an allocation of pie. When they look through the dining hall window in
the morning they can see the slices of pie lying on the plates: the only
problem is that no child knows which plate he or she will receive. Taking
the space of all possible pie distributions as a complete description of all
the possible social states for these schoolchildren, and assuming that ex
ante there are equal chances of any one child receiving any one of the
plates discuss how a von Neumann-Morgenstern utility function may be
used as a simple social-welfare function.
5. What determines the degree of inequality aversion of this social-welfare
function?
6. Consider the possible problems in using this approach as a general method
of specifying a social-welfare function.
Outline Answer:
000
1. We …nd that majority voting produces the ranking 0
but also
000
0
0 000
0
; i.e. the states f ; ; g form a cycle. (ii)
would be strictly
preferred to the state of perfect equality . (iii) Since the probability of
being assigned any one of the three identities is 13 , the utility payo¤s are:
c Frank Cowell 2006
140
Microeconomics
0
00
000
a
3
1
5
2
b
3
4
1
6
c
3
4
3
1
Expected Utility
1
3 log(27)
1
3 log(16)
1
3 log(15)
1
3 log(12)
2. Now consider ordering the states by income inequality
0
00
000
Distribution
[3,3,3]
[1,4,4]
[1,3,5]
[1,2,6]
Expected Utility
1
3 log(27)
1
3 log(16)
1
3 log(15)
1
3 log(12)
3. Concavity of the utility function implies (in a primordial ignorance about
identity) a preference for equality.
4. Let there be nh schoolchildren, and let the allocation of pie received by
child h be y h ; h = 1; 2; :::nh . The set of all possible pie distributions is
(
)
nh
X
1 2
nh
h
h
y ; y ; :::; y
:y
0;
y
1
h=1
If each child perceives an equal chance of being assigned any plate the
expected utility of the perceived pie distribution is
nh
1 X
u yh
nh
h=1
where u is an increasing function. If u is concave then this utility function will rank more equal distributions as being preferable to less equal
distributions.
5. Inequality aversion is identical to risk aversion.
6. In practice the problem of receiving a particular pie allocation will not be
the same individuals; personal risk-aversion may not be an appropriate
basis for inequality aversion with reference to life chances.
c Frank Cowell 2006
141
Microeconomics
CHAPTER 9. WELFARE
Exercise 9.4 Table 9.1 shows the preferences over three social states for two
groups of voters; the row marked “#” gives the number of voters with each set
of preferences; preferences are listed in row order, most preferred at the top.
1. Find the Condorcet winner among right-handed voters only.
2. Show that there is a cycle among left-handed voters only.
3. Suppose that the cycle among the left-handed voters is broken by ignoring
the vote that has the smallest winner. Show that the winner is then the
same as that among the right-handed voters.
4. Show that if the two groups are merged there is a Condorcet winner but is
di¤ erent from the winners found for the left-handers and the right-handers
separately!
5. Would the above paradox occur if one used de Borda voting?
Left-handers
#
10
6
0
0
00
6
0
Right-handers
12
18
00
00
00
00
17
00
0
0
0
Table 9.1: Left-handed and right-handed voters
Outline Answer:
Use to denote “beats in a bilateral vote.”
0
00
1. Clearly, from the right-hand part of the table
and
, both
by a majority of 18 to 17. So is unambiguously the Condorcet winner
among the right-hand voters
2. Consider pairwise votes among the left-hand group:
versus 0 . There are 10+12 votes for
0
by 22 to 12
0
0
versus 00 . There are 10+6+6 votes for
00
by 22 to 12.
versus
So 00
00
. There are 10+6 votes for
by 18 to 16
So there is a cycle
0
00
against 6+6 votes for
0
against 12 votes for
0
. So
00
against 6+12 votes for
.So
00
.
.
3. The “weakest link” in the above cycle is 00
; here 00 wins by only
two votes as against the margin of 10 votes in the other two cases. If we
remove this link it is then clear that the winner is as with the right-hand
voters
c Frank Cowell 2006
142
Microeconomics
4. For the two groups together:
versus 0 . There are 10+12+18+17 votes for
0
for 0 . So
by 57 to 12
0
00
versus
for 00 .So
. There are 10+6+6 votes for
0
by 47 to 22.
against 12+18+17 votes
00
versus 00 . There are 10+6+18 votes for
for 00 . So 00
by 35 to 34
So now
0
against 6+6 votes
00
0
unambiguously.
00
against 6+12+17 votes
is the clear winner!
5. Borda voting where 3 is the score given to the best alternative, 1 to the
worst. Votes are as in Table 9.2. wins in each subgroup and overall.
Left-handers
0
00
10
3
2
1
6
2
3
1
6
1
3
2
12
2
1
3
tot
72
68
64
Right-handers
17 tot
18
3
1
2
2
1
3
88
35
87
Table 9.2: Borda votes
c Frank Cowell 2006
143
Both groups
160
103
151
Microeconomics
CHAPTER 9. WELFARE
Exercise 9.5 Suppose social welfare is related to individual incomes y h thus:
W =
nh
X
(y h )
h=1
where ( ) has the form
(x) =
and
x1
1
1
is a non-negative parameter.
1. What form does
take for
= 1? [Hint, use l’Hôpital’s rule].
2. What is relative inequality aversion for this W ?
3. Draw the contours of the social welfare function for the cases = 1, ! 0,
! 1. What is equally-distributed-equivalent income in each case?
4. If, instead of a …nite population f1; ::; nh g, there is a continuum of individuals distributed on R with density at income y given by f (y) write down
the equivalent form of the social welfare function W in general and in the
particular cases cited in part 3.
Outline Answer:
1
1. Clearly the denominator and the numerator of x 1 1 both vanish as
! 1. L’Hôpital’s rule implies that the limiting value can be found by
taking the ratio of the …rst derivatives of the denominator and numerator.
Evaluating the derivatives we have
x1
!1
1
lim
2. Di¤erentiating
1
= lim
x1
!1
log x
= log x.
1
we get
d (x)
dx
d2 (x)
dx2
= x
=
1
x
and so (x) = .
3. For the case = 1 contours will be rectangular hyperbolas and ede-income
is the geometric mean. For = 0 contours will be diagonal straight lines
and ede-income is the arithmetic mean. For = 1 contours will be Lshapes and ede-income is the smallest of the incomes y h .
4.
W
=
=
=
=
c Frank Cowell 2006
Z
Z
Z
(y)f (y)dy
yf (y)dy, if
=0
log yf (y)dy, if
inffy : f (y) > 0g, if
144
=1
=1
Microeconomics
Exercise 9.6 In a two-commodity exchange economy there are two groups of
people: type a have the utility function 2 log(xa1 ) + log(xa2 ) and an endowment
of 30 units of commodity 1 and k units of commodity 2; type b have the utility
function log(xb1 ) + 2 log(xb2 ) and an endowment of 60 units of commodity 1 and
210 k units of commodity 2.
1. Show that the equilibrium price, , of good 1 in terms of good 2 is
[Hint: use the answer to Exercise 7.4].
210+k
150 .
2. What are the individuals’incomes (y a ; y b ) in equilibrium as a function of
k? As a function of ?
3. Suppose it is possible for the government to carry out lump-sum transfers
of commodity 2, but impossible to transfer commodity 1. Use the previous
answer to show the set of income distributions that can be achieved through
such transfers. Draw this in a diagram.
4. If the government has the social welfare function
W (y a ; y b ) = log(y a ) + log(y b )
…nd the optimal distribution of income using the transfers mentioned in
part 3. [hint use the diagram constructed earlier].
5. If instead the government has the social welfare function
W (y a ; y b ) = y a + y b
…nd the optimal distribution of income using transfers. Comment on the
result.
Outline Answer:
1. Let us use commodity 2 as numéraire. From the details in the question
we …nd that incomes for the two types of people are:
ya
yb
=
=
30 + k
60 + [210
(9.3)
(9.4)
k]
where is the price of commodity 1 in terms of commodity 2. Again we
have a Cobb-Douglas utility function (Cf the answer to Exercise 7.4), and
so the demands by a for the two commodities are
2
3
1
3
30 +k
[30 + k]
and for b the demands are
1
3
2
3
60 +210 k
[60 + 210
k]
This means that the excess demand for commodity 2 at price must
1
10 + k + 40 + 140
3
c Frank Cowell 2006
145
2
k
3
210
be
Microeconomics
CHAPTER 9. WELFARE
The equilibrium value of is found by setting this equal to 0 (remember
that Walras’ Law will ensure that the other market also has zero excess
demand). Doing this we get the value speci…ed in the question.
2. Substituting back into (9.3) and (9.4) we get:
ya
=
yb
=
210 + 6k
5
1470 3k
5
(9.5)
(9.6)
or, using the formula for the equilibrium price, we have equivalently:
ya
yb
=
=
180
420
210
90
(9.7)
(9.8)
3. Equations (9.5) and (9.6) imply that there is a straight-line frontier on
the set of income pairs (y a ; y b ) mapped out by letting k vary from 0 to
210 (or by using the price equations), in other words a straight line from
(42,294) to (294,168). The equation of this line segment is
y b = 315
1 a
y :
2
This is depicted as the solid line segment in Figure 9.1 are the incomes
that the government could generate by choosing k –in e¤ect a lump-sum
transfer between the two persons. The shaded area gives all the possible
combination of incomes if income can be thrown away.
yb
300
(42,294)
200
(294,168)
100
ya
0
100
200
300
Figure 9.1: Income possibility set
c Frank Cowell 2006
146
Microeconomics
4. If the government tries to maximise
W = log(y a ) + log(y b )
subject to the (truncated) straight line given in …gure 9.1, the best it can
achieve is a corner solution giving all of resource 2 to person a.
^ = y a + y b . So even
5. But this of course is exactly what happens with W
though the SWF W exhibits inequality aversion, you get the same outcome
^ which ignores distributional issues and just seeks to
as with the SWF W
maximise total income.
c Frank Cowell 2006
147
Microeconomics
CHAPTER 9. WELFARE
Exercise 9.7 This is an example of rent seeking. In a certain industry it is
known that monopoly pro…ts are available. There are N …rms that are lobbying
to get the right to run this monopoly. Firm f spends an amount cf on lobbying;
the probability that …rm f is successful in its lobbying activity is given by
f
cf
:= PN
j=1
(9.9)
cj
1. Suppose …rm f makes the same assumptions about other …rms’ activities
as in Exercise 3.2 . It chooses cf so as to maximise expected returns to
lobbying assuming the other …rms’ lobbying expenditures are given. What
is the …rst-order condition for a maximum?
2. If the …rms are identical show that the total lobbying costs chosen by the
…rms must be given by
1
1
Nc =
N
3. If lobbying costs are considered to contribute nothing to society what is the
implication for the measurement of “waste” attributable to monopoly?
Outline Answer:
1. Making the behavioural assumption of Exercise 3.2 the probability of lobbying success (9.9) becomes
f
=
where
cf
K + cf
N
X
K :=
(9.10)
cj :
j=1;j6=f
Firm f maximises expected returns max f
choose lobbying expenditure cf to maximise:
cf
K + cf
cf so that the problem is,
cf
The FOC for a maximum is
K+
cf
cf
K + cf
1
1 = 0:
(9.11)
2. If the …rms are identical then the optimal lobbying expenditure is the same
c for each …rm f . So K + cf is just N c and (9.11) becomes
Nc
c
Nc
1
1 = 0:
(9.12)
:
(9.13)
This implies
Nc =
1
N
3. Clearly there is an additional component to the waste generated by a
monopoly over and above deadweight loss; the additional component is
given by (9.13).
c Frank Cowell 2006
148
Microeconomics
Exercise 9.8 In an economy there are n commodities and nh individuals, and
there is uncertainty: each individual may have good or poor health. The state
of health is an independently distributed random variable for each individual
and occurs after the allocation of goods has taken place. Individual h gets the
following utility in state-of-the-world !:
uh xh ; ! := ah xh1 ; ! +
n
X
bh xhi
i=2
where xh := xh1 ; xh2 ; :::; xhn , xhi is the amount of commodity i consumed by h,
the functions ah ,bh are increasing and concave in consumption, and ! takes one
of the two values “poor health” or “good health” for each individual; good 1 is
health-care services.
1. The government estimates that for each individual the probability of stateof-the-world ! is ! . If aggregate production possibilities are described
by the production constraint (x) = 0 (where x := (x1 ; x2 ; :::; xn ) and xi
is the aggregate consumption of commodity i) and the government has a
social-welfare function
nh X
X
!u
h
xh ; !
h=1 !
…nd the …rst-order conditions for a social optimum.
2. The government also has the ability to tax or subsidise commodities at
di¤ erent rates for di¤ erent individuals: so individual h faces a price phi
for commodity i. If the person has an income y h and estimates that the
probability of state-of-the-world ! is h! , and if he maximises expected
utility, write down the …rst-order conditions for a maximum.
3. Show that the solutions in parts 1 and 2 can only coincide if
phi
=
phj
ph1
=
phj
(x)
j (x)
1
(x)
; i; j = 2; :::; n
j (x)
i
"P
!
P
!
h h
! a1
h
! a1
xh1 ; !
xh1 ; !
#
; j = 2; :::; n
Is there a case for subsidising health-care? Is there a case for subsidising
any other commodity?
Outline Answer:
1. The Lagrangean for the social optimum is
L x1 ; x2 ; :::; xm ;
c Frank Cowell 2006
:=
m X
X
h=1 !
149
(!)
h
xh ; !
m
X
h=1
xh
!
Microeconomics
CHAPTER 9. WELFARE
so that the …rst-order conditions for the social optimum are
X
(!) h1 xh1 ; ! =
1 (x)
(9.14)
!
h
i
where the subscripts on
sumption.
xhi
and
=
i
(x) ; i = 2; :::; n
(9.15)
denote derivatives with respect to con-
2. The Lagrangean for the consumer’s optimum is
"
X
h
h
h
h
h
L x ;
:=
(!)
x ;! +
Mh
!
n
X
phi xhi
i=1
#
so that the …rst-order conditions for the consumer’s optimum are
X
h
(!) h1 xh1 ; ! = ph1
(9.16)
!
h
i
xhi = phi
(9.17)
3. From the (9.15) and (9.17) we get
(x)
=
j (x)
i
h
i
h
j
xhi
=
xhj
phi
; i; j = 2; :::; n
phj
and from the (9.14) and (9.16) we get
P
h
h
1 (x)
! (!) 1 x1 ; !
=
; j = 2; :::; n
h
h
j (x)
j xj
P h
(!) h1 xh1 ; !
ph1
!
=
; j = 2; :::; n
h
h
h
pj
j xj
from which we have
1=
1=
h
j
xhj
=
h
j
xhj
=
1
j
(x)
1
P
(x) ! (!) h1 xh1 ; !
ph1
P
phj !
1
h (!) h
1
xh1 ; !
so that the relationship in part 3 holds. If personal perceptions of the risk
of poor health di¤er from the government’s assessment of risk the price
of health-care may need to be adjusted for that individual. This does not
arise with reference to pairs of goods that do not involve health-care: there
the price ratio can be the same for everyone.
c Frank Cowell 2006
150
Microeconomics
Exercise 9.9 Revisit the economy of San Serrife (Exercise 4.11) Heterogeneity
amongst the inhabitants of San Serrife was ignored in Exercise 4.11. However, it
is now known that although all San Serrife residents have preferences of form A
in Exercise 4.2 they di¤ er in their tastes: Northern San Serrifeans spend 34%
of their budget on milk and only 2% on wine, while Southern San Serrifeans
spend just 4% of their budget on milk and 32% on wine. The question of entry
to the EU is to be reviewed; the consequences for the prices of milk and wine of
entry to the EU are as in Exercise 4.11.
1. Assume that there are eight times as many Southerners as Northerners in
the San Serrife population, but that the average income of a Northerner is
four times that of a Southerner. On the basis of the potential-superiority
criterion, should San Serrife enter the EU?
2. Suppose Northerners and Southerners had equal incomes. Should San Serrife enter the EU?
3. What would be the outcome of a straight vote on entry to the EU?
Outline Answer:
1. Given that the i have the interpretation of expenditure shares it is clear
that Northerners would lose by EU entry and Southerners would gain. So
what should happen? The total CV (summed over all San Serrife) is
X
CVh (p ! p
^ ) = [1
(9.18)
N ]nN yN + [1
S ]nS yS
h
where the subscripts refer to North and South, nj is the population in
region j and yj is the income in region j . Given the information in the
question we …nd that N = 23 0:34 0:02 = 2 and S = 23 0:04 0:32 = 2 0:2
and that (9.18) is proportional to
4[1
N ] + 8[1
= 4 8 2 0:2
= 4[1 20:8 ] < 0
S]
There appears to be no potential Pareto gain in entering the EU - the
total gain to the poor Southerners is less than the loss experienced by the
rich Northerners. A conventional “cost-bene…t” rule would thus indicate
that San Serrife should not enter.
2. It is interesting to note that had Northerners and Southerners had the
same percapita income then the conclusion would have been the opposite:
1 + 8[1
2
0:04
]=7
22:96 > 0:
3. Simple voting would have given the opposite answer to the “potential
Pareto gain” rule.
c Frank Cowell 2006
151
Microeconomics
c Frank Cowell 2006
CHAPTER 9. WELFARE
152
Chapter 10
Strategic Behaviour
Exercise 10.1 Table 10.1 is the strategic form representation of a simultaneous
move game in which strategies are actions.
sa1
sa2
sa3
sb1
0; 2
2; 4
1; 1
sb2
3; 1
0; 3
2; 0
sb3
4; 3
3; 2
2; 1
Table 10.1: Elimination and equilibrium
1. Is there a dominant strategy for either of the two agents?
2. Which strategies can always be eliminated because they are dominated?
3. Which strategies can be eliminated if it is common knowledge that both
players are rational?
4. What are the Nash equilibria in pure strategies?
Outline Answer:
1. No player has a dominant strategy.
2. Both sa3 and sb2 can be eliminated as individually irrational.
3. With common knowledge of rationality we can eliminate the dominated
strategies: sa3 and sb2 :
4. The Nash Equilibria in pure strategies are (sa2 ; sb1 ) and (sa1 ; sb3 )
153
Microeconomics
CHAPTER 10. STRATEGIC BEHAVIOUR
Exercise 10.2 Table 10.2 again represents a simultaneous move game in which
strategies are actions.
sa1
sa2
sa3
sb1
0; 2
2; 0
1; 3
sb2
2; 0
0; 2
1; 3
sb3
3; 1
3; 1
4; 4
Table 10.2: Pure-strategy Nash equilibria
1. Identify the best responses for each of the players a, b.
2. Is there a Nash equilibrium in pure strategies?
Outline Answer
1. For player A the best reply is sa2 if player B plays sb1 , sa1 if B plays sb2 , sa3
if B plays sb3 :For player B the best reply is sb1 if A plays sa1 , sb2 if A plays
sa2 , sb3 if A plays sa3
2. The unique Nash Equilibrium is (sa3 ; sb3 )
c Frank Cowell 2006
154
Microeconomics
Exercise 10.3 A taxpayer has income y that should be reported in full to the
tax authority. There is a ‡at (proportional) tax rate on income. The reporting
technology means that that taxpayer must report income in full or zero income.
The tax authority can choose whether or not to audit the taxpayer. Each audit
costs an amount ' and if the audit uncovers under-reporting then the taxpayer
is required to pay the full amount of tax owed plus a …ne F .
1. Set the problem out as a game in strategic form where each agent (taxpayer,
tax-authority) has two pure strategies.
2. Explain why there is no simultaneous-move equilibrium in pure strategies.
3. Find the mixed-strategy equilibrium. How will the equilibrium respond to
changes in the parameters , ' and F ?
Outline Answer
1. See Table 10.3.
sa1
conceal
sa2
report
Tax-Authority
sb1
sb2
Audit
Not audit
] y F; y + F '
y; 0
[1
Taxpayer
[1
] y;
y
'
[1
] y;
y
Table 10.3: The tax audit game
2. Consider the best responses:
Tax-Authority’s best response to “conceal” is “audit”
Taxpayer’s best response to “audit” is “report”
Tax-Authority’s best response to “report” is “not audit”
Taxpayer’s best response to “not audit” is “conceal”
a
3. Suppose the taxpayer conceals with probability
audits with probability b .
and the tax authority
(a) Expected payo¤ to the taxpayer is
a
a
=
b
[[1
]y
a
+ [1
]
b
b
F] + 1
[1
y
b
]y + 1
[1
which, on simplifying, gives
a
So we have
c Frank Cowell 2006
= [1
]y +
d
d
a
a
= 1
155
a
1
b
b
y
a b
y
b
F
F:
]y ;
Microeconomics
CHAPTER 10. STRATEGIC BEHAVIOUR
πb
1
taxpayer
reaction
•
π*b
tax authority
reaction
0
πa
π*a
1
Figure 10.1: The tax audit game
It is clear that
d
d
a
a
T 0 as
S
b
b
b
:=
where
y
:
y+F
(10.1)
So the taxpayer’s optimal strategy is to conceal with probability 1
if the probability of audit is too low ( b < b ) and to conceal with
probability zero if the probability of audit is high.
(b) Expected payo¤ to the tax-authority is
b
b
[ a[ y+F
b
+ 1
[
=
a
'] + [1
a
[0] + [1
a
] [ y ']]
] [ y]]
which, on simplifying, gives
b
a
= [1
So we have
d
d
It is clear that
d
d
b
b
] y+
b
b
T 0 as
a
=
a
[ y + F]
[ y + F]
T
a
a b
:=
a
b
'
'
where
'
:
y+F
(10.2)
So the tax authority’s optimal strategy is to audit with probability
0 if the probability of the taxpayer concealing is low ( a < a ) and
to audit with probability 1 if the probability of concealment is high.
(c) This yields a unique mixed-strategy equilibrium
trated in Figure 10.1.
c Frank Cowell 2006
156
a
;
b
as illus-
Microeconomics
(d) The e¤ect of a change in any of the model parameters on the equilibrium can be found by di¤erentiating the expressions (10.1) and
(10.2). we have
@
@
a
=
'y
2
[ y + F]
> 0;
@ b
Fy
=
2 > 0:
@'
[ y + F]
@ a
1
=
> 0;
@'
y+F
@ a
=
@F
c Frank Cowell 2006
'
2
[ y + F]
< 0;
157
@ b
= 0:
@'
@ b
=
@F
y
2
[ y + F]
< 0:
Microeconomics
CHAPTER 10. STRATEGIC BEHAVIOUR
Exercise 10.4 Take the “battle-of-the-sexes” game in Table 10.4
.
sa1
sa2
sb1
sb2
[West]
[East]
2,1
0,0
0,0
1,2
[West]
[East]
Table 10.4: “Battle of the sexes” –strategic form
1. Show that, in addition to the pure strategy Nash equilibria there is also a
mixed strategy equilibrium.
2. Construct the payo¤ -possibility frontier . Why is the interpretation of this
frontier in the battle-of-the-sexes context rather unusual in comparison
with the Cournot-oligopoly case?
