Microeconomic Theory -1- Uncertainty
© John Riley October 20, 2016 revised
Choice Under Uncertainty
A. Introduction to choice under uncertainty 2
B. Expected utility and risk aversion 8
C. Favorable gambles 15
D. Measures of risk aversion 20
E. Small favorable gambles 25
F. Portfolio choice 28
G. Gains from exchange 37
H. Efficient risk sharing 47
I. Expected Utility Theorem 63
72 slides
Microeconomic Theory -2- Uncertainty
© John Riley October 20, 2016 revised
A. Introduction to choice under uncertainty (two states)
Let X be a closed bounded set of possible outcomes (“states of the world”).
An element of X might be a consumption vector, health status, inches of rainfall etc.
Initially, simply think of each element of X as a consumption bundle. Let x be the most preferred
element of X and let x be the least preferred element.
**
Microeconomic Theory -3- Uncertainty
© John Riley October 20, 2016 revised
A. Introduction to choice under uncertainty (two states)
Let X be a closed bounded set of possible outcomes (“states of the world”).
An element of X might be a consumption vector, health status, inches of rainfall etc.
Initially, simply think of each element of X as a consumption bundle. Let x be the most preferred
element of X and let x be the least preferred element.
Consumption prospects
Suppose that there are only two states of the world. 1 2{ , }X x x Let
1 be the probability that the
state is 1x so that 2 11 is the probability that the state is 2x .
We write this “consumption prospect” as follows:
1 2 1 2( ; ) ( , ; , )x x x
*
Microeconomic Theory -4- Uncertainty
© John Riley October 20, 2016 revised
A. Introduction to choice under uncertainty (two states)
Let X be a closed bounded set of possible outcomes (“states of the world”).
An element of X might be a consumption vector, health status, inches of rainfall etc.
Initially, simply think of each element of X as a consumption bundle. Let x be the most preferred
element of X and let x be the least preferred element.
Consumption prospects
Suppose that there are only two states of the world. 1 2{ , }X x x Let
1 be the probability that the
state is 1x so that 2 11 is the probability that the state is 2x .
We write this “consumption prospect” as follows:
1 2 1 2( ; ) ( , ; , )x x x
If we make the usual assumptions about preferences, but now on prospects, it follows that
preferences over prospects can be represented by a continuous utility function
1 2 1 2( , , , )U x x .
Microeconomic Theory -5- Uncertainty
© John Riley October 20, 2016 revised
An example:
A consumer with wealth w is offered a “fair gamble” . With probability 1 / 2 his wealth will be w x
and with probability ½ his wealth will be w x . If he rejects the gamble his wealth remains w . Note
that this is equivalent to a prospect with 0x
**
Microeconomic Theory -6- Uncertainty
© John Riley October 20, 2016 revised
An example:
A consumer with wealth w is offered a “fair gamble” . With probability 12
his wealth will be w x
and with probability 12
his wealth will be w x . If he rejects the gamble his wealth remains w . Note
that this is equivalent to a prospect with 0x
In prospect notation the two alternatives are
1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( , ; , )w w w w
and
1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( , ; , )w w w x w x .
These are depicted in the figure assuming 0x .
*
set of
acceptable
gambles
Microeconomic Theory -7- Uncertainty
© John Riley October 20, 2016 revised
An example:
A consumer with wealth w is offered a “fair gamble” . With probability 12
his wealth will be w x
and with probability 12
his wealth will be w x . If he rejects the gamble his wealth remains w . Note
that this is equivalent to a prospect with 0x
In prospect notation the two alternatives are
1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( , ; , )w w w w
and
1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( , ; , )w w w x w x .
These are depicted in the figure assuming 0x .
The expected value of the risky prospect is the
probability weighted average
1 1 2 2[ ]w w w
1 12 2ˆ ˆ( ) ( )w x w x
w
Thus the expected value of the risky prospect is the value of the riskless prospect.
set of
acceptable
gambles
Microeconomic Theory -8- Uncertainty
© John Riley October 20, 2016 revised
Convex preferences
The two prospects are depicted opposite.
The level set for 1 11 2 2 2
( , ; , )U w w through the riskless
prospect N is depicted.
Note that the superlevel set
1 1 1 11 2 2 2 2 2
ˆ ˆ( , ; , ) ( , ; , )U w w U w w
is a convex set.
*
set of
acceptable
gambles
Microeconomic Theory -9- Uncertainty
© John Riley October 20, 2016 revised
Convex preferences
The two prospects are depicted opposite.
The level set for 1 11 2 2 2
( , ; , )U w w through the riskless
prospect N is depicted.
Note that the superlevel set
1 1 1 11 2 2 2 2 2
ˆ ˆ( , ; , ) ( , ; , )U w w U w w
is a convex set.
This is the set of acceptable gambles for the consumer.
As depicted the consumer strictly prefers the riskless prospect N to the risky prospect R .
Most individuals, when offered such a gamble (say over $5 ) will not take this gamble.
set of
acceptable
gambles
Microeconomic Theory -10- Uncertainty
© John Riley October 20, 2016 revised
Class Discussion: Which alternative would you choose?
