uncertainty and riskdarp.lse.ac.uk/pdflse/ec202/ec202_classes/... · chapter 8 uncertainty and risk...

Chapter 8

Uncertainty and Risk

Exercise 8.1 Suppose you have to pay $2 for a ticket to enter a competition.The prize is $19 and the probability that you win is 1

3 . You have an expectedutility function with u(x) = log x and your current wealth is $10.

1. What is the certainty equivalent of this competition?

2. What is the risk premium?

3. Should you enter the competition?

Outline Answer:

1. Given the probabilities and payoffs we have the following expected utilityif the person enters the lottery

Eu(x) =1

3log (10− 2 + 19) +

2

3log (10− 2)

=1

3log (27) +

2

3log (8)

= log 3 + log 4 = log 12

So the certainty equivalent is $12.

2. The expected wealth at the end of the period is

Ex =1

3[10− 2 + 19] +

2

3[10− 2]

=27

3+

16

3=

43

3= 14

1

3

So the risk premium is $14 13 − $12 = $21

3 .

3. If the person does not enter the lottery he has just his initial wealth, $10.So, in view of the answer to part (a) it makes sense to enter the lottery.

115

Microeconomics CHAPTER 8. UNCERTAINTY AND RISK

Exercise 8.2 You are sending a package worth 10 000AC. You estimate thatthere is a 0.1 percent chance that the package will be lost or destroyed in tran-sit. An insurance company offers you insurance against this eventuality for apremium of 15AC. If you are risk-neutral, should you buy insurance?

Outline Answer:No you should not. Your expected loss is 10 euros whereas the premium is

15 euros.

c©Frank Cowell 2006 116

Microeconomics

Exercise 8.3 Consider the following definition of risk aversion. Let P :=(xω, πω) : ω ∈ Ω be a random prospect, where xω is the payoff in state ωand πω is the (subjective) probability of state ω , and let Ex :=

∑ω∈Ω πωxω,

the mean of the prospect, and let Pλ := (λxω + [1 − λ]Ex, πω) : ω ∈ Ω be a“mixture” of the original prospect with the mean. Define an individual as riskaverse if he always prefers Pλ to P for 0 < λ < 1.

1. Illustrate this concept in (xred, xblue)-space and contrast it with the conceptof risk aversion used in the text

2. Show that this definition of risk aversion need not imply convex-to-the-origin indifference curves.

Outline Answer:

1. See Figure 8.1.

xBLUE

xRED

P

P

Pλ

Figure 8.1: Nonconvex indifference curve

2. Let P be the prospect and P its mean. Pλ can be any point in the linejoining them. The definition implies that moving along this line towardsP puts the person on a successively higher indifference curves. In Figure8.1 it is clear that this condition is consistent with there being indifferencecurves that violate the convex-to-the-origin property locally.



Exercise 8.4 Suppose you are asked to choose between two lotteries. In onecase the choice is between P1 and P2,and in the other case the choice offered isbetween P3 and P4, as specified below:

P1 : $1, 000, 000 with probability 1

P2 :

$5, 000, 000$1, 000, 000

$0

with probability 0.1with probability 0.89with probability 0.01

P3 :

$5, 000, 000

$0with probability 0.1with probability 0.9

P4 :

$1, 000, 000

$0with probability 0.11with probability 0.89

It is often the case that people prefer P1 to P2 and then also prefer P3 to P4.Show that these preferences violate the independence axiom.

Outline Answer:Let there be only three possible states of the world: red, blue and green,

with probabilities 0.01, 0.10, 0.89 respectively. Then the payoffs in the fourprospects can be written

red blue greenP1 1 1 1P2 0 5 1P3 0 5 0P4 1 1 0

where all the entries in the table are in millions of dollars. Note that P1 and P2

have the same payoff in the green state; P3 and P4 form a similar pair, exceptthat the payoff in the green state is 0. Axiom 8.2 states that if P1 is preferredto P2 than any other similar pair of prospects (1, 1, z) and (0, 5, z) ought alsoto be ranked in the same order, for arbitrary z: but this would imply that P4

is preferred to P3, the opposite of the preferences as stated.Note also that if the preferences had been such that P4 was preferred to P3

then the independence axiom would imply that P1 was preferred to P2.