3. Show that the mixed-strategy equilibrium lies strictly inside the frontier.
4. Suppose the two players adopt the same randomisation device, observable
by both of them: they know that the speci…ed random variable takes the
value 1 with probability and 2 with probability 1
; they agree to play
sa1 ; sb1 with probability and sa2 ; sb2 with probability 1
; show that
this correlated mixed strategy always produces a payo¤ on the frontier.
Outline Answer
1. Suppose a plays [West] with probability
ability b . The expected payo¤ to a is
a
a
=
=
=
2
b
a b
a
+ [1
a
1
b
a
d
d
b
=
b
=
a b
T 0 as
a
a
[
2
2
b
a
2
d
d
b
c Frank Cowell 2006
a
b
[1]
(10.3)
=
a
1+3
b
a b
=
[
a
[0] + [1
a
] [2]]
b
:
(10.4)
b
b
b
] [0]] + 1
] 1
+3
And so
It is clear that dd b T 0 as
where a ; b = 32 ; 13 .
[0] + 1
b
a
a
a
b
b
]
T 13 . The expected payo¤ to b is
[1] + [1
+ 2 [1
a
a b
d
d
It is clear that
and b plays [West] with prob-
[0] + [1
] 1
+3
So we have
=
b
[2] + 1
a
2+3
a
T 23 . So there is a mixed-strategy equilibrium
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2. See Figure 10.2. Note that, unlike oligopoly where the payo¤ (pro…t)
is transferable, in this interpretation the payo¤ (utility) is not so the
frontier has not been extended beyond the points (2,1) and (1,2). The
lightly shaded area depicts all the points in the attainable set of utility
can be “thrown away.” The heavily shaded area in Figure 10.2 shows the
expected-utility outcomes achievable by randomisation. The frontier is
given by the broken line joining the points (2,1) and (1,2).
Figure 10.2: Battle-of-sexes: payo¤s
3. The utility associated with the mixed-strategy equilibrium is
clearly lies inside the frontier in Figure 10.2.
2 2
3; 3
and
4. Given that the probability of playing [West] is , the expected utility for
each player is
a
= 2 + [1
]=1+
b
=
+ 2 [1
]=2
If we allow to take any value in [0; 1] this picks out the points on the
broken line in Figure 10.2.
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CHAPTER 10. STRATEGIC BEHAVIOUR
Exercise 10.5 Rework Exercise 10.4 for the case of the Chicken game in Table
10.5.
sb1
2; 2
3; 1
sa1
sa2
sb2
1; 3
0; 0
Table 10.5: “Chicken” –strategic form
Outline Answer
υb
•
3
•
2
•
(1½, 1½)
•
1
υa
0
1
2
3
Figure 10.3: Chicken: payo¤s
1. Suppose a plays sa1 with probability
The expected payo¤ to a is
a
=
a
=
a
b
b
[2] + 1
b
+3
2
d
d
d
d
b
=
b
=
b
a
a
[
T 0 as
a
b
+3
2
b
]
[3] + 1
b
.
[0]
(10.5)
a
a
a
=1
2
b
b
] [1]] + 1
a b
And so
d
d
c Frank Cowell 2006
a
[1] + [1
b
S 12 . The expected payo¤ to b is
[2] + [1
a
and b plays sb1 with probability
a b
So we have
It is clear that
a
[
a
[3] + [1
a
] [0]]
(10.6)
b
b
=1
160
2
a
Microeconomics
b
It is clear that dd b T 0 as
where a ; b = 21 ; 12 .
a
S 12 . So there is a mixed-strategy equilibrium
2. See Figure 10.3. The lightly shaded area depicts all the points in the
attainable set of utility can be “thrown away.” The heavily shaded area
shows the expected-utility outcomes achievable by randomisation
3. The utility associated with the mixed-strategy equilibrium is 1 12 ; 1 21 and
clearly lies inside the frontier.
4. Once again a correlated strategy would produce an outcome on the broken
line.
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CHAPTER 10. STRATEGIC BEHAVIOUR
Alf
[RIGHT]
[LEFT]
Bill
[left]
[right]
[left]
[right]
Charlie
[L]
0
1 
3
[M]
2
2 
2
[R]
[L]
0 0
1  0 
0 0
[M]
0
0 
0
[R]
[L]
0 1
0  1 
0 1
[M]
2
2 
0
[R]
[L]
1 1
1  0 
0 0
[M]
2
2 
2
[R]
1
0 
3
Figure 10.4: Bene…ts of restricting information
Exercise 10.6 Consider the three-person game depicted in Figure 10.4 where
strategies are actions. For each strategy combination, the column of …gures in
parentheses denotes the payo¤ s to Alf, Bill and Charlie, respectively.
1. For the simultaneous-move game shown in Figure 10.4 show that there is
a unique pure-strategy Nash equilibrium.
2. Suppose the game is changed. Alf and Bill agree to coordinate their actions by tossing a coin and playing [LEFT],[left] if heads comes up and
[RIGHT],[right] if tails comes up. Charlie is not told the outcome of the
spin of the coin before making his move. What is Charlie’s best response?
Compare your answer to part 1.
3. Now take the version of part 2 but suppose that Charlie knows the outcome of the coin toss before making his choice. What is his best response?
Compare your answer to parts 1 and 2. Does this mean that restricting
information can be socially bene…cial?
Outline Answer
1. The strategic form of the game can be represented as in Table 10.6 from
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Microeconomics
sb1
sb2
sc1
L
0; 1; 3
0; 0; 0
[left]
[right]
sa1 :[LEFT]
sc2
sc3
M
R
2; 2; 2 0; 1:0
0; 0; 0 0; 0; 0
sa2 :[RIGHT]
sc1
sc2
sc3
L
M
R
1; 1; 1 2; 2; 0 1; 1; 0
1; 0; 0 2; 2; 2 1; 0; 3
Table 10.6: Alf, Bill, Charlie –Simultaneous move
Heads
Tails
[left,LEFT]
[right,RIGHT]
sc1
L
0; 1; 3
1; 0; 0
sc2
M
2; 2; 2
2; 2; 2
sc3
R
0; 1:0
1; 0; 3
Table 10.7: Alf, Bill correlate their play
which it is clear that the best responses for the three players are as follows:
BRa (left; L)
BRa (left; M)
BRa (left; R)
BRa (right; L)
BRa (right; M)
BRa (right; R)
=
=
=
=
=
=
RIGHT
fLEFT; RIGHTg
fLEFT; RIGHTg
RIGHT
RIGHT
RIGHT
BRb (LEFT; L)
BRb (LEFT; M)
BRb (LEFT; R)
BRb (RIGHT; L)
BRb (RIGHT; M)
BRb (RIGHT; R)
BRc (LEFT; left)
BRc (LEFT; right)
BRc (RIGHT; left)
BRc (RIGHT; right)
=
=
=
=
=
=
left
left
left
left
fleft; rightg
left
= L
= fL; M; Rg
= L
= R
it is clear that (RIGHT; left; L) is the unique Nash equilibrium. Everyone
gets a payo¤ of 1 at the Nash equilibrium: total payo¤ is 3.
2. Charlie knows the coordination rule but not the outcome of the coin toss.
The payo¤s are now as in Table 10.7. Note that neither of the possible
action combinations by Alf and Bill would have emerged under the Nash
equilibrium in part 1. It is clear that now the expected payo¤ to Charlie
of playing L is 1:5; the expected payo¤ of playing R is also 1:5. But
the expected payo¤ of playing M is 2. So Charlie’s best response is M
Everybody gets a payo¤ of 2 with certainty: total payo¤ is 6.
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CHAPTER 10. STRATEGIC BEHAVIOUR
3. Charlie now knows both the coordination rule and the outcome of the coin
toss. From Table 10.7 it is clear that his best response is L if it is “heads”
and R if it is “tails”. Now he gets a payo¤ of 3 and the others get an
equal chance of 0 or 1: total payo¤ is 4, less than that under part 2 but
more than under part 1.
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Exercise 10.7 Consider a duopoly with identical …rms. The cost function for
…rm f is
C0 + cq f ; f = 1; 2:
The inverse demand function is
q
0
where C0 , c,
q = q1 + q2 .
0
and
are all positive numbers and total output is given by
1. Find the isopro…t contour and the reaction function for …rm 2.
2. Find the Cournot-Nash equilibrium for the industry and illustrate it in
q 1 ; q 2 -space.
3. Find the joint-pro…t maximising solution for the industry and illustrate it
on the same diagram.
4. If …rm 1 acts as leader and …rm 2 as a follower …nd the Stackelberg solution.
5. Draw the set of payo¤ possibilities and plot the payo¤ s for cases 2-4 and
for the case where there is a monopoly.
Outline Answer
1. Firm 2’s pro…ts are given by
2
= pq 2
=
C0 + cq 2
q1 + q2
0
q2
C0 + cq 2
So it is clear that a typical isopro…t contour is given by the locus of q 1 ; q 2
satisfying
c
q 1 + q 2 q 2 = constant
0
see Figure 10.5. The FOC for a maximum of 2 with respect to q 2 keeping
q 1 constant is
q 1 + 2q 2
c=0
0
which yields the Cournot reaction function for …rm 2
q2 =
2
0
q1 =
c
2
1 1
q
2
(10.7)
– a straight line. Note that this relationship holds wherever …rm 2 can
make positive pro…ts. See Figure 10.6 which shows the locus of points
that maximise 2 for various given values of q 1 .
2. By symmetry the reaction function for …rm 1 is
q1 =
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2
165
c
1 2
q
2
(10.8)
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CHAPTER 10. STRATEGIC BEHAVIOUR
q2
prof
it
q1
Figure 10.5: Iso-pro…t curves for …rm 2
q2
χ2(·)
q1
Figure 10.6: Reaction function for …rm 2
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The Cournot-Nash solution is where (10.7) and (10.8) hold simultaneously,
i.e. where
c 1
c 1 1
0
q1 = 0
q
(10.9)
2
2
2
2
The solution is at q 1 = q 2 = qC where
qC =
–see Figure 10.7. The price is
2
3
0
0
c
(10.10)
3
+ 13 c.
q2
χ1(·)
qC
•
χ2(·)
q1
Figure 10.7: Cournot-Nash equilibrium
3. Writing q = q 1 + q 2 , the two …rms’joint pro…ts are given by
2
= pq [2C0 + cq]
= [ 0
q] q [2C0 + cq]
The FOC for a maximum is
0
c
2 q=0
which gives the collusive monopoly solution as
qM =
0
2
c
:
(10.11)
with the corresponding price 12 [ 0 + c]. However, the break-down into
outputs q 1 and q 2 is in principle unde…ned. Examine Figure 10.8. The
points (qM ; 0) and (0; qM ) are the endpoints of the two reaction functions
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CHAPTER 10. STRATEGIC BEHAVIOUR
(each indicates the amount that one …rm would produce if it knew that the
other was producing zero). The solution lies somewhere on the line joining
these two points. In particular the symmetric joint-pro…t maximising
outcome (qJ ; qJ ) lies exactly at the midpoint where the isopro…t contour
of …rm 1 is tangent to the isopro…t contour of …rm 2.
q2
χ1(·)
(0,qM)
qC
•
•qJ
χ2(·)
q1
(qM,0)
Figure 10.8: Joint-pro…t maximisation
4. If …rm 1 is the leader and …rm 2 is the follower then …rm 1 can predict
…rm 2’s output using the reaction function (10.7) and build this into its
optimisation problem. The leader’s pro…ts are therefore given as
0
q1 +
2
q1
0
c
q1
C0 + cq 1
which, using (10.7), becomes
0
q1 +
1 1
q
2
2
q1
1
c
q 1 q 1 C0
2 0
The FOC for the leader’s problem is
=
1
[
2
so that the leader’s output is
0
qS1 =
(10.12)
q1 = 0
c]
0
c
2
and, using (10.7), the follower’s output must be
qS2 =
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0
4
C0 + cq 1
c
Microeconomics
output
Cournot
0
c
1
3
1
2
1
4
1
4
3
Joint pro…t max
0
c
4
Stackelberg leader
Stackelberg follower
0
c
2
0
price
c
4
0
[
0
0
[
+ c]
[
0
3
4c
3
4c
[
0
[
0
+
0
pro…t
2
3 c.
+
+
0
9
8
8
c]2
c]2
c]2
c]2
16
C0
C0
C0
C0
Table 10.8: Outcomes of quantity competition –linear model
–see Figure 10.9. The price is
1
4
0
+ 34 c.
q2
qC
• •qS
χ2(·)
q1
(qM,0)
Figure 10.9: Firm 1 as Stackelberg leader
5. The outcomes of the various models are given in Table 10.8.and the possible payo¤s are illustrated in Figure 10.10. Note that maximum total pro…t
on the boundary of the triangle is exactly twice the entry in the“Joint
2
pro…t max”row, namely 41 [ 0 c] =
2C0 . This holds as long as there
are two …rms present –i.e. right up to a point arbitraily close to either of
the end-points. But if one …rm is closed down (so that the other becomes
a monopolist) then its …xed costs are no longer incurred and the monop2
C0 . In Figure 10.10 the point
olist makes pro…t M := 14 [ 0 c] =
marked “ ” is where both …rms are in operation but …rm 1 is getting all
of the joint pro…t and the point ( M ; 0) is the situation where …rm 1 is
operating on its own.
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CHAPTER 10. STRATEGIC BEHAVIOUR
Π2
(0, ΠM)
•
•(Π ,Π )
J
J
(ΠC,ΠC)
•(Π ,Π )
1
S
2
S
{
C0
° (Π•,0)
0
M
Figure 10.10: Possible payo¤s
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Exercise 10.8 An oligopoly contains N identical …rms. The cost function is
convex in output. Show that if the …rms act as Cournot competitors then as N
increases the market price will approach the competitive price.
Outline Answer
The assumption of convex costs will ensure that there is no minimum viable
size of …rm. Pro…ts for a typical …rm are given by
p qf + K qf
where
K :=
N
X
C qf
(10.13)
qj
j=1
j6=f
is the total output of all the other …rms, which of course …rm f takes to be
constant under the Cournot assumption. Maximising this by choice of q f gives
the FOC for an interior solution
pq q f + K q f + p q f + K
Cq q f = 0
(10.14)
Given that all the …rms are identical we may rewrite condition (10.14) as
pq (q)
q
+ p (q)
N
Cq = 0
(10.15)
where q is industry output. This in turn can be rewritten as
p (q) =
where
:=
Cq
1 + 1N
(10.16)
p (q)
qpq (q)
is the elasticity of demand. The result follows immediately: as N becomes large
(10.16) approaches
p (q) = Cq
(10.17)
.
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CHAPTER 10. STRATEGIC BEHAVIOUR
Exercise 10.9 Two identical …rms consider entering a new market; setting up
in the new market incurs a once-for-all cost K > 0; production involves constant
marginal cost c. If both …rms enter the market Bertrand competition then takes
place afterwards. If the …rms make their entry decision sequentially, what is the
equilibrium?
Outline Answer
1. The …rms …rst decide whether to enter (and hence incur the …xed cost K),
then they play the Bertrand pricing game. K is thus considered a sunk
cost when the second-stage game is played. When both …rms decide to
enter, the unique Nash equilibrium of the Bertrand pricing game is to set
prices equal to marginal cost, (p1 ; p2 ) = (c; c). This yields overall pro…ts
for the two …rms ( 1 ; 2 ) = ( K; K).
2. The extensive form is shown in Figure 10.11.
Figure 10.11: The entry game
3. To …nd the Subgame Perfect Nash Equilibrium, solve the game by backwards induction. If …rm 1 decides to enter, …rm 2’s optimal strategy is
not to enter (pro…t of 0 compared to K). If …rm one decides not to
enter, then …rm 2 should enter. Firm two can observe the action of …rm
1, thus it can form history-dependent strategies. The optimal strategy is
not to enter if …rm 1 entered, and to enter if …rm 1 did not enter the market, (not enter;enter). Thus, the decision for …rm 1 is between entering
and receiving pro…ts of M K, or not entering and receiving 0. For K
“small”, …rm 1 will decide to enter. The unique Subgame Perfect Nash
Equilibrium in strategies is thus
(enter; (not enter,enter));
yielding the equilibrium outcome (enter;not enter).
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(10.18)
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4. Clearly, there is a …rst-mover advantage, since even a small …xed cost leads
to the monopoly outcome and hence strictly positive pro…ts for the …rst
mover.
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CHAPTER 10. STRATEGIC BEHAVIOUR
Exercise 10.10 Two …rms have inherited capacity from the past so that production must take place subject to the constraint
qf
q f ; f = 1; 2
There are zero marginal costs. Let ( ) be the Cournot (quantity-competition)
reaction function for each …rm. If the …rms compete on prices show that the
following must be true in a pure-strategy equilibrium:
1. Both …rms will charge the same price p.
2. p = p q 1 + q 2 .
3. p
4. q 1
1
p
1
q2 + q2 .
q2 .
Outline Answer
1. If p1 < p2 then, if q 1 = q 1 …rm 1 could make higher pro…ts by raising its
price. Otherwise …rm 1 would be undercutting …rm 2 so that …rm 2 would
be forced to reduce its price or lose all its sales to …rm 1. Hence we must
have p1 = p2 = p in equilibrium.
2. Consider two cases:
(a) If p > p q 1 + q 2 .then for one or both …rms it must be the case that
q f < q f – the price is too high to exhaust capacity. In which case
one of the …rms could reduce the price slightly, capture sales from
the other …rm and increase pro…ts.
(b) If p < p q 1 + q 2 .then both …rms are producing to capacity. Each
could increase pro…ts just by raising prices to its (rationed) customers.
Therefore
p = p q1 + q2
3. If p < p
1
(10.19)
q 2 + q 2 .one of the …rms must be capacity constrained:
(a) If …rm 1 is capacity constrained it could raise its price by
make additional pro…ts pq 1 .
p and
(b) Otherwise, if …rm 2 is capacity constrained then …rm 1’s pro…ts are
q1 p q1 + q2
(10.20)
If …rm 1 is also capacity constrained then it could increase pro…ts
by raising its price. So we may take …rm 1 as not being capacity
constrained. Given the de…nition of 1 ( ) as …rm 1’s best response
function (10.20) can be written:
1
c Frank Cowell 2006
q2 p
174
1
q2 + q2
Microeconomics
with
q1 =
1
q2
(10.21)
and the price must be
p=p
1
q2 + q2
(10.22)
4. Suppose
q1 >
1
q2 :
If there is a pure strategy equilibrium then, by (10.19)
p = p q1 + q2 < p
which contradicts (10.22)
c Frank Cowell 2006
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1
q2 + q2
(10.23)
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CHAPTER 10. STRATEGIC BEHAVIOUR
Exercise 10.11 In winter two identical ice-cream …rms have to choose the capacity that they plan to use in the summer. To install capacity q costs an amount
k q where k is a positive constant. Production in the summer takes place subject
to
q f q f ; f = 1; 2
where q f is the capacity that was chosen in the previous winter. Once capacity is
installed there is zero marginal cost. The market for ice-cream is characterised
by the inverse demand function p q 1 + q 2 . There are thus two views: the
“before” problem when the decision on capacity has not yet been taken; the
“after” problem (in the summer) once capacity has been installed.
1. Let ( ) be the Cournot reaction function for either …rm in the “after”
problem (as in Exercise 10.10). In the context of a diagram such as Figure
10.6 explain why this must lie strictly above the Cournot reaction function
for the “before” problem.
2. Let qC be the Cournot-equilibrium quantity for the “before”problem. Write
down the de…nition of this in terms of the present model.
3. Suppose that in the summer competition between the …rms takes place in
terms of prices (as in Exercise 10.10). Show that a pure-strategy Bertrand
equilibrium for the overall problem is where both …rms produce qC .
Outline Answer
q2
low
hig
hc
cos
t
ost
q1
1.
Figure 10.12: Low-cost and high-cost reaction functions for …rm 2
First consider how the reaction function of …rm 2 would di¤er if the constant marginal cost were lower than its current value. Given a linear
inverse demand function
q
0
and constant marginal costs c, the reaction function for …rm 2 is
q2 =
c Frank Cowell 2006
0
2
176
c
1 1
q
2
(10.24)
Microeconomics
q2
afte
r
bef
ore
afte
bef
r
ore
q1
Figure 10.13: Reaction functions before and after capacity costs are sunk
–see Exercise 10.7. So for high and low values of c we have the situation
depicted in Figure 10.12. Before installation capacity costs are proportional to the amount of capacity (marginal cost of capacity is k). But
once the capacity has been installed it represents sunk costs so, from the
…rm’s point of view it is as though the marginal cost of production has
been cut. So the after-installation reaction functions must be as in Figure
10.13.
q2
χ(q2) − ½k/β
qC
•
χ(q1) − ½k/β
q1
2.
Figure 10.14: Equilibrium capacity
Clearly if (10.24) is the “after” reaction function for …rm 2, so that q 2 =
q 1 once capacity cost is sunk, the “before”reaction functions for …rms
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CHAPTER 10. STRATEGIC BEHAVIOUR
1 and 2 are, respectively
q2 =
0
q1 =
0
c
k
1 1
q
2
(10.25)
c
k
1 2
q
2
(10.26)
2
2
so that the “before” reaction function for each …rm is
()
k
:
2
The value qC is found from setting q 1 = q 2 = qC in (10.25) and (10.26) to
give
c k
qC = 0
:
(10.27)
3
This gives the solution to the amount of capacity q 1 ; q 2 to install in the
…rst period
3. If there is price competition in the second period then, from the solution
to Exercise 10.10
p = p q1 + q2 ;
we have the solution to the ice-cream sellers’Summer price problem as
p = p (2qC ) :
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Exercise 10.12 There is a cake of size 1 to be divided between Alf and Bill.
In period t = 1 Alf o¤ ers Bill a share: Bill may accept now (in which case the
game ends), or reject. If Bill rejects then, in period t = 2 Alf again makes an
o¤ er, which Bill can accept (game ends) or reject. If Bill rejects, the game ends
one period later with exogenously …xed payo¤ s of to Alf and 1
to Bill.
Assume that Alf and Bill’s payo¤ s are linear in cake and that both persons have
the same, time-invariant discount factor < 1.
1. What is the backwards induction outcome in the two-period model?
2. How does the answer change if the time horizon increases but is …nite?
3. What would happen if the horizon were in…nite?
Alf
[offer 1 − γ1]
period 1
Bill
[accept]
[reject]
(γ1, 1 − γ1)
Alf
period 2
[offer 1 − γ2]
Bill
[accept]
[reject]
(δγ2, δ[1 − γ2])
(δ2γ, δ2[1 − γ ])
period 3
Figure 10.15: One-sided bargaining game
1. Begin by drawing the extensive form game tree for this bargaining game.
Note that payo¤s can accrue either in period 1 (if Bill accepts immediately), in period 2 (if Bill accepts the second o¤er), or in period 3 (Bill
rejects both o¤ers). Using this time frame to discount all payo¤s back
to period 1 we …nd the game tree shown in Figure 10.15. We can solve
this game using backwards induction. Assume the game has reached period 2 where Alf makes an o¤er of 1
2 to Bill (keeping 2 for himself).