N : 1 2 1 2 1 2
ˆ ˆ( , ; , ) ( , ; , )w w w w R : 1 2 1 2 1 2
ˆ ˆ( , ; , ) ( , ; , )w w w x w x where 1
50
100
What if the gamble were “favorable” rather than “fair”
R : 1 2 1 2 1 2
ˆ ˆ( , ; , ) ( , ; , )w w w x w x where (i) 1
55
100 (ii) 1
60
100 (iii) 1
75
100
*
Microeconomic Theory -11- Uncertainty
© John Riley October 20, 2016 revised
Class Discussion: Which alternative would you choose?
N : 1 2 1 2 1 2
ˆ ˆ( , ; , ) ( , ; , )w w w w R : 1 2 1 2 1 2
ˆ ˆ( , ; , ) ( , ; , )w w w x w x where 1
50
100
What if the gamble were “favorable” rather than “fair”
R : 1 2 1 2 1 2
ˆ ˆ( , ; , ) ( , ; , )w w w x w x where (i) 1
55
100 (ii) 1
60
100 (iii) 1
75
100
What is the smallest integer n such that you would gamble if 1100
n ?
Preference elicitation
In an attempt to elicit your preferences write down your number n (and your first name) on a piece
of paper. The two participants with the lowest number n will be given the riskless opportunity.
Let the three lowest integers be 1 2 3, ,n n n . The win probability will not be 1
100
n or 2
100
n. Both will get
the higher win probability 3
100
n .
Microeconomic Theory -12- Uncertainty
© John Riley October 20, 2016 revised
B. Expected utility representation of preferences
We will argue below that one representation of preferences is the following “expected utility
function”.
1 1 2 2( , ) ( ) ( )U x u x u x
Note that utility is the probability weighted expectation
of u, that is, the expected value of u
( , ) [ ]U x u
*
Microeconomic Theory -13- Uncertainty
© John Riley October 20, 2016 revised
B. Expected utility representation of preferences
We will argue below that one representation of preferences is the following “expected utility
function”.
1 1 2 2( , ) ( ) ( )U x u x u x
Note that utility is the probability weighted expectation
of u, that is, the expected value of u
( , ) [ ]U x u
Risk preferring consumer
Consider the two wealth levels 1x and
2 1x x .
1 1 2 2 1 1 2 2( ) ( ) ( )u x x u x u x
If ( )u x is convex, then the slope of ( )u x
is strictly increasing as shown in the top figure.
Consumer prefers risk
Microeconomic Theory -14- Uncertainty
© John Riley October 20, 2016 revised
1 1 2 2( , ) ( ) ( )U x u x u x
Risk averse consumer
1 1 2 2 1 1 2 2( ) ( ) ( )u x x u x u x .
In the lower figure ( )u x is strictly concave so that
1 1 2 2 1 1 2 2( ) ( ) ( ) [ ]u x x u x u x u .
In practice consumers exhibit aversion to such a risk.
Thus we will (almost) always assume that the
expected utility function ( )u x is a strictly increasing
strictly concave function.
Class Discussion:
If consumers are risk averse why do they go to Las Vegas?
Risk averse consumer
Consumer prefers risk
Microeconomic Theory -15- Uncertainty
© John Riley October 20, 2016 revised
C. Favorable gambles: Improving the odds to make the gamble just acceptable.
New risky alternative: 1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( , ; , )w w w x w x .
Choose so that the consumer is indifferent between gambling and not gambling.
****
Microeconomic Theory -16- Uncertainty
© John Riley October 20, 2016 revised
C. Favorable gambles: Improving the odds to make the gamble just acceptable.
New risky alternative: 1 11 2 1 2 2 2
( , ; , ) ( , ; , )W W W x W x .
Choose so that the consumer is indifferent between gambling and not gambling.
For small x we can use the quadratic approximation of the utility function
212
ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )au w x u w u w x u w x
Note that the value and the first two derivatives of ˆ( )u w x and ˆ( )au w x are equal at 0x .
***
Microeconomic Theory -17- Uncertainty
© John Riley October 20, 2016 revised
C. Favorable gamble: Improving the odds to make the gamble just acceptable.
New risky alternative: 1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( , ; , )w w w x w x .
Choose so that the consumer is indifferent between gambling and not gambling.
For small x we can use the quadratic approximation of the utility function
212
ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )au w x u w u w x u w x
Note that the value and the first two derivatives of ˆ( )u w x and ˆ( )au w x are equal at 0x .
The expected value utility of the risky alternative is
1 12 2
ˆ ˆ( ) ( ) ( ) ( )u w x u w x
1 12 2
ˆ ˆ( ) ( ) ( ) ( )a au w x u w x
**
Microeconomic Theory -18- Uncertainty
© John Riley October 20, 2016 revised
C. Favorable gamble: Improving the odds to make the gamble just acceptable.
New risky alternative: 1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( , ; , )w w w x w x .
Choose so that the consumer is indifferent between gambling and not gambling.
For small x we can use the quadratic approximation of the utility function
212
ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )au w x u w u w x u w x
Note that the value and the first two derivatives of ˆ( )u w x and ˆ( )au w x are equal at 0x .
The expected utility of the risky alternative is
1 12 2
ˆ ˆ( ) ( ) ( ) ( )u w x u w x
1 12 2
ˆ ˆ( ) ( ) ( ) ( )a au w x u w x
2 21 1 1 12 2 2 2
ˆ ˆ ˆ ˆ ˆ ˆ( )[ ( ) ( ) ( ) ] ( )[ ( ) ( )( ) ( )( ) ]u w u w x u w x u w u w x u w x
*
Microeconomic Theory -19- Uncertainty
© John Riley October 20, 2016 revised
C. Favorable gamble: Improving the odds to make the gamble just acceptable.
New risky alternative: 1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( , ; , )w w w x w x .
