Absolute Risk Aversion, Stochastic Dominance,
and Introduction to General Equilibrium
Econ 2100
Lecture 15, October 26
Outline
1
Absolute Risk Aversion
2
Stochastic Dominance
3
General Equilibrium Notation
Fall 2015
From Last Class: Utility of Money
A cumulative distribution function (cdf) F : R ! [0; 1] satis…es:
x y implies F (x )
limx !1 F (x ) = 1.
x
F (y ); limy #x F (y ) = F (x ) limx !
(
0
yields x with certainty: x (z) =
1
1
F (x ) = 0 and
if z < x
.
if z x
R
denotes the mean (expected value) of F , i.e. F = x dF (x).
von Neumann–Morgenstern preference representation: there exists a utility
function U on distributions, and a continuous index v : R ! R over wealth
Z
Z
such that
F % G () U(F ) = v dF
v dG = U(G )
F
0
If F is di¤erentiable, the expectation
is
Z
Z computed using the density f = F :
v dF = v (x )f (x ) dx :
The preference relation % is risk averse if, for all F , F % F .
Given a strictly increasing and continuous vNM index v over wealth,
the certainty equivalent of F (c(F ; v )) is de…ned by v (c(F ; v )) =
the risk premium of F , denoted r (F ; v ) is de…ned by
r (F ; v ) = F c(F ; v ):
% is risk averse, v is concave, r (F ; v )
0
R
v ( ) dF .
Another Application: Asset Demand
An asset is a divisible claim to a …nancial return in the future.
Asset Demand
An agent has initial wealth w ; she can invest in a safe asset that returns $1
per dollar invested, or in a risky asset that returns $z per dollar invested.
The general version has N assets each yielding a return zn per unit invested.
The risky return has cdf F (z), and assume
R
z dF > 1.
Let and be the amounts invested in the risky and safe asset respectively.
Then, one can think of ( ; ) as a portfolio allocation that pays z + .
R
;
0 and + = w
The agent solves max v ( z + ) dF s. t.
The …rst oder conditions for this optimal portfolio problem is
Z
(z 1)v 0 ( (z 1) + w ) dF = 0
If the decision maker is risk averse, this expression is decreasing (in ) because
of the concavity of v .
One can use this fact to verify that if DM1 is more risk averse than DM2 then
her optimal 1 is smaller than the corresponding 2 : the more risk averse
consumer invests less in the risky asset.
How to Measure Risk Aversion
Since concavity of v re‡ects risk aversion, v 00 is a natural candidate measure
of risk aversion.
Unfortunately, v 00 is not appropriate since it is not robust to strictly increasing
linear transformations.
De…nition
Suppose % is a preference relation represented by the twice di¤erentiable vNM
index v : R ! R. The Arrow–Pratt measure of absolute risk aversion : R ! R is
de…ned by
v 00 (x)
(x) =
:
v 0 (x)
The second derivative is normalized to measure risk aversion properly.
Notice that by integrating (x) twice one could recover the utility function.
How about the constants of integration?
Absolute Risk Aversion
Proposition
Suppose %1 and %2 are preference relations represented by the twice di¤erentiable
vNM indices v1 and v2 . Then
%1 is more risk averse than %2
,
1 (x)
2 (x)
for all x 2 R
This con…rms that the Arrow-Pratt coe¢ cient is the correct measure of
increasing absolute risk aversion.
We can add this to the characterizations of more risk averse than that you
proved in the homework.
%1 is more risk averse than %2
1 (x)
,
2 (x)
for all x 2 R
Proof.
We know v1 = (v2 ) for some strictly increasing
(by the homework).
Di¤erentiating
v10 (x) =
0
(v2 (x))v20 (x)
and v100 (x) =
0
(v2 (x))v200 (x) +
Dividing by v10 > 0 we have
0
00
v100 (x)
(v2 (x))v200 (x)
(v2 (x))v20 (x)
=
+
v10 (x)
v10 (x)
v10 (x)
Subsitute the …rst equation
00
v100 (x)
v200 (x)
(v2 (x))v20 (x)
=
+
0
0
v1 (x)
v2 (x)
v10 (x)
or
00
v100 (x)
v200 (x)
(v2 (x))v20 (x)
=
v10 (x)
v20 (x)
v10 (x)
using the de…nition of Arrow-Pratt:
1 (x)
since
=
2 (x)
is concave and v is increasing.
+ something
00
(v2 (x))v20 (x)
First Order Stochastic Dominance (FOSD)
What kind of relationship must exist between lotteries F and G to ensure that
anyone, regardless of her attitude to risk, will prefer F to G so long as she
likes more wealth than less?