Bill can decide whether to accept or reject the o¤er made by Alf.: the
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CHAPTER 10. STRATEGIC BEHAVIOUR
best-response function for Bill is
[accept]
[reject]
if 1
2
otherwise
[1
]
Since Alf wants to maximize his own payo¤, he would not o¤er more than
[1
] to Bill, leaving him (Alf) with 2 =1
+ . The other option
is to o¤er less today and receive tomorrow, discounted back to date 2.
But since < 1,
1
+
>
and hence Alf would o¤er 1
2
= [1
] to Bill, which is accepted.
Thus, going back to date 1, where Alf would o¤er Bill 1
1 for himself), the best-response function for Bill is now
[accept]
[reject]
if 1
1
otherwise
2
[1
1
(keeping
]
By a similar argument as before, Alf would not o¤er more than 2 [1
]
2
in period 1, and thus has the choice between receiving 1 = 1
+ 2
in period 1, or receiving 1
+
in period 2. But again, since < 1, we
…nd
2
1
+ 2 > [1
+ ]
2
and hence Alf will o¤er 1
[1
] to Bill in period 1, which is
1 =
2
accepted; Alf’s equilibirum share is 1 = 1
[1
].
2. Now consider a longer, but …nite time horizon. The structure of the backwards induction solution outlined above shows that as the time horizon
increases from T = 2 bargaining rounds to T = T 0 , the o¤er made by Alf
reduces to
T0
1
[1
]
1 =
which is accepted immediately by Bill; Alf’s share is
1
=1
T0
[1
].
3. Now consider an in…nite time horizon. The solution for the …nite case
would suggest that as T ! 1, ( 1 ; 1
1 )!(1; 0). However, this reasoning is inappropriate for the in…nite case, since there is no last period from
which a backwards induction outcome can be obtained. Instead, we use
the crucial insight that the continuation game after each period, i.e. the
game played if Bill rejects the o¤er made by Alf, looks identical to the
game just played. In both games, there is a potentially in…nite number of
future periods. This insight enables us to …nd the equilibrium outcome of
this game. Assume that the continuation game that follows if Bill rejects
has a solution with allocation ( ; 1
). Then, in the current period, Bill
will accept Alf’s o¤er 1
if
1
[1
], as before. Thus, given
1
1
a solution ( ; 1
), Alf would o¤er 1
=
[1
]. But if 1
is a
1
solution to the continuation game, it has to be a solution to the current
game as well, and hence 1
. It follows that
1 =1
1
Alf will o¤er 1
c Frank Cowell 2006
=
[1
= 1
]
= 0 to Bill, and Bill accepts immediately.
180
Microeconomics
Exercise 10.13 Take the game that begins at the node marked “*” in Figure
10.16.
1
[NOT
INVEST]
*
[INVEST]
2
[In]
2
[In]
[Out]
1
[FIGHT`]
(ΠF ,0)
**
[Out]
1
[CONCEDE]
_
(ΠM, Π)
[FIGHT`]
(ΠJ, ΠJ)
(ΠF ,0)
[CONCEDE]
(ΠM –k, Π)
_
(ΠJ –k, ΠJ)
Figure 10.16: Entry deterrence
1. Show that if M >
to a challenger.
J
>
F
then the incumbent …rm will always concede
2. Now suppose that the incumbent operates a chain of N stores, each in a
separate location. It faces a challenge to each of the N stores: in each
location there is a …rm that would like to enter the local market). The
challenges take place sequentially, location by location; at each point the
potential entrant knows the outcomes of all previous challenges. The payo¤ s in each location are as in part 1 and the incumbent’s overall payo¤ is
the undiscounted sum of the payo¤ s over all locations. Show that, however
large N is, all the challengers will enter and the incumbent never …ghts.
Outline Answer
1. Consider the concept of an equilibrium here.
(a) First note that there are several Nash equilibria as we can see from the
strategic form in Table 10.9, where the …rst part of the monopolist’s
strategy speci…es the action after the entrant played [in], while the
second speci…es the action after [out].
c Frank Cowell 2006
181
Microeconomics
CHAPTER 10. STRATEGIC BEHAVIOUR
([concede]; [concede])
([concede]; [ ght])
([ ght]; [concede])
([ ght]; [ ght])
1 (incumbent)
2 (Entrant)
[in]
[out]
J; J
M;
J; J
M;
F; 0
M;
F; 0
M;
Table 10.9: Weak monopolist: strategic form
We …nd immediately that there are four Nash equilibria:
([concede]; [concede]); [in]
([concede]; [ ght]); [in]
([ ght]; [concede]); [out]
([ ght]; [ ght]); [out]
Note that the outcome ([ ght]; [in]), where the entrant enters and
the incumbent …ghts, is not an equilibrium outcome.
Ei
[In]
[Out]
Ii
[FIGHT`]
(ΠF ,0)
[CONCEDE]
(ΠM, Π)
_
(ΠJ, ΠJ)
Figure 10.17: Challenge i
(b) To …nd the Subgame Perfect Nash equilibrium, we have to …nd the
strategy combinations that are Nash equilibria in every subgame.
There are two subgames here at …rm 1’s decision nodes. In the case
that the entrant (…rm 2) chose [in], the best response is to choose
[concede], while if the entrant chose [out], the best response is to
choose either [concede] or [ ght]. But the best response of the entrant
to those best responses is to choose [in]. Thus, the Subgame-Perfect
Nash Equilibria are
(([concede]; [concede]); [in])
and
(([concede]; [ ght]); [in])
c Frank Cowell 2006
182
Microeconomics
which both yield the backwards induction outcome. Hence, we …nd
that the threat to …ght after entry is not a credible strategy. However, if there were a precommitment device, such that the threat of
…ghting became credible, then it would be better for the entrant
not to enter, so the equilibrium outcome would be ([ ght]; [out])
or ([concede]; [out]), which imply Subgame-Perfect Nash equilibrium
strategies (([ ght]; [concede]); [out]) or (([ ght]; [ ght]); [out]).
2. We may now consider an extension of essentially the same model. The
incumbent has stores I1 ; I2 ; :::; IN in local markets 1; 2; :::; N . There is sequence of challenges in each market from potential entrants E1 ; E2 ; :::; EN
A typical encounter is depicted in Figure 10.17. The outcome in this market is independent of actions in markets 1; 2; :::; i 1. So the equilibrium
behaviour of Ii and Ei is determined by the situation in local market i
alone. By the result in part 1 the outcome is
([concede]; [in]):
c Frank Cowell 2006
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Microeconomics
CHAPTER 10. STRATEGIC BEHAVIOUR
Exercise 10.14 In a monopolistic industry …rm 1, the incumbent, is considering whether to install extra capacity in order to deter the potential entry of
…rm 2. Marginal capacity installation costs, and marginal production costs (for
production in excess of capacity) are equal and constant. Excess capacity cannot
be sold. The potential entrant incurs a …xed cost k in the event of entry.
1. Let qS1 be …rm 1’s output level at the Stackelberg solution if …rm 2 enters.
Suppose qS1 6= qM , where qM is …rm 1’s output if its monopolistic position is
unassailable (i.e. if entry-deterrence is inevitable). Show that this implies
that market demand must be nonlinear.
2. Let q 1 be the incumbent’s output level for which the potential entrant’s best
response yields zero pro…ts for the entrant. In the case where entry deterrence is possible but not inevitable, show that if qS1 > q 1 , then it is more
pro…table for …rm 1 to deter entry than to accommodate the challenger.
Outline Answer
1. We begin by modelling the use of capacity as deterrence.
q2
2 ;0)
χ(q
q1 =
2 ;c)
χ(q
q1 =
q
•C
q2 =
χ(q 1
;0
)
q1
qM,
Figure 10.18: Quantity and capacity choices
(a) Suppose the two …rms were to choose capacity z 1 and z 2 simultaneously –a minor variation on the standard Cournot model. Firm 1’s
problem is
1
max
= p q 1 + q 2 q 1 cz 1
(10.28)
1 1
q ;z
subject to
q1
c Frank Cowell 2006
184
z1
(10.29)
Microeconomics
with solution
q1 = z1 =
q2 ; c
(10.30)
where the dependence of the reaction function on the marginal cost
parameter is made explicit–see Figure 10.18. Note that it never pays
to leave capacity unused in this version of the story. Also note that
if Firm 1’s marginal cost were cut from c to 0 then the schedule
would be shifted to the right in Figure 10.18 –compare the schedule
( ; 0) with ( ; c) and also Figure 10.12 in the answer to Exercise
10.11.
(b) Firm 2’s problem is similar:
max
2 2
q ;z
1
= p q1 + q2
cz 2
k
(10.31)
subject to
q2
z2
(10.32)
Note that the term k in (10.31), being a …xed entry cost, will not affect the …rst order condition that characterises …rm 2’s best-response
output, given that it enters the market. So its behaviour conditional
on entry is also given by the reaction function :
q2 = z2 =
q1 ; c .
(10.33)
Taking (10.30) and (10.33) together we get the symmetric CournotNash solution (qC ; qC ) –see point qC .in Figure 10.18.
q2
q2 =
χ(q 1
;0
)
q1
•
z
Figure 10.19: Incumbent’s best response with precommitted capacity
(c) But if …rm 1’s capacity were …xed in advance (and cannot be sold
o¤) then the term cz 1 in.(10.28) would be treated as a sunk cost,
irrelevant to the optimisation problem. The optimisation problem
for …rm 1 would e¤ectively become that of (10.28, 10.29) with c = 0.
So its behaviour would be characterised by
q1 =
c Frank Cowell 2006
185
1
q2 ; 0 :
(10.34)
Microeconomics
CHAPTER 10. STRATEGIC BEHAVIOUR
Suppose …rm 1’s advance capacity is …xed at z. Then, there are in
principle two regimes that can apply when at the output-choice stage
of the game:
q1
z. Output can be met from pre-existing capacity and so
the best response of …rm 1 (the incumbent) is given by (10.34)
in this region.
q 1 > z. Extra capacity will have to be installed at marginal cost
c; the best response of …rm 1 is given by (10.30) in this region.
So the combined best responses will look like Figure 10.19.
q2
q2
q1
•
z_
_•
z
q1
Figure 10.20: Limits on incumbent’s capacity precommitment
(d) Now consider …rm 1’s choice when determining advance capacity to
be installed in advance as a possible deterrent. Let z be the capacity
level corresponding to qC –the solution to (10.30) and (10.33). The
incumbent would not precommit to an amount less than z because it
is pointless –both …rm 1 and …rm 2 can always install extra capacity
during the output-choice stage. Let z be the capacity level corresponding to the solution to (10.34) and (10.33) –see the right-hand
intersection in Figure 10.18. The incumbent could not credibly precommit to an amount greater than z because this would imply that
it would have capacity that would never be used in the output-choice
stage. So the capacity precommitment must be in the range [z; z] –
see Figure 10.20.
(e) Choice of z within the range [z; z] depends on the size of the entry
cost for …rm 2. Note that, in the usual interpretation of a Cournot
diagram …rm 2’s pro…ts rise as one moves North-west along …rm 2.s
reaction function – see Figure 10.5 in the answer to Exercise 10.7.
Now consider the implications of the size of q 1 , the output level of
…rm1 for which the …rm 2’s best response yields it zero pro…ts.
q1
z. Firm 2 can always make a pro…t for any any capacity
choice made by …rm 1 such that z 2 [z; z]. The best that …rm
1 can do is to accommodate …rm 1’s entry and will act as a
Stackelberg leader.
c Frank Cowell 2006
186
Microeconomics
q 1 2 [z; z]. Firm 2 will make a loss for any capacity choice by …rm
1 such that q 1 < z z. Here …rm 1 may either (a) be assured of
deterring …rm 2’s entry and and choose monopoly output or (b)
choose capacity z so as to deter entry or (c) accommodate entry
as above
q2
q
•S
χ2(·)
z_
q1
qM•_
q1
_
z
Figure 10.21: Capacity choice of incumbent
(f) Compare the Stackelberg and monopoly solutions in the case where
there is a linear demand curve
0
q
In the case where …rm 1 can blockade entry and act as a monopoly
case pro…ts are
q 1 q 1 cq 1
0
and the solution as
c
0
q 1 = qM =
(10.35)
2
–see equation (10.11) in the outline answer to Exercise 10.7.
Take the case where …rm 1 accommodates and acts as a Stackelberg leader. Firm 2’s reaction as follower is given by
q2 =
2
q1 =
0
2
c
1 1
q
2
(10.36)
–see equation (10.7) in Exercise 10.7. Firm 1’s pro…ts as leader
are
1
c
q1 q1
(10.37)
2 0
c Frank Cowell 2006
187
Microeconomics
CHAPTER 10. STRATEGIC BEHAVIOUR
–see equation (10.12) in Exercise 10.7 and the pro…t-maximising
output is
c
:
q 1 = qS1 = 0
2
So, if the demand function were linear we must have qS1 = qM
which contradicts what is stated in the question. Therefore the
demand function must be nonlinear.
2. See Figure 10.21. If q 1
qM then …rm 1 sets q 1 = qM sells qM and
the potential entrant does not enter: blockading of entry is inevitable.
By assumption entry deterrence is not inevitable so q 1 > qM . If q 1 > z
then q 1 = q 1 is not credible, it is outside the range [z; z] of credible precommitments. Since the pro…ts of …rm 2 increase as one moves Northwest
along …rm 2’s reaction function it follows that it is not possible to deter
z. Also, qS1 > q 1 by assumption. Hence
entry. So, by assumption, q 1
1
1
qS ; z > q > qM . In Figure 10.21 it has been assumed that qS1 < z. We
q1 .
have entry accommodation if q 1 < q 1 and entry deterrence if q 1
1
1
In this case set q = q to get on to the lowest (and so higher pro…ts)
isopro…t contour possible. Of course this isopro…t contour is below the
one through qS1 ; qS2 which, in turn, is below the one which is applicable
if q 1 < q 1 . This establishes the conclusion. Note that if qS1 > z then q 1
can be larger or smaller than z. If it is smaller, we get the same solution
as above. If it is larger then entry deterrence is not possible for q 1 = q 1 is
outside the range of credible pre-commitments. So now …rm 1 sets q 1 = z
and concedes entry.
c Frank Cowell 2006
188
Microeconomics
Exercise 10.15 Two …rms in a duopolistic industry have constant and equal
marginal costs c and face market demand schedule given by p = k q where
k > c and q is total output..
1. What would be the solution to the Bertrand price setting game?
2. Compute the joint-pro…t maximising solution for this industry.
3. Consider an in…nitely repeated game based on the Bertrand stage game
when both …rms have the discount factor < 1. What trigger strategy,
based on punishment levels p = c; will generate the outcome in part 2?
For what values of do these trigger strategies constitute a subgame-perfect
Nash equilibrium?
Outline Answer
1. Suppose …rm 2 sets a price p2 > c. Firm 1 then has three options: It can
set a price p1 >p2 , it can match the price of …rm 2, p1 = p2 , or it can
undercut, since there exists an > 0 such that p1 = p2
> c. The pro…ts
for …rm 1 in the three cases are:
8
if p1 > p2
< 0
2
1
k
p
2
=
if p1 = p2
p
c 2
: 2
2
p
c k p
if p1 = p2
For su¢ ciently small pro…ts in the last case exceed those in the other
two, and hence …rm 1 will choose to undercut …rm two by a small and
capture the whole market. The …rms will choose to share the market if
they are playing a one-shot simultaneous move game, where they will set
p1 = p2 = c
2. If the …rms maximise joint pro…ts the problem becomes choose k to
max [k
q] q
cq
The FOC is
k
2q
c=0
which implies that pro…t-maximising output is
qM =
k
c
2
so that the price and the (joint) pro…t are, respectively
pM =
M
c Frank Cowell 2006
=
k+c
2
1
[k
4
189
2
c]
Microeconomics
CHAPTER 10. STRATEGIC BEHAVIOUR
3. The trigger strategy is to play p = pM at each stage of the game if the
other …rm does not deviate before this stage. But if the other …rm does
deviate then in all subsequent stages set p = c. In the accompanying table
…rm 2 deviates at t = 3 by setting
p = pM
" < pM
so that …rm 1 responds with p = c to which the best response by …rm 2 is
also p = c.
1
pM
pM
…rm 1:
…rm 2:
2
pM
pM
3
pM
p
4
c
c
5
c
c
:::t
If " is small then for that one period …rm 2 would get the whole market so 2 = M . Thereafter 2 = 0. If the …rm had cooperated it would
have got 2 = 12 M . The present discounted value of the net gain from
defecting is
1
2
if and only if
M
1
2
c Frank Cowell 2006
1
2
M
+
2
+
1.
190
3
+ ::: =
1
2
M
1 2
1
0
Microeconomics
Exercise 10.16 Consider a market with a very large number of consumers in
which a …rm faces a …xed cost of entry F . In period 0, N …rms enter and in
period 1 each …rm chooses the quality of its product to be High, which costs
c > 0, or Low, which costs 0. Consumers choose which …rms to buy from,
choosing randomly if they are indi¤ erent. Only after purchasing the commodity
can consumers observe the quality. In subsequent time periods the stage game
just described is repeated inde…nitely. The market demand function is given by
8
< '(p) if quality is believed to be High
q=
:
0
otherwise
where ' ( ) is a strictly decreasing function and p is the price of the commodity.
The discount rate is zero.
1. Specify a trigger strategy for consumers which induces …rms always to
choose High quality. Hence determine the subgame-perfect equilibrium.
What price will be charged in equilibrium?
2. What is the equilibrium number of …rms, and each …rm’s output level in
a long-run equilibrium with free entry and exit?
3. What would happen if F = 0?
Outline Answer
1. For a given price p pro…ts for i are given by
#
"
1
1
+
+ :::
= qi [p c]
1 + r [1 + r]2
F
qi [p c]
F
(10.38)
r
if it produces High quality for ever and
qi p
F
1+r
if it produces low quality in one period and then never sells any product
thereafter. The trigger strategy is to stay with the …rm unless it is observed
to have changed its quality to Low, in which case the …rm will never again
have customers. This punishment ensures SPNE (Note the zero demand
for Low quality.) To ensure that this strategy is incentive-compatible,
and thus consistent with SPNE, we need
"
#
1
1
qi p
qi [p c] 1 +
+
+ :::
1 + r [1 + r]2
=
=
1+r
qi [p
r
c]
(10.39)
[1 + r] c
(10.40)
Hence
p
in equilibrium. The equilibrium price is su¢ ciently high that the …rm is
unwilling to sacri…ce its future pro…ts for a one-o¤ gain from producing
Low quality and selling it at a high price.
c Frank Cowell 2006
191
Microeconomics
CHAPTER 10. STRATEGIC BEHAVIOUR
2. From (10.38) the long-run zero-pro…t condition implies
q(p) [p
r
c]
F
(10.41)
We have the market-clearing condition
N qi = q(p)
(10.42)
This (10.42) determines the equilibrium number of …rms
N = int q(p)
p c
rF
(10.43)
From (10.40) and (10.41) we get
qi =
1 rF
Np c
(10.44)
3. Without entry cost, but with demand elastic, q(p) is determined, but qi
and N are indeterminate.
c Frank Cowell 2006
192
Microeconomics
Exercise 10.17 In a duopoly both …rms have constant marginal cost. It is
common knowledge that this is 1 for …rm 1 and that for …rm 2 it is either 43
or 1 14 . It is common knowledge that …rm 1 believes that …rm 2 is low cost with
probability 21 . The inverse demand function is
2
q
where q is total output. The …rms choose output simultaneously. What is the
equilibrium in pure strategies?
Outline Answer
Let marginal cost be denoted by c. Firm 1’s and …rm 2’s pro…ts are given
by, respectively:
1
:= 2 q 1 q 2 q 1 C1 q 1
(10.45)
2
q1
:= 2
q2 q2
cq 2
C2
2
(10.46)
1
Maximising (10.46) with respect to q , given q we have the FOC
q1
2
2q 2
c=0
which yields the Cournot reaction function for …rm 2
q2 =
2
q1 ; c = 1
1 1
q
2
1
c
2
(10.47)
(compare the answer for Exercise 10.7). The two cases (low-cost and high-cost)
for …rm 2 are therefore
qL2
=
2
qH
=
5
8
3
8
1 1
q
2
1 1
q
2
(10.48)
(10.49)
–compare this with Figure 10.12 in Exercise 10.11.
In the light of the above the expected pro…ts for …rm 1 are
1
2
2
q1
qL2 q 1
2
q1 +
C1
q1
1 2
q
2 L
1
2
2
1 2 1
q q
2 H
q1
2
qH
q1
C1
C1
q1
q1
2
Under the Cournot assumption …rm 1 takes qL2 and qH
as given. So the FOC
for …rm 1 is
1 2 1 2
2 2q 1
q
q
1=0
2 L 2 H
which implies
1 1 2
2
q + qH
2 4 L
as the best response of …rm 1 to …rm 2.
Substituting from (10.48) and (10.49) into (10.50) we have
q1 =
q1 =
c Frank Cowell 2006
1
2
1 5
4 8
1 1 3
q +
2
8
193
1 1
q
2
(10.50)
(10.51)
Microeconomics
CHAPTER 10. STRATEGIC BEHAVIOUR
which simpli…es to
q1
=
=
1 1
1
2 4
1 1 1
+ q
4 4
q1
From which we …nd the solution as
c Frank Cowell 2006
q1
=
qL2
=
2
qH
=
1
3
5
8
3
8
194
1
29
=
6
48
1
17
=
6
48
(10.52)
(10.53)
Chapter 11
Information
Exercise 11.1 A …rm sells a single good to a group of customers. Each customer either buys zero or exactly one unit of the good; the good cannot be divided
or resold. However, it can be delivered as either a high-quality or a low-quality
good. The quality is characterised by a non-negative number q; the cost of producing one unit of good at quality q is C(q) where C is an increasing and strictly
convex function. The taste of customer h is h – the marginal willingness to
pay for quality. Utility for h is
U h (q; x) =
h
q+x
where h is a positive taste parameter and x is the quantity consumed of all
other goods.
1. If F is the fee required as payment for the good write down the budget
constraint for the individual customer.
2. If there are two types of customer show that the single-crossing condition
is satis…ed and establish the conditions for a full-information solution.
3. Show that the second-best solution must satisfy the no-distortion-at-the-top
principle.
4. Derive the second-best optimum.
Outline Answer
1. If the consumer has income y then the budget constraint is
x + F (q)
y
where is a variable taking the values 0 or 1, representing the cases “not
buy” and “buy”.
2. Assume that each person’s type is common knowledge.
(a) If there are two taste types
a
b
;
a
with
>
b
The preferences are as shown in Figure 11.1.