Choose so that the consumer is indifferent between gambling and not gambling.
For small x we can use the quadratic approximation of the utility function
212
ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )au w x u w u w x u w x
Note that the value and the first two derivatives of ˆ( )u w x and ˆ( )au w x are equal at 0x .
The expected utility of the risky alternative is
1 12 2
ˆ ˆ( ) ( ) ( ) ( )u w x u w x
1 12 2
ˆ ˆ( ) ( ) ( ) ( )a au w x u w x
2 21 1 1 12 2 2 2
ˆ ˆ ˆ ˆ ˆ ˆ( )[ ( ) ( ) ( ) ] ( )[ ( ) ( )( ) ( )( ) ]u w u w x u w x u w u w x u w x
212
ˆ ˆ ˆ( ) 2 ( ) ( )u w u w x u w x
The utility of the riskless alternative is ˆ( )u w . Therefore the consumer is indifferent if
212
ˆ ˆ2 ( ) ( ) 0u w x u w x
i.e. ˆ( )
( ) 0ˆ( ) 4
u w x
u w
.
Microeconomic Theory -20- Uncertainty
© John Riley October 20, 2016 revised
D. Measure of risk aversion
Absolute aversion to risk
The bigger is ( )
( )( )
u wA w
u w
the bigger is
( )( ) ( )
( ) 4 4
u w x xA w
u w
.
Thus an individual with a higher ( )A w requires the odds of a favorable outcome to be moved more.
Thus ( )A w is a measure of an individual’s aversion to risk.
( )A w degree of absolute risk aversion
Microeconomic Theory -21- Uncertainty
© John Riley October 20, 2016 revised
Relative risk aversion
Betting on a small percentage of wealth
New risky alternative: 1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( (1 ), (1 ); , )w w w w .
Choose so that the consumer is indifferent between gambling and not gambling.
Note that we can rewrite the risky alternative as follows:
1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( , ; , )w w w x w x where ˆx w .
**
Microeconomic Theory -22- Uncertainty
© John Riley October 20, 2016 revised
Relative risk aversion
Betting on a small percentage of wealth
New risky alternative: 1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( (1 ), (1 ); , )w w w w .
Choose so that the consumer is indifferent between gambling and not gambling.
Note that we can rewrite the risky alternative as follows:
1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( , ; , )w w w x w x where ˆx w .
From our earlier argument,
The expected utility of the risky alternative is
1 12 2
ˆ ˆ( ) ( ) ( ) ( )u w x u w x
1 12 2
ˆ ˆ( ) ( ) ( ) ( )a au w x u w x
212
ˆ ˆ ˆ( ) 2 ( ) ( )u w u w x u w x
*
Microeconomic Theory -23- Uncertainty
© John Riley October 20, 2016 revised
Relative risk aversion
Betting on a small percentage of wealth
New risky alternative: 1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( (1 ), (1 ); , )w w w w .
Choose so that the consumer is indifferent between gambling and not gambling.
Note that we can rewrite the risky alternative as follows:
1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( , ; , )w w w x w x where ˆx w .
From our earlier argument,
The expected utility of the risky alternative is
1 12 2
ˆ ˆ( ) ( ) ( ) ( )u w x u w x
1 12 2
ˆ ˆ( ) ( ) ( ) ( )a au w x u w x
212
ˆ ˆ ˆ( ) 2 ( ) ( )u w u w x u w x
The utility of the riskless alternative is ˆ( )u w . Therefore the consumer is indifferent if
212
ˆ ˆ2 ( ) ( ) 0u w x u w x
i.e. ˆ ˆ ˆ ˆ( ) ( ) ( )
ˆ( ) ( ) ( )ˆ ˆ ˆ( ) 4 ( ) 4 ( ) 4
u w x u w w u ww
u w u w u w
.
Microeconomic Theory -24- Uncertainty
© John Riley October 20, 2016 revised
Relative aversion to risk
The bigger is ( )
( )( )
u wR w w
u w
the bigger is
( )( ) ( )
( ) 4 4
u ww R w
u w
.
Thus an individual with a higher ( )R w requires the odds of a favorable outcome to be moved more.
Thus ( )R w is a measure of an individual’s aversion to risk.
( )R w degree of relative risk aversion
Remark on estimates of relative risk aversion
Microeconomic Theory -25- Uncertainty
© John Riley October 20, 2016 revised
E. Small favorable gambles:
New risky alternative: 1 11 2 1 2 2 2
ˆ ˆ( , ; , ) ( , ; , )w w w y w x , where (1 )y x for some 0
It is tempting to believe that a highly risk averse consumer would not accept such a favorable gamble
unless is sufficiently large.
We show that this intuition is false.