Microeconomics

Exercise 8.5 This is an example to illustrate disappointment. Suppose thepayoffs are as followsx′′ weekend for two in your favourite holiday locationx′ book of photographs of the same locationx fish-and-chip supperYour preferences under certainty are x′′ x′ x. Now consider the followingtwo prospects

P1 :

x′′

x′

x

with probability 0.99with probability 0with probability 0.01

P2 :

x′′

x′

x

with probability 0.99with probability 0.01with probability 0

Suppose a person expresses a preference for P1 over P2. Briefly explain whythis might be the case in practice. Which of the three axioms State Irrelevance,Independence, Revealed Likelihood, is violated by such preferences?

Outline Answer:It is possible that, given the information that the first event (with payoff x′′)

has not happened you would then prefer x to x′: photographs of your favouriteholiday spot may be too painful once you know that the holiday is not going tohappen. So you may prefer P1 over P2.These preferences violate the independence axiom. To see this, note that,

by the revealed likelihood axiom, since x′ is strictly preferred to x, it must bethe case that P ′2 is strictly preferred to P

′1, where

P ′1 :

x′

xwith probability 0with probability 1

P ′2 :

x′

xwith probability 0.01with probability 0.99

But P ′1 and P′2 can be written equivalently as

P ′1 :

x

x′

x

with probability 0.99with probability 0with probability 0.01

P ′2 :

x

x′

x

with probability 0.99with probability 0.01with probability 0

By the independence axiom if P ′2 is strictly preferred to P′1, then P2 must be

strictly preferred to P1.



Exercise 8.6 An example to illustrate regret. Let

P := (xω, πω) : ω ∈ Ω

P ′ := (x′ω, πω) : ω ∈ Ω

be two prospects available to an individual. Define the expected regret if theperson chooses P rather than P ′ as∑

ω∈Ω

πω max x′ω − xω, 0 (8.1)

Now consider the choices amongst prospects presented in Exercise 8.4. Showthat if a person is concerned to minimise expected regret as measured by (8.1),then it is reasonable that the person select P2 when P1 is also available and thenalso select P4 when P3 is available.

Outline Answer:Denote the regret in (8.1) by r (P, P ′).If I choose P2 when P1 is also available then the regret is

r (P2, P1) = 0.1 [0] + 0.89 [0] + .01 [1]

= 10, 000

Whereas, had I chosen P1 when P2 was available, then the regret would havebeen

r (P1, P2) = 0.1 [4, 000, 000] + 0.89 [0] + .01 [0]

= 400, 000

If I choose P4 when P3 is also available then the regret is

r (P4, P3) = 0.1 [0] + 0.89 [0] + .01 [0]

= 0

Whereas, had I chosen P3 when P4 was available, then the regret would havebeen

r (P3, P4) = 0.1 [0] + 0.89 [0] + .01 [5, 000, 000]

= 50, 000


Microeconomics

Exercise 8.7 An example of the Ellsberg paradox . There are two urns markedLeft and Right each of which contains 100 balls. You know that in Urn Lthere exactly 49 white balls and the rest are black and that in Urn R there areblack and white balls, but in unknown proportions. Consider the following twoexperiments:

1. One ball is to be drawn from each of L and R. The person must choosebetween L and R before the draw is made. If the ball drawn from the chosenurn is black there is a prize of $1000, otherwise nothing.

2. Again one ball is to be drawn from each of L and R; again the person mustchoose between L and R before the draw. Now if the ball drawn from thechosen urn is white there is a prize of $1000, otherwise nothing.

You observe a person choose Urn L in both experiments. Show that thisviolates the Revealed Likelihood Axiom.

Outline AnswerThe implication of the revealed likelihood axiom is that there exist subjective

probabilities πω. The result is proved by showing that it the stated behaviouris inconsistent with the existence of subjective probabilities.In this case the revealed likelihood axiom implies that for each urn there is

a given subjective probability of drawing a black ball πL (left-hand urn) and πR

(right-hand urn) such that preferences can be represented as

πvblack (xblack, xwhite) + [1− π] vwhite (xblack, xwhite) (8.2)

where π = πL or πR and xblack and xwhite are the payoffs if a black ball or awhite ball are drawn respectively.Note that the representation (8.2) does not impose either the State Irrel-

evance Axiom (which would require that vblack (·) and vwhite (·) be the samefunction) or the Independence axiom (which would require that vblack (·) be afunction only of xblack etc.). Nor does it impose the common-sense requirementthat πL = 0.49. All we need below is the very weak assumption that preferencesare not perverse:

vblack (1000, 0) > vwhite (1000, 0) (8.3)

andvblack (0, 1000) < vwhite (0, 1000) (8.4)