195
(11.1)
Microeconomics
CHAPTER 11. INFORMATION
x
τa
τb
q
quality
Figure 11.1: Preferences: quality
F
Π2 = F2 − C(q)
inc
pro reas
fit ing
Π1 = F1 − C(q)
Π0 = F0 − C(q)
q
quality
Figure 11.2: Isopro…t curves: quality
c Frank Cowell 2006
196
Microeconomics
F
τaq
•
F*a
τbq
•
F*b
q
q*b
q*a
quality
Figure 11.3: Full-information solution: quality
(b) Given that each person’s type can be observed the …rm can tailor
the fee schedule exactly to personal characteristics, charging F a to
an a-type and F b to an b-type The …rm’s pro…ts from each of the two
groups of consumers are
Fa
Fb
C (q a )
C qb
(11.2)
(11.3)
and the isopro…t curves are as shown in Figure 11.2.
(c) If a person chooses not to buy the good then his utility is just y. So
the …rm chooses q a and F a to maximise (11.2) subject to the a-type’s
participation constraint
a a
q
Fa + y
y
or, equivalently
a a
q
b
Fa
0;
(11.4)
b
it likewise chooses q and F to maximise (11.3) subject to the btype’s participation constraint
b b
q
Fb
0:
(11.5)
The full information solution is found at the tangency of an isopro…t curve with a reservation indi¤erence for each of the two types,
as shown in …gure 11.3 Clearly, at the optimum (q a ; F a ) and
q b ; F b , each of the participation constraints (11.4) and (11.5) is
binding and each person just gets his reservation utility y.
3. Now it is no longer possible to condition the fee schedule directly on a
person’s type.
c Frank Cowell 2006
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Microeconomics
CHAPTER 11. INFORMATION
F
τaq
pre
fer
en
ce
F*a
•
τbq
•
F*b
q
q*b
q*a
quality
Figure 11.4: Type a prefers type b contract
(a) If the full-information contracts from part 2 were available an a-type
person would want to take a b-type contract since his utility would
then be a q b F b +y which, in view of the fact that b q b F b = 0,
becomes
a
b
q b+y
which is strictly greater than y. See Figure 11.4.
(b) We need to …nd the second-best contracts that take account of this
incentive-compatibility problem. Incentive compatibility requires that,
for a:
a a
a b
q
Fa
q
Fb
(11.6)
and, for b:
b b
Fb
q
b a
q
Fa
(11.7)
Suppose it is known that there is a proportion , 1
of a-types
and b-types, respectively. Now the …rm’s problem is to choose q a , q b ,
F a and F b to maximise
[F a
C (q a )] + [1
] Fb
C qb
(11.8)
subject to the participation constraints (11.4) and (11.5) and the
incentive-compatibility constraints (11.6) and (11.7). As in the text
we can simplify the problem by determining which constraints are
slack and which are binding. Note that:
(11.6) and (11.1) imply
a a
q
Fa
a b
q
Fb
b b
q
Fb
(11.9)
This implies that if constraint (11.5) were slack, then constraint
(11.4) would also be slack; this cannot be true at the optimum
c Frank Cowell 2006
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Microeconomics
since it would then be possible for the …rm to increase both F a
and F b and increase pro…ts. Hence (11.5) must be binding.
Given that F b > 0 at the optimum (11.5) then implies that
q b > 0: However (11.1). (11.9) and q b > 0 imply
a a
q
Fa
a b
Fb >
q
b b
q
Fb = 0
(11.10)
which implies that
a a
Fa > 0
q
(11.11)
So constraint (11.4) is slack and can be ignored.
If (11.6) were slack then, by (11.11) it would be possible to increase F a without violating the constraint. So (11.6) must be
binding.
a a
q
F a = a qb F b
(11.12)
If (11.7) were binding, then this and (11.12) would imply
qb
a
b
= qa
a
b
but, given (11.1).this can only be true if
qb = qa :
But this implies a pooling outcome and we know that the …rm
can do better than a pooling outcome by forcing the high-value
consumers to reveal themselves. This implies that constraint
(11.7) is slack and can be ignored.
F
τaq
pre
fer
en
ce
Fa
•
τbq
Fb
q
qb
q*a
quality
Figure 11.5: Second-best solution: quality
(c) The Lagrangean is therefore
[F a
c Frank Cowell 2006
C (q a )]
+ [1
+
+
199
] F b C qb
q
Fb
a a
a b
q
Fa
q + Fb
b b
(11.13)
Microeconomics
CHAPTER 11. INFORMATION
where and are the Lagrange multipliers for the constraints (11.5,
11.6) respectively. The …rst-order conditions are
[1
] Cq
Cq (q a ) +
q + b
1
From (11.16) and (11.17) and we have
values in (11.14) and (11.15) we have
Cq (q a )
Cq q
b
a
+
=
=
=
=
=
and
a
b
b
=
a
=
b
a
0
0
0
0
(11.14)
(11.15)
(11.16)
(11.17)
= 1. Using these
(11.18)
1
<
b
(11.19)
Clearly the a-type’s consumption is at the point where marginal cost
- marginal willingness to pay for quality –see Fig 11.5.
c Frank Cowell 2006
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Microeconomics
Exercise 11.2 An employee’s type can take
The bene…t of the employee’s services to his
amount of education that the employee has
years of education for an employee of type
C (z; ) = ze
the value 1 or 2 , where 2 > 1 .
employer is proportional to z, the
received. The cost of obtaining z
is given by
:
The employee’s utility function is
U (y; z) =
e
y
C (z; )
where y is the payment received from his employer. The risk-neutral employer
designs contracts contingent on the observed gross bene…t, to maximise his expected pro…ts.
1. If the employer knows the employee’s type, what contracts will be o¤ ered?
If he does not know the employee’s type, which type will self-select the
“wrong” contract?
2. Show how to determine the second-best contracts. Which constraints bind?
How will the solution to the second-best problem compare with that in part
1?
Outline Answer
This is a problem of hidden information: The type of the employee is
exogenously given, but private information. The problem for the employer is to
design a contract that leads the employee to reveal his true type.
The employer is interested in truthful revelation since it is less costly for employees of type 2 to attain a given level of education. Hence, the employer would
have to reward 2 types by less to get the same level of education. However,
type 2 would like to pretend to be of type 1 to receive the higher payment.
The reservation utility of the employee corresponds to the case where the
employee receives 0 years of education, for which he receives 0 from the employer:
=
1:
This yields participation constraints for each type:
1
e
y
ze
0
For further reference, we can deduce the shape of the indi¤erence curves for a
given type. The …rst and second derivatives, respectively, are:
dy
dz
d2 y
dz 2
= ey
i
= ey
i
>0
dy
= e2[y
dz
(11.20)
i]
>0
See Figure 11.6 for an illustration of the reservation indi¤erence curves for the
two types, which go through the origin. The shaded areas are the acceptance
sets for the two types of individuals. Note that as dy
dz is independent of z, the
indi¤erence curves for a given type must be horizontally parallel in (z; y) space.
Also note that the single crossing property is satis…ed.
c Frank Cowell 2006
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Microeconomics
CHAPTER 11. INFORMATION
υ
_1
y
υ
_2
•
y2*
•
y1*
z
0
z1*
z2*
Figure 11.6: Full-information contracts
1. If the employer knows the employee’s type, then, as before, the employer
can implement full-information contracts, maximising the return from
each type separately. Let be the type. Then the problem of the employer
is to
max := z y
z;y
subject to the participation constraint
1
y
e
ze
0
(11.21)
We can now set up the Lagrangean associated with this maximization
problem
L (z; y; ) := z y + 1 e y ze
where
yield:
is the Lagrange multiplier for the constraint (11.21). The FOCs
@L
@z
@L
@y
@L
@
=
1
=
=
e
1+ e
1
e
y
=0
(11.22)
y
(11.23)
=0
ze
0
(11.24)
Combining equations (11.22) and (11.23), we see that e = ey and hence
that
y =
Those equations also imply that > 0, and hence that the participation
constraint (11.24) must bind. Hence
z
=
1
= e
c Frank Cowell 2006
202
e
y
e
1>0
Microeconomics
Thus the employer o¤ers two di¤erent contracts to the two types, where
the level of education is zi = e i 1, and the compensation is yi = i .
Since the slope of the indi¤erence curve in Figure 11.6 was given by
equation 11.20, we see that the contracts are located where the slope of
the reservation indi¤erence curves are equal to 1.
2. Now assume that the employer cannot observe the type of the employee.
(a) Given the contracts found in part 1 an employee of type 2 will selfselect the wrong contract. To see this, compare the utility of type 2
when choosing the type-2 contract
2
=
e
[e
2
1] e
2
2
to that when choosing the type-1 contract:
2
=
e
[e
1
1] e
1
2
Since 1 < 2 it is clear that 2 > 2 . Hence, type 2 would self-select
the type 1 contract, and hence incentive compatibility is violated.
(b) If there is an incentive for type 2 to self-select the wrong contract,
then any second-best contract has to insure that the incentive compatibility constraint for type 2 is not violated. Type 1 will never have
an incentive to select the type 2 contract, hence the incentive compatibility constraint for type 1 will not be binding. Furthermore, since
any non-trivial contract for type 1 would enable type 2 to achieve a
utility level greater than his reservation utility level, the participation constraint for type 2 cannot be binding. However, we can keep
type 1 on his reservation utility. Hence, we expect that the two constraints which will be binding in the second-best contract will be the
participation constraint for type 1
1
e
y1
z1 e
0
1
(11.25)
and the incentive-compatibility constraint for type 2:
e
y2
z2 e
2
e
y1
z1 e
(11.26)
2
This is as in the standard adverse selection model:
(c) Assume that the probability of encountering a type 1 employee is
2 (0; 1). We may now set up the maximization problem for the
monopolist, which consists of choosing z1 ; y1 ; z2 ; y2 to maximise :
:=
[z1
y1 ] + [1
] [z2
y2 ]
subject to (11.25) and (11.26). The Lagrangean is
L (z1 ; y1 ; z2 ; y2 ; ; ) :=
c Frank Cowell 2006
[z1 y1 ] + [1
] [z2
+ [1 e y1 z1 e 1 ]
+ [e y1 e y2 + z1 e
203
y2 ]
2
z2 e
2
]
Microeconomics
CHAPTER 11. INFORMATION
where and are the Lagrange multipliers associated with the constraints (11.25) and (11.26), respectively. Maximising this yields the
FOCs
@L
=
e 1+ e 2 =0
(11.27)
@z1
@L
(11.28)
=
+[
] e y1 = 0
@y1
@L
(11.29)
= [1
]
e 2 =0
@z2
@L
(11.30)
=
[1
] + e y2 = 0
@y2
As
0 by Kuhn-Tucker conditions, and
> 0, equation 11.27
implies that > 0. Hence, constraint (11.25) will bind. Considering equation 11.29, we see similarly that > 0, so that constraint
(11.26) will bind. Since both constraints will bind, and the …rst order
conditions are satis…ed, we have a system with six equations and six
variables. Call the solutions to this system (^
z1 ; y^1 ) and (^
z2 ; y^2 ).
(d) Since the incentive-compatibility constraint (11.26) is binding, these
two contracts will lie on the same type 2 indi¤erence curve. But the
type-1 incentive compatibility constraint is slack, so type 1 strictly
prefers (^
z1 ; y^1 ) to (^
z2 ; y^2 ). Given the Single Crossing Property and
the fact that the type 2 indi¤erence curves are ‡atter that of type
1, we must have z^1 < z^2 and y^1 < y^2 . Consider equations 11.29
and 11.30. We …nd that y^2 = 2 , the “no distortion at the top“
result. Clearly, the marginal rate of substitution for type 2 at this
point is
e 2 =e y2 = 1;
which is the same as the slope of the isopro…t contour. Now compare
the second-best contracts with the full-information solution. Type 2
was on his reservation indi¤erence curve 2 under the full-information
contract. But under the second-best, the participation constraint for
type 2 is slack, and hence he is now on an indi¤erence curve I2 above
2 . But since
y^2 = y2 = 2 ;
this must mean that z^2 < z2 .
(e) Finally, how far has the solution moved from 2 ? There are two
possibilities: The new (^
z1 ; y^1 ) could be below and to the left of the
full-information (z1 ; y1 ), or above and to the right. We will show
that z^1 < z1 and y^1 < y1 (recall that (11.25) is still binding, so the
new solution will be on the reservation indi¤erence curve for type 1).
The slope of the indi¤erence curve 1 at (^
z1 ; y^1 ) is
e 1
:
e y^1
From equations (11.27) and (11.28) we have
e
e
c Frank Cowell 2006
y^1
1
204
=
=
+ e
+ e
y^1
2
Microeconomics
hence the slope of the indi¤erence curve is
+ e
+ e
2
y^1
<1
(11.31)
This follows because y^1 < y^2 = 2 which implies e 2 < e y^1 . We
found that the slope of the type 1 indi¤erence curve at (z1 ; y1 ) was 1,
so condition (11.31) implies that z^1 < z1 and y^1 < y1 . See …gure 11.7.
υ
_1
y
υ
_2
^y
2
•
•
^y
1
0
z
^z
2
^z
1
Figure 11.7: Second-best contracts
c Frank Cowell 2006
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Microeconomics
CHAPTER 11. INFORMATION
Exercise 11.3 A large risk-neutral …rm employs a number of lawyers. For a
lawyer of type the required time to produce an amount x of legal services is
given by
x
z=
The lawyer may be a high-productivity a-type lawyer or a low-productivity b-type:
a
> b > 0. Let y be the payment to the lawyer. The lawyer’s utility function
is
1
y2 z
and his reservation level of utility is 0. The lawyer knows his type and the …rm
cannot observe his action z: The price of legal services is 1.
1. If the …rm knows the lawyer’s type what contract will it o¤ er? Is it e¢ cient?
2. Suppose the …rm believes that the probability that the lawyer has low productivity is : Assume b
[1
] a : In what way would the …rm then
modify the set of contracts on o¤ er if it does not know the lawyer’s type
and cannot observe his action?
Outline Answer.
The problem is one of adverse selection with hidden information.
υ
_b
y
υ
_a
•
y*a = 1
slope = 1
• slope = 1
y*b = ¼
0
x
x*a = 2
x*b = ½
Figure 11.8: Full-information contracts
1. Full-information.
(a) The principal knows the type and so maximises x
p
y
x
where
=0
c Frank Cowell 2006
206
y subject to
Microeconomics
for each individual type. We know that the participation constraint
binds and that there is no distortion. So
=
p
x
y
(11.32)
Di¤erentiate to …nd the slope of the indi¤erence curve:
p
2 y
dy
=
dx =
(11.33)
Since there is no distortion (11.33) must be equal to 1 which implies
1
4
y =
Using the fact that
get
=
2
:
(11.34)
and substituting (11.34) into (11.32) we
1
2
x =
2
:
(11.35)
In this case the optimal contracts are (x a ; y a ) and x b ; y
shown in Figure 11.8.
b
as
(b) Since there is no distortion the solution is e¢ cient. There is no
change which could make one person better o¤ without making the
other worse o¤.
2. Types unknown.
(a) If it is impossible to monitor the lawyer’s type it is no longer viable
to o¤er the e¢ cient contracts (x a ; y a ) and x b ; y b from part 1. If
a type-a lawyer accepts the e¢ cient contract meant for him he gets
utility
1
a 2
p a x a
1 a
2 [ ]
y
=
=0
a
a
2
But if a type-a lawyer were to get a type-b contract he would get
utility
p
y
b
x
b
a
=
1
2
b
b 2
1
2
a
=
1
2
b
b
1
a
> 0:
So a type a would prefer to take a type-b contract.
(b) Given this problem the best that the …rm can do is to maximise
expected pro…ts subject to an incentive-compatibility constraint for
the a types:
p
p a xa
xb
b
y
y
a
a
Let
be the the probability that the lawyer is of type a. Then
expected pro…ts are
[xa
c Frank Cowell 2006
y a ] + [1
207
] xb
yb
Microeconomics
CHAPTER 11. INFORMATION
In the problem of maximising expected pro…ts under these conditions
we know from previous exercises (such as Exercise 11.2) that the
participation constraint (11.32) for type b will be binding
p
yb
xb
b
=
b
(11.36)
as is the incentive-compatibility constraint for type:
p
xa
ya
=
a
So the relevant Lagrangean is
L xa ; y a ; xb ; y b ; ;
:=
p
xb
yb
a
:
(11.37)
[xha y a ] + [1i
] xb
p
b
+
y b xb
hp
p
a
+
y a xa
yb +
9
yb >
>
=
xb
a
i
>
>
;
(11.38)
where and are the Lagrange multipliers for the constraints (11.36)
and (11.37) respectively.
(c) Let the solution values for maximising (11.38) be denoted x
^a , y^a , x
^b ,
b
y^ . Di¤erentiating (11.38) the FOC are:
[1
+ p a
2 y^
=
0;
(11.39)
p
2 y^b
=
0;
(11.40)
a
=
0;
(11.41)
+
a
=
0:
(11.42)
]+ p
2 y^b
1
b
From (11.41) and (11.42) we have
a
=
=
b
;
and substituting these values in (11.39) and (11.40) we get
a
+ p a
2 y
b
[1
which implies
b
y^ =
c Frank Cowell 2006
208
0
=
0
a
]+ p
2 y^b
y^a =
=
1
[
4
b
2 [1
p
2 y^b
a 2
]
2
a
]
Microeconomics
And so, using the participation constraint (11.32) we have
p
b
x
^b =
y^b
b
=
2 [1
b
a
]
which is non-negative by assumption. Finally, from the incentivecompatibility constraint we get
x
^a =
[
a 2
]
a
b
b
2
x
^b :
υ
_b
y
υ
_a
^ya
^yb
0
•
•
x
^xa
^xb
Figure 11.9: The second-best solution in the adverse selection problem
(d) Note that
x
^a
x
^b
y^a
y^b
< x
< x
= y
< y
a
b
a
b
– see Figure 11.9. This is the no-distortion-at-the-top result. Note
that indi¤erence curves for a given type are horizontally parallel because utility is linear in x. So y^a = y a is immediate.
c Frank Cowell 2006
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Microeconomics
CHAPTER 11. INFORMATION
Exercise 11.4 The analysis of insurance in the text (section 11.2.6) was based
on the assumption that the insurance market is competitive. Show how the
principles established in section 11.2.4 for a monopolist can be applied to the
insurance market:
1. In the case where full information about individuals’risk types is available.
2. Where individuals’ risk types are unknown to the monopolist.
Outline Answer
1. See Figure 11.10
xBLUE
s, π
b
pr
of
its
,π
b
profit
_
•
ya
y−L
•
κa
0
xRED
_
y
ya
Figure 11.10: Insurance: iso-pro…t
(a) The endowment point for both types of individual is at (y; y L).
Given that the probability of an accident for high-risk type a is a
an insurance …rm would break even if it sold insurance against the
loss L for a premium a where
a
=
a
L:
(11.43)
De…ne
y a := y
a
a
L=
a
:
(11.44)
The line with slope 1 a passing through the endowment point and
the point (y a ; y a ) is an isopro…t contour for the …rm when dealing
with the high-risk types. Pro…ts must increase to the “South-West”
(consider the impact on pro…ts if the …rm were able to charge a higher
premium , all other things being equal).
c Frank Cowell 2006
210
Microeconomics
(b) For the low-risk types the isopro…t contours are a family of lines with
b
slope 1 b .
(c) Clearly the reservation indi¤erence curve for each type of person is
given as the contour passing through the endowment point. It has
h
slope 1 h ; h = a; b where it intersects the 45 line – see Figure
11.11.
xBLUE
_ _
(yb, yb)
•
_ _
(ya, ya)
•
•(y, y − L)
xRED
0
Figure 11.11: Insurance: monopoly with full information
(d) So the full-information outcome is where the high-risk types are located at (y a ; y a ) and the low-risk types at y b ; y b : the monopoly
insurance …rm rationally o¤ers better insurance terms to the lowrisk.
2. Take the case where the individual’s risk type is unknown to the insurer.
(a) Given that there is imperfect information it is clear that a high-risk
type would like to masquerade as a low-risk type and so take advantage of the more favourable terms. In Figure 11.12 see the a-type inyb ; yb .
di¤erence
curve
passing
through
the
point
The monopolist must take account of this possibility in setting up
the second-best optimisation problem.
(b) The solution to this problem will be of the following form: restrict the
low-risk b-types in the amount of insurance that they can purchase so
that they choose the prospect P~ which lies on the b-type reservation
indi¤erence curve in Figure 11.12; o¤er full insurance at point (~
y a ; y~a )
c Frank Cowell 2006
211
Microeconomics
CHAPTER 11. INFORMATION
xBLUE
_ _
(yb, yb)
•
~ a ~a
(y , y )
•
•
~
b
•P
•(y, y − L)
xRED
0
Figure 11.12: Insurance: monopoly second-best solution
–only the a-types will wish to take up this o¤er. The formal analysis
closely follows that of section 11.2.4.
c Frank Cowell 2006
212
Microeconomics
Exercise 11.5 Good second-hand cars are worth a1 to the buyer and a0 to
the seller where a1 > a0 . Bad cars are worth b1 to the buyer and b0 to the
seller where b1 > b0 . It is common knowledge that the proportion of bad cars is
. There is a …xed stock of cars and e¤ ectively an in…nite number of potential
buyers
1. If there were perfect information about quality, why would cars be traded
in equilibrium? What would be pa and pb , the equilibrium prices of good
cars and of bad cars respectively?
2. If neither buyers nor sellers have any information about the quality of an
individual car what is p, the equilibrium price of cars?
3. If the seller is perfectly informed about quality and the buyer is uninformed
show that good cars are only sold in the market if the equilibrium price is
above a0 .
4. Show that in the asymmetric-information situation in part 3 there are only
two possible equilibria
The case where pb <
The case where p
a
b
0 : equilibrium price is p .
a
0 : equilibrium price is p.
Outline Answer.
1. If there is perfect information about quality then:
(a) Cars of whatever quality will always be traded if the value of the
buyer is greater than that to the seller, as in the question.
(b) Equilibrium prices are
pa
pb
a
1
b
1
=
=
2. Given that neither party can verify the quality of a car ex ante, but both
know that the probability of a bad car is :
(a) The expected value of a car to the buyer is
[1
]
a
1
b
1
+
(11.45)
and the expected value of a car to the seller is
[1
]
a
0
b
0
+
(11.46)
(b) Given that (11.45) is greater than (11.46) the equilibrium price is
p = [1
]
a
1
+
b
1
(11.47)
3. Sellers will only be willing to supply good cars to the market if the price
is at least as great as their private valuation a0 .
c Frank Cowell 2006
213
Microeconomics
CHAPTER 11. INFORMATION
4. Consider two cases.
If the price is less than a0 then, from part 3, the sellers will only supply bad cars and the buyers will be well aware of this. The fact that,
after purchase, the quality is revealed as bad con…rms the buyers’
beliefs. Hence we have an equilibrium with price set at the buyers’
valuation of bad cars:
p = b1
If the price is not less than a0 then both types of car may be supplied
to the market. Once again the buyers will be aware of this and, in
the absence of further information will estimate the value as given
by (11.45). We have an equilibrium with price
p=p
where p is given by (11.47) if this price is at least as high as the
sellers’valuation of good cars
[1
c Frank Cowell 2006
]
214
a
1
+
b
1
a
0:
(11.48)
Microeconomics
Exercise 11.6 In an economy there are two types of worker: type-a workers
have productivity 2 and type-b workers have productivity 1. Workers productivities are unobservable by …rms but workers can spend their own resources to
acquire educational certi…cates in order to signal their productivity. It is common knowledge that the cost of acquiring an education level z equals z for type-b
workers and 21 z for type-a workers.