Microeconomic Theory -26- Uncertainty
© John Riley October 20, 2016 revised
Small favorable gambles:
Consider a small x and small y x where (1 )y x . Since x is small, we can then use the
quadratic approximation of the utility function
The expected value of the risky alternative is
1 12 2
ˆ ˆ( ) ( )u w y u w x
1 12 2
ˆ ˆ( ) ( )a au w y u w x
2 21 1 1 12 2 2 2
ˆ ˆ ˆ ˆ ˆ ˆ[ ( ) ( ) ( ) ] [ ( ) ( )( ) ( )( ) ]u w u w y u w y u w u w x u w x
2 21 12 4
ˆ ˆ ˆ( ) ( )( ) ( )( )u w u w y x u w y x
*
Microeconomic Theory -27- Uncertainty
© John Riley October 20, 2016 revised
Small favorable gambles:
Consider a small x and small y x where (1 )y x . Since x is small, we can then use the
quadratic approximation of the utility function
The expected value of the risky alternative is
1 12 2
ˆ ˆ( ) ( )u w y u w x
1 12 2
ˆ ˆ( ) ( )a au w y u w x
2 21 1 1 12 2 2 2
ˆ ˆ ˆ ˆ ˆ ˆ[ ( ) ( ) ( ) ] [ ( ) ( )( ) ( )( ) ]u w u w y u w y u w u w x u w x
2 21 12 4
ˆ ˆ ˆ( ) ( )( ) ( )( )u w u w y x u w y x
The value of the riskless alternative is ˆ( )u w . Therefore the consumer is better off taking the risk if
2 21 12 4
ˆ ˆ( )( ) ( )( )u w y x u w y x
2 2 2 21 12 2
ˆ ˆ( )[ ( )((1 ) )]u w x A w x x
2 21 12 2
ˆ ˆ( )[ ( )((1 ) )]xu w xA w
0 for any x that is sufficiently small
Microeconomic Theory -28- Uncertainty
© John Riley October 20, 2016 revised
F. Portfolio choice
An investor with wealth W chooses how much to invest in a risky asset and how much in a riskless
asset. Let 1+ 0r be the return on each dollar invested in the risky asset and let 1 r be the return on
the risky asset (a random variable.) If the investor spends x on the risky asset (and so W x on the
riskless asset) her final wealth is
0ˆ( )(1 ) (1 )W W x r x r
**
Microeconomic Theory -29- Uncertainty
© John Riley October 20, 2016 revised
F. Portfolio choice
An investor with wealth W chooses how much to invest in a risky asset and how much in a riskless
asset. Let 1+ 0r be the return on each dollar invested in the risky asset and let 1 r be the return on
the risky asset (a random variable.) If the investor spends x on the risky asset (and so W x on the
riskless asset) her final wealth is
0ˆ( )(1 ) (1 )W W x r x r
0 0ˆ (1 ) ( )W r x r r
*
Microeconomic Theory -30- Uncertainty
© John Riley October 20, 2016 revised
F. Portfolio choice
An investor with wealth W chooses how much to invest in a risky asset and how much in a riskless
asset. Let 1+ 0r be the return on each dollar invested in the risky asset and let 1 r be the return on
the risky asset (a random variable.) If the investor spends x on the risky asset (and so W x on the
riskless asset) her final wealth is
0ˆ( )(1 ) (1 )W W x r x r
0 0ˆ (1 ) ( )W r x r r
0ˆ (1 )W r x where 0r r .
Class exercise:
What is the simplest possible model that we can use to analyze the investor’s decision?
Microeconomic Theory -31- Uncertainty
© John Riley October 20, 2016 revised
Two state model
Wealth in state , 1,2s s
0ˆ( )(1 ) (1 )s sW W x r x r
0 0ˆ (1 ) ( )sW r x r r
ˆ (1 ) sW r x where 0s sr r .
**
Microeconomic Theory -32- Uncertainty
© John Riley October 20, 2016 revised
Two state model
Wealth in state , 1,2s s
0ˆ( )(1 ) (1 )s sW W x r x r
0 0ˆ (1 ) ( )sW r x r r
0ˆ (1 ) sW r x where 0s sr r .
0x
ˆ (1 )sW W r
ˆx W
ˆ ˆ(1 )s sW W r W
*
Microeconomic Theory -33- Uncertainty
© John Riley October 20, 2016 revised
Two state model
Wealth in state , 1,2s s
0ˆ( )(1 ) (1 )s sW W x r x r
0 0ˆ (1 ) ( )sW r x r r
ˆ (1 ) sW r x where 0s sr r .
0x
ˆ (1 )sW W r
ˆx W
ˆ ˆ(1 )s sW W r W
Expected utility of the investor
1 1 2 2( , ) ( ) ( )U w u W u W
When will the investor purchase some of the risky asset?
Microeconomic Theory -34- Uncertainty
© John Riley October 20, 2016 revised
Two state model
ˆ (1 )s sW W r x where 0s sr r .
The steepness of the boundary of the
set of feasible outcomes is 2
1
**
Microeconomic Theory -35- Uncertainty
© John Riley October 20, 2016 revised
Two state model
ˆ (1 )s sW W r x where 0s sr r .
The steepness of the boundary of the
set of feasible outcomes is 2
1
1 1 2 2( , ) ( ) ( )U W u W u W
The steepness of the level set through
the no risk portfolio is
1
2
NMRS
*
Microeconomic Theory -36- Uncertainty
© John Riley October 20, 2016 revised
Two state model
ˆ (1 )s sW W r x where 0s sr r .