In general, if a consumer’s utility of wealth is increasing, but its functional
form unknown, we do not have enough information to know her rankings
among all distributions...but we know how she ranks some pairs; namely, those
comparable with respect to the following transitive, but incomplete, binary
relation.
De…nition
F …rst-order stochastically dominates G , denoted F
Z
Z
v dF
v dG ;
FOSD
G , if
for every nondecreasing function v : R ! R.
If F
FOSD
G , then anyone who prefers more
money
to less prefers F to G .
R
R
This follows because F % G , U(F ) =
Proposition
F
FOSD
G if and only if F (x)
v dF
G (x) for all x 2 R.
v dG = U(G ).
First Order Stochastic Dominance (FOSD)
F
FOSD
G , if
R
v dF
R
v dG ; for every nondecreasing function v : R ! R.
FOSD is characterized by only looking at distribution functions.
Proposition
F
FOSD
G if and only if F (x)
G (x) for all x 2 R.
This follows from “integration by parts”
Z b
b
v (x) f (x) dx = v (x) F (x)ja
a
= v (b)
= v (b)
1
Z
b
v 0 (x) F (x) dx
a
v (0)
0
Z
b
v 0 (x) F (x) dx
a
Z
b
v 0 (x) F (x) dx
a
Thus
Z
a
b
v dF
Z
a
b
v dG =
Z
b
v 0 (x) [G (x)
a
and you can …ll in the blanks for a formal proof.
F (x)] dx
Second Order Stochastic Dominance (SOSD)
If one also knows that the decision maker is risk averse (her utility index for wealth
is concave), we know how she ranks more pairs.
De…nition
F second-order stochastically dominates G , denoted F
Z
Z
v dF
v dG ;
SOSD
G , if
for every nondecreasing concave function v : R ! R.
If F SOSD G , then anyone who prefers more money to less and is risk averse
prefers F to G .
By construction, the set of distributions ranked by FOSD is a subset of those
ranked by SOSD
F
FOSD
G implies F
Proposition
F
SOSD
G if and only if
Rx
SOSD
1
G.
F (t) dt
Rx
1
G (t) dt
for all x 2 R.
SOSD is also characterized by looking at distribution functions.
What if F SOSD G an DM chooses G ? Is this a ‘reasonable’choice?
Preferences and Lotteries Over Money
So far, we have looked at expected utility preferences over sums of money.
Dollar bills cannot be eaten, so where do these preferences come from?
There are L commodities and % ranks lotteries on X = RL+ .
The expected utility axioms hold: the consumer is an expected utility
maximzer, and u : X ! R is her vNM utility function.
u ( ) is also a utility function over realized consumption (consumer theory).
Fix prices p 2 RL++ ; let w 2 [0; 1) be income, x (p; w ) be the Walrasian
demand corresponding to u ( ), and v (p; w ) the indirect utility function:
v (p; w ) = u (x) for x 2 x (p; w ).
Suppose all uncertainty is resolved so that the consumers learns how much
money she has before she goes to the markets to buy x.
How does the consumer rank lotteries over income?
A lottery
(w ) is a probability distributions on [0; 1).
P
y 2support( ) (w ) v (p; w ); and
X
(w ) v (p; w )
(w ) v (p; w )
The expected utility of is
X
% ,
y 2support( )
y 2support( )
The indirect utility function v ( ) is a vNM utility function over money.
How do properties of % transfer to v ( )? Think about the answer.
General Equilibrium
General Equilibirum Theory
Theory of market interactions between anonymous agents who take prices as given
(everyone is “small” relative to the market).
Nothing is ‘exogenous’, hence general (as opposed to partial) equilibirum.
1
How do competitive markets allocate resources?
1
2
2
Important questions:
1
2
3
4
5
6
7
3
main assumption: agents are ‘price-taking optimizers’;
main outcomes of interest: competitive equilibrium (everyone makes optimal
choices and prices are such that demand and supply are equal), and Pareto
e¢ cient allocations (cannot be redistributed to improve consumers welfare).
when is a competitive equilibirum Pareto e¢ cient? First Welfare Theorem;
when is a Pareto e¢ cient outcome an equilibirum? Second Welfare Theorem;
when does a competitive equilibirum exhist? Arrow-Debreu-McKenzie Theorem;
when is a competitive equilibrium unique?
is a competitive equilibrium robust to small perturbations of the parameters?
how do we get to a competitive equilibrium and what happens away from it?
what happens if we add time or uncertainty?
Applications: …nance, macro, labor, etc.
The Economy
An economy consists of I individuals and J …rms who consume and produce L
perfectly divisible commodities.