Condition (8.3) simply says that if the $1000 prize is attached to a black ballthen the utility to be derived from having selected a black ball is higher thanselecting a white ball; condition (8.4) is the counterpart when the prize attachesto the white ball..Experiment 1 suggests that

πLvblack (1000, 0) + [1− πL] vwhite (1000, 0)

> πRvblack (1000, 0) + [1− πR] vwhite (1000, 0) (8.5)



while experiment 2 suggests that

πLvblack (0, 1000) + [1− πL] vwhite (0, 1000)

> πRvblack (0, 1000) + [1− πR] vwhite (0, 1000) (8.6)

We can see that (8.5) implies

πL [vblack (1000, 0)− vwhite (1000, 0)] > πR [vblack (1000, 0)− vwhite (1000, 0)]

from which we deduce that, given (8.3), πL > πR. However (8.5) implies

πL [vblack (0, 1000)− vwhite (0, 1000)] > πR [vblack (0, 1000)− vwhite (0, 1000)] .

So that, given (8.4), we would have πL < πR —a contradiction. Therefore therevealed likelihood axiom must be violated.


Microeconomics

Exercise 8.8 An individual faces a prospect with a monetary payoff representedby a random variable x that is distributed over the bounded interval of the realline [a, a]. He has a utility function Eu(x) where

u(x) = a0 + a1x−1

2a2x

2

and a0, a1, a2 are all positive numbers.

1. Show that the individual’s utility function can also be written as ϕ(Ex, var(x)).Sketch the indifference curves in a diagram with Ex and var(x) on theaxes, and discuss the effect on the indifference map altering (i) the para-meter a1, (ii) the parameter a2.

2. For the model to make sense, what value must a have? [Hint: examinethe first derivative of u.]

3. Show that both absolute and relative risk aversion increase with x.

Outline Answer:

1. Clearly

Eu(x) = a0 + a1E(x) + a21

2[(E(x))2 − var(x)].

Marginal utility is a1 − a2x.

2. For this to be non-negative we must have E(x) ≤ a1/a2 hence the in-difference curves are depicted with E(x) as good, var(x) as bad andMRS = −2 [a1/a2 − E(x)] .

3.

ux (x) = a1 + a2x

uxx (x) = a2

α(x) = −uxx (x)

ux (x)= − a2

a1 + a2x

α(x) =1

xmax − x

%(x) =1

xmax/x− 1

where and 0 ≤ x ≤ xmax := −a1a2 .



Exercise 8.9 A person lives for 1 or 2 periods. If he lives for both periods hehas a utility function given by

U (x1, x2) = u (x1) + δu (x2) (8.7)

where the parameter δ is the pure rate of time preference. The probability ofsurvival to period 2 is γ, and the person’s utility in period 2 if he does notsurvive is 0.

1. Show that if the person’s preferences in the face of uncertainty are repre-sented by the expected-utility functional form∑

ω∈Ω

πωu (xω) (8.8)

then the person’s utility can be written as

u (x1) + δ′u (x2) . (8.9)

What is the value of the parameter δ′?

2. What is the appropriate form of the utility function if the person could livefor an indefinite number of periods, the rate of time preference is the samefor any adjacent pair of periods, and the probability of survival to the nextperiod given survival to the current period remains constant?

Outline Answer:

1. Consider the person’s lifetime utility with the consumption x1 and x2

in the two periods. If the person survives into the second period utilityis given by u (x1) + δu (x2) otherwise it is just u (x1). Given that theprobability of the event “survive to second period”is γ expected lifetimeutility is

γ [u (x1) + δu (x2)] + [1− γ]u (x1) .

On rearranging we getu (x1) + γδu (x2) , (8.10)

in other words the form (8.9) with δ′ = γδ.