The steepness of the boundary of the
set of feasible outcomes is 2
1
1 1 2 2( , ) ( ) ( )U W u W u W
The steepness of the level set through
the no risk portfolio is
1
2
NMRS
Purchase some of the risky asset as long as 2
1
1
2
i.e.
1 1 2 2 0 .
The risky asset has a higher expected return
Microeconomic Theory -37- Uncertainty
© John Riley October 20, 2016 revised
G. Gains from exchange
Preliminary observation
Consider the standard utility maximization problem
with two commodities.
If the solution 0x then the marginal utility per
dollar must be the same for each commodity
1 1 2 2
1 1( ) ( )
U Ux x
p x p x
.
**
slope =
Microeconomic Theory -38- Uncertainty
© John Riley October 20, 2016 revised
G. Gains from exchange
Preliminary observation
Consider the standard utility maximization problem
with two commodities.
If the solution 0x then the marginal utility per
dollar must be the same for each commodity
1 1 2 2
1 1( ) ( )
U Ux x
p x p x
.
Equivalently the marginal rate of substitution satisfies
1 11 2
2
2
( , )
U
x pMRS x x
U p
x
*
slope =
Microeconomic Theory -39- Uncertainty
© John Riley October 20, 2016 revised
G. Gains from exchange
Preliminary observation
Consider the standard utility maximization problem
with two commodities.
If the solution 0x then the marginal utility per
dollar must be the same for each commodity
1 1 2 2
1 1( ) ( )
U Ux x
p x p x
.
Equivalently the marginal rate of substitution satisfies
1 11 2
2
2
( , )
U
x pMRS x x
U p
x
In the figure the slope of the budget line is 1
2
p
p .
At the maximum this slope is the same as the slope of the indifference curve.
Therefore 1 2( , )MRS x x is the slope of the indifference curve.
slope =
Microeconomic Theory -40- Uncertainty
© John Riley October 20, 2016 revised
Pareto Efficient allocation in a 2 person 2 commodity economy
An allocation ˆ Ax and ˆBx is not a PE allocation if there
Is an exchange of commodities 1 2( , )e e e such that
ˆ ˆ( ) ( )A A
A AU x e U x and ˆ ˆ( ) ( )B B
B BU x e U x
**
Microeconomic Theory -41- Uncertainty
© John Riley October 20, 2016 revised
Pareto Efficient allocation in a 2 person 2 commodity economy
An allocation ˆ Ax and ˆBx is not a PE allocation if there
Is an exchange of commodities 1 2( , )e e e such that
ˆ ˆ( ) ( )A A
A AU x e U x and ˆ ˆ( ) ( )B B
A AU x e U x
Proposition: If ˆ 0Ax and ˆ 0Bx then a necessary
condition for an allocation to be a PE allocation is that
marginal rates of substitution are equal.
*
Microeconomic Theory -42- Uncertainty
© John Riley October 20, 2016 revised
Pareto Efficient allocation in a 2 person 2 commodity economy
An allocation ˆ Ax and ˆBx is not a PE allocation if there
Is an exchange of commodities 1 2( , )e e e such that
ˆ ˆ( ) ( )A A
A AU x e U x and ˆ ˆ( ) ( )B B
A AU x e U x
Proposition: If ˆ 0Ax and ˆ 0Bx then a necessary
condition for an allocation to be a PE allocation is that
marginal rates of substitution are equal.
Suppose instead that, as depicted,
ˆ ˆ( ) ( )A B
A BMRS x MRS x
Consider a proposal by Alex of 1 2( , )e e e where
1 20e e
and the exchange rate lies between the
two marginal rates of substitution
slope =
slope =
Microeconomic Theory -43- Uncertainty
© John Riley October 20, 2016 revised
Such an exchange is depicted.
On the margin, Alex is willing to give up more
of commodity 2 In exchange for commodity 1.
Therefore Alex offers Bev some of commodity 2
In exchange for commodity 1.
*
Microeconomic Theory -44- Uncertainty
© John Riley October 20, 2016 revised
Such an exchange is depicted.
On the margin, Alex is willing to give up more
of commodity 2 In exchange for commodity 1.
Therefore Alex offers Bev some of commodity 2
In exchange for commodity 1.
If the proposed trade is too large it may not be better
for both consumers due to the curvature of the level sets.
But for all sufficiently small , the proposed
trade e must raise the utility of both consumers.
So the initial allocation ˆ ˆ,A Bx x is not a Pareto efficient
allocation.
Microeconomic Theory -45- Uncertainty
© John Riley October 20, 2016 revised
What if there are more than two commodities?
For all possible allocations we can, in principle
compute the utilities and hence the set of feasible
utilities.
For any point in the interior of this set there is
another allocation such that Bev is no worse off
and Alex is strictly better off.
*
Pareto preferred
allocation
Microeconomic Theory -46- Uncertainty
© John Riley October 20, 2016 revised
What if there are more than two commodities?
For all possible allocations we can, in principle
compute the utilities and hence the set of feasible
utilities.
For any point in the interior of this set there is
another allocation such that Bev is no worse off
and Alex is strictly better off.
Consider the following maximization problem.
ˆ ˆ ˆ{ ( ) | ( ) ( )}B B
A B Be
Max U x e U x e U x
Class Exercise
What exchange *e solves this problem if the allocation
ˆ ˆ,A Bx x is Pareto efficient?
