CHAPTER 7: UNCERTAINTY - Essential Microeconomicsessentialmicroeconomics.com/ChapterY7/SlideChapter7A.pdf · Essential Microeconomics-1- Sections 7.1 and 7.2 © John Riley CHAPTER

Essential Microeconomics -1- Sections 7.1 and 7.2

© John Riley

CHAPTER 7: UNCERTAINTY

Ranking risky prospects 2

Expected utility 7

Allais Paradox 11

Acceptable gambles 19

Risk-neutrality and risk aversion 20

Degrees of risk aversion 23

Absolute and relative risk aversion 24

Jensen’s Inequality 29

Trading in markets for state claims 31

Portfolio Choice 35


© John Riley

RISKY CHOICES

Key ideas: prospects, compound prospects, independence axiom, expected utility, paradoxes

We now consider the choices of a consumer who is uncertain as to the consequences of his or her

action. As we shall see, the basic choice model can be readily extended as long as the consumer is able

to characterize fully the set of possible consequences and assign a probability to each.

Initially we consider decisions when the number of possible outcomes or “states” is finite. The

decision maker then assigns a probability to each state.

Let sx be the consequence in state s.

Let sπ be the probability that the individual assigns to this state.

Then a complete description of the uncertain outcome or "prospect" is the 2 S× vector

1 1( ; ) (( ,..., );( ,..., ))S Sx x xπ π π= .

Under the axioms of consumer choice, there exists a continuous utility function ( , )U xπ over prospects.


© John Riley

Because our focus is initially on the choice over probabilities,

it is convenient to fix the S consequences and simply write the

prospect as the probability vector π . For the case of three states,

preferences over prospects 1 2 3( , , )π π π π= are illustrated.

We assume that 1 2 3x x x≺ ≺ . The figure shows only the

probabilities of the best and worst states.

If 1 1π = the outcome is 1x with probability 1. If 3 1π = , the outcome is 3x with probability 1.

Because probabilities add up to 1, the set of possible probability vectors is represented by the shaded

region. In particular, at the origin 1 3 0π π= = , so this is the certain outcome 2x .

Note that the probability of state 2 is constant along the dotted line so the prospect 1π has a higher

probability of the best outcome and a lower probability of the worst outcome.

1

1


© John Riley

Suppose you are indifferent between two prospects !π and 2π . You are about to choose one of them

when you are offered the opportunity to randomize. If you accept, a spinner will be used that will

select prospect 1 with probability 1p and prospect 2 with probability 2 11p p= − .

We write this new "compound prospect" as follows:

1 2 1 1 2 21 2 1 1ˆ ( ; , ) (( , );( ,,,. ),( ,,,. ))S Sp p pπ π π π π π π= = .

*

1

1


© John Riley

Suppose you are indifferent between two prospects !π and 2π . You are about to choose one of them

when you are offered the opportunity to randomize. If you accept, a spinner will be used that will

select prospect 1 with probability 1p and prospect 2 with probability 2 11p p= − .

We write this new "compound prospect" as follows:

1 2 1 1 2 21 2 1 1ˆ ( ; , ) (( , );( ,,,. ),( ,,,. ))S Sp p pπ π π π π π π= = .

Note that the probability of state s in this compound lottery is

1 21 2

ˆs s sp pπ π π= + . Thus, 1 2

1 1ˆ (1 )p pπ π π= + − is a convex

combination of 1π and 2π . In the figure, this is a point on

the line joining the two prospects 1π and 2π .

Given that you are indifferent between the prospects 1π and 2π .

How will you feel about randomizing between them? When asked this question, respondents

overwhelmingly indicate that they will be indifferent between all three prospects. Formally,

1 2 1 ~π π π⇒∼ 1 21 2( , ; , ),p p π π where 1 2, 0p p > and 1 2 1p p+ = .

It follows that indifference curves must be linear as depicted.

1

1


© John Riley

For the indifference curves to be parallel, as depicted in this

diagram, we need a further assumption.

Consider the prospect 1 1(1 , ; , )q rλ λ π= − , a convex combination

of 1π and some alternative prospect r.

Similarly define 2 2(1 , ; , )q rλ λ π= − . These compound prospects

can also be depicted in tree diagrams as shown below.