2. Apply the argument to one more period. Now there is consumptionx1,x2, x3 in the three periods and the probability of surviving into periodt + 1 given that you have made it to period t is still γ. Consider thesituation of someone who survives to period 2. The person gets utility

u (x2) + δu (x3) (8.11)

if he survives to period 3 and u (x2) otherwise. His expected utility forthe rest of his lifetime, contingent on having reached period 2 is therefore

γ [u (x2) + δu (x3)] + [1− γ]u (x2)

= u (x2) + γδu (x3) (8.12)


Microeconomics

So now view the situation from the position of the beginning of the lifetime.The person gets utility

u (x1) + δ [u (x2) + γδu (x3)] (8.13)

if he makes it through to period 2, where the expression in square bracketsin (8.13) is just the rest-of-lifetime expected utility if you get to period 2,taken from (8.12); of course if the person does not survive period 1 he getsjust u (x1). So, using the same reasoning as before, from the standpointof period 1 lifetime expected utility is now

γ [u (x1) + δ [u (x2) + γδu (x3)]]

+ [1− γ]u (x1) .

Rearranging this we have

u (x1) + γδu (x2) + γ2δ2u (x2) . (8.14)

It is clear that the same argument could be applied to T > 2 periods andthat the resulting utility function would be of the form

u (x1) + γδu (x2) + γ2δ2u (x2) + ...+ γT δTu (x2) . (8.15)

In other words we have the standard intertemporal utility function withthe pure rate of time preference δ replaced by the modified rate of timepreference δ′ := γδ.



Exercise 8.10 A person has an objective function Eu(y) where u is an increas-ing, strictly concave, twice-differentiable function, and y is the monetary valueof his final wealth after tax. He has an initial stock of assets K which he maykeep either in the form of bonds, where they earn a return at a stochastic rater, or in the form of cash where they earn a return of zero. Assume that Er > 0and that Prr < 0 > 0.

1. If he invests an amount β in bonds (0 < β < K) and is taxed at rate ton his income, write down the expression for his disposable final wealth y,assuming full loss offset of the tax.

2. Find the first-order condition which determines his optimal bond portfolioβ∗.

3. Examine the way in which a small increase in t will affect β∗.

4. What would be the effect of basing the tax on the person’s wealth ratherthan income?

Outline Answer:

1. Suppose the person puts an amount β in bonds leaving the remainingK − β of assets in cash. Then, given that the rate of return on cash iszero and on bonds is the stochastic variable r, income is

[K − β] 0 + βr = βr

If the tax rate is t then, given that full loss offset implies that losses andgains are treated symmetrically, disposable income is

[1− t]βr

and (disposable) final wealth is

x = [K − β][cash]

+ β[value of bonds]

+ [1− t]βr[income]

= K + [1− t]βr. (8.16)

Note that x is a stochastic variable and could be greater or less than initialwealth K.

2. The individual’s optimisation problem is to choose β to maximise Eu(x).Using (8.16) the FOC for an interior solution is

E (ux(x) [1− t] r) = 0,

which impliesE (ux(x)r) = 0. (8.17)

Solving this determines β∗ = β∗ (t,K), the optimal bond purchases thatdepends on the tax rate and initial wealth as well as the distribution ofreturns and risk aversion.


Microeconomics

3. Take the FOC (8.17). Substituting for x from (8.16) and differentiatingwith respect to t we get

E(uxx(x)

[−β∗r +

∂β∗

∂t[1− t] r

]r

)= 0,

E(uxx(x)r2

[−β∗ +

∂β∗

∂t[1− t]

])= 0.

so that

−β∗ +∂β∗

∂t[1− t] = 0

∂β∗

∂t=

β∗

1− t .

An increase in the tax rate increases the demand for bonds.

4. Final wealth is initial wealth plus income. If the tax is on wealth thendisposable final wealth is

x = [1− t]K + [1− t]βr (8.18)

instead of (8.16). Clearly the FOC (8.17) remains essentially unaltered(the new tax just reduces total wealth). Differentiating the FOC with xdefined by (8.18) we now find

E(uxx(x)

[−K − β∗r +

∂β∗

∂t[1− t] r

]r

)= 0,

−KE (uxx(x)r) + E(uxx(x)r2

[−β∗ +

∂β∗

∂t[1− t]

])= 0.

This implies

−β∗ −K E (uxx(x)r)

E (uxx(x)r2)+∂β∗

∂t[1− t] = 0.

∂β∗

∂t[1− t] = β∗ +K

E (uxx(x)r)

E (uxx(x)r2).

∂β∗

∂t=

β∗

1− t +K

1− tE (uxx(x)r)

E (uxx(x)r2).