Pareto preferred
allocation
Microeconomic Theory -47- Uncertainty
© John Riley October 20, 2016 revised
H. Sharing the risk on a South Pacific Island
Alex lives on the East end of the island and has 600 coconut palm trees. Bev lives on the West end
and has 800 coconut palm trees. If the hurricane approaching the island makes landfall on the west
end it will wipe out 400 of Bev’s palm trees. We will call this outcome “state 1”. Then in state 1 Bev
will have 400 coconut palm trees. If instead the hurricane makes landfall on the East end of the island
(state 2) it will wipe out 400 of Alex’s coconut palms. So in state 1 Alex will have 600 and in state 2
Alex will have 200 coconut palm trees. The probability of each event is 0.5.
Then the risk facing Alex is 1 12 2
(600,200; , ) while the risk facing Bev is 1 12 2
(400,800; , ) .
What should they do?
What would be the WE outcome if they could trade state contingent commodities?
Microeconomic Theory -48- Uncertainty
© John Riley October 20, 2016 revised
Let ( )Bu be Bev’s utility function so that
her expected utility is
1 1 2 2( ) ( ) ( )B B B B B BU x u x u x .
where s is the probability of state s .
In state 1 Bev’s “endowment “ is 1 800B
In state 2 the endowment is 2 400B .
*
line
400
800
Microeconomic Theory -49- Uncertainty
© John Riley October 20, 2016 revised
Let ( )Bu be Bev’s utility function so that
her expected utility is
1 1 2 2( ) ( ) ( )B B B B B BU x u x u x .
where s is the probability of state s .
In state 1 Bev’s “endowment “ is 1 800B
In state 2 the endowment is 2 400B .
The level set for ( )B BU x through the
endowment point B is depicted.
At a point ˆBx in the level set the steepness
of the level set is
1 1 1 1
2 2 2
2
ˆ( )ˆ( )
ˆ( )
B
B BB B B
B B
BB
U
MU x u xMRS x
UMU u x
x
.
Note that along the 45 line the MRS is the probability ratio 1
2
(in this case 1).
line
400
800
slope =
Microeconomic Theory -50- Uncertainty
© John Riley October 20, 2016 revised
The level set for Alex is also depicted.
At each 45 line the steepness of the
Respective sets are both 1.
Therefore
( ) 1 ( )B B A AMRS MRS
Therefore there are gains to be made from
trading state claims.
The consumers will reject any proposed exchange
that does not lie in their shaded superlevel sets.
line
400
800
line
600
200
Microeconomic Theory -51- Uncertainty
© John Riley October 20, 2016 revised
Equivalently, Bev will reject any proposed
exchange that is in the shaded sublevel set.
Since the total supply of coconut palms is
1000 in each state, the set of potentially
acceptable trades must be the unshaded
region in the red “Edgeworth Box”
*
400
800
Microeconomic Theory -52- Uncertainty
© John Riley October 20, 2016 revised
Equivalently, Bev will reject any proposed
exchange that is in the shaded sublevel set.
Since the total supply of coconut palms is
1000 in each state, the set of potentially
acceptable trades must be the unshaded
region in the red “Edgeworth Box”
Note also that
A Bx x
Thus if Bev’s allocation is ˆBx then
Alex has the allocation ˆ ˆA Bx x .
We can then rotate the box 180 to analyze the choices of Alex.
400
800
Microeconomic Theory -53- Uncertainty
© John Riley October 20, 2016 revised
The rotated Edgeworth Box
Note that A B and ˆ ˆA Bx x
Also added to the figure is the green level set
for Alex’s utility function through A .
**
400
800
200
600
Microeconomic Theory -54- Uncertainty
© John Riley October 20, 2016 revised
The rotated Edgeworth Box
Note that A B and ˆ ˆA Bx x
Also added to the figure is the green level set
for Alex’s utility function through A .
Any exchange must be preferred by both consumers
over the no trade allocation (the endowments).
Such an exchange must lie in the lens shaped
region to the right of Alex’s level set and to the left
of Bev’s level set.
*
400
800
200
600
Microeconomic Theory -55- Uncertainty
© John Riley October 20, 2016 revised
The rotated Edgeworth Box
Note that A B and ˆ ˆA Bx x
Also added to the figure is the green level set
for Alex’s utility function through A .
Any exchange must be preferred by both consumers
over the no trade allocation (the endowment).
Such an exchange must lie in the lens shaped
region to the right of Alex’s level set and to the left
of Bev’s level set.
Pareto preferred allocations
If the proposed allocation is weakly preferred by both consumers and strictly preferred by at least
one of the two consumers the new allocation is said to be Pareto preferred.
In the figure both ˆ Ax and ˆ Ax (in the lens shaped region) are Pareto preferred to A since Alex is
strictly better off and Bev is no worse off.
400
800
200
600
Microeconomic Theory -56- Uncertainty
© John Riley October 20, 2016 revised
Consider any allocation such as ˆ Ax
Where the marginal rates of substitution
differ. From the figure there are exchanges that
the two consumers can make and both
have a higher utility.
400
800
200
600
Microeconomic Theory -57- Uncertainty
© John Riley October 20, 2016 revised
Consider any allocation such as ˆ Ax
Where the marginal rates of substitution
differ. From the figure there are exchanges that
the two consumers can make and both
have a higher utility.