Figure 7.1-2: Tree Diagrams of Compound

N

N

1

1


© John Riley

Suppose 1 2π π . Does it follow that 1 2q q ? When asked what they will do, respondents

overwhelmingly conclude that this is the case. Intuitively, because the outcome r is an independent

event, that occurs with probability λ in both compound prospects, the ranking should be the same

regardless of whether λ is zero or positive.

This idea is formalized in the following axiom.

From the geometry of the diagram, it follows that indifference curves are parallel.

Independence Axiom (IA)

If , then for any prospect r and probabilities

where ,

1

1


© John Riley

Expected Utility

Consider the diagram. Note that for every feasible prospect π ,

there is a prospect ( ( ),0,1 ( ))v vπ π− on the line 1 3 1π π+ = such that

~ ( ( ),0,1 ( ))v vπ π π− .

This is a prospect in which there is positive probability on only the

best and worst outcomes. Thus, one possible utility representation

of the consumer's preferences is the win probability ( )v π in this extreme lottery.

For 1x the win probability 1( ) 0v x = and for 3x , the win probability is 3( ) 1v x = . For the intermediate

consequence 2x the win probability 2( )v x lies between zero and one. This is true for any consequence

x such that 3 1x x x . Thus the probability ( )v x is a utility representation of preferences over certain

consequences.

The great advantage of using this utility function is that for any prospect 1 1( ; ) ( ,..., , ,..., )S Sp x p p x x= ,

the consumer is indifferent between ( ; )p x and playing the extreme lottery with a win probability of

1( )

S

s ss

p v x=∑ .

1

1


© John Riley

Thus, we can represent the consumer's preferences in terms of expected win probabilities in the

extreme lotteries, or expected utility.

Expected Utility Rule

Preferences over prospects 1 1( ; ) ( ,..., ; ,..., )S Sp x p p x x= can be represented by the von Neumann-

Morgenstern utility function 1

( , ) ( )S

s ss

u p x p v x=

= ∑ .

To prove this result we appeal to the following restatement of the Independence Axiom.

( )′Independence Axiom IA

If ˆ , 1,...,m m m Mπ π =∼ , then for any probability vector 1( ,..., )Mp p p=

1 11 1 ˆ ˆ( ,..., : ,..., ) ( ,..., : ,..., )M M

M Mp p p pπ π π π∼ .

Remark: Clearly ′IA implies IA . To prove the converse see the exercises (and answers) in EM.


© John Riley

To derive the expected utility rule,

consider any S consequences where

*x is the most preferred and *x the least.

For each consequence there must be

some probability ( )sv x of winning in

the extreme lottery. That is,

**~ ( ( ),1 ( ) : , )s

s s sx e v x v x x x≡ − .

By ′IA , it follows that

11( : ) ~ ( ,..., : ,..., )S

Sp x p p e e .

The compound prospect is depicted in the

tree diagram in Figure 7.1-3.

N

.

Figure 7.1-3: Deriving the expected utility rule

N

N

N


© John Riley

Although there are 2S terminal nodes, each of these is either *x or *x . Thus the big tree and the little

tree in Figure 7.1-3 are equivalent.

That is,

1 *1 *( : ) ~ ( ,..., : ,..., ) ~ ( ( , ),1 ( , ) : , )s

Sp x p p e e u p x u p x x x− .

Although the assumptions underlying this rule seem plausible, experiments show that there are

systematic deviations from the rule. Perhaps the most famous is an example of Allais.


© John Riley

Allais Paradox

In Figure 7.1-4 the payoff in state 1 is zero,

in state 2 it is $1 million, and in state 3 it is $5 million.

A prospect is a point in the triangle. Consider

the four prospects A,B,C and D. Note that the slope

of the line CD is (1 )(1 ) / (1 ) (1 ) /a b a b a a− − − = − .

This is also the slope of the line AB. We have seen that

the independence axiom implies that indifference curves are parallel lines. Thus, if D C , then

expected utility must be rising along the line AB in the direction of B. Conversely, if D C≺ , then

expected utility must be decreasing along the line AB .

Suppose 0.9 and 0.1a b= = . Then the four prospects are as follows.

(0,1,0) (0.9,0,0.1) (0,0.1,0.9) (0.09,0,0.91)A B C D= = = =

$0

$1 1

1 $5


© John Riley

A or B

If offered $1 million for sure (A) or a 90% chance of $5 million (B) would you choose A?