The first term on the right-hand side is positive; as for the second term,the denominator is negative and the numerator is positive, given DARA.So the impact of tax on bond-holding is now ambiguous.



Exercise 8.11 An individual taxpayer has an income y that he should report tothe tax authority. Tax is payable at a constant proportionate rate t. The taxpayerreports x where 0 ≤ x ≤ y and is aware that the tax authority audits some taxreturns. Assume that the probability that the taxpayer’s report is audited isπ, that when an audit is carried out the true taxable income becomes publicknowledge and that, if x < y, the taxpayer must pay both the underpaid tax anda surcharge of s times the underpaid tax.

1. If the taxpayer chooses x < y, show that disposable income c in the twopossible states-of-the-world is given by

cnoaudit = y − tx,caudit = [1− t− st] y + stx.

2. Assume that the individual chooses x so as to maximise the utility function

[1− π]u (cnoaudit) + πu (caudit) .

where u is increasing and strictly concave.

(a) Write down the FOC for an interior maximum.

(b) Show that if 1− π − πs > 0 then the individual will definitely under-report income.

3. If the optimal income report x∗ satisfies 0 < x∗ < y:

(a) Show that if the surcharge is raised then under-reported income willdecrease.

(b) If true income increases will under-reported income increase or de-crease?

Outline Answer:If the individual reports x then he pays tax tx —i.e. he underpays an amount

t [y − x]. So

1. If the under-reporting remains undetected then

cnoaudit = y − tx= y − ty + t [y − x]

and if the audit takes place then

caudit = y − tx− [1 + s] t [y − x]

= [1− t− st] y + stx

2. The individual maximises

Eu(c) = [1− π]u (y − tx) + πu ([1− t− st] y + stx)

Differentiating this we have

∂Eu(c)

∂x= −t [1− π]uc (y − tx) + stπuc ([1− t− st] y + stx)

where uc (·) denotes the first derivative of u.


Microeconomics

(a) If there is an interior maximum at x∗ then the following FOC musthold

[1− π]uc (y − tx∗) = sπuc ([1− t− st] y + stx∗) .

(b) If the person reports fully then

∂Eu(c)

∂x

∣∣∣∣x=y

= −t [1− π]uc (y − ty) + stπuc ([1− t] y)

= − [1− π − sπ] tuc (y − ty)

Given that t and uc are positive it is clear that the above expressionis negative if 1 − π − sπ > 0. Therefore the individual’s expectedutility would increase if he reduced x below y.

3. Differentiating the FOC with respect to s and rearranging we get

−t [1− π]ucc (y − tx∗) ∂x∗

∂s− s2tπucc ([1− t− st] y + stx∗)

∂x∗

∂s= πuc ([1− t− st] y + stx∗) + st [x∗ − y]πucc ([1− t− st] y + stx∗)

(a) Therefore

∂x∗

∂s= π

uc (caudit) + st [x∗ − y]ucc (caudit)

t∆(8.19)

where

∆ := − [1− π]ucc (y − tx∗)− s2πucc ([1− t− st] y + stx∗) > 0

Given that uc > 0, x∗ < y and ucc < 0 it is clear that the numeratorof (8.19) is positive.x∗ increases with s so t [y − x∗] decreases.

(b) Differentiating the FOC with respect to y we get

[1− π]ucc (y − tx∗)[1− t∂x

∗

∂y

]= sπucc ([1− t− st] y + stx∗)

[[1− t− st] + st

∂x∗

∂y

].

Therefore we have

∂x∗

∂y=πs [1− t− st]ucc (caudit)− [1− π]ucc (cnoaudit)

−t [[1− π]ucc (cnoaudit) + πs2ucc (caudit)]

∂ [y − x∗]∂y

= 1 +πs [1− t− st]ucc (caudit)− [1− π]ucc (cnoaudit)

[[1− π] tucc (cnoaudit) + πs2tucc (caudit)]

∂ [y − x∗]∂y

= −1− tt

[1− π]ucc (cnoaudit)− πsucc (caudit)

[[1− π]ucc (cnoaudit) + πs2ucc (caudit)]

This is of ambiguous sign unless we assume DARA in which case itis positive.



Exercise 8.12 A risk-averse person has wealth y0 and faces a risk of lossL < y0 with probability π. An insurance company offers cover of the loss ata premium κ > πL. It is possible to take out partial cover on a pro-rata basis,so that an amount tL of the loss can be covered at cost tκ where 0 < t < 1.