Pareto Efficient Allocations
It follows that for an allocation
Ax and B Ax x
to be Pareto efficient (i.e. no Pareto improving allocations)
( ) ( )A A B BMRS x MRS x
Along the 45 line 1
2
( ) ( )A A B BMRS x MRS x
.
Thus the Pareto Efficient allocations are all the allocations along 45 degree line.
Pareto Efficient exchange eliminates all individual risk.
400
800
200
600
Microeconomic Theory -58- Uncertainty
© John Riley October 20, 2016 revised
Walrasian Equilibrium?
Suppose that there are markets for state claims. Let sp be the price that a consumer must pay for
delivery of a unit in state s , i.e. the price of “claim” in state s.
A consumer’s endowment 1 2( , ) , thus has a market value of
1 1 2 2p p p . The consumer
can then choose any outcome 1 2( , )x x satisfying
p x p
Given a utility function ( )h su x , the consumer chooses hx to solve
{ ( , ) | }h h
hx
Max U x p x p
i.e.
1 1 2 2{ ( ) ( ) | }h h h h
h hhx
Max u x u x p x p
FOC:
1 11 1
2 22 2
( )( )
( )
hh h
hh
h
u xMU pMRS x
MU pu x
Microeconomic Theory -59- Uncertainty
© John Riley October 20, 2016 revised
We have seen that
1 11 1
2 22 2
( )( )
( )
hh h
hh
h
u xMU pMRS x
MU pu x
.
Thus in the WE for Alex and Bev
1 1 1
22 2
( )( )
( )
AA A
AA
A
u x pMRS x
pu x
and 1 1 1
22 2
( )( )
( )
BB B
BB
B
u x pMRS x
pu x
.
Class Question: What does the First Welfare Theorem tell us about the WE allocation?
Given this, what must be the WE price ratio.
Microeconomic Theory -60- Uncertainty
© John Riley October 20, 2016 revised
Exercises (for the TA session)
1. Consumer choice
(a) If ( ) lns su x x what is the consumer’s degree of relative risk aversion?
(b) If there are two states, the consumer’s endowment is and the state claims price vector is p ,
solve for the expected utility maximizing consumption.
(c) Confirm that if 1 1
2 2
p
p
then the consumer will purchase more state 2 claims than state 1 claims.
2 . Consumer choice
(a), (b), (c) as in Exercise 1 except that 1/2( )s su x x .
(d) Try to compare the state claims consumption ratio in Exercise 1 with that in Exercise 2.
(e) Provide the intuition for your conclusion.
Microeconomic Theory -61- Uncertainty
© John Riley October 20, 2016 revised
3. Equilibrium with social risk.
Suppose that both consumers have the same expected utility function
1 1 2 2( , ) ln lnh h
hU x x x .
The aggregate endowment is 1 2( , ) where
1 2 .
(a) Solve for the WE price ratio 1
2
p
p .
(b) Explain why 1 1
2 2
p
p
.
4. Equilibrium with social risk.
Suppose that both consumers have the same expected utility function
1/2 1/2
1 1 2 2( , ) ( ) ( )h h
hU x x x .
The aggregate endowment is 1 2( , ) where
1 2 .
(a) Solve for the WE price ratio 1
2
p
p .
(b) Compare the equilibrium price ratio and allocations in this and the previous exercise and provide
some intuition.
Microeconomic Theory -62- Uncertainty
© John Riley October 20, 2016 revised
5. Portfolio choice
Suppose that the utility function of the investor is ( )u a W . Her expected utility is
2
1 1 2 2 0
1
ˆ( ) ( ) ( (1 ) )s s
s
U u W u W u a W r x
where 0s sr r .
Define 0ˆ (1 )b a W r . Then
2
1
( ) ln( )s s
s
U x b x
(a) Explain why ( )U x is a strictly concave function
(b) Solve for the expected utility maximizing choice of x
Microeconomic Theory -63- Uncertainty
© John Riley October 20, 2016 revised
I. Expected utility theorem
Reference lottery
( , ; ,1 )x x u u
We argue that for any certain outcome x a consumer must be indifferent between x and a
“reference lottery” in which the two outcomes are x and x , i.e. the most and least preferred.
**
Microeconomic Theory -64- Uncertainty
© John Riley October 20, 2016 revised
Expected utility theorem
Reference lottery
( , ; ,1 )x x u u
We argue that for any certain outcome x a consumer must be indifferent between x and a
“reference lottery” in which the two outcomes are x and x , i.e. the most and least preferred.
Step 1:
If ( , ;1,0)x x x then ( ) 1u x and we are done. Similarly, if ( , ;0,1)x x x then ( ) 0u x and we are
done.
Step 2:
If not consider 2 12
u . If 1 12 2
( , ; , )x x x then 12
( )u x and we are done.
If 1 12 2
( , ; , )x x x then if there is a u it must be in 12
( ,1) . Then consider 3 34
u .
If 1 12 2
( , ; , )x x x then if there is a u it must be in 12
(0, ) . Then consider 3 14
u .
. . . . . . . . . .
Microeconomic Theory -65- Uncertainty
© John Riley October 20, 2016 revised
Expected utility theorem
Reference lottery
( , ; ,1 )x x u u
We argue that for any certain outcome x a consumer must be indifferent between x and a
“reference lottery” in which the two outcomes are x and x , i.e. the most and least preferred.