C or D

If offered a 10% chance of $1 million (C) or a 9% chance of $5 million (D) would you choose D?

The most common rankings are A B and D C 1

1 In his original experiment, Allais used the convex combination (0.9,0.1; , )E A B= rather than B. (Adding up probabilities, it is readily checked that (0.09,0.9,0.01)E = ).


© John Riley

Given such inconsistencies, there have been many attempts to provide alternatives to the Independence

Axiom. One such approach proposed by Machina (1982) allows for indifference curves to be linear but

replaces the parallel indifference lines with indifference lines that fan out as depicted in Figure 7.1-5.

Note that with fanning, it is no longer inconsistent for an individual to prefer D to C and A to B.

$0

$1 1

1 $5


© John Riley

Discussion: Instinct or reflection?

When individuals must make very quick decisions, experimenters have shown that there are systematic

deviations from expected utility maximization. The “instinctive” brain moulded on the plains of Africa

takes over. But given time to reflect and discuss, these deviations are much less pronounced. For what

kinds of economic decisions is it most important to model non-expected utility maximizing behavior?


© John Riley

Continuous probability distributions

In the previous discussion the number of states is finite. The decision maker assigns a probability

sπ to each of the states in {1,,..., }S=S . An alternative modeling strategy is to assume that the set of

states are points in some closed set [ , ]α β=S . We define the increasing “cumulative distribution

function” (c.d.f.) ( ) Pr{ }F t s t= ≤ where ( ) 0F α = and ( ) 1F β = . The probability of being in any sub-

interval [ , ]C s s′= is then ( ) ( ) ( )C F s F sπ ′= − . This is known as the probability measure of C. The

laws of probability then apply to the intersection and union of these closed intervals.

If for every neighborhood ( , )N x δ of x , ( ( , )) 0N xπ δ > , the point x is said to be in the support of

the distribution. Suppose for example that [1,3]=S and

121212

, 0 1( ) , 1 2

( 1), 2 3F s

θ θθ

θ θ

≤ ≤⎧⎪= < <⎨⎪ − ≤ ≤⎩

.

Then ( )F θ is strictly increasing on the sub-intervals [0,1]and [2,3] so the support of the distribution is

the union of these intervals.


© John Riley

The continuous model is easily generalized to higher dimensions. Let C be a closed convex

hypercube in n . The decision-maker then assigns a probability measure over all closed hyper cubes in

C.

There are some mathematical subtleties when using infinite state spaces to prove theorems.

Despite this, as a modeling strategy, it is often more convenient to work with continuous distributions

than with finite state spaces.


© John Riley

7.2 ATTITUDES TOWARD RISK Key ideas: risk aversion, acceptable gambles, measures of risk aversion, Jensen’s Inequality, trading in state claims

Consider an individual choosing over prospects 1 1( ,..., ; ,..., )S Sx xπ π , where sx ∈ . If his VNM

utility function ( )u x is linear there is no loss of generality in defining ( )u x x= so his expected utility is

0

1( )

Ss s

sU x xπ

=∑=

This individual is indifferent between prospects with the same expected value regardless of how risky.

That is, he is risk neutral.

Holding the consequences fixed, we now examine more systematically choices over probability

distributions. We begin by considering changes to the probabilities of three outcomes. We re-label the

S outcomes so that the first three are the ones with changing probabilities and let 1x be the worst

outcome, 2x be the intermediate outcome, and 3x be the best outcome. First consider an individual

whose expected utility function is

0

1( )

Ss s

sU x xπ

=∑= .


© John Riley

Suppose that the initial and final probability vectors are

π̂ and ˆπ π π= + Δ .

Then the change in the expected payoff is

3

0

1s s

sU xπ

=

Δ = Δ∑ .

Also, because probabilities sum to 1

3

10s

sπ

=∑ Δ = .

Using this equation to substitute for 2πΔ ,

we can write the change in expected payoff as follows: 1 2 3x x x< <

01 1 2 3 2 3( ) ( )U x x x xπ πΔ = Δ − − Δ − .

Indifference curves are depicted in Figure 7.2-1. Note that the slope of each indifference curve is

0 3 1 2

1 2 3

x xmx x

ππ

Δ −= =Δ −

.