1. Find the least-cost separating equilibrium.
2. Suppose the proportion of type-b workers is . For what values of
the no-signalling outcome dominate any separating equilibrium?
3. Suppose
will
= 14 . What values of z are consistent with a pooling equilibrium?
Outline Answer
1. There is some z such that …rms believe a worker to be of type a if z z
and type b otherwise. Beliefs are self-con…rming if type-a workers choose
z = z and type-b workers choose z = 0. This is satis…ed if
1
z
2
2
1
1
2
z
in other words
1
z
2:
Any z in [1; 2] does the job so that there is a continuum of separating
equilibria. z = 1 is the least-cost separating equilibrium. The net payo¤
to an a-type in the least-cost separating equilibrium is 2 21 z = 1 12 .
2. Given the de…nition of
worker is
the expected productivity of a randomly chosen
+ 2 [1
]=2
:
This would give a better payo¤ to a-types in the separating equilibrium if
2
so that
>1
1
2
< 12 .
3. If z is education in the pooling equilibrium then the net payo¤ to a-type
workers is
1
3 1
z=1
z
2
2
4 2
and for b-type workers is
2
z=1
3
4
z
The b-type workers are better o¤ in the pooling equilibrium if
1
c Frank Cowell 2006
3
4
z
> 1
z
<
215
3
4
Microeconomics
CHAPTER 11. INFORMATION
The a-type workers are better o¤ in the pooling equilibrium if
1
3
4
1
z
2
z
c Frank Cowell 2006
216
> 1
<
1
2
1
2
Microeconomics
Exercise 11.7 A worker’s productivity is given by an ability parameter >
0. Firms pay workers on the basis of how much education, z, they have: the
wage o¤ ered to a person with education z is w (z) and the cost to the worker of
acquiring an amount of education z is ze .
1. Find the …rst-order condition for a type person and show that it must
satisfy
dw (z )
(11.49)
= log
dz
2. If people come to the labour market having the productivity that the employers expect on the basis of their education show that the optimal wage
schedule must satisfy
w (z) = log (z + k)
(11.50)
where k is a constant.
3. Compare incomes net of educational cost with incomes that would prevail
if it were possible to observe directly.
Outline Answer
1. Individual income, net of educational costs is
w (z)
ze
:
Maximising this with respect to z gives the FOC
wz (z )
e
=0
from which (11.49) immediately follows.
2. If the employer’s expectations are ful…lled then the revealed marginal product equals the wage paid w (z). Using this result in (11.49) we obtain
log
dw (z)
dz
= w (z)
(11.51)
which is a …rst-order di¤erential equation in z. Rearranging we have
ew(z)
dw (z)
=1
dz
(11.52)
Integrating over z we get
ew(z) = z + k
(11.53)
where k is a constant of integration. From this (11.50) follows.
3. In this model the costs of education ze are a net loss to the workers who
would have been paid according to their type anyway, if only could
have been observed.
c Frank Cowell 2006
217
Microeconomics
CHAPTER 11. INFORMATION
Exercise 11.8 The manager of a …rm can exert a high e¤ ort level z = 2 or
a low e¤ ort level z = 1. The gross pro…t of the …rm is either 1 = 16 or
2 = 2. The manager’s choice a¤ ects the probability of a particular pro…t
outcome occurring. If he chooses z, then 1 occurs with probability = 34 , but if
he chooses z then that probability is only = 14 . The risk neutral owner designs
contracts which specify a payment yi to the manager contingent on gross pro…t
1=2
z, and his reservation
i . The utility function of the manager is u(y; z) = y
utility = 0.
1. Solve for the full-information contract.
2. Con…rm that the owner would like to induce the manager to take action
z.
3. Solve for the second-best contracts in the event that the owner cannot
observe the manager’s action.
4. Comment on the implications for risk sharing.
Outline Answer.
1. Under the full-information contract, both manager and owner can observe
the action of the manager. So we can solve the optimization problem for
the owner for both high and low e¤ort levels separately, and then let the
owner choose the one that yields higher expected pro…ts.
(a) Formally, denoting the expected pro…t to the owner under high and
low e¤ort levels by the manager as and respectively, we have the
following pair of optimisation problems for the owner:
max
y1 ;y2
=
[
1
y1 ] + [1
][
y2 ]
2
(11.54)
subject to
u (y1 ; z) + [1
] u (y2 ; z)
0
(11.55)
in the high-e¤ort case and
max
y1 ;y2
=
i
[
1
y1 ] + [1
][
y2 ]
2
(11.56)
subject to
u (y1 ; z) + [1
] u (y2 ; z)
0
(11.57)
in the low-e¤ort case, where (11.55) and (11.57) are the participation
constraints that will be binding at the optimum.
(b) Since the owner can observe the manager’s action, and since the
manager is risk-averse, the owner will set y1 = y2 = y in the solution
to (11.54, 11.55) and we can solve for y simply by setting
u (y; z) = 0
which implies
y 1=2
z=0
Likewise in the solution to (11.56,11.57) we can solve for y from the
equation
y 1=2 z = 0
c Frank Cowell 2006
218
Microeconomics
(c) Using the numerical values given in the question, we obtain
y
y
=
=
4
1
2. From part 1 the optimal payments in the case of high and low e¤ort yield
expected pro…ts to the owner of
= 8:5
= 4:5
Thus, the owner would indeed like to induce the manager to take action
z. By construction, the manager is indi¤erent between taking z or z, since
he receives his reservation utility in either case.
3. However, if the manager could convince the owner that he was using z,
and get the owner to pay him y, while in fact only using z, his payo¤
would be
u(y; z) = y 1=2 z = 1 > 0
and hence the incentive-compatibility constraint would be violated. Thus,
we now consider the second-best contract, where the owner cannot observe
the action of the manager, but can induce him to take the ’right’ e¤ort
level.
(a) Under the second-best contract, the owner has the choice of inducing
the manager to choose either high or low e¤ort levels. Since there
is no incentive compatibility problem with the low e¤ort level, the
full-information solution continues to hold. The interesting case is
inducing the manager to take the high e¤ort level with a second-best
contract.
(b) Under the second-best contract, the participation constraint continues to hold. In addition, however, we have an incentive compatibility
constraint that guarantees that the manager would choose z over z,
given the contract. Thus, the owner has to solve the problem
max
y1 ;y2
=
[
y1 ] + [1
1
][
2
y2 ]
subject to the participation constraint
1
1
y12 + [1
which becomes
] y22
1
z
0
1
3y12 + y22
8
(11.58)
and the incentive-compatibility constraint
1
y12 + [1
which becomes
c Frank Cowell 2006
1
1
] y22
z
y12 + [1
1
1
y12
y22
219
2
1
] y22
z
(11.59)
Microeconomics
CHAPTER 11. INFORMATION
(c) We may set up the appropriate Lagrangean as:
h 1
1
3
1
L(y1 ; y2 ; ; ) = [16 y1 ]+ [2 y2 ]+ 3y12 + y22
4
4
i
h 1
8 + y12
1
y22
where and are the Lagrange multipliers on constraints (11.58) and
(11.59) respectively. Using the Kuhn-Tucker conditions the FOC for
a maximum we have
1
1
@L
3
3 21
(11.60)
=
+
y1 +
y1 2 = 0 if y1 > 0
@y1
4
2
2
@L
=
@y2
1
1
1 21
1
+
y
y 2 = 0 if y2 > 0
4
2 2
2 2
h 1
i
1
@L
= 3y12 + y22 8 = 0
@
h 1
i
1
@L
=
y12 y22 2
@
From (11.60) and (11.61) we have
[6 + 2 ] y1
2[
] y2
1
2
1
2
(11.61)
(11.62)
(11.63)
=
3
(11.64)
=
1
(11.65)
We need to determine whether the constraints will be binding. Consider = 0. By equations (11.64) and (11.65), this implies that
1
2
and hence that y1 = y2 . But this would violates the incentivecompatibility constraint, as before. Thus,
> 0, and, by (11.62)
the incentive-compatibility constraint (11.59) must bind. But equation (11.65) implies that
> , and hence (11.58) must bind as
well. We can now solve for y1 and y2 using the participation and
incentive-compatibility constraints, and obtain that
y1
1
2
= y2
1
2
=
25
4
1
y2 =
4
It can be shown that the expected pro…ts associated with this contract are higher than those under a …rst best contract with low e¤ort
level, but lower than those under …rst best contract with high e¤ort
level.
y1
=
4. We observe that y1 > y > y2 . The manager receives more than under
…rst best in the good outcome, but less in the bad outcome. While it was
rational for the owner to bear all the risk under …rst best, given that he
was risk-neutral while the manager was risk-averse, under the second-best
contract the risk is shared between the owner and manager. This induces
the manager to take the action which yields a higher probability of a good
outcome.
c Frank Cowell 2006
220
2
i
Microeconomics
Exercise 11.9 The manager of a …rm can exert an e¤ ort level z = 34 or z = 1
and gross pro…ts are either 1 = 3z 2 or 2 = 3z. The outcome 1 occurs with
probability = 23 if action z is taken, and with probability = 13 otherwise. The
manager’s utility function is u(y; z) = log y z, and his reservation utility is
= 0. The risk neutral owner designs contracts which specify a payment yi to
the manager, contingent on obtaining gross pro…ts i .
1. Solve for the full-information contracts. Which action does the owner wish
the manager to take?
2. Solve for the second-best contracts. What is the agency cost of the asymmetric information?
3. In part 1, the manager’s action can be observed. Are the full-information
contracts equivalent to contracts which specify payments contingent on effort?
Outline Answer
1. The structure of the problem is identical to that of Exercise 11.8 so we
may follow the same logic.
(a) Since the owner can observe the manager’s action, and since the
manager is risk-averse, the owner will set y1 = y2 = y in the solution
to the counterpart to (11.54, 11.55) and we can solve for y by setting
u (y; z) = 0:
In the present case, this implies
log y
z=0
Likewise we can solve for y from the equation
log y
z=0
Substituting in the speci…c values given in the questions, we immediately obtain that the full-information contract puts
y = e4=3
(11.66)
y = e:
(11.67)
and
(b) To ascertain which action the owner would like the manager to take,
compare expected pro…ts under the two actions:
=
and
=
c Frank Cowell 2006
2
3
1
3
221
1
+
1
1
3
+
2
3
2
2
e4=3
(11.68)
e :
(11.69)
Microeconomics
CHAPTER 11. INFORMATION
Clearly
h
i
1
4=3
[ 1
]
e
e
2
3
which is positive. Hence, the owner would like the manager to take
z.
=
2. To solve for the second-best contract, we again use the structure developed
in the Exercise 11.8.
(a) The goal is to …nd a contract that induces the manager to take the
high e¤ort level. From the Exercise 11.8, we know that both the
participation and incentive-compatibility constraints will be binding.
Hence, denoting v(y) = log(y), we have the participation constraint:
log (y1 ) + [1
] log (y2 )
z=0
(11.70)
and the incentive-compatibility constraint:
log (y1 ) + [1
] log (y2 )
z=
log (y1 ) + [1
] log (y2 ) z
(11.71)
Rearranging equations (11.70) and (11.71), we …nd
log (y1 )
log (y2 ) =
z
z
(11.72)
Substituting in the values given in the question, the right-hand side
of (11.72) is 1 and so we have
log (y1 ) = 1 + log (y2 )
(11.73)
Hence, using (11.70) we have
[1 + log (y2 )] + [1
] log (y2 )
z=0
Substituting in the values given in the question we have
2
y2 = e 3
(11.74)
Using this value in (11.73) we get
5
y1 = e 3
[
(11.75)
y1 ] + [1
] [ 2 y2 )]
i
h
i
5
2
1
3z 2 e 3 +
3z e 3
3
3
1
2h
(b) The agency cost of asymmetric information is given by the di¤erence
of pro…ts under a full information contract, given by (11.68) and
those under a second-best contract. So the agency cost is
|
[
> 0
c Frank Cowell 2006
1
y] + [1
{z
][
2
[under full information]
222
y]
}
|
[
1
y1 ] + [1
{z
][
2
[under second-best]
y2 ]
}
Microeconomics
which becomes
[y1
y] + [1
] [y2
y]
substituting in from (11.66),(11.67), (11.74) and (11.75) we get
2h 5
e3
3
i 1h 2
4
e3 +
e3
3
i 2 5
4
1 2
e3 = e3 + e3
3
3
4
e3
which is positive. Hence, the agency cost of asymmetric information
is greater than zero, as expected. The owner incurs an agency cost,
which arises from delegating the decision making to the manager and
not observing the manager’s e¤ort level.
3. Full-information contracts are equivalent to contracts which specify payments contingent on e¤ort, since in both kinds of contracts, the owner can
induce the manager to implement the high e¤ort level, and expected gross
pro…ts are the same. However, the high-e¤ort manager extracts a positive
rent, so we do not have equivalence in payo¤ structure.
c Frank Cowell 2006
223
Microeconomics
CHAPTER 11. INFORMATION
Exercise 11.10 A risk-neutral …rm can undertake one of two investment projects
each requiring an investment of z. The outcome of project i is xi with probability
i and 0 otherwise, where
1 x1
x2
1
>
>
>
2 x2
>z
x1 > 0
2 > 0:
The project requires credit from a monopolistic, risk-neutral bank. There is
limited liability, so that the bank gets nothing if the project fails.
1. If the bank stipulates repayment y from any successful project what is the
expected payo¤ to the …rm and to the bank if the …rm selects project i?
2. What would be the outcome if there were perfect information?
3. Now assume that the bank cannot monitor which project the …rm chooses.
Show that the …rm will choose project 1 if y y where
1 x1
y :=
2 x2
1
2
4. Plot the graph of the bank’s expected pro…ts against y. Show that the bank
will set y = y if 1 y > 2 x2 and y = x2 otherwise.
5. Suppose there are N such …rms and that the bank has a …xed amount M
available to fund credit to the …rms where
z < M < Nz
Show that if 1 y >
tioning otherwise.
2 x2
there will be credit rationing but no credit ra-
Outline Answer.
1. The expected payo¤ to the …rm if project i is selected is given by
i
[xi
y]
(11.76)
The expected pro…t for the bank is
iy
z
2. If there were perfect information then the bank can observe which project
is carried out and whether or not it succeeds. Only project 1 (with the
higher probability of success) will be funded and carried out and the bank
will require a repayment y = x1 . The …rm is e¤ectively forced on to its
reservation indi¤erence curve so that no …rm gets less in the absence of
credit than with credit; there is no credit rationing.
c Frank Cowell 2006
224
Microeconomics
Π
Π∗(y)
•
_
0
y
x2
y
π1
π2
Figure 11.13: Credit rationing
3. The …rm will choose the project that gives the greatest pro…ts so that,
given z, project 1 is chosen if and only if:
1
[x1
y]
2
which is equivalent to the condition y
1 x1
y :=
1
4. The bank’s pro…ts are given by
8
< 1y
(y) :=
:
2y
[x2
y]
y where
2 x2
2
z
if y
z
if y < y
y
x2
– see Figure 11.13. There are clearly two local maxima for
bank would set y = y if
1 y > 2 x2
and the
(11.77)
and set y = x2 if
1
<
2 x2 :
(11.78)
5. Distinguish between the situations at the two local maxima
Take the case (11.78). From (11.76) the …rm’s expected payo¤ from
the loan is
x2 ] = 0:
2 [x2
So …rms are indi¤erent between taking and not taking the loan.
There is no credit rationing.
c Frank Cowell 2006
225
Microeconomics
CHAPTER 11. INFORMATION
Take the case (11.77). From (11.76) the …rm’s expected payo¤ from
the loan is
y] > 0:
1 [x1
All …rms would wish to apply for the loan, so that the total demand
is N z. However, the amount available is M so that there is credit
rationing.
c Frank Cowell 2006
226
Microeconomics
Exercise 11.11 The tax authority employs an inspector to audit tax returns.
The dollar amount of tax evasion revealed by the audit is x 2 fx1 ; x2 g. It
depends on the inspector’s e¤ ort level z and the random complexity of the tax
return. The probability that x = xi conditional on e¤ ort z is i (z) > 0 i = 1; 2.
The tax authority o¤ ers the inspector a wage rate wi = w(x), contingent on
the result achieved and obtains the bene…t B (x w). The inspector’s utility
function is
U (w; z) = u(w) v(z)
and his reservation level of utility is . Assume
B 0 ( ) > 0; B 00 ( )
0; u0 ( ) > 0; u00 ( )
0; v 0 ( ) > 0; v 00 ( )
0:
where primes denote derivatives. Information is symmetric unless otherwise
speci…ed.
1. For each possible e¤ ort level …nd the …rst-order conditions characterising
the optimal contract wi i = 1; :::; n.
2. What is the form of the optimal contract when the tax-authority is riskneutral and the inspector is risk-averse? Comment on your solution and
illustrate it in a box diagram.
3. How does this optimal contract change if the inspector is risk-neutral and
the tax-authority is risk-averse? Characterise the e¤ ort level that the tax
authority will induce. State clearly any additional assumptions you wish
to make.
4. As in part 2 assume that the tax authority is risk-neutral and the tax
inspector is risk-averse. E¤ ort can only take two possible values z or z
with z > z. The e¤ ort level is no longer veri…able. Because the agency
cost of enforcing z is too high the tax authority is content to induce z.
What is the optimal contract?
Outline Answer.
1. The problem is
max
2
X
i (z)B(xi
wi )
i=1
over z and wi such that
X
i (z)u(wi )
v(z)
:
First …x z, write down the Lagrangian, and di¤erentiate with respect to
wi to obtain
0
wi ) + i (z)u0 (wi ) = 0
i (z)B (xi
so
8i :
=
B 0 (xi wi )
>0
u0 (wi )
so that the participation constraint binds.
c Frank Cowell 2006
227
Microeconomics
CHAPTER 11. INFORMATION
x1- w1
Authority
x2
w2
x2- w2
υ
45o
Inspector
x1
w1
Figure 11.14: Inspector-authority equilibrium
2. Risk neutrality implies that B 0 ( ) is constant and so u0 (wi ) is constant.
As u00 ( ) 6= 0 this implies w(x1 ) = w(x2 ) = w, say. The tax authority
bears all the risk –see Figure 11.14. Therefore
X
v(z) =
i (z)u(w)
w=u
1
( + v(z))
Of course the wage rate does depend on the e¤ort demanded.
3. In this case we have
u0 ( ) = constant
so B 0 (xi
wi ) is constant 8i. Therefore
xi
wi = constant > 0
which implies
w(xi ) = xi
k
This is a franchise contract where the inspector keeps the result xi but
pays a …xed amount k for the privilege independent of the results. The
participation constraint implies
X
k(z) =
v(z)
i (z)xi
Thus
X
c Frank Cowell 2006
i (z)B(xi
wi )
= B(k(z))
= B(k(z))
228
X
i (z)
Microeconomics
x1- w1
Authority
x2
x2- w2
w2
υ
45o
Inspector
x1
w1
Figure 11.15: Inspector is risk-neutral
As B 0 ( ) > 0 the tax authority seeks to induce an e¤ort level which maximises k. This implies that it chooses z to maximise
X
v(z)
i (z)xi
i
so
X
0
i (z)xi
v 0 (z) = 0
i
which has the interpretation “expected marginal payo¤ = marginal cost”.
The second-order condition is
X 00
v 00 (z) 0
i (z)xi
i
and so a su¢ cient condition is
X
00
i
(z)xi
0
Note that this is trivially satis…ed if z is discrete. See 11.15.
4. O¤er wL = u 1 ( + v(zL )) irrespective of xi :This guarantees the inspector
his reservation level of utility. As
u(wL )
v(zL )
u(wL )
v(zH )
the incentive compatibility constraint is satis…ed.
hazard problem here.
c Frank Cowell 2006
229
There is no moral
Microeconomics
c Frank Cowell 2006
CHAPTER 11. INFORMATION
230
Chapter 12
Design
Exercise 12.1 In a two-good exchange economy there are two persons a and b
with utility functions, respectively:
h
i
a
a
1 e x1 + xa2
b b
x1
+ xb2
where xhi is the amount of good i consumed by person h, and h > 0 is a taste
parameter for person h (h = a; b), where b < a . Person b owns the entire
stock of good 1, R1 ; a and b each own half of the stock of good 2, R2 .
1. Assuming that both persons act as price-takers:
(a) Find the o¤ er curve for person a.
(b) Describe the competitive equilibrium allocation. What would have
happened if b > a ?
2. Now suppose that person b can act as a simple monopolist in the supply of
good 1 while person a continues to act as a price-taker. Show that b will
set a price strictly greater than b ; comment on the outcome in terms of
e¢ ciency.
3. Now suppose that person b can set a fee (…xed charge) for the right to
purchase good 1 as well as setting the unit price of good 1. For any given
price of good 1 what is the maximum fee that person b can set so that a is
still willing to trade? Find b’s optimal fee and price of good 1; comment
on the outcome in terms of e¢ ciency.
Outline Answer
1. Given the utility function
a
and a’s budget constraint
h
1
e
xa
1
i
xa1 + xa2 =
231
+ xa2
(12.1)
1
R2
2
(12.2)
Microeconomics
CHAPTER 12. DESIGN
where is the price of good 1 in terms of good 2, the demand by person
a for good 1 is
a
xa1 ( ) = log
:
(12.3)
Using (12.3) we …nd
xa2 ( ) =
a
1
R2
2
log
(12.4)
(a) Person a’s o¤er curve is given by the locus of (xa1 ( ); xa2 ( )), using
(12.3) and (12.4).
(b) Given the utility function
b b
x1
+ xb2
(12.5)
for b it is clear that MRSb21 = b , a constant. However, from (12.1) we
a
have MRSa21 = a e x1 . It is clear that the competitive equilibrium
price
must be b and, given that MRS will be equal in CE, we
must have
a
x1 a
=
b
x1a
=
log
x2a
=
1
R2
2
e
a
b
x1a =
1
R2
2
b
a
log
b
a
x1b
= R1
x2b
=
log
1
R2 +
2
b
b
a
log
b
a
where “*” denotes CE values. Obviously we require b
for this
b
a
solution; if it had been the case that
>
no trade would have
taken place.
