Microeconomic Theory -1- Uncertainty - UCLA Econ · Microeconomic Theory-1- Uncertainty © John Riley

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



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

*








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 .



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

**



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


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



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


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



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



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



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

*



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 .



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

*



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



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



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.

****



C. Favorable gambles: Improving the odds to make the gamble just acceptable.


( , ; , ) ( , ; , )W W W x W x .


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 .

***



C. Favorable gamble: Improving the odds to make the gamble just acceptable.


ˆ ˆ( , ; , ) ( , ; , )w w w x w x .



212



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

**





ˆ ˆ( , ; , ) ( , ; , )w w w x w x .



212



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

*





ˆ ˆ( , ; , ) ( , ; , )w w w x w x .



212




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

.



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



Relative risk aversion

Betting on a small percentage of wealth


ˆ ˆ( , ; , ) ( (1 ), (1 ); , )w w w w .


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 .

**






ˆ ˆ( , ; , ) ( (1 ), (1 ); , )w w w w .



1 11 2 1 2 2 2


From our earlier argument,


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

*






ˆ ˆ( , ; , ) ( (1 ), (1 ); , )w w w w .



1 11 2 1 2 2 2


From our earlier argument,


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

.



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



E. Small favorable gambles:


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.



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

*



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



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

**



F. Portfolio choice





0ˆ( )(1 ) (1 )W W x r x r

0 0ˆ (1 ) ( )W r x r r

*



F. Portfolio choice





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?



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 .

**



Two state model


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

*



Two state model


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?



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

**



Two state model




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

*



Two state model




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



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 =









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 =









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 =



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

**






ˆ ˆ( ) ( )A A


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.

*






ˆ ˆ( ) ( )A A


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 =



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.

*



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.



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



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



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?



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



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 =



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



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



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



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







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







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



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



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



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



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.



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.



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.



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



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.

**



Expected utility theorem

Reference lottery

( , ; ,1 )x x u u



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


(0, ) . Then consider 3 14

u .

. . . . . . . . . .



Expected utility theorem

Reference lottery

( , ; ,1 )x x u u



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


( ,1) . Then consider 3 34

u .

If 1 12 2


(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



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

**





Independence Axiom



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

*





Independence Axiom



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 .



By the Independence axiom, since 1

1x L ,

1

1 2 2( , ; ,1 ) ( , ; ,1 )x x p p L x p p .

***




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 .

**




1x L ,

1

1 2 2( , ; ,1 ) ( , ; ,1 )x x p p L x p p .


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 .

*




1x L ,

1

1 2 2( , ; ,1 ) ( , ; ,1 )x x p p L x p p .


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 .



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

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Microeconomic Theory -1- Uncertainty - UCLA Econ · Microeconomic Theory-1- Uncertainty © John Riley