1

1

Fig. 7.2-1: Acceptable gambles for a risk-neutral individual


© John Riley

Consider the indifference curve through the origin.

At O , both 1π and 3π are zero so this is the certain

consequence 2x . All the points in the shaded region

on or below the indifference line have a positive expected

payoff. Suppose that the individual begins with 2x .

Then the shaded region 0G is the set of acceptable risky

outcomes or “acceptable gambles” for a risk- neutral

individual. We will define a risk-averse individual to

be one who will accept fewer gambles than a risk-neutral 1 2 3x x x< <

individual.

1

1

Fig. 7.2-1: Acceptable gambles for a risk-neutral individual


© John Riley

Let ( )s su u x= be the von Neumann Morgenstern

utility function of a risk averse agent.

Arguing as earlier, if the probability vector changes

from π̂ to π , the change in expected utility is

1 2 1 3 3 21

ˆ( ) ( ) ( )S

s s ss

U u u u u uπ π π π=∑Δ = − = −Δ − + Δ − .

Along an indifference line the slope is

3 2 1

1 3 2

u umu u

ππ

Δ −= =Δ −

.

This is depicted in Figure 7.2-2. The set of acceptable gambles G is the set of points on or above the

indifference line through the origin. Thus 0G G⊂ and 0G G≠ if and only if 0m m> , that is

02 1 2 1

3 2 3 1

u u x xm mu u x x

− −= > =

− −.

Slope

Figure 7.2-2: Acceptable gambles for a risk-averse individual

1

1


© John Riley

Rearranging, 0G G⊂ and 0G G≠ if and only if 2 31 2

1 2 2 3

u uu ux x x x

−−>

− −. (7.2-1)

This is illustrated in Figure 7.2-3.

The function ( )u ⋅ is strictly concave if and only if for all 2 1 3( , )x x x∈ , the point 2A lies above the chord

1 3A A . Equivalently, the slope of the chord 1 2A A exceeds the slope of the chord 2 3A A . This illustrates an

important result. An expected-utility maximizing individual is risk averse if and only the utility

function is concave.

Lemma 7.2-1: Strictly concave function1

is strictly concave if and only if for any ,

.

Figure 7.2-3: Strictly concave function


© John Riley

Greater Aversion to Risk

Next consider two individuals, Victor and Ursula with utility functions ( )v ⋅ and ( )u ⋅ ,

respectively. Suppose that the former is a strictly concave transformation of the latter; that is, ( )v g u=

where g is an increasing strictly concave function. We now argue that Victor must have a smaller set of

acceptable gambles. In other words, Victor is more risk averse than Ursula.

Appealing to Lemma 7.2-1, the function g is strictly concave if and only if, for all 1 3 1,x x x> and

convex combination 2x ,

2 1 2 1

3 2 3 2

( ) ( )( ) ( )

g u g u u ug u g u u u

− −>

− −,

That is,

2 1 2 1

3 2 3 2

v uv v u um mv v u u− −

= > =− −

.

Slope

Figure 7.2-2: Acceptable gambles for a risk-averse individual

1

1


© John Riley

Measures of absolute risk aversion

As we have just argued, Victor is everywhere more risk averse than Ursula if his utility function

is a strictly increasing concave transformation of Ursula’s. For any pair of increasing functions ( )v x

and ( )u x there is some increasing mapping ( )v g u= . Then

( ) ( ( )) ( )v x g u x u x′ ′ ′= .

Taking logs,

ln ( ) ln ( ( )) ln ( )v x g u x u x′ ′ ′= + .

Differentiating again and rearranging,

( ) ( ) ( )( ) ( ) ( )

v x g u u xv x g u u x′′ ′′ ′′

− = − −′ ′ ′

.

Note that because g is strictly increasing and concave, the first term on the right-hand side is positive.

Thus, Victor is more risk averse if and only if


© John Riley

( ) ( )( ) ( )( ) ( )

v uv x u xA x A xv x u x′′ ′′

≡ − ≥ − =′ ′

.

This ratio is known as the degree of absolute risk aversion.