2. Person b now assumes that he can announce the price and that a will
continue to act as a price-taker. His goal is therefore to maximise (12.5)
under this regime and therefore to choose so as to maximise
( ) :=
b
[R1
xa1 ( )] + R2
xa2 ( )
(12.6)
Using (12.3) and (12.4) in (12.6) we get
( )
=
b
=
b
a
R1
1
+ R2 + log
2
log
1
R1 + R2 +
2
b
Di¤erentiating (12.7) with respect to
@ ( )
= log
@
c Frank Cowell 2006
232
a
a
(12.7)
log
we have
a
b
1+
(12.8)
Microeconomics
Given that b < a it is clear that (12.8) is strictly positive at = b
and will become negative for su¢ ciently large: so the value of which
maximises ( ) must be strictly greater than b . Given that MRSa21 =
and MRSb21 = b this implies that the monopoly solution is Paretoine¢ cient, as we would expect.
3. Notice that inserting the …xed charge F in (12.2) leaves a’s demand for
good 1 unaltered (12.3), but that a’s demand for good 2 changes from
(12.4) to
a
1
log
F
(12.9)
xa2 ( ; F ) = R2
2
Person a is just willing to trade if the attainable utility with trade equals
the attainable utility without trade. This requires
a
h
1
e
xa
1
i
+ xa2 =
1
R2
2
(12.10)
with (xa1 ; xa2 ) given by (12.3,12.9). This implies
F = F ( ) := [
a
a
]
(12.11)
log
For any price it is obviously in person b’s interest to charge as high a
value of F as will ensure that a is still willing to trade. This requires
F = F ( ), and so, using (12.6) and (12.7), we …nd that utility for b under
this regime can be written
( ) :=
b
xa1 ( )] + R2
[R1
xa2 ( ; F ( ))
(12.12)
Substituting (12.9) and (12.11) in (12.12) we get:
( )=
b
[R1
log (
a
1
)] + R2 +
2
a
+
b
log ( )
(12.13)
Di¤erentiating (12.13) we have:
@ ( )
=
@
b
1
(12.14)
Clearly person b maximises utility precisely where = b ; so MRSa21 =
MRSb21 . The solution is e¢ cient, even though b fully exploits a.
c Frank Cowell 2006
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Microeconomics
CHAPTER 12. DESIGN
Exercise 12.2 Consider a collective choice problem where the set of options is
given by the interval
:= ; . Three agents, Alf, Bill and Charlie are to
select a particular 2 . They all have single-peaked preferences over
and
h
is the element of
at which agent h has peak preference, h = a; b; c. The
mechanism designer does not know the preferences of the agents and sets out
to design a direct revelation mechanism. Show that the following mechanism is
manipulable:
The designer chooses
0
<
such that that
0
<
0
preference
If is e¢ cient, given the preferences announced by the three agents, then
it is selected
n
o
Otherwise the selected alternative is min a ; b ; c .
Alf
Charlie
Bill
12:00
11:00
10:00
13:00
Figure 12.1: Alf, Bill, Charlie and the taxi
Outline Answer
First note that if
0
does not lie between min
then it cannot be e¢ cient. For if, say,
0
0
n
a
> max
n
o
and max a ; b ; c
o
a b c
; ;
then choosing a
; b;
n
c
o
<
would be strictly preferred by all agents – by switching from 0 to you
are movingnin the direction
of each person’s peak. A similar argument follows
o
for 0 < min a ; b ; c
Now let utility functions v a ; v b ; v c be such that a < b < c < 0 and
0
v
> v c ( a ). Clearly 0 is not e¢ cient because it does not lie between the
min and the max of the peaks. So if agents truthfully reveal their preferences
the outcome is a . However, by misrepresenting preferences and announcing v~c
c
c Frank Cowell 2006
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Microeconomics
c
such that ^ = 0 , Charlie can obtain 0 instead of a , which is preferred by him.
Therefore, truth-telling is not a dominant strategy: this rule is manipulable.
The result can be illustrated by the example in The three agents want to
share a taxi; for personal reasons they have divergent views as to when they
would like the taxi to pick them up and their preferences for a pickup time are
represented by the curves in Figure 12.1. The taxi company is going to call back
to suggest a pickup time and the three agents have agreed that if the suggested
time lies outside the range of their top preferences –here the range is 11:00 to
12:00 –then they will ask it to pick them up at 11:00. Now consider two cases:
1. The taxi company calls and suggests 12:45. This suggestion is not e¢ cient
since Alf Bill and Charlie would all have been happier if 12:30 had been
suggested instead. So the rule would require that they request a pickup
time of 11:00 and Figure 12.1 makes it clear that they all prefer this to
the taxi company’s suggestion.
2. The taxi company calls and suggests 12:15. Again this suggestion is ine¢ cient – it still lies outside the range of the peaks – and again the rule
suggests the 11:00 pickup. But Charlie now has an interest in misrepresenting his preferences. He would prefer a pickup time of 12:15 to that
of 11:00. So, knowing the rule, he can get want he wants by pretending
that his peak is indeed at 12:15. If his stated peak preference of 12:15
is believed then the taxi company’s suggested pickup time is e¢ cient and
the agents accept it. Given this possibility of fruitful misrepresentation
the rule is clearly ine¢ cient.
c Frank Cowell 2006
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Microeconomics
CHAPTER 12. DESIGN
0 00 000
Exercise 12.3 Two voters are to select one element from
; ;
: Each
voter has a strict preference over the alternatives. Show that the following rule
is manipulable: if 0 is e¢ cient, then it is selected; if 0 is not e¢ cient, but 00
is, then 00 is selected; if neither 0 nor 00 is e¢ cient, then 000 is selected.
Outline Answer
1. One approach to solving the problem is by de…ning the voting rule extensively. This gives Table 12.1.
voter 2
voter 1
0 00 000
0 000 00
00 0 000
00 000 0
000 0 00
000 00 0
0 00 000
0
0
0
0
0
0
0 000 00
0
0
0
0
0
0
00 0 000
0
0
00
00
0
00
00 000 0
0
0
00
00
00
00
000 0 00
0
0
0
00
000
000
000 00 0
0
0
00
00
000
000
Table 12.1: Selection under alternative combinations of voter preferences
For example when voter 1 is of type 000 0 00 and voter 2 is of type 00 000 0 ;
the selected alternative is 00 ; whereas it would be 0 if voter 1 says that he
has preferences 0 00 000 : Therefore he has an incentive to misrepresent his
preferences in that case, and truth-telling is not a dominant strategy. Also
if voter 1 is of type 00 0 000 and 2 is of type 000 0 00 ; the selected alternative
is 0 ; whereas it would be 00 if he says that has preferences 00 000 0 .
2. An alternative approach to the problem is to consider directly the possible ways of misrepresenting one’s preferences to one’s own advantage.
Consider three cases:
(a)
(b)
(c)
0
is voter 1’s most preferred alternative. In this case 0 is e¢ cient,
and so voter 1 clearly has no incentive to misrepresent his preferences
as he will get 0 .
00
is voter 1’s most preferred alternative and 00 is selected. Given
that 00 is selected, then clearly voter 1 has an incentive to tell the
truth.
00
is voter 1’s most preferred alternative and 0 is selected because
it is e¢ cient. But if 0 is e¢ cient, is it possible to still obtain 00
by misrepresenting his preferences? For this to work it must be true
that by announcing 0 lower in his ranking than is actually the case he
can make 0 become ine¢ cient ( 00 is still kept as the most preferred
alternative). In other words it can work if, by announcing 00 000 0
instead of 00 0 000 , voter 1 forces 0 to be ine¢ cient. This is possible
in the circumstance where voter 2’s most preferred alternative is 000
and 0 is not dominated by 00 – in other words if voter 2 is of type
000 0 00
. Under such circumstances by announcing 00 000 0 instead of
c Frank Cowell 2006
236
Microeconomics
00 0 000
, voter 1 makes 0 dominated by 000 : As 0 is no longer e¢ cient,
but
is e¢ cient, 00 is the new selection. So voter 1 has an interest
in declaring that he is of type 00 000 0 when he is actually of type
00 0 000
and voter 2 is of type 000 0 00 :
00
c Frank Cowell 2006
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Microeconomics
CHAPTER 12. DESIGN
Exercise 12.4 Two …rms are considering whether to undertake a joint project.
It is common knowledge that the value of the project to …rm 1 must be either
200 or 700 and that the value to …rm 2 must be either 400 or 700. However,
information about the precise value is private to the …rm. The …rms would like
to undertake the project if the sum of their values exceeds the cost of the project
which is known to be 1000. The …rms use the pivotal mechanism.
1. Find the amount each …rm will have to pay as a function of its announcement.
2. Show that the mechanism cannot be manipulated.
3. Check that there will always be enough money if they decide to undertake
the project.
Outline Answer
1. The project will be undertaken if (v1 ; v2 ) = (700; 400) or (700; 700). Consider the outcome in each case.
(a) Case (200; 700). It is clear that …rm 1 is pivotal in this case In fact
…rm 2’s valuation is greater than the amount necessary on average
to undertake the project even though the project is not undertaken.
Under the pivotal mechanism case …rm 1 pays
X
vj [n 1] c = 700 500 = 200:
j6=i
(b) Case (700; 400). Firm 1 is also pivotal here. In fact …rm 2’s valuation
is lower than the cost of the project on average, even though the
project is undertaken. In this case, …rm 1 pays
X
nc
vj = 1000 400 = 600:
j6=i
(c) Other cases. Firm 1 pays the average cost. Firm 2 is never pivotal.
(d) So the payments in the various cases are as in Table 12.2
Firm 1
v1 = 200
v1 = 700
Firm 2
v2 = 400
v2 = 700
(0; 0)
(200; 0)
(600; 500) (500; 500)
Table 12.2: Payments under the pivotal mechanism
2. Consider the payo¤s to each …rm under the possible combinations of declarations
c Frank Cowell 2006
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Microeconomics
v^1 = 200
v^1 = 700
(a) True value low
v^2 = 400 v^2 = 700
0
200
400
300
(b) True value high
v^2 = 400 v^2 = 700
0
200
100
200
Table 12.3: Payo¤s to …rm 1 under all combinations of declarations
Table 12.3 gives the payo¤s to …rm 1 in the two cases (a) when its
true valuation is low (v1 = 200), (b) when its true valuation is high
(v1 = 700).
It is clear that when v1 = 200 (left-hand side of Table 12.3) declaring
v^1 = 200 produces higher payo¤s than declaring v^1 = 700: telling the
truth is dominant. Likewise when v1 = 700 (right-hand side of Table
12.3) declaring v^1 = 700 produces higher payo¤s. Truth-telling is a
dominant strategy.
Table 12.4 gives the payo¤s to …rm 2 in the two cases (a) when its
true valuation is low (v2 = 400), (b) when its true valuation is high
(v2 = 700).
Again it is clear that when v2 = 400 (left-hand side of Table 12.4)
v^1 = 200
v^1 = 700
(a) True value low
v^2 = 400 v^2 = 700
0
0
100
100
(b) True value high
v^2 = 400 v^2 = 700
0
0
200
200
Table 12.4: Payo¤s to …rm 2 under all combinations of declarations
declaring v^2 = 400 produces higher payo¤s than declaring v^2 = 700:
telling the truth is dominant. Likewise when v2 = 700 (right-hand
side of Table 12.4) declaring v^2 = 700 produces higher payo¤s. Truthtelling is a again dominant strategy.
In the light of the result about truth-telling by both parties the mechanism cannot be manipulated.
3. The total payments by the …rms are 1100 in the case (700; 400), and 1000
in the case, (700; 700) so that the cost is always covered.
c Frank Cowell 2006
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Microeconomics
CHAPTER 12. DESIGN
Exercise 12.5 There is a …xed number of bidders N in a …rst-price (Dutch)
auction.
1. Suppose the values are drawn from the same rectangular distribution with
support [ ; ]. What is the density function and the distribution function
for ?
2. Agent h assumes that all the other agents’ bids are determined by their
type: so that a rival with value bids ( ) where is a strictly increasing
function. On this assumption show that, if agent h bids a price p, the
probability that he wins the auction is
N 1
' (p)
;
where ' ( ) is the inverse function of
( ).
3. Suppose the agent is risk neutral and that = 0, = 1. Again assuming
that the other agents’bids are determined by the same bid function ( ) ,
show that the agent’s best response is determined by the following equation:
[p
' (p)] [N
1]
d' (p)
+ ' (p) = 0
dp
(12.15)
4. Show that condition (12.15) implies
' (p) =
N
N
1
p
(12.16)
and that the equilibrium bid function takes the form
( )=
N
1
N
(12.17)
5. Use the result that the expected value of the kth smallest member of a
sample of size N drawn from a rectangular distribution on [0; 1] is
k
N +1
(12.18)
to show that the expected price received by the seller in the special case
= 0, = 1 is
N 1
:
p (N ) =
N +1
6. Use this to …nd the optimal bid function and the expected price received by
the seller for a rectangular distribution with support [ ; ].
c Frank Cowell 2006
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Microeconomics
Outline Answer
1. The density function is
1
f( )=
and the distribution function is
:
F( )=
2. Take the case where there are just two bidders, agent h and one other
agent. If the other agent’s value is and his bid is determined by his type
then agent h wins the auction if he bids p > ( ) where is the other
agent’s bid function. This is equivalent to requiring
' (p) >
If the other agent’s value is unknown, but is assumed to be rectangularly
distributed with support [ ; ] then the probability that agent h wins the
auction is
:= F (' (p))
which, from part 1, is
' (p)
:
If there are two other, agents each with values independently drawn from
the same distribution, then the probability of winning is just 2 . So, if
there are N 1 other agents, all with values independently drawn from
the same distribution then the probability of winning is N 1 , in other
words
N 1
' (p)
Pr (win) =
:
3. If agent h’s value is
then the expected return to bidding p is
[
Given that
= 0,
expected return is
p] Pr (win)
= 1 then the result from part 2 shows that this
[
p]
N 1
=[
N 1
p] [' (p)]
(12.19)
Agent h seeks to choose p so as to maximise (12.19). Di¤erentiating, the
FOC for an interior maximum is
[
p] [N
N 2
1] [' (p)]
d' (p)
dp
N 1
[' (p)]
= 0:
On rearranging we have:
[p
c Frank Cowell 2006
] [N
1]
d' (p)
+ ' (p) = 0
dp
241
(12.20)
Microeconomics
CHAPTER 12. DESIGN
This determines the agent’s bid function
p=
such that
( )
If all agents are identical then this bid function applies to all N agents
and so it must be true that
= ' (p)
in equation (12.20). Making this substitution we have (12.15) as required.
4. Given that (12.15) is homogenous in ' and p, try the solution of value
proportional to bid:
' (p) = kp
for which we have
d' (p)
= k:
dp
This means that (12.15) becomes
[p
kp] [N
1] k + kp = 0
(12.21)
which implies
1
(12.22)
N 1
from which the result (12.16) follows. However, by de…nition ' is just the
inverse of the bid function ; so (12.16) implies
k
1=
=
N
N
1
p
(12.23)
which implies
p=
N
1
(12.24)
N
which is (12.17).
5. If there are N agents we can represent their values as
[1]
:::
[2]
[N ]
where [k] is the kth order statistic of the given distribution. Obviously
the person with the highest valuation gets the object at the price
p=
N
1
(12.25)
[N ]
N
The expected value of the price is
p (N )
= Ep
N 1
=
E
N
[N ]
Using the result on order statistics stated in the question we have
p (N ) =
N
1
N
from which the required result follows.
c Frank Cowell 2006
242
N
N +1
Microeconomics
6. Instead of (12.19) the expected return to bidding p is now
[
' (p)
p]
N 1
(12.26)
and so, on di¤erentiating and rearranging we have:
[p
' (p)] [N
1]
d' (p)
+ ' (p)
dp
=0
(12.27)
instead of (12.15). By analogous working to part 4 the solution to (12.27)
is
N
N
]+ =
' (p) =
[p
p
N 1
N 1
N 1
and so the optimal bid function is now
( )=
N
1
N
[
]+ :
(12.28)
Note that the random variable characterising individual values can be
written = [
] + where is rectangularly distributed with support
[0; 1]. So now
k
E [k] = [
+
]
(12.29)
N +1
and so
N 1
p (N ) =
E [N ]
+
N
Using the order-statistics result again we have
p (N ) =
c Frank Cowell 2006
N 1
[
N +1
243
]+
Microeconomics
CHAPTER 12. DESIGN
Exercise 12.6 Suppose that in the model of exercise 12.5 a second-price (English) auction were used rather than a …rst-price auction
1. What is the optimal bid function?
2. Use the result in (12.18) to determine the expected price paid by the winner.
3. Use the results on order statistics in Appendix A to draw the distribution
of the price paid to the seller in the case of the …rst and the second price
auction.
Outline Answer
1. In a second-price auction we know that truth-telling is a dominant strategy
and so the optimal bid function is just
( )= :
2. Given a sample of N from the distribution of values the price paid under
the English auction is [N 1] . So, given a rectangular distribution with
support [0; 1] we have expected value of the price is
p (N ) = E
[N 1]
Using the result on order statistics stated in Exercise 12.5 we have immediately that
N 1
:
p (N ) =
N +1
if the support of the rectangular distribution were [ ; ] then
E
where
[N 1]
=[
]E
[N 1]
+
is rectangularly distributed with support [0; 1]. So in this case
p (N ) =
N 1
[
N +1
]+ :
(12.30)
3. Given that there are N bidders with values [1]
:::
[2]
[N ] drawn
from a rectangular distribution with support [0; 1] the price in the two
auctions is as follows. For the auction we have:
p=
N
1
N
[N ]
(12.31)
and for the English auction:
p=
The values
[k]
c Frank Cowell 2006
[N 1]
(12.32)
can be taken as the order statistics for a sample of size N .
244
Microeconomics
F(p)
1
English
Dutch
p
0
½
1
Figure 12.2: Distribution of auction prices
So, use the result (given in Appendix A) that, if has the distribution
F then F[k] ( ), the distribution of the kth order statistic is given by
F[k] ( ) =
N
X
N
F ( )j [1
j
N j
F ( )]
:
(12.33)
j=k
In the case of the rectangular distribution with support [0; 1] this
simpli…es to
N
X
N j
N j
F[k] ( ) =
[1
]
:
(12.34)
j
j=k
so that, for the particular cases in which we are interested, we have
F[N ] ( ) =
N
:
(12.35)
and
F[N
1] (
)
= N
N 1
= N
N 1
[1
]: +
: + [1
N]
N
:
N
:
(12.36)
For the Dutch auction we have, from (12.31)
[N ]
=
N
N
1
p
(12.37)
Using (12.35), therefore, the distribution of price for the Dutch auction is given by
)
(
N
N
N
F (p) = min
p ;1
N 1
c Frank Cowell 2006
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Microeconomics
CHAPTER 12. DESIGN
and, in the case where N = 2, this becomes
F (p) =
4p2
1
0
1
2
p
p
1
2
1
For the English auction in the case N = 2 we have, using (12.32) and
(12.36), the following:
F (p) = N p: + [1
N ] p2 : = 2p:
p2 :
The two distributions are illustrated in Figure 12.2 for the case N =
2.
c Frank Cowell 2006
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Microeconomics
Exercise 12.7 Alf and Bill are bidding in a conventional English auction for
an object of unknown market value: Alf ’s valuation of the object is a , Bill’s
valuation is b and the true total value is expected to be a + b , but at the
time of the auction neither bidder knows the other’s valuation. However, it is
common knowledge that a and b are drawn from a rectangular distribution
with support [ ; ].
1. What is the expected value of the object?
2. From Alf ’s point of view, what is the expected value of the object, conditional on his winning the object?
a
3. Show that the price 2 min
b
;
is an equilibrium.
4. Suppose Alf followed a policy of bidding up to a + E b and believed that
Bill was following the same type of policy. Why might this bidding policy
lead to an unfavourable outcome for the winner? (a phenomenon known
as “the winner’s curse”).
Outline Answer
1. If a and b are each drawn from the same distribution then the expected
total value is
E
a
b
+
=
=
=
2E (
a
)
1
[ + ]
2
2
+ :
2. An informal argument is as follows. Suppose Bill drops out of the bidding
when the price is p. Alf would then win the object at that price and he
knows that, on the assumption that Bill was bidding rationally, Bill had
valued the object at p (or a fraction less). So, given this information, the
expected value of Bill’s private valuation E b is 12 p. So the expected
value of the object, conditional on Alf’s winning the object is a + 12 p. Note
that this result is independent of the distribution of . A more rigorous
argument is provided in the next part.
3. The equilibrium would be generated if each bidder were following a bidding
strategy ( ) given by
( )=2 :
(12.38)
and ( ) is a bidder’s best response if the other bidder is using ( ). So,
suppose Bill plays using ( ); if Bill stops bidding at p then Alf knows
that, in the light of (12.38)
1
p
2
So Alf’s estimate of the expected value v a of the object should be
E
va
=
b
j
= E
=
c Frank Cowell 2006
b
a
+
a
=p =
b
b
j
+E
j
1
a
+ p
2
247
b
=p
b
=p
(12.39)
Microeconomics
CHAPTER 12. DESIGN
Clearly Alf would carry on bidding if
va > p
i.e. if p < 2
a
. By the same argument Alf would stop bidding where
p=2
a
:
The same applies to Bill which establishes that ( ) is indeed a best response. So the equilibrium is indeed at the point where
a
p = 2 min
;
b
(12.40)
Again, this result is independent of the distribution of . However, in
the case where is uniformly distributed on [ ; ] we know from equation
(12.30) in the answer to Exercise 12.6 that, in the case of two bidders
E min
a
b
;
=
1
[
3
]+
so that the expected equilibrium price in this case is
2
[
3
]+2
which is less than the expected value of the the object found in part 1.
4. This bidding strategy will imply that the average winning price will exceed
the value of the object.
c Frank Cowell 2006
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Microeconomics
Exercise 12.8 A Principal hires an Agent to manage a …rm. The Agent produces an outcome q given by
q = (z; )
where z is e¤ ort and is the talent of the Agent that may or may not be observable by the Principal; the function
is twice di¤ erentiable, increasing in
both arguments, concave in z, and has a positive cross derivative. The Principal
contracts to pay the Agent an amount of income y in return for output q:The
Principal’s utility is
q y
and the Agent’s utility is
# (y)
z
where # is an increasing, strictly concave function such that # (0) = 0.
1. Using the production function show that the the indi¤ erence curves in
(q; y)-space for Agents of di¤ erent types must satisfy the single-crossing
condition.
2. If the Principal can costlessly observe the talent of any potential Agent
at the time of hiring and knows that the utility available to any Agent
in alternative employment is 0, …nd the conditions that characterise the
full-information solution.
3. It is known that there are exactly two types of Agent and the proportion of
high-talent agents is . Write down the incentive-compatibility constraint
for the second-best problem.
4. Find the conditions that characterise the second-best equilibrium and show
that the no-distortion-at-the-top principle holds.
