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Microeconomic Theory -1- Uncertainty

© John Riley November 9, 2017

Choice under uncertainty

A. Introduction to choice under uncertainty 2

B. Risk aversion 11

C. Favorable gambles 15

D. Measures of risk aversion 20

E. Insurance 26

F. Small favorable gambles (2017 skip) 28

G. Portfolio choice 33

H. Efficient risk sharing 43

55 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 .



Prospect or ‘Lottery”

1 2 1( , ,...., ; ,..., )S SL x x x

(outcomes; probabilities)

Consider two prospects or “lotteries”, AL and

BL

1 2 1( , ,...., ; ,..., )A A

A S SL x x x 1 2 1( , ,...., ; ,..., )B B

B S SL c c c

Independence Axiom (axiom of complex gambles)

Suppose that a consumer is indifferent between these two prospects (we write A BL L ).

Then for any probabilities 1 and

2 summing to 1 and any other lottery CL

1 2 1 2( , ; , ) ( , ; , )A C B CL L L L

Tree representation



This axiom can be used to prove the following Implication

Suppose that a consumer is indifferent between the prospects AL and

BL

and is also indifferent between the two prospects CL and

DL ,

i.e. A BL L and

C DL L

Then for any probabilities 1 and

2 summing to 1,

1 2 1 2( , ; , ) ( , ; , )A C B DL L L L

Tree representation



Expected utility

Consider some great outcome x and bad outcome x and outcomes 1x and

2x satisfying

1x x x and 2x x x

Reference lottery

( ) ( , , ,1 )R v w w v v so v is the probability of the great outcome.

1(0) (1)R x R and 2(0) (1)R x R

Then for some probabilities 1( )v x and

2( )v x

1 1 1 1( ( )) ( , ; ( ),1 ( ))x R v x x x v x v x and 2 2 2 2( ( )) ( , ; ( ),1 ( ))x R v x x x v x v x

Then by the independence axiom

1 2 1 2 1 2 1 2( , ; , ) ( ( ( )), ( ( )); , )x x R v x R v x



We have just argued that by the independence axiom

1 2 1 2 1 2 1 2( , ; , ) ( ( ( )), ( ( )); , )x x R v x R v x

Define 1 1 2 2[ ] ( ) ( )v v x v x .

Class exercise

Show that 1 2 1 2( ( ( )), ( ( )); , ) ( , ; [ ],1 [ ])R v x R v x x x v v

Hence 1 2 1 2( , ; , ) ( , ; [ ],1 [ ])x x x x v v

Thus the expected win probability in the reference lottery is a representation of preferences over

prospects.



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 .

Expected utility

1 2 1 2 1 1 2 2( , , , ) [ ] ( ) ( )U w w v v w v w

Class discussion

MRS if ( )v w is a concave function

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



B. Risk aversion

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. Risk preferences

1 1 2 2( , ) ( ) ( )U x v x v x or ( , ) [ ]U x v

Risk preferring consumer

Consider the two wealth levels 1x and

2 1x x .

1 1 2 2 1 1 2 2( ) ( ) ( )v x x v x v x

If ( )v x is convex, then the slope of ( )v x

is strictly increasing as shown in the top figure.

Consumer prefers risk



1 1 2 2( , ) ( ) ( )U x v x v x

Risk averse consumer

1 1 2 2 1 1 2 2( ) ( ) ( )v x x v x v x .

In the lower figure ( )u x is strictly concave so that

1 1 2 2 1 1 2 2( ) ( ) ( ) [ ]v x x v x v x v .

In practice consumers exhibit aversion to such a risk.

Thus we will (almost) always assume that the

expected utility function ( )v 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.


ˆ ˆ( , ; , ) ( , ; , )w w w x w x .


For small x we can use the quadratic approximation of the utility function

212

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )av w x v w v w x v w x

Class Exercise: Confirm that the value and the first two derivatives of ˆ( )v w x and ˆ( )av 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


Class Exercise: Confirm that the value and the first two derivatives of ˆ( )v w x and ˆ( )av w x are

equal at 0x .

The expected value utility of the risky alternative is

1 12 2

ˆ ˆ( ) ( ) ( ) ( )v w x v w x

1 12 2

ˆ ˆ( ) ( ) ( ) ( )a av w x v w x

**





ˆ ˆ( , ; , ) ( , ; , )w w w x w x .



212


Class Exercise: Confirm that 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( ) ( ) ( ) ( )v w x v w x

1 12 2

ˆ ˆ( ) ( ) ( ) ( )a av w x v w x

2 21 1 1 12 2 2 2

ˆ ˆ ˆ ˆ ˆ ˆ( )[ ( ) ( ) ( ) ] ( )[ ( ) ( )( ) ( )( ) ]v w v w x v w x v w v w x v w x

*





ˆ ˆ( , ; , ) ( , ; , )w w w x w x .