Small risks

Consider the special case in which

1 2 3( , , ) ( , , 2 )x x x w w z w z= + + . Suppose the probabilities

are chosen so that the individual is indifferent between

1 3( ,0, )π π and (0,1,0) . That is, the individual is indifferent

between receiving z for sure and receiving 0 with probability 1π

and 2z with probability 3π . This is depicted in Figure 7.2-4.

The individual is indifferent between the origin and the gamble 1 3( , )π π . Thus the slope of the

indifference line is

32 1

3 2 1

( ) u um zu u

ππ

−≡ =

− .

Slope

Figure 7.2-4: Shifting the odds

1

1


© John Riley

If the individual is risk neutral,

3

1

( ) ( ) ( ) ( )( ) 1( 2 ) ( ) ( 2 ) ( )u w z u w w z wm z

u w z u w z w z w zππ

+ − + −= = = =

+ − + + − +.

Thus for any risk-averse individual, the odds must be greater than 1. As a measure of aversion to small

risks we ask how rapidly the odds that an individual will accept increase as the size of the gamble

increases from zero.

Define 2 1( )n z u u= − and 3 2( )d z u u= − .

Then

( ) ( ) ( )( )( 2 ) ( ) ( )u w z u w n zm z

u w z u w z d z+ −

= =+ − +

.

It is easy to check that (0) (0) 0n d= = , (0) (0) ( )n d u w′ ′ ′= = , (0) ( )n u w′′ ′′= and (0) 3 ( )d u w′′ ′′=

As z approaches zero both numerator and denominator approach zero. By l’Hôpital’s Rule, the limit of

the right hand side is the ratio of the first derivatives.

0

( )(0) lim 1( )x

n zmd z→

′= =

′.


© John Riley

Then

( ) (0) ( ) 1 ( ) ( )( )

m z m m z n z d zz z zd z− − −

= = .

Both the numerator and the denominator and their first derivatives are zero at 0z = . By l’Hôpital’s

Rule the limit of the right-hand side is the ratio of the second derivatives. The limit of the left hand

side is the derivative of m. Differentiating and collecting non-zero terms,

(0) (0) ( )(0)2 (0) ( )

n d u wmd u w

′′ ′′ ′′−′ = = −′ ′

.

Thus, for any small z the odds 3 1/π π needed for the individual to accept the gamble must be

( ) (0) (0) 1 ( )m z m m z A w z′≈ + = − .

The measure of risk aversion ( )A w is therefore a measure of aversion to a small absolute risk.


© John Riley

Relative risk aversion

Economists are often interested in responses to changes in proportional or “relative” risk, rather

than absolute risk. We can use our results to get a measure of aversion to such risk. Suppose that the

gamble is over a fraction θ of the individual’s wealth; that is x wθ= . For each fraction θ , let ( )Rm θ

be the slope of the indifference line (the acceptable odds) .

Then ( ) ( )Rm m wθ θ= . Differentiating by θ , ( ) ( )Rm wm wθ θ′ ′= . Hence,

( )(0) ( )( )R

u wm w R wu w′′′ = − ≡′

.

Thus, ( )R w is a measure of aversion to relative risk and is called the degree of relative risk aversion.


© John Riley

State Claims

Thus far, our focus has been on the ranking of probability vectors. We now switch our attention

to the choice of consumption bundles. It is helpful to think about a prospect as being Nature’s choice

of one of the S states (e.g., inches of rainfall). The probability of state s is sπ . In state s the

individual’s consumption is sx . Thus our focus shifts from varying the probabilities of the different

states to varying the consumption in these states.

We first compare a risky prospect 1 1( ,..., ; ,..., )S Sx x π π with its expectation 1

Ss s

sx xπ

=∑= . Intuitively,

an individual who is risk averse (or equivalently has a concave utility function), will strictly prefer the

certain expectation x over the risky prospect.

Jensen’s Inequality

If ( )u ⋅ is strictly concave, then 1 1

( ) ( )S S

s s s ss s

u x u xπ π= =

≤∑ ∑ and the inequality is strict unless 1 ... Sx x= = .


© John Riley

Proof: Re-label states so that 1 2x x≠ . Then for 2S = , Jensen’s Inequality is a restatement of the

definition of a strictly concave function. We appeal to concavity to show that the proposition holds for

3S = .

From the strict concavity of ( )u ⋅ ,

1 2 1 1 2 21 2

1 2 1 2 1 2

( ) ( ) ( )x xu x u x uπ π π ππ π π π π π

++ <

+ + +.