Step 1:
If ( , ;1,0)x x x then ( ) 1u x and we are done. Similarly, if ( , ;0,1)x x x then ( ) 0u x and we are
done.
Step 2:
If not consider 2 12
u . If 1 12 2
( , ; , )x x x then 12
( )u x and we are done.
If 1 12 2
( , ; , )x x x then if there is a u it must be in 12
( ,1) . Then consider 3 34
u .
If 1 12 2
( , ; , )x x x then if there is a u it must be in 12
(0, ) . Then consider 3 14
u .
. . . . . . . . . .
Continue this process. Either there is some t -th step such that ( , ; ,1 )t tx x x u u or there is an
infinite sequence sequence { }tu converging to some 0u .
In the latter case, given the continuity axiom, 0 0( , ; ,1 )x x x u u
Microeconomic Theory -66- Uncertainty
© John Riley October 20, 2016 revised
We have established the existence of a utility function ( )u x over the certain outcomes.
Extending this to prospects requires a further assumption:
Independence Axiom
Suppose that 1 ( , )L x , 2 ˆ ˆ( , )L x and 1 2L L i.e. a consumer prefers 1 ( , )L x over 2 ˆ ˆ( , )L x .
Let 3L be any other lottery. Then
1 3 2 3( , ; ,1 ) ( , ; ,1 )L L p p L L p p
**
Microeconomic Theory -67- Uncertainty
© John Riley October 20, 2016 revised
We have established the existence of a utility function ( )u x over the certain outcomes.
Extending this to prospects requires a further assumption:
Independence Axiom
Suppose that 1 ( , )L x , 2 ˆ ˆ( , )L x and 1 2L L i.e. a consumer prefers 1 ( , )L x over 2 ˆ ˆ( , )L x .
Let 3L be any other lottery. Then
1 3 2 3( , ; ,1 ) ( , ; ,1 )L L p p L L p p
Note that if 1 2L L 2 ˆ ˆ( , )L x it follows from two applications of the Axiom that
1 3 2 3( , ; ,1 ) ( , ; ,1 )L L p p L L p p
*
Microeconomic Theory -68- Uncertainty
© John Riley October 20, 2016 revised
We have established the existence of a utility function ( )u x over the certain outcomes.
Extending this to prospects requires a further assumption:
Independence Axiom
Suppose that 1 ( , )L x , 2 ˆ ˆ( , )L x and 1 2L L i.e. a consumer prefers 1 ( , )L x over 2 ˆ ˆ( , )L x .
Let 3L be any other lottery. Then
1 3 2 3( , ; ,1 ) ( , ; ,1 )L L p p L L p p
Note that if 1 2L L 2 ˆ ˆ( , )L x it follows from two applications of the Axiom that
1 3 2 3( , ; ,1 ) ( , ; ,1 )L L p p L L p p
Let 1L be the reference lottery for 1x and let 2L be the reference lottery for
2x
i.e. 1
1 1 1( , , ( ),1 ( ))x L x x u x u x and 2
2 2 2( , , ( ),1 ( ))x L x x u x u x .
Microeconomic Theory -69- Uncertainty
© John Riley October 20, 2016 revised
By the Independence axiom, since 1
1x L ,
1
1 2 2( , ; ,1 ) ( , ; ,1 )x x p p L x p p .
***
Microeconomic Theory -70- Uncertainty
© John Riley October 20, 2016 revised
By the Independence axiom, since 1
1x L ,
1
1 2 2( , ; ,1 ) ( , ; ,1 )x x p p L x p p .
Again by the Independence Axiom, since 2
2x L
1 1 2
2( , ; ,1 ) ( , ; ,1 )L x p p L L p p .
**
Microeconomic Theory -71- Uncertainty
© John Riley October 20, 2016 revised
By the Independence axiom, since 1
1x L ,
1
1 2 2( , ; ,1 ) ( , ; ,1 )x x p p L x p p .
Again by the Independence Axiom, since 2
2x L
1 1 2
2( , ; ,1 ) ( , ; ,1 )L x p p L L p p .
Therefore
1 2
1 2( , ; ,1 ) ( , ; ,1 )x x p p L L p p .
*
Microeconomic Theory -72- Uncertainty
© John Riley October 20, 2016 revised
By the Independence axiom, since 1
1x L ,
1
1 2 2( , ; ,1 ) ( , ; ,1 )x x p p L x p p .
Again by the Independence Axiom, since 2
2x L
1 1 2
2( , ; ,1 ) ( , ; ,1 )L x p p L L p p .
Therefore
1 2
1 2( , ; ,1 ) ( , ; ,1 )x x p p L L p p .
Consider the lottery on the right hand side. The two possible outcomes are x and x .
With probability p the consumer plays lottery 1 and receives the favorable outcome with probability
1( )u x .
With probability 1 p the consumer plays lottery 2 where the probability of the favorable outcome is
2( )u x .
Thus the joint probability of the favorable outcome is
1 2( , ,1 ) ( ) (1 ) ( )U x p p pu x p u x .
Microeconomic Theory -73- Uncertainty
© John Riley October 20, 2016 revised
With a little work this argument can be extended to the following lottery over S outcomes.
1 1( , ) ( ,..., ; ,..., )S Sx x x .
The joint probability of winning the reference lottery is
1 1( , ) ( ) ... ( )S SU x u x u x