Outline Answer
1. We need to …nd the e¤ect of a change in the parameter
the indi¤erence curve.
on the slope of
(a) The Agent’s utility function can be written as
# (y)
z
or equivalently as
# (y)
(q; )
(12.41)
where ( ) is the inverse of the function ( ) yielding the amount of
z required to produce a particular amount of q for a speci…ed value
of talent . Because it is an inverse we have
q=
(z; ) =
( (q; ) ; ) :
So, di¤erentiating (12.42) with respect to q:
1=
c Frank Cowell 2006
z
( (q; ) ; )
249
q
(q; )
(12.42)
Microeconomics
CHAPTER 12. DESIGN
which implies
q
1
>0
(
(q;
); )
z
(q; ) =
(12.43)
Also, di¤erentiating (12.42) with respect to :
0=
z
( (q; ) ; )
(q; ) +
( (q; ) ; ) :
which implies
( (q; ) ; )
< 0:
(
(q; ) ; )
z
(q; ) =
(12.44)
Also note that
q
(q; ) =
zz
( (q; ) ; )
z
(q; ) +
z
2
( (q; ) ; )
( (q; ) ; )
<0
(12.45)
(b) Now …nd the slope of the indi¤erence curve in (q; y)-space. Di¤erentiating (12.41) we have
0 = #y (y) dy
so that we have
q
(q; ) dq
dy
q (q; )
=
dq
#y (y)
Using (12.43) this gives
dq
= #y (y)
dy
Consider the e¤ect of changing
d
d
dq
dy
= #y (y) [
z
z
( (q; ) ; )
(12.46)
on (12.46):
( (q; ) ; ) +
zz
( (q; ) ; )
(q; )]
(12.47)
From the question we have z > 0 and zz < 0; from (12.44) we
have
< 0. Therefore it is clear from (12.47) that an increase in
will increase dq=dy; so an increase in will decrease dy=dq. The
indi¤erence curves therefore look like those depicted in Figure 12.3
where a and b are the reservation indi¤erence curves for two talent
types, with a representing the higher talent.
2. The Principal’s objective is to maximise q
constraint for an individual of type :
# (y)
(q; )
y subject to the participation
0;
(12.48)
the shaded areas in Figure 12.3 represent contracts that satisfy this constraint for the two talent types. The optimisation problem can be represented by the Lagrangean
L (q; y; ; ) := q
c Frank Cowell 2006
250
y + [# (y)
(q; )]
(12.49)
Microeconomics
where is the Lagrange multiplier for the participation constraint (12.48).
The FOCs for an interior maximum of (12.49) with respect to q; y; are
@L
@q
@L
@y
@L
@
=
1
=
q
1
(q ; ) = 0
#y (y ) = 0
= # (y )
(q ; ) = 0
.so the full-information solution (q ; y ) is characterised by the condition
(q ; )
=1
#y (y )
q
(12.50)
for each talent type and the requirement that (12.48) be binding. Condition (12.50) characterises an e¢ cient outcome. The solution is illustrated
by points (q a ; y a ) and q b ; y b in Figure 12.3.
υ
_b
υ
_a
y
•
y*a
•
~
ya
slope = 1
slope = 1
slope = 1
•
y*b
~
yb
0
~
qb q*b
q
~
qa
q*a
Figure 12.3: Principal with Agents of di¤erent talents
3. It is clear that if the Principal cannot observe the Agent’s type then the
e¢ cient solution in part 1 cannot be implemented.
(a) Consider two talent types a and b where
person accepts the contract designed for =
# (y a )
c Frank Cowell 2006
251
(q a ;
a
) = 0:
a
a
> b . If a type-a
then utility is
(12.51)
Microeconomics
CHAPTER 12. DESIGN
Likewise if a type-b person accepts the contract designed for = b
then utility is
q b ; b = 0:
(12.52)
# y b
But if an a-type gets a b-type contract then utility is
# y
b
q b;
a
>0
(12.53)
which follows because of condition (12.44) and the fact that a > b .
So any a-type has an incentive to misrepresent himself as a b-type in
order to obtain a more favourable contract.
(b) In view of the above the above the incentive-compatibility constraint
is
# (y a )
(q a ; a ) # y b
qb ; a
(12.54)
4. In the asymmetric-information case it may be assumed that the probability
that a potential manager is of type a is .
(a) The second-best problem for the Principal is to choose q a , q b , y a , y b
to maximise expected utility
[q a
y a ] + [1
] qb
yb
subject to the participation constraint (12.48) for type b and the
incentive-compatibility constraint (12.54) for type a; from previous
exercises we know that both these constraints will be binding. The
Lagrangean is therefore
L qa ; qb ; ya ; yb ; ;
where and
and (12.54).
:=
[q a y a ] + [1
] qb yb
b
b b
+ # y
q ;
+ # (y a )
(q a ; a ) # y b +
qb ; a
(12.55)
are the Lagrange multipliers for the constraints (12.48)
(b) Let q~a ; q~b ; y~a ; y~b denote the values for that maximise (12.55). The
FOCs for a maximum are
@L
@q a
@L
@q b
@L
@y a
@L
@y b
=
=
=
=
q
(~
qa ;
1
q
a
)=0
q~b ;
b
(12.56)
+
q
a
+ #y (~
ya ) = 0
1+
+ #y y~b
=
=
#y y~b = 0
#y (~
ya )
1
+
#y (~
y b ) #y (~
ya )
252
=0
(12.57)
(12.58)
From (12.58) and (12.59) we have
c Frank Cowell 2006
q~b ;
(12.59)
9
=
;
Microeconomics
So that, from (12.56),
(~
qa ; a )
=1
#y (~
ya )
q
(12.60)
and, from (12.57),
q~b ; b
=1+
#y (~
yb )
1
q
"
q
q~b ;
a
q
#y (~
ya )
q~b ;
b
#
<1
(12.61)
where the last part of (12.61) follows from (12.45) and the fact that
a
> b.
(c) The solution is illustrated by points (~
q a ; y~a ) and q~b ; y~b in Figure
12.3. Condition (12.60) establishes the point that there is no distortion at the top –note that the MRS=1 as in the full-information
case..
c Frank Cowell 2006
253
Microeconomics
CHAPTER 12. DESIGN
Exercise 12.9 Consider a good that is produced by a monopoly that is known
to have constant marginal cost c and a …xed cost C0 ; marginal cost is given by
c=
1
(12.62)
where is a parameter that characterises the …rm’s e¢ ciency. If the monopoly
sells a quantity q of output in the market then the price that it can command
is given by p (q) where p ( ) is a known, decreasing function. A government authority wants to regulate the behaviour of the monopoly. The regulation speci…es
an output level q and a subsidy (fee) F that is intended to ensure that the …rm
does not make a loss. The authority has been instructed to take into account the
interests of both consumers and the owners of the …rm. Consumers’ interests
are assumed to be given by consumer’s surplus; the …rms’interests are assumed
to be given by pro…ts including the subsidy F .
1. If the authority places a weight 1 on consumer’s surplus and a weight
(where 0 <
1) on pro…ts show that the authority’s objective function
can be expressed as
V (q; R) :=
Zq
p (x) dx
[1
]R
C0
q
0
where R is total revenue for the …rm.
2. If the parameters of the cost function C0 and
are common knowledge
(a) What is the relevant participation constraint and Lagrangean for the
problem?
(b) How will the authority set q and F ?
(c) Interpret this full-information result.
3. Assume that C0 is known but marginal cost is unknown; however it is
known that the e¢ ciency of the …rm is either a or b where
a
>
b
:
and marginal cost is again given by (12.62).The authority wants to use
the same kind of regulatory régime but it does not know for certain which
type of …rm, low-cost (i.e. a or high-e¢ ciency) or high-cost (i.e. b or
low-e¢ ciency), it is dealing with.
(a) What is the incentive compatibility constraint that the authority should
take into consideration?
(b) What form does the participation constraint now take?
(c) If the authority faces a low-cost type …rm with probability and it
intends to maximise expected welfare what is the Lagrangean for the
problem?
(d) Find and interpret the second-best solution.
c Frank Cowell 2006
254
Microeconomics
Outline Answer
1. Consumer’s surplus is given by:
CS(q) :=
Zq
p (x) dx
pq
F
0
Pro…ts are given by
pq + F
C0
cq
So the objective function is
Zq
p (x) dx
pq
F+
[pq + F
C0
cq]
0
which can be written as
V (q; R) :=
Zq
p (x) dx
[1
]R
C0
cq
(12.63)
0
where R is total revenue for the …rm (sales pq plus fee F ). Note that
Vq (q; R) = p (q)
VR (q; R) =
1
c
(12.64)
(12.65)
2. Maximising the objective function with respect to q and F is equivalent
to maximising V with respect to q and R.
(a) The participation constraint requires that non-negative pro…ts be
made:
R C0 cq 0
(12.66)
and the Lagrangean is
L (q; R; ) := q
V (q; R) + [R
C0
cq]
(12.67)
where is the Lagrange multiplier for the constraint (12.66).
(b) Denote by q ; R the values that maximise (12.67). The FOCs are
Vq (q ; R )
c = 0
VR (q ; R ) +
= 0
[R
C0 cq ] = 0
Therefore
p (q )
=
c+ c
= 1
and so
p (q ) = c + c
c Frank Cowell 2006
255
c=c
(12.68)
Microeconomics
CHAPTER 12. DESIGN
so that
q =p
Given
< 1 we have
1
(c) :
(12.69)
> 0 and so
R
C0
cq = 0
(12.70)
Therefore
R
F
= C0 + cq
= C0 + cq
p (q ) q = C0
(12.71)
(12.72)
(c) It is clear that condition (12.68) characterises e¢ ciency (set output
at the point where price equals marginal cost) and condition (12.72)
sets the fee at the point where it just covers …xed costs. The results
are independent of the value of the weighting parameter .
3. Now assume that c is unknown to the regulator although the distribution
of c is known.
(a) The above e¢ cient regulation scheme cannot be implemented given
this information setup. The reason is evident if we consider the …rm’s
pro…ts under the scheme. Write (q (^
c) ; R (^
c)) for the regulatory
regime that would be implied by (12.69) and (12.71) if the …rm’s
marginal cost is believed to be c^. Suppose that the …rm is invited to
declare its marginal cost to the regulator. If its true marginal cost is
c and it declares c^ then its pro…ts are given by
(^
c; c) := [R (^
c)
C0
cq (^
c)]
which can be rewritten as
[R (^
c)
C0
c^q (^
c)] + [^
c
c] q (^
c)
(12.73)
The …rst term in (12.73) is zero by condition (12.70) so that (c; c) =
0: truth-telling yields zero pro…ts. It is also clear that the …rm could
make positive pro…ts by falsely declaring c^ > c. The design problem
must thus take account of this possibility of misrepresentation. If it
is known that there are just two values for marginal cost ca = 1= a
and cb = 1= b where ca < cb and the regulator intends to impose the
(quantity, revenue)-regime (q a ; Ra ) on a low-cost type of …rm and
q b ; Rb on a high-cost type then, in order to ensure truth-telling on
the part of a low-cost …rm it must be the case that
Ra
C0
ca q a
Rb
C0
Ra
ca q a
Rb
ca q b
ca q b
or, more simply:
(12.74)
This is the incentive-compatibility constraint.
(b) However, the regulatory regime must be such that even if the …rm is
genuinely high-cost, it will not make a loss:
Rb
C0
cb q b
0
This is the relevant participation constraint.
c Frank Cowell 2006
256
(12.75)
Microeconomics
(c) If it is known that there is a probability that the …rm is low cost
then the rational thing for the regulator to do is to choose (q a ; Ra )
and q b ; Rb so as to maximise the expected value of V (q; R) de…ned
in (12.63). The relevant Lagrangean is
9
L q a ; q b ; Ra ; Rb ; ;
:=
V (q a ; Ra ) + [1
] V q b ; Rb =
+ Rb C0 cb q b
;
+ Ra ca q a Rb + ca q b
(12.76)
where is the Lagrange multiplier for constraint (12.75) and is
the Lagrange multiplier for constraint (12.74), both of which will be
binding at the optimum.
(d) Let the (quantity, revenue)-values that maximise (12.76) be denoted
~ a and q~b ; R
~ b . Then the FOCs for the maximum are
by q~a ; R
given by
@L
@q a
@L
@q b
@L
@Ra
@L
@Rb
~a
Vq q~a ; R
=
=
ca = 0
~b
] Vq q~b ; R
[1
cb + ca = 0
~a +
VR q~a ; R
=
=
=0
~b +
] V q~b ; R
[1
=0
Substituting in for the partial derivatives of V ( ) from (12.64) and
(12.65) we have
[p (~
qa )
] p q~
cb
ca ]
ca
b
c + ca
[
1] +
1] +
b
[1
[1
][
= 0
= 0
= 0
= 0
(12.77)
(12.78)
(12.79)
(12.80)
From (12.79) we have
=
[1
]
and from (12.80) we have
= [1
] [1
Substituting in these values for
get
[1
b
] p q~
[p (~
qa )
cb
[1
]+
and
=1
into (12.77) and (12.78) we
ca ]
[1
b
] c + [1
] ca
] ca
= 0
= 0
So that the second-best pricing rule is
p (~
q a ) = ca
p q~b
= cb +
c Frank Cowell 2006
257
(12.81)
(12.82)
Microeconomics
CHAPTER 12. DESIGN
where
[1
] cb ca
(12.83)
1
This implies that price equals marginal cost for the low-cost …rm but
is above marginal cost for the high-cost …rm. Note that the size of
the distortion depends on the proportion of low-cost …rms, the cost
di¤erence and the weight attached to pro…ts by the regulator. Also,
from the constraint (12.75) we have
:=
~b
R
cb q~b = 0
C0
(12.84)
so that
F~ b = C0
q~b p q~b
cb = C0
q~b
and from the constraint (12.74) we have
~ a = C0 + q~b cb
R
ca + ca q~a
(12.85)
which yields
F~ a = C0 + q~b cb
ca
(12.86)
Hence, under the second-best policy we have
q~a
F~ a
b
q~
~
Fb
c Frank Cowell 2006
= p 1 (ca )
= C0 + cb
1
= p
a
c
p
b
c +
= C0
258
p
1
1
b
c +
(12.87)
(12.88)
(12.89)
b
c +
(12.90)
Microeconomics
Exercise 12.10 An economy consists of two equal-sized groups of people. The
gifted, with an ability parameter = 2 and the deprived, for whom = 1. All
persons have a utility function given by
U (x; z) = log x + log (1
z)
where x is consumption and z is e¤ ort in the labour market, 0
sumption is given by
x=w+T
z
1. Con-
where w is the market wage given by
w(z; ) = z
and T is a transfer from the government (if positive) or a tax (if negative). The
government would like to set T = > 0 for deprived individuals and T =
for gifted people.
1. If utility is observable and each person chooses z so as to maximise utility,
draw the graph of utility against earnings. for a gifted person and for a
deprived person, for a given value of . What is the optimum value of
earnings for each type of person?
2. Assume utility is not observable. The government plans the following
transfer scheme
8
if w w
<
T =
:
if w > w
where w is the optimum earnings for deprived people in part 1. Show that
gifted people may …nd it in their interest to pretend to be deprived. In the
diagram for part 1 plot the utility of someone carrying out this pretence.
3. Use the above diagram to show the case where a gifted person is just indi¤ erent between acting as though utility were observable and pretending
to be deprived.
4. Show that deprived people’s utility could be increased by restricting the
amount that they are allowed to earn.
Outline Answer
1. Using w as the control variable and noting that z = w= the problem is
to maximise
w
log (w + T ) + log 1
(12.91)
This has the solution at
w = max 0;
c Frank Cowell 2006
259
1
[
2
T]
Microeconomics
CHAPTER 12. DESIGN
υ
Gifted
pretending to
be deprived
Gifted
Deprived
w
w1 = w*
w2
Figure 12.4: Masquerading in income support
υ
∆υ2
∆υ1
w
w**
w2
Figure 12.5: Design of income support
c Frank Cowell 2006
260
Microeconomics
2. It is clear from (12.91) that utility at …rst rises with earnings and then
falls; also the graph of utility against earnings is shifted upwards by an
increase in ability and by an increase in transfer T . So for individuals
who are able (and so who should have T < 0) but in fact pretend to be
deprived and get T > 0 we have a graph such as the broken line in Figure
12.4.
3. See Figure 12.4.
4. The situation in Figure 12.4 means that resources that should support
the deprived will be siphoned o¤ by some of the gifted. This could be
prevented if the earnings of the deprived were limited to w in Figure 12.5
– this reduces their utility in…nitesimally ( 1 ' 0) but it substantially
reduces the utility of the gifted who are masquerading ( 2 < 0) – see
Figure 12.5. So the policy forces the gifted to reveal themselves and
enables the income support scheme for the deprived to be viable.
c Frank Cowell 2006
261
Microeconomics
CHAPTER 12. DESIGN
Exercise 12.11 The government has to raise a …xed sum K through income
tax. It is known that there are two types of worker a and b in the economy and
that the output (gross income) produced by each is given by
qj =
j j
z ; j = a; b
j
where z is the amount of e¤ ort supplied by a person of type j and
is also known that workers have preferences given by the function
j
= qj
Tj +
a
>
b
. It
zj
where T j is the tax on a worker of type j and is a decreasing concave function.
The government has no interest in the inequality of utility outcomes and so just
seeks to maximise
a
+ [1
] b
where
is the proportion of a-type workers.
1. What is the government’s budget constraint?
2. Write the government’s objective function in terms of q a , q b ,
and K.
j
3. If the value of
for each worker is private and unknown to the government, write down the Lagrangean for the government’s optimisation
problem.
4. Show that the second-best solution in this case is identical to the fullinformation solution.
5. Set up the problem in an alternative, equivalent way where the government’s budget constraint is modelled as a separate constraint in the Lagrangean. What must be the value of the Lagrange multiplier on this constraint at the optimum?
Outline Answer
1. If K is the amount to be raised per person in the economy the government’s
budget constraint is
K
T a + [1
]Tb
(12.92)
At the optimum we can ignore the “<” part of (12.92) – if the sum of
taxes exceeded K you could increase individual utility and therefore social
welfare by cutting the taxes can always. So we may write the constraint
as
K = T a + [1
] T b:
(12.93)
a
2. By de…nition the objective function is
+ [1
the individual expressions for utility we have
[q a
Ta +
(z a )] + [1
] qb
]
Tb +
b
. Substituting in
zb
which may be rewritten as
[q a +
(z a )] + [1
] qb +
zb
T a + [1
]Tb
(12.94)
Substituting in (12.94) from the budget constraint we have
[q a +
c Frank Cowell 2006
(z a )] + [1
262
] qb +
zb
K:
(12.95)
Microeconomics
3. The government’s problem is to maximise (12.95) by choice of T a and
T b . Each individual chooses the amount of e¤ort z. The government can
in‡uence the choice of e¤ort by choice of the pairs (q a ; y a ) and q b ; y b
but, given that the values of a and b are private information, the values
of these pairs must be chosen so as to satisfy the participation constraint
for the b-types
qb
yb +
b
(12.96)
b
and the incentive-compatibility constraint for the a-types
qa
ya +
qb
yb
a
(12.97)
a
Then the government’s problem can be written
max
fq a ;q b ;y a ;y b g
h
qa +
h
+ yb +
h
+ ya +
qa
a
qb
i
b
b
q
h
] qb +
+ [1
a
i
q
yb
a
b
a
qb
b
i
i
K
(12.98)
where and are the Lagrange multipliers for the constraints (12.96) and
(12.97) which will be binding at the optimum.
4. The FOC for a maximum of (12.98) are
[1
] 1+
b
z
z
a
qb
1
qa
1
1+
a
qa
1
a
z
qb
+
b
+
z
b
a
qb
b
a
z
a
= (12.99)
0
= (12.100)
0
= (12.101)
0
= (12.102)
0
Using (12.101) and (12.102) to substitute into (12.101) and (12.102) we
…nd
1+
[1
] 1+
qa
1
z
a
a
qb
1
b
z
b
=
0
=
0
In other words we get the full-information solution
qa
z
a
qb
z
c Frank Cowell 2006
b
263
=
a
=
b
:
Microeconomics
CHAPTER 12. DESIGN
5. If there are just two types and again then the problem can be written
h
i
h
i
qa
qb
max
ya +
+ [1
] yb +
a
b
fq a ;q b ;y a ;y b g
+ h [q a
+ yb +
h
+ ya +
y a ] + [1 i ] q b
b
q
yb
K
(12.103)
b
b
qa
yb
a
qb
a
i
where
is the Lagrange multiplier for the binding budget constraint
(12.93) and and have the same interpretation as before. The FOC
for maximising (12.103) are
qa
1
a
[1
]
qb
1
z
b
b
+ [1
z
]+
+
a
qb
1
b
+
z
qa
1
a
z
qb
1
b
a
a
z
a
+
1
[1
]+
=
0
=
0
= 0
= 0
From these FOC we …nd the following relationships among the Lagrange
multipliers.
=
=
[
1]
1
Substituting these back into the other FOC and rearranging we have:
qa
z
a
qb
z
b
=
a
=
b
1
1
b
b
+
a
qb
z
a
But what is the value of ? If
= 1 then we are back at the fullinformation solution. However = 1 makes sense because the objective
function is linear in income and the marginal cost (in social-welfare terms)
of the revenue requirement in the budget constraint is therefore exactly 1.
c Frank Cowell 2006
264
Chapter 13
Government and the
Individual
Exercise 13.1 A local public good is one that is speci…c to a particular municipality: within the municipality good 1 is provided as a public good, but the
consumer has to be resident there to bene…t from good 1. Residence in a municipality also determines liability to provide for the public good. Residents can
choose which municipality to live in. Suppose that exactly one unit of private
good 2 is required to produce one unit of the local public good 1 and that, in
a given municipality of N people, total output is determined by a production
function
q = (N ) :
where is a strictly concave function. Each individual in a municipality gets
utility U (x1 ; x2 ) where x1 is the amount of the local public good, x2 is the
individual’s consumption of the private good and U is a utility function with the
usual properties.
1. What are the transformation curve and the production-possibility set for a
single municipality of size N ?
2. If the residents of the municipality could choose both the proportion of total
output devoted to the local public good and the size of the municipality,
explain how these would be determined. Show that if individuals are paid
their marginal product then the amount required to …nance the public good
is exactly the di¤ erence between the wage bill and the total output.
3. If the size of a municipality N can take any value between 1 and N inclusive, where N is the total population in the economy, use your answer
to part 1 to explain what the production possibility set is when N is allowed to vary. Hence show that the optimal size of municipality could be
multivalued.
4. Assuming that there is an interior solution to the optimal value of N sketch
the relationship between utility and size of municipality.
5. In an economy there are two municipalities with identical production conditions. Individuals can migrate costlessly from one community to another
265
Microeconomics
CHAPTER 13. GOVERNMENT AND THE INDIVIDUAL
if they would achieve higher utility by doing so. Using the diagram from
part 4 show that this migration mechanism may lead to multiple equilibria,
some of which will be unstable.
6. Show that the stable equilibria in part 5 may be ine¢ cient.
Outline Answer.