Again appealing to the concavity of ( )u ⋅ ,

1 1 2 2 1 1 2 21 2 3 3 1 2 3 3 1 1 2 2 3 3

1 2 1 2

( ) ( ) ( ) (( )( ) ) ( )x x x xu u x u x u x x xπ π π ππ π π π π π π π ππ π π π+ +

+ + ≤ + + = + ++ +

Multiplying the first inequality by 1 2π π+ and adding the two inequalities, it follows that

1 1 2 2 3 3 1 1 2 2 3 3( ) ( ) ( ) ( )u x u x u x u x x xπ π π π π π+ + < + + .

Thus, the theorem holds for the three-outcome case. An inductive argument extends this to the S-

outcome case.

Q.E.D.


© John Riley

Trading in Markets for State Claims

Suppose Bev will have a higher

income in state 1 (Republican candidate wins)

than state 2 (Democrat wins).

That is, Bev’s state-contingent income vector

is 1 2( , )ω ω , where 1 2ω ω> . Before the election

she can trade in the Iowa Presidential futures market.2

Let 1p be the current price of 1 unit of consumption if

the Republican candidate wins and let 2p be the corresponding price if the Democrat wins.

Bev’s budget constraint is

1 1 2 2 1 1 2 2p x p x p pω ω+ = + .

Her budget set is depicted in Figure 7.2-5.

2 You can place a modest bet at http://www.biz.uiowa.edu/iem/

Figure 7.2-5: Partial insurance against a Democratic victory

Consumption in state 1

(Republican Victory)


(Democratic Victory)


© John Riley

Bev’s expected utility is

1 2 1 1 2 2( , ) ( ) ( )U x x u x u xπ π= +

Therefore her marginal rate of substitution is

2 1 1

1 2 2

( )( )( )

U U

dx u xMRS xdx u x

ππ

=

′= − =

′ .

Then along the 45 line 2 11 1

1 2

( , )U U

dxMRS x xdx

ππ

=

= − =

Thus, if the market price ratio is equal to her subjective odds, Bev will choose a bet that moves her to

the 45 line. In other words, Bev will trade in the presidential futures market in order to completely

insure herself against the unfavorable outcome (victory by the Democratic Party candidate). As

depicted, the price is less favorable so Bev partially insures against a Democratic victory.







© John Riley

Wealth effects

How would the riskiness of her optimal consumption

bundle change if her wealth were to increase?

Consider the effect on the marginal rate of substitution

as consumption rises from 1 2( , )x x to 1 2( , )x z x z+ + ,

a point on the line parallel to the 45 line. The MRS is

1 11 2

2 2

( )( , )( )

v x zM x z x zv x z

ππ

′ ++ + =

′ +

Therefore 1 2 1 2 1 2ln ( , ) ln ( ) ln ( ) ln( / )M x z x z v x z v x z π π′ ′+ + = + − + + .

Hence,

1 2

1 2

1 ( ) ( )( ) ( )

dM v x z v x zM dz v x z v x z

′′ ′′+ += −

′ ′+ +.

It follows immediately that if the degree of absolute risk aversion is constant, then the marginal rate of

substitution is unchanged. Thus in this case Bev’s consumption increase is the same in each state.







© John Riley

If the degree of absolute risk aversion decreases

with x, then below the 45 certainty line

1 2

1 2

( ) ( )( ) ( )

v x z v x zv x z v x z′′ ′′+ +

− < −′ ′+ +

Hence

1 2

1 2

1 ( ) ( ) 0( ) ( )

dM v x z v x zM dz v x z v x z

′′ ′′+ += − >

′ ′+ +

Thus, in this case the marginal rate of substitution

increases and the optimal consumption bundle is

therefore farther from the certainty line. Therefore greater wealth leads to a consumption bundle

exhibiting greater absolute risk.







© John Riley

Portfolio Choice

Finally, we consider a simple portfolio decision. Ursula must decide how much to invest in a riskless

asset that has a gross yield of 11 r+ and how much to invest in a risky asset (or mutual fund) with a

gross yield 21 r+ .

Simple intuition: If Ursula is risk averse she will need to be paid some premium to invest in the risky

asset.