1. Explain why the person will not choose full insurance

2. Find the conditions that will determine t∗, the optimal value of t.

3. Show how t will change as y0 increases if all other parameters remainunchanged.

Outline Answer:

1. Consider the person’s wealth after taking out (partial) insurance coverusing the two-state model (no loss,loss). If the person remained unin-sured it would be (y0, y0 − L); if he insures fully it is (y0 − κ, y0 − κ). Soif he insures a proportion t for the pro-rata premium wealth in the twostates will be

([1− t] y0 + t [y0 − κ] , [1− t] [y0 − L] + t [y0 − κ])

which becomes(y0 − tκ, y0 − tκ− [1− t]L)

So expected utility is given by

Eu = [1− π]u (y0 − tκ) + πu (y0 − tκ− [1− t]L)

Therefore

∂Eu∂t

= − [1− π]κuy (y0 − tκ) + [L− κ]πuy (y0 − tκ− [1− t]L)

Consider what happens in the neighbourhood of t = 1 (full insurance).We get

∂Eu∂t

∣∣∣∣t=1

= − [1− π]κuy (y0 − κ) + [L− κ]πuy (y0 − κ)

= [Lπ − κ]uy (y0 − κ)

We know that uy (y0 − κ) > 0 (positive marginal utility of wealth) and, byassumption, Lπ < κ. Therefore this expression is strictly negative whichmeans that in the neighbourhood of full insurance (t = 1) the individualcould increase expected utility by cutting down on the insurance cover.

2. For an interior maximum we have

∂Eu∂t

= 0

which means that the optimal t∗ is given as the solution to the equation

− [1− π]κuy (y0 − t∗κ) + [L− κ]πuy (y0 − t∗κ− [1− t∗]L) = 0


Microeconomics

3. Differentiating the above equation with respect to y0 we get

− [1− π]κuyy (y0 − t∗κ)

[1− κ ∂t

∗

∂y0

]+[L− κ]πuyy (y0 − t∗κ− [1− t∗]L)

[1− [κ− L]

∂t∗

∂y0

]= 0

which gives

∂t∗

∂y0=

[1− π]κuyy (y0 − t∗κ)− [L− κ]πuyy (y0 − t∗κ− [1− t∗]L)

[1− π]uyy (y0 − t∗κ)κ2 + [L− κ]2πuyy (y0 − t∗κ− [1− t∗]L)

The denominator of this must be negative: uyy (·) is everywhere negativeand the other terms are positive. The numerator is positive if DARAholds: therefore an increase in wealth reduces the demand for insurancecoverage.



Exercise 8.13 Consider a competitive, price-taking firm that confronts one ofthe following two situations:

• “uncertainty”: price p is a random variable with expectation p.

• “certainty”: price is fixed at p.

It has a cost function C(q) where q is output and it seeks to maximise theexpected utility of profit.

1. Suppose that the firm must choose the level of output before the particularrealisation of p is announced. Set up the firm’s optimisation problem andderive the first- and second-order conditions for a maximum. Show that,if the firm is risk averse, then increasing marginal cost is not a necessarycondition for a maximum, and that it strictly prefers “certainty” to “un-certainty”. Show that if the firm is risk neutral then the firm is indifferentas between “certainty”and “uncertainty”.

2. Now suppose that the firm can select q after the realisation of p is an-nounced, and that marginal cost is strictly increasing. Using the firm’scompetitive supply function write down profit as a function of p and showthat this profit function is convex. Hence show that a risk-neutral firmwould strictly prefer “uncertainty” to “certainty”.

Outline Answer:

1. Profit is given byΠ := pq − C(q)

where p is a random variable. Maximising expected utility of profit Eu(Π)by choice of q requires the FOC

E(uΠ(Π)p)− E(uΠ(Π))Cq = 0

where uΠ(·) is the first derivative of u(·). This will represent a maximumif

d2Eudq2

< 0.

We find that this implies

E(uΠΠ[p− Cq]2)− E(uΠΠ)Cqq < 0.