1. For a given size N the maximum amount of (locally public) good 1 that
could be available if all resources were used to produce it is (N ); the corresponding maximum amount per person of the private good is (N ) =N .
So production possibilities are given by the shaded area in Figure 13.1.
x2
φ(N)/N
φ(N)
x1
Figure 13.1: Production possibilities in municipality of given size
2. In this case the problem is e¤ectively just a version of the club model.
The simple materials balance condition in this economy is given by
(N )
| {z }
total output
=
x1
|{z}
+
am ount of lo cal public go o d
total am ount of private go o d
so that the amount of private good per person is
x2 =
1
(N )
N
x2 N
|{z}
x1
N
Given the standard utility function in the question the individual seeks to
maximise
1
x1
(N )
(13.1)
U x1 ;
N
N
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The FOCs for a maximum give
1
1
U2 x1 ;
(N )
N
N
(N )
N
(N )
N2
N
x1
N
+ U1 x1 ;
1
(N )
N
x1
1
U2 x1 ;
(N )
2
N
N
x1
N
x1
N
=0
=0
(13.2)
(13.3)
which implies
1=N
U1 (x1 ; x)
U2 (x1 ; x)
(13.4)
and
x1 =
(N )
N
N
(N )
(13.5)
If workers are paid their marginal products then the total wage-bill is
N N (N ). So the amount required to …nance the public good is exactly
the di¤erence between the wage bill and the total product.
3. Consider the extension of the production possibilities in Figure 13.1. to a
case where there are three possible sizes of community: this is depicted as
the intersection of the triangular shapes in the left-hand panel of Figure
13.2. By extension the general case is as in the right-hand panel of Figure
13.2. Given the non-convexity of the attainable set it is clear that for some
types of indi¤erence curve one could have multiple solutions at (0; (1))
N ;0 .
and
x2
x2
(a)
(b)
x1
Figure 13.2: Production possibilities for (a) three values of N (b) many values
of N
4. If there is a well-de…ned interior solution then the optimum must look like
that depicted in Figure 13.3. Clearly this implies that utility at the point
(0; (1)), where N = 1, and at
N ; 0 , where N = N , is less than
at the optimum. So maximised utility for a representative individual as
a function of the size of the municipality (N ) must look like one of the
two cases in Figure 13.4.
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x1
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CHAPTER 13. GOVERNMENT AND THE INDIVIDUAL
x2
x1
Figure 13.3: Optimal size of municipality
υ(N)
υ(N)
–
N
N
Figure 13.4: Utility and size of municipality
c Frank Cowell 2006
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–
N
N
–
υ(N –N )
υ(N)
?
–
½N
Utility in second municipality
Utility in first municipality
Microeconomics
N
Figure 13.5: Equilibrium with migration
5. Given that there are two municipalities we may plot the maximised utility
for each on the same diagram as (N ) and
N N – see Figure 13.5
where ( ) is assumed to have the form illustrated on the left-hand side
of Figure 13.4. Now consider the migration mechanism: individuals will
move to the municipality where there is the higher utility. Now, on the
left-hand side of Figure 13.5 we have
N
N >
(N )
so that N must be falling (migration leads to an increase in the size of
the second municipality N N ) the migration mechanism continues until
N = 0. On the right-hand side of the …gure
(N ) >
N
N
and so the reverse happens –N is rising and carries on rising until N = N .
In this case, therefore, it is clear that there are three equilibrium values
of N , namely 0, 21 N and N where the cases N = 0 and N = N are locally
stable and the case N = 12 N is unstable. On the other hand if ( ) is
assumed to have the form illustrated on the right-hand side of Figure 13.4
then the migration diagram becomes as in Figure 13.6; clearly there is a
single stable equilibrium where both municipalities have size 12 N .
1
6. From Figure 13.5 it is clear that
N , establishing the
2 N > (0) =
point that the locally stable equilibria at N = 0 and N = N are ine¢ cient.
.
.
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CHAPTER 13. GOVERNMENT AND THE INDIVIDUAL
Figure 13.6: Equilibrium with migration 2
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Exercise 13.2 (A continuation of Exercise 4.13) There is a single …rm producing good 1 with costs …xed cost
C0 + cq
(13.6)
where the constant marginal cost c is assumed to be large as in Exercise 4.13;
all consumers are identical with preferences given by
1
U (x1 ; x2 ) = x12 + x2
(13.7)
as in Exercise 4.12. The government allows the …rm to charge what price it likes
but o¤ ers to pay the …rm a subsidy equal to the consumer’s surplus generated by
the price that it charges.
1. Is the regulation mechanism e¢ cient?
2. Does the government as regulator need to know (a) the cost function? (b)
the utility function?
3. Show that this mechanism allows the …rm to exploit consumers completely
.
Outline Answer
1. Given (13.6) the …rm’s pro…ts can be written
= pq
C0
cq + B
(13.8)
where p is the price that the monopoly charges for good 1 and B is the
subsidy. Given (13.7) the consumer seeks to maximise
1
x12 + [y
px1 ]
where y is the consumer’s income and the price of good 2 has been normalised at 1. The FOC for an interior maximum is
1
x
2 1
1
2
p=0
(13.9)
from which we obtain the consumer’s demand for good 1 as
2
x1 =
(13.10)
4p2
(the condition on the cost function in Exercise 4.12 will ensure that the
corner solution, where the person spends everything on good 1, is irrelevant), From (13.9) the inverse demand function is
p (x1 ) =
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1
x
2 1
1
2
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CHAPTER 13. GOVERNMENT AND THE INDIVIDUAL
So the consumer’s surplus is
CS
Z
=
x1
p (t) dt
p (x1 ) x1
Z0 x1
1 12
x
2 1
1
1
t 2 dt
2
0
1
1 12
x12
x
2 1
1 12
x
2 1
=
=
=
2
=
(13.11)
4p
From (13.8) …rm’s pro…ts can be written
[p
c] q
C0 + B:
If the size of the market is standardised at 1 for convenience (there is a
very large number of small consumers with total mass 1) then q = x1
and so, setting B = CV and substituting from (13.10) and (13.11), the
expression for pro…ts becomes
2
[p
c]
2
C0 +
4p2
4p
Simplifying this, the …rm’s problem can be represented as choosing p to
maximise
2
2
=
The FOC is
c
2p
2
2p2
C0 :
4p2
(13.12)
2
+c
2p3
=0
which on simplifying yields p = c. In other words we have price equal
to marginal cost, the necessary condition for an e¢ cient outcome. This
outcome will be chosen by the …rm if pro…ts are non-negative. Under the
above conditions (13.12) shows that this requires
2
2p
2
p
4p2
C0
0;
(13.13)
in other words
CS
C0
(13.14)
But this is exactly the condition under which it is e¢ cient to produce
good 1 rather than not produce the good.
2. The only information required is the ordinary demand function (13.10).
3. Suppose the monopolist were empowered to charge an entrance fee F for
the right to buy commodity 1. Pro…ts would be
= pq
c Frank Cowell 2006
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272
cq + F
(13.15)
Microeconomics
and the consumer’s budget constraint (if he chooses to buy good 1) would
be
px1 + x2 F y
(13.16)
Given that the person’s utility if he does not consume good 1 (he spends
all of his income on good 2) is simply y then the …rm knows that the
consumer’s participation constraint is
1
x12 + x2
y
(13.17)
Combining (13.16) and (13.17) we have the constraint
1
x12
px1
F
0
(13.18)
Clearly, for any p, the pro…t-maximising …rm will increase F to ensure
that (13.18) is satis…ed with equality. The consumer’s demand for good 1
is given by (13.10) and so (13.18) becomes
2
F =
2p
x
p
4p2
(13.19)
in other words
F = CS:
Hence the scheme whereby the …rm is subsidised by the amount of the
consumer’s surplus (and this subsidy is raised from the consumers as a
lump-sum tax) is equivalent to the scheme where the …rm is allowed to
charge an entry fee. The consumer is forced back on to reservation utility
level where (13.16) and (13.18) are binding.
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CHAPTER 13. GOVERNMENT AND THE INDIVIDUAL
Exercise 13.3 The government of the tiny island of Mugg is considering whether
it would be a good idea to install piped gas. Once the gas distribution system
is installed each unit of gas (commodity 1) costs a …xed amount m of other
goods (commodity 2); there is additionally a …xed cost F incurred in setting up
a distribution system on the island. Before the system is installed Mugg enjoys
a total amount R2 of commodity 2. The residents of Mugg are assumed to be
identical in every respect and their tastes are represented by the utility function
1
e
x1
+ x2
where (x1 ; x2 ) represent the quantities consumed of the two goods and
non-negative parameter.
is a
1. What is the maximum number of units of gas that could be a¤ orded on
Mugg?
2. Draw the production-possibility set; draw the indi¤ erence curves in two
cases: where is large and where is small.
3. Use the diagram to show that whether it is a Pareto improvement to install
the gas system depends on the value of .
4. If the installation of gas on Mugg is Pareto-improving, describe the Paretoe¢ cient allocation of goods and suggest a scheme by which the publiclyowned corporation MuggGas could implement this allocation if it knows
the willingness to pay of the residents.
Outline Answer
1. Because setting up the gas supply costs F and each unit costs m, the
supply of x1 units will cost F + mx1 units of commodity 2. Given that
the total amount of commodity 2 available is R2 , the maximum supply of
gas is
R2 F
x1 =
m
2. In Figure 13.7 the feasible set is represented by the solid triangle with the
spike on top: the height of the spike is F and the slope of the boundary
is m. The MRS for the speci…ed family of preferences is given by
dx2
dx1
= e
x1
:
=const
which means that they are “vertically displaced” in (x1 ; x2 )-space, they
get steeper as the parameter increases and they can touch the x2 -axis .
Clearly picks up the “strength” of the willingness-to-pay for gas.
3. If < mex1 then there can be no solution on the x1 -axis. There remain
two di¤erent types of solution, depending on the values of :
The high-valuation tastes (high- indi¤erence curves) have steep intercepts on the vertical axis: clearly the optimum in this case would
be as in Figure 13.8.
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x2
R2
F
m
x1
[R2−F]/m
Figure 13.7: Gas: production possibilities
The low-valuation tastes are given by the ‡at indi¤erence curves and
the optimum is as in Figure 13.9 – no gas should be supplied and
everyone consumes (0; R2 ). The no-gas utility level is obviously just
R2 .
Clearly it is Pareto-optimal to provide gas if
m
m log
m
satis…es
> F:
4. If the installation is undertaken, the maximum utility that can then be
obtained is at the point where the indi¤erence curve touches the production possibility set. Since the slope of the production possibility frontier
is m this means that
x1
=
log
x2
= R2
F
m log
m + R2
F
m log
m
;
m
:
At (x1 ; x2 ) utility is
m
:
Evidently this yields greater utility than the no-installation case. The gassupply corporation can induce the (x1 ; x2 ) outcome by making consumers
pay F as a …xed charge and then a (marginal) price of m per unit of gas.
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CHAPTER 13. GOVERNMENT AND THE INDIVIDUAL
x2
R2
x1
Figure 13.8: High valuation: optimum production is positive
x2
R2
x1
Figure 13.9: Low valuation: optimum production is zero
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Exercise 13.4 Take the model of the Island of Mugg (Exercise 13.3) again.
Suppose that the government of Mugg has a horror of public enterprise and decides to delegate the decision on installation and supply by selling o¤ MuggGas.
1. Will this generate an improvement on the no-gas situation?
2. Will it generate an e¢ cient allocation?
3. How would your answers be a¤ ected if MuggGas were split into a number
of private companies, or if consumers were allowed to resell gas to each
other?
Outline Answer
1. The answer depends on whether the …rm is allowed to make a …xed charge
to cover the …xed cost. If the …rm is restricted to charging a uniform price
to all customers then privatisation here could be a disaster. There are two
possibilities:
x2
R2
x1
Figure 13.10: Gas: simple pro…t maximisation
If the …rm (or …rms) were to set price equal to marginal cost at the
e¢ cient outcome it would make a loss.
If there were a monopoly with a uniform price for all units of gas
and no …xed charge then the result is still bound to be ine¢ cient.
First note that given gas-price p a Mugg consumer will choose x1 to
maximise
1 e x1 + [R2 px1 ]
The …rst-order condition for an interior maximum is
e
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x1
p=0
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CHAPTER 13. GOVERNMENT AND THE INDIVIDUAL
which implies
x1 (p) = log
p
as the consumer’s demand for commodity 1, given the (uniform) price
p. Second, note that maximum pro…ts for the …rm are found by
choosing p to maximise
(p) := px1
F
mx1 :
This becomes
(p)
= px1 (p)
=
[p
F
m] log
mx1 (p)
F
p
(13.20)
First-order conditions for an interior maximum are
@ (p)
= log
@p
p
p
m
p
=0
(13.21)
This implies that pro…ts at the optimum are given by
(p) =
[p
2
m]
p
F
This may imply a Pareto-improvement over the no-gas case: the
consumer will be better o¤ than at B if:
h
i
1 e x1 (p)
px1 (p) > 0:
in other words:
p
p log
) log
p
>0
p
which implies
<
1
p
(13.22)
This is satis…ed if p < . However the consumer cannot be better o¤
than at A. For this to happen we would need :
p
p log
p
>
m
F
m log
m
which implies
h
m 1
p
p i
+ log
> [p
m
m
m] log
p
F
(13.23)
Given that p > m, the left-hand side of (13.23) must be negative
– Cf (13.22); but the expression for pro…ts (13.20) implies that the
right-hand side of (13.23) must be nonnegative –a contradiction.
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2. If the …rm were sold of as a simple monopoly then clearly the solution
would not be e¢ cient. If it were a regulated monopoly (as in Exercise
13.3) or could make a …xed charge then it is possible to achieve an e¢ cient
outcome.
3. Assume that the residents of Mugg are all high-valuation consumers.
Given that they can resell gas to each other (and if it is possible to do so
at low cost) there is no possibility of one of the privatised MuggGas …rms
imposing a …xed charge upon them. If gas is supplied the price would be
driven down to the marginal cost m, but the pro…t-maximising solution
will not be at point A but at B. No gas will be supplied, even though this
is a Pareto-inferior outcome to A (see the broken line in Figure 13.10).
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CHAPTER 13. GOVERNMENT AND THE INDIVIDUAL
Exercise 13.5 In an economy there are two …rms each producing a single output from a single non-produced resource according to
p
q1 = z
q2 = max
p
R
z
q1 ; 0
where qi is the amount produced of good i, z is the amount of the resource used
in the production of good 1, R is the total stock of the resource and
is a
parameter.
1. What phenomenon does this model represent?
2. Draw the production-possibility set.
3. Assuming that all consumers are identical, sketch a set of indi¤ erence
curves for which (a) an e¢ cient allocation may be supported by a pseudo
market in externalities; (b) a pseudo market is not possible.
4. What role does the parameter
play in the answer to the previous parts?
Outline Answer
1. There is a simple production externality: production of good 1 reduces
the amount of good 2 that can be produced for a given amount of input
devoted to the production of good 2.
Figure 13.11: Externality: two cases
2. Note that production of good 2 is positive if z < z0 where
z0 :=
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R
1+
2
(13.24)
Microeconomics
The production-possibility set is as illustrated in the two panels of Figure
p
p
13.11.Note that the line segment L from
z0 ; 0 to
R; 0 belongs to
the set. The set is non-convex.
3. In panel (a) the solution is at the point of tangency and can be decentralised using prices shown by the broken line. However in panel (b) this
decentralisation is not possible. If the prices given by the broken line are
used then production takes place, not at the point of tangency, but at
p
R; 0 .
point
4. From equation (13.24) shows that as gets larger so too does the line L
and the problem raised by the non-convexity becomes more salient.
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CHAPTER 13. GOVERNMENT AND THE INDIVIDUAL
Exercise 13.6 In a large economy all agents have a utility function of the form
(x1 ) + xh2
where x1 is the amount provided of a public good and xh2 is agent h’s consumption
of a private good. All agents are endowed with the same amount of private good
R2h = 1. Each individual can choose whether to contribute to the public good:
z2h =
1
0
“contribute”
“not contribute”
The unit contribution costs an amount ch to agent h; the individual costs are
unobservable but the distribution function F ( ) of costs is known. The production
of the public good is given by
x1 = ( )
where
is the proportion of contributing individuals.
1. Show that an e¢ cient outcome implies that there is cost level c0 such that
z2h =
1
0
ch c0
ch > c0
2. The government introduces a tax-subsidy scheme based on individual actions as follows. Each contributor receives a subsidy s and each noncontributor has to pay t. Given c0 and the distribution of costs F what is
the condition for a balanced budget if agents behave as in part 1?
3. Under the conditions of part 2 what is the utility of someone with ch < c0 ?
Of someone with ch > c0 ?
4. By requiring that someone with ch = c0 be indi¤ erent between contributing and non-contributing show that this tax-subsidy scheme induces an
e¢ cient equilibrium.
5. How much of the public good is provided and what is the tax rate and
subsidy rate?
Outline Answer
1. Pick a given level of provision of the public good x1 . This then determines
a given number representing the proportion of the population that must
contribute. Consider the situation where there is some c0 such that z2h = 0
if and only if ch > c0 . Under this scheme
= F c0
x1 = (F c0 ):
Take a high-cost type individual that does not contribute (ch > c0 ) and
a low-cost type individual who does contribute (c` < c0 ). Now change the
situation described above so that h contributes and ` does not. The utility
loss to h is ch which is strictly greater than the utility gain c` to individual
`. Clearly no such perturbation can lead to a Pareto improvement.
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2. Under the above scheme there is a proportion F c0 who are contributing,
so that the total subsidy required is given by sF c0 . Given that there
is a proportion 1 F c0 who are not contributing, the tax raised is
t 1 F c0 . So the budget constraint is
sF c0 = t 1
F c0
(13.25)
3. Consider someone with exactly cost c0 . This person is indi¤erent between
contributing and not contributing if
|
c0 + s =
}
(x1 ) + 1
{z
utility if contribute
|
(x1 ) + 1
{z
t
}
(13.26)
utility if don’t contribute
where x1 = F (c0 ). Clearly if (13.26) holds then
(x1 ) + 1
c` + s >
(x1 ) + 1
t
(x1 ) + 1
ch + s <
(x1 ) + 1
t
for c` < c0 , and
for ch > c0 . It is also clear that c0 is publicly available information
given that F is known. So this scheme is indeed an equilibrium and is
implementable
4. Condition (13.26) implies
s + t = c0
(13.27)
Combining the budget constraint (13.25) with (13.27) we get
c0
t F c0 = t 1
F c0
and so the required tax rate is
t = c0 F (c0 );
(13.28)
the subsidy rate is
s = c0 1
F (c0 ) ;
(13.29)
and the amount of public good provided is
F c0 . It remains
P to
0
determine c . This follows from the standard …rst-order condition MRS
=MRT. In the present case this gives
F c0
1
F c0
=
1
F c0
(13.30)
where 1 and 1 denote the …rst derivatives of
and
respectively.
Given particular functional forms for F , and the value of c0 can be
determined from (13.30).
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CHAPTER 13. GOVERNMENT AND THE INDIVIDUAL
Exercise 13.7 An economy consists of individuals whose income is only from
labour: each person is endowed with a speci…c level of ability which is re‡ected
in his or her market wage w, and chooses `, the amount of time he or she
works 0
`
1. The minimum value of w in the population is w0 , and
the mean value is w0 , where > 1. The government imposes a tax-transfer
scheme such that a person with pretax cash income y has after-tax cash income
of x = [1 t][y y0 ] + y0 .
1. Interpret the parameters t and y0 .
2. Assume that every individual’s preferences can be represented by the utility
1
function x [1 `]
. What is optimal labour supply as a function of w,
t and y0 ?
3. If the government wants to ensure that everyone works what constraint will
this impose on the values of t and y0 ? Assuming that everyone does work,
and that the tax raised is used solely for the purposes of redistribution,
show that this implies that t and y0 must satisfy the constraint:
y0 =
w0
1
1
t
t
4. If the government seeks to maximise the after-tax cash income of the poorest person (the welfare function with = 1 in Exercise 9.5) subject to the
above constraints show that the optimal tax rate is
t =
1
r
1
1
Interpret this result.
Outline Answer
1. The operation of the tax and transfer system is illustrated in Figure 13.12:
t is the marginal tax rate, y0 is the “break-even” income and ty0 gives a
minimum guaranteed income.
2. The consumer’s problem is exactly the same as that of Exercise 5.7 with
the e¤ective wage rate [1 t] w and nonwage income ty0 . So we can just
adapt the result from there to get
` =
[1
] 1 t t yw0 , if w
0 , otherwise
w1
:
(13.31)
where
w1 := y0
t
1
1
t
(13.32)
is the “reservation wage”. Equation (13.31) gives labour supply as a function of w, t and y0 .
c Frank Cowell 2006
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Microeconomics
disposable income
x
τ y0
1−τ
pre-tax income
y0
y
Figure 13.12: The tax-transfer system and disposable income
3. To ensure that everyone works the government must set the tax parameters so that w0 > w1 in (13.31). From (13.32) this requires y0 < y1 :=
1 t
w0 : Net revenue raised in this system is given by
t 1
Z
t[w` y0 ]dF (w)
(13.33)
and if the tax is purely redistributive, then (13.33) should be zero. If
everyone works then (13.31) and (13.33) imply
Z
t
t w
[1
]
y0 y0 dF (w) = 0
(13.34)
1 t
Simplifying we get
Z
w dF (w) =
Z
1
1
t
y0 dF (w)
t
(13.35)
Using the de…nition of the mean of the unknown distribution F we get
immediately
1 t
y0 = w0
:
(13.36)
1
t
Choosing the tax rate t then automatically …xes the guaranteed income
(ty0 ) that can be a¤orded.
4. The after-tax cash income of the poorest person is given by [1 t]w0 ` + ty0
which, if the person works, becomes [1 t]w0 + ty0 using the expression
for ` in (13.31); in view of the net-revenue constraint (13.36) this in turn
becomes
1 t
[1 t]w0 + 2 t
w0 :
(13.37)
1
t
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CHAPTER 13. GOVERNMENT AND THE INDIVIDUAL
Expression (13.37) then becomes the objective function for a government
with “Rawls”type objectives. To …nd the optimum we di¤erentiate (13.37)
with respect to t so as to get
2
w0 +
1
1
2t
w0 +
t
3
t
1
t
w0 = 0
2
[1
t]
(13.38)
which implies
1=
1
1
2t
+
t
2
t
1
[1
t
2
t]
:
(13.39)
Simplifying this …rst-order condition yields a quadratic in t:
2
[
1] t2
2t [
1] +
1 = 0:
Applying the standard solution algorithm we get:
q
2
2[
[
1]
1] + 2 [
1] [1
t =
2[
1]
(13.40)
]
(13.41)
If we rearrange (13.41) and ignore the irrelevant root we get the result.
Note that the optimal tax rate increases with : the larger is the mean
wage (relative to the lowest wage) the more well-o¤ people there are to
pay for transfers.
c Frank Cowell 2006
286