212


Note that the value and the first two derivatives of ˆ( )v w x and ˆ( )av w x are equal at 0x .

The expected utility of the risky alternative is

1 12 2

ˆ ˆ( ) ( ) ( ) ( )v w x v w x

1 12 2

ˆ ˆ( ) ( ) ( ) ( )a av w x v w x

2 21 1 1 12 2 2 2

ˆ ˆ ˆ ˆ ˆ ˆ( )[ ( ) ( ) ( ) ] ( )[ ( ) ( )( ) ( )( ) ]v w v w x v w x v w v w x v w x

212

ˆ ˆ ˆ( ) 2 ( ) ( )v w v w x v w x

The utility of the riskless alternative is ˆ( )v w . Therefore the consumer is indifferent if

212

ˆ ˆ2 ( ) ( ) 0v w x v w x

i.e. ˆ( )

( )ˆ( ) 4

v w x

v w

.



D. Measure of risk aversion

Absolute aversion to risk

The bigger is ( )

( )( )

v wARA w

v w

the bigger is

( )( ) ( )

( ) 4 4

v w x xARA w

v w

.

Thus an individual with a higher ( )ARA w requires the odds of a favorable outcome to be moved

more. Thus ( )ARA w is a measure of an individual’s aversion to risk.

( )ARA 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


*






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



1 11 2 1 2 2 2


From our earlier argument,

ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )

( ) ( ) ( )ˆ ˆ ˆ( ) 4 ( ) 4 ( ) 4

v w x v w w wv w

v w v w v w

.

Relative aversion to risk

The bigger is ( )

( )( )

wv wRRA w

v w

the bigger is

( )( ) ( )

( ) 4 4

wv wRRA w

v w

.

Thus an individual with a higher ( )RRA w requires the odds of a favorable outcome to be moved

more. Thus ( )RRA w is a measure of an individual’s aversion to risk.

( )RRA w degree of relative risk aversion



Remark on estimates of relative risk aversion

( )( )

( )

wv wRRA w

v w

. Typical estimate between 1 and 2

Remark on estimates of absolute risk aversion

( ) 1( ) ( )

( )

v wARA w RRA w

v w w

Thus ARA is very small for anyone with significant life-time wealth



E. Insurance

A consumer with a wealth w has a financial loss of L with probability 1 . We shall call this outcome

the “loss state” and label it state 1. With probability 2 11 the consumer is in the “no loss state”

an label it state 2.

With no exchange the consumer’s state contingent wealth is

1 2ˆ ˆ( , ) ( , )x x w L w .

This consumer wishes to exchange wealth in state 2 for wealth in state 1. Suppose there is a market

in which such an exchange can take place. For each dollar of coverage in the loss state, the consumer

must pay

1

2

p

p dollars in the no loss state.



Suppose that the consumer purchases q units.

1ˆx w L q

1p 1 1 1 1

ˆ( )p x p w L p q

12

2

ˆp

x w qp

2p 2 2 2 1

ˆp x p w p q

Adding these equations,

1 1 2 2 1 2

ˆ ˆ( )p x p x p w L p w

*

Slope =1

2

p

p



Suppose that the consumer purchases q units.

1ˆx w L q

1p 1 1 1 1

ˆ( )p x p w L p q

12

2

ˆp

x w qp

2p 2 2 2 1

ˆp x p w p q

Adding these equations,

1 1 2 2 1 2

ˆ ˆ( )p x p x p w L p w

The consumer’s expected utility is

1 1 2 2( ) ( )U v x v x

We have argued that the consumer’s choices are constrained to satisfy the following budget

constraint

1 1 2 2 1 2ˆ ˆ( )p x p x p w L p w .

This is the line depicted in the figure.