We now show that this is not (quite) correct.

Converting to the state claim formulation, let the gross yield of the risky asset be 21 sr+ in state s, and

the probability of state s be , 1,...,s s Sπ = . Suppose that Ursula invests q dollars in the risky asset and

her remaining wealth W q− in the riskless asset. Her final consumption in state s is then

1 2 1( )(1 ) (1 ) (1 )s s sx W q r q r W r qθ= − + + + = + + where 2 1s sr rθ = − . (7.2-2)

Substituting from (7.2-2), Ursula has an expected utility of

11 1

( ) ( ) ( (1 ) )S S

s s s ss s

U q u x u W r qπ π θ= =∑ ∑= = + + .


© John Riley

11 1

( ) ( ) ( (1 ) )S S

s s s ss s

U q u x u W r qπ π θ= =∑ ∑= = + + .

Then her marginal gain from increasing q is

11 1

( ) ( (1 ) ) ( )S S

s s s s s ss s

U q u W r q u xπ θ θ π θ= =

′ ′ ′= + + =∑ ∑ . (7.2-3)

Differentiating again, it is easily checked that the second derivative is negative and so there is a single

turning point. Moreover, at 0q = ,

11 1

(0) ( (1 )) 0 0S S

s s s ss s

U u W r π θ π θ= =

′ ′= + > ⇔ >∑ ∑ .

Thus unless Ursula is infinitely risk averse, she will purchase some of the risky asset.

To understand why this must be the case, consider equation (7.2-3). Ursula weights her marginal

claims in state s by her marginal utility in that state and then takes the expectation of marginal utilities.

But; when she is taking no risk, each of the marginal utility weights is the same. Thus, the decision

whether to invest at all is the same as the decision of a risk-neutral individual. Of course, for any

positive investment in the risky asset, the marginal utility weights change. Intuitively, the marginal

utility weights change more if the degree of risk aversion is greater.


© John Riley

Proposition: A more risk-averse individual invests less in the risky asset

If Victor is more risk averse than Ursula, then his utility function ( )v x is a concave function of Ursula’s

utility function; that is ( ) ( ( ))v x g u x= , where ( )g ⋅ is an increasing strictly concave function.

Let *z be optimal for Ursula and define * *(1 )s s sx W r qθ= + + . Then

* *

1( ) ( ) 0

S

s s ss

U q u xπ θ=

′′ = =∑ .

Since ( )U ⋅ is a strictly concave function we need to show that if Victor chooses the same portfolio his

expected marginal utility of increasing q is negative at *q .

Order states so that 1 2 ... Sθ θ θ> > > . Let t be the largest state for which sθ is positive. Then * *, 1,...,s tx x s t≥ = , and * *, 1,...,s tx x s t S< = + .

Define 1

( ) ( )S

s ss

V q v xπ=

=∑ . Then

* * * *

1 1( ) ( ) ( ( )) ( )

S S

s s s s s s ss s

V q v x g u x u xπ θ π θ= =∑ ∑′ ′ ′′= =


© John Riley

* * * *

1 1( ( ) ( ) ( ( )) ( )( )

t S

s s s s s s s ss s t

g u x u x g u x u xπ θ π θ= = +

′ ′′ ′= − −∑ ∑ .

Each term in the two summations is non-negative. For each of the terms in the first summation, the

strict concavity of ( )g ⋅ implies that * *( ( )) ( ( )s tg u x g u x′ ′≤ and the inequality is strict for 1s = (otherwise

there is no risk.). For each of the terms in the second summation, * *( ( )) ( ( ))s tg u x g x x′ ′≥ .

Hence,

* * * * *

1 1( ) ( ( )) ( ) ( ( )) ( )( )

t S

s t s s s t s ss s t

V q g u x u x g u x u xπ θ π θ= = +∑ ∑′ ′′ ′ ′< − −

* * * *

1

( ( ) ( ) ( ( )) ( ) 0S

t s s s ts

g u x u x g u x U qπ θ=

′ ′′ ′= = =∑

Q.E.D.

Documents

CHAPTER 7: UNCERTAINTY - Essential Microeconomicsessentialmicroeconomics.com/ChapterY7/SlideChapter7A.pdf · Essential Microeconomics-1- Sections 7.1 and 7.2 © John Riley CHAPTER