Notice that since the first term is negative for a risk-averse firm then thecondition can be satisfied not only if Cqq > 0 but also if Cqq < 0 and |Cqq|is not too large. Now consider transforming p to p thus: p = (1−λ)p+λpthen p has the same mean as p but is less dispersed. Maximised utility forthe random variable p is

Eu([(1− λ)p+ λp]q∗ − C(q∗))

where q∗ is the output satisfying the first-order conditions for a maximum.Differentiate this expected utility with respect to λ

∂Eu∗∂λ

= [E(uΠ[p− p])]q∗ + [E(uΠ[p− Cq])]∂q∗

∂λ


Microeconomics

where the last term vanishes because of the first order condition. So∂Eu∗∂λ has the sign of E(uΠ[p − p]). But this must be positive if uΠ isdecreasing with Π and will be zero if uΠ is constant. Hence the firm strictlyprefers certainty if it is risk averse and is indifferent between certainty anduncertainty if it is risk neutral.

2. For any known realization p we may write q = S(p) where S is the com-petitive firm supply curve. Profits as a function of P may thus be written:

Π(p) = pS(p)− C(S(p))

which implies

dΠ(p)

dp= [p− Cq]Sp(p) + S(p) = S(p) (8.20)

where Sp(p) is the slope of the supply curve at p, a positive number.Therefore, differentiating (8.20) we have

d2Π(p)

dp2= Sp(p) > 0.

HenceΠ(·) is increasing and convex. So it is immediate that EΠ(p) > Π(p).



Exercise 8.14 Every year Alf sells apples from his orchard. Although the mar-ket price of apples remains constant (and equal to 1), the output of Alf’s orchardis variable yielding an amount R1, R2 in good and poor years respectively; theprobability of good and poor years is known to be 1 − π and π respectively. Abuyer, Bill, offers Alf a contract for his apple crop which stipulates a down pay-ment (irrespective of whether the year is good or poor) and a bonus if the yearturns out to be good.

1. Assuming Alf is risk averse, use an Edgeworth box diagram to sketch theset of such contracts which he would be prepared to accept. Assuming thatBill is also risk averse, sketch his indifference curves in the same diagram.

2. Assuming that Bill knows the shape of Alf’s acceptance set, illustrate theoptimum contract on the diagram. Write down the first-order conditionsfor this in terms of Alf’s and Bill’s utility functions.

0a

0 b

R1

xREDbxREDb

xBLUEbxBLUEb

xBLUEaxBLUEa

xREDaxREDa

•R2D

Figure 8.2: Acceptable contracts

Outline Answer:

1. In Figure 8.2 the contours represent Alf’s indifference curves: note thatthey are convex to the point 0a (risk aversion) and that they have the sameslope [1− π] /π where they cross the 45 ray through 0a (consequence ofvon-Neumann utility function). Point D represents the initial endowment;Alf’s endowment is (R1, R2). Alf’s indifference curve through point Drepresents the boundary of the set of consumptions that Alf would regardas being at least as good as the initial endowment: the shaded area is hisacceptance set. The buyer (Bill) has an endowment K that is independentof the state of the world —see Figure 8.3. Note that the indifference curvesfor Bill also have the slope [1− π] /π where they cross the 45 ray through0b.

2. Point E in Figure ?? represents the optimum contract (from Bill’s pointof view) since it is a point of common tangency of two indifference curves.


Microeconomics

0a

0 bxREDbxREDb

xBLUEbxBLUEb

xBLUEaxBLUEa

xREDaxREDa

D•

K

K

Figure 8.3: Buyer’s situation

0a

0 b

E

xREDbxREDb

xBLUEbxBLUEb

xBLUEaxBLUEa

xREDaxREDa

•

R1

R2D•

Figure 8.4: Optimal contract

At E we have thatua′(xa1)

ua′(xa2)=ub′(xb1)

ub′(xb2).



Exercise 8.15 In exercise 8.14, what would be the effect on the contract if (i)Bill were risk neutral; (ii) Alf risk neutral?

Outline Answer:In case (i) Bill’s indifference curves become lines with slope [1− π] /π and

the optimum is at E in Figure 8.5. In case (ii) Alf’s indifference curves becomelines with slope [1− π] /π and the optimum is at the endowment point D.

0a

0 b

E

xREDbxREDb

xBLUEbxBLUEb

xBLUEaxBLUEa

•

R1

R2D•

Figure 8.5: Optimal contract: risk-neutral buyer


uncertainty and riskdarp.lse.ac.uk/pdflse/ec202/ec202_classes/... · chapter 8 uncertainty and risk...

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