Group Exercise: What must be the price ratio if the consumer purchase full coverage? (i.e. 1 2x x )

Slope =1

2

p

p



F. Small favorable gambles (skip):


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( ) ( )v w y v w x

1 12 2

ˆ ˆ( ) ( )a av w y v w x

2 21 1 1 12 2 2 2

ˆ ˆ ˆ ˆ ˆ ˆ[ ( ) ( ) ( ) ] [ ( ) ( )( ) ( )( ) ]v w v w y v w y v w v w x v w x

2 21 12 4

ˆ ˆ ˆ( ) ( )( ) ( )( )v w v w y x v 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( ) ( )v w y v w x

1 12 2

ˆ ˆ( ) ( )a av w y v w x

2 21 1 1 12 2 2 2

ˆ ˆ ˆ ˆ ˆ ˆ[ ( ) ( ) ( ) ] [ ( ) ( )( ) ( )( ) ]v w v w y v w y v w v w x v w x

2 21 12 4

ˆ ˆ ˆ( ) ( )( ) ( )( )v w v w y x v w y x

The value of the riskless alternative is ˆ( )v w . Therefore the consumer is better off taking the risk if

2 21 12 4

ˆ ˆ( )( ) ( )( )v w y x v w y x

2 2 2 21 12 2

ˆ ˆ( )[ ( )((1 ) )]v w x ARA w x x

2 21 12 2

ˆ ˆ( )[ ( )((1 ) )]xv w xARA 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 riskless 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


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 q on the risky asset (and so W q on the


0ˆ( )(1 ) (1 )W W q r q r

0 0ˆ (1 ) ( )W r q r r

*



G. Portfolio choice


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 q on the risky asset (and so W q on the


0ˆ( )(1 ) (1 )W W q r q r

0 0ˆ (1 ) ( )W r q r r

0ˆ (1 )W r q 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 q r q r

0 0ˆ (1 ) ( )sW r q r r

ˆ (1 ) sW r q where 0s sr r .

**



Two state model


0ˆ( )(1 ) (1 )s sW W q r q r

0 0ˆ (1 ) ( )sW r q r r

0ˆ (1 ) sW r q where 0s sr r .

0q

ˆ (1 )sW W r

ˆq W

ˆ ˆ(1 )s sW W r W

*



Two state model


0ˆ( )(1 ) (1 )s sW W q r q r

0 0ˆ (1 ) ( )sW r q r r

ˆ (1 ) sW r q

where 1 1 0 2 0 20r r r r .

0q

ˆ (1 )sW W r

ˆq 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 q 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 v W v 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 v W v 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



Calculus approach

1 1 2 2 1 0 1 2 0 2( ) ( ) ( ) ( (1 ) ) ( (1 ) )U q v W v W v W r q v W r q

Where

1 1 0r r and 2 2 0r r

1 1 0 1 2 2 0 2( ) ( (1 ) ) ( (1 ) )U q v W r q v W r q

*



Calculus approach

1 1 2 2 1 0 1 2 0 2( ) ( ) ( ) ( (1 ) ) ( (1 ) )U q v W v W v W r q v W r q

Where

1 1 0r r and 2 2 0r r

1 1 0 1 2 2 0 2( ) ( (1 ) ) ( (1 ) )U q v W r q v W r q

Therefore

1 1 0 2 2 0 1 1 2 2 0(0) ( (1 )) ( (1 ) ( ) ( (1 ))U v W r v W r v W r

Thus if 0q , then the marginal gain to investing in the risky asset is strictly positive if and only if

1 1 2 2 0 ,

i.e.

1 1 0 2 2 0 1 1 2 2 0( ) ( ) 0r r r r r r r

i.e. the expected payoff is strictly greater for the risky asset

Class Exercise: Is this still true with more than two states



H. Sharing the risk on a South Pacific Island

Alex lives on the west end of the island and has 600 coconut palm trees. Bev lives on the East 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 Alex’s palm trees. If instead the hurricane makes landfall on the East end of

the island it will wipe out 400 of Bev’s coconut palms. The probability of each event is 0.5.

Let the West end landing be state 1 and let the East end landing be state 2. Then the risk facing Alex

is 1 12 2

(200,600; , ) while the risk facing Bev is 1 12 2

(800,400; , ) .

What should they do?

What would be the WE outcome if they could trade state contingent commodities?



Let ( )Bv be Bev’s utility function so that

her expected utility is

1 1 2 2( ) ( ) ( )B B B B B BU x v x v 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 ( )Bv 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 v xMRS x

UMU v x

x

.

Note that along the 45 line the MRS is the probability ratio 1

2

(equal probabilities so ratio is 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 v x v x p x p

FOC:

1 11 1

2 22 2

( )( )

( )

hh h

hh

h

v xMU pMRS x

MU pv x



We have seen that

1 11 1

2 22 2

( )( )

( )

hh h

hh

h

v xMU pMRS x

MU pv x

.

Thus in the WE for Alex and Bev

1 1 1

22 2

( )( )

( )

AA A

AA

A

v x pMRS x

pv x

and 1 1 1

22 2

( )( )

( )

BB B

BB

B

v x pMRS x

pv 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.

microeconomic theory -1- uncertainty - ucla econ · microeconomic theory-1- uncertainty © john...

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