Each …rm j = 1; :::; J is described by a production set Yj RL .
Each consumer i = 1; :::; I is described by four objects:
a consumption set Xi
RL ;
preferences %i ;
an individual endowment given by a vector of commodities ! i 2 Xi ; and
a non-negative share ij of the pro…ts of each …rm j, with ij 2 [0; 1] for all j.
De…nition
An economy (with private ownership) is a tuple
n
o
I
J
fXi ; %i ; ! i ; ( i 1 ; :::; iJ )gi =1 ; fYj gj =1
The resources available to the economy are given by the aggregate endowment:
I
X
!=
!i
i =1
If we do not need to tracknwho owns what, we can ode…ne an economy as
I
J
fXi ; %i gi =1 ; fYj gj =1 ; !
where ! 2 RL is the aggregate endowment.
Exchange Economies and Edgeworth Boxes
Exchange Economy
In an exchange economy there is no production; all consumers can do is trade
with each other.
This is described by assuming that there is one …rm with a single technology:
free disposal.
Formally, J = 1, and Yj =
RL+ .
This is a simple environment in which many results are easier to prove, and
that most of the time has the same intuition as an economy with production.
Edgeworth Box
An Edgeworth Box is an exchange economy with two consumers and two
goods.
Formally L = 2, I = 2, X1 = X2 = R2+ , J = 1, and Yj =
Draw an Edgeworth box.
R2+ .
Allocations and Feasibility
De…nition
An allocation speci…es a consumption vector xi 2 Xi for each consumer i = 1; :::; I ,
and a production vector yj 2 Yj for each …rm j = 1; :::; J; an allocation is
(x; y ) = (x1 ; :::; xI ; y1 ; :::; yJ ) 2 RL(I +J )
This lists what everyone consumes and what every …rm produces.
De…nition
An allocation (x; y ) is feasible if
X
xi
!+
i
X
yj
j
Consumption and production are compatible with endowment.
De…nition
The set of feasible allocations is denoted by
f(x; y ) 2 X1
:::
XI
Y1
:::
YJ :
X
i
xi
!+
X
j
yj g
RL(I +J )
Feasible Allocations: Examples
Example
In an exchange economy the set of feasible allocations is
X
fx 2 X1 ::: XI :
xi !g RLI
i
Example
In an Edgeworth Box, any point in the box represents a feasible allocation.
Feasible Allocations: Conditions
Proposition
If the following conditions are satis…ed, the set of feasible allocations is closed and
bounded.a
1
2
3
Xi is closed and bounded below for each i = 1; :::; I .b
Yj is closed for each j = 1; :::; J.
P
Y = j Yj is convex and satis…es
1
2
3
0L 2 Y (inaction),
Y \ RL+ f0L g (no free-lunch), and
if y 2 Y and y 6= 0L then y 2
= Y (irreversibility).
Moreover, if RL+ Yj (free-disposal) and there are xi 2 Xi for each i such that
P
!, then the set of feasible allocations is also non empty.
i xi
a Closed and bounded means there exists an r > 0 such that for each l = 1; :::; L, i = 1; :::; I ,
and j = 1; :::; J , we have xl ;i < r and yl ;j < r for each (x ; y ) 2 A.
b Bounded below means there exists an r > 0 such that x
l ;i > r for each l = 1; :::; L.
When these conditions are not satis…ed, the latter in particular, we may have
nothing to talk about. So we take them for granted.
Problem Set 9 (Incomplete)
Due Monday 2 November
1
Suppose there are two goods and the consumer has a vNM utility index over
consumption bundles of v (x) = f (x1 + x2 ) where f : R ! R is a strictly increasing
function. Let p = (1; 3) and p 0 = (3; 1). Let q = :5p + :5p 0 = (2; 2). In Regime 1,
there is a 1=2 probability that the realized price will be p and a 1=2 probability that
the realized price will be p 0 , so her ex-ante utility is
:5 max f (x1 + x2 ) + :5 max f (x1 + x2 ):
x 2B p;w
x 2B p 0 ;w
In Regime 2, the price is q with certainty, so her ex-ante utility is
max f (x1 + x2 ):
x 2B q;w
2
3
4
Prove that, for any strictly increasing f , the consumer is ex-ante better o¤ in
Regime 1.
Kreps 6.4
Kreps 6.14.
Suppose a consumer’s utility for CDFs is
R
where Var (F ) = (x
1
2
U(F ) =
F
Var (F )
2
F ) dF is the variance of F .
Prove that if > 0 then the preference corresponding to
the one corresponding to 0 .
Prove that U violates expected utility.
is more risk averse than