Page 1 of 10

Problem Set 1

Micro Analysis, S. Wang

Question 1.1. A farm produces yams using capital , labor , and land according to the

production technology described by:

The firm faces prices for

(a) Suppose that, in the short run, and are fixed. Derive the short-run supply and profit

functions of the firm.

(b) Suppose that, in the long run, and are marketable but is fixed. Derive the long-run

supply and profit functions. If there were a market for land, how much would the firm be

willing to pay for one more unit of land (the internal price of land)?

(c) Suppose that, in the long run, all the factors and are marketable. Does this produc-

tion function exhibit diminishing, constant, or increasing returns to scale? Suppose that

competitive conditions ensure zero profits. Derive the long-run supply and demand func-

tions.

Question 1.2. Show that implies

”

Question 1.3. Use a Lagrange function to solve for the following problem:

,

Question 1.4. Use a graph to solve the cost function for the following problem:

,

Page 2 of 10

Question 1.5. Find the cost function for the following problem:

,

Question 1.6. In the short run, assume is fixed: Find STC, FC, SVC, SAC, SAVC,

SAFC, SMC, LC, LAC, and LMC for the following problem:

,

Question 1.7. Prove the first two properties of the cost function.

Question 1.8. Prove the three properties of the demand and supply functions in Proposition

1.10.

Question 1.9. Consider the factor demand system:

where are parameters. Find the condition(s) on the parameters so that

this demand system is consistent with cost minimizing behavior. What is the cost function

then corresponding to the above factor demand system?

Question 1.10. Show that if satisfies Assumptions 1.1 and 1.2 and

then also satisfies Assumptions 1.1 and 1.2.

Question 1.11. A firm buys inputs at levels and on competitive markets and uses them

to produce a level of output Its technology is such that the minimum cost of producing at

input prices and is given by the cost function

Page 3 of 10

where and are constant parameters.

(a) What parameter condition does the homogeneity of this cost function imply?

(b) Derive the conditional demand functions and Verify that the

cross price effects are symmetric for these demand functions.

(c) Show that the MC curve is upward sloping and that the AC curve is U-shaped (convex).

Question 1.12. The Ace Transformation Company can produce guns ( ), or butter ( ), or

both; using labor ( ), as the sole input to the production process. Feasible production is repre-

sented by a production possibility set with a frontier

(a) Write the production function on the implicit form Does satisfy As-

sumptions 1.1 and 1.2?

(b) Suppose that the company faces the following union demands. In the next year it must

purchase exactly units of labor at a wage rate or no labor will be supplied in the next

year. If the company knows that it can sell unlimited quantities of guns and butter at prices

and respectively, and chooses to maximize next year's profits, what is its optimal

production plan?

Page 4 of 10

Answer Set 1

Answer 1.1. (a) The short-run profit is

implying

implying

implying

implying

(b) The Long-run profit is

,The FOC's are:

implying

Substituting this solution into the first FOC, we can solve for

implying

implying

The internal price of land will then be

Page 5 of 10

(c) By the definition, the production exhibits CRS. The Long-run cost function is

Take Then the FOC's are

implying

which imply that and Substituting these into the constraint, we can solve for

and then and

implying

implying

Competitive market ensures zero profit, which requires that

in the long run. This means that no matter how much the firm produces the profit is always

zero. Therefore, the output is indeterminate, meaning that the firm may produce any

amount.

Answer 1.2. For any and let and We then have

Therefore, where the equality for is already given.

Answer 1.3. See Varian (2nd ed.) p.31–33, or Varian (3rd ed.) p.55–56.

Page 6 of 10

Answer 1.4. From Figure 1.2, we see that the minimum point is or depending

on the ratio of Therefore, the cost is or That is,

.cxwxw =+ 2211

ybxax =+ 21Isoquant

2x

1x/y a

Figure 1.2. Cost Minimization with Linear Technology

Answer 1.5. Since the production is not differentiable, we cannot use FOC to solve the prob-

lem. One way to do is to use a graph.

1x

2x

/y a

yb

( )y f x=

1 2ax bx=

Figure 1.3. Cost Minimization with Leontief Technology

From Figure 1.3, we see that the minimum point is Therefore, the cost function is:

Answer 1.6. See Varian, Example 2.16, p.55 and p.66.

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Answer 1.7. The cost function and the expenditure function in consumer theory are mathe-

matically the same.

Answer 1.8. (1) Since is linearly homogeneous and since ( , )

is homogeneous of degree in Similarly for

(2) By Hotelling’s lemma, we have

This immediately implies

which gives the second property.

(3) By the symmetry of the matrix we immediately have

Answer 1.9. If the demand system is a solution of a cost minimization problem, then it must

satisfy the properties listed in Proposition 1.6. Property (1) in the proposition is obviously

satisfied. Property (2) requires symmetric cross-price effects, that is,

or

Therefore, With the substitution matrix is

We have

Page 8 of 10

and

Thus, the substitution matrix is negative semi-definite. Finally, property (4) is implied by the

fact that the substitution matrix is negative semi-definite. Therefore, to be consistent with cost

minimization, we need and only need condition:

Let Then the cost function is

Answer 1.10. Since Assumption 1.1 is satisfied. Since

we have

Multiply the first column of the right determinant by and then add what you have got

to the jth column. This operation won’t affect the value of the determinant. Thus,

for any Therefore, also satisfies Assumption 1.2.

Answer 1.11. (a) By

we immediately see that the linear homogeneity of cost function implies that

(b) We have

Then,

Page 9 of 10

When the functions are differentiable, taking derivatives is often the easiest way to find mono-

tonicity and convexity.

Therefore, is upward sloping and is U-shaped.

Answer 1.12.

(a) The production set is defined by

which means that if the firm wants to produce it needs at least amount of labor.

Since the labor is an input, it should be negative in the definition of implicit production

function. This means that we can choose and define

The production process is then defined by for We first have

thus Assumption 1.1 is satisfied. The 2nd order conditions are

/ // /

and

Page 10 of 10

Therefore, Assumptions 1.2 is satisfied.

(b) The problem is

The solution is:

Therefore, the supplies are:

Page 1 of 10

Problem Set 2

Micro Analysis, S. Wang

Question 2.1. Show that strong monotonicity implies local nonsatiation but not vice versa.

Question 2.2. A consumer has a utility function

(a) Compute the ordinary demand functions.

(b) Show that the indirect utility function is

(c) Compute the expenditure function.

(d) Compute the compensated demand functions.

Question 2.3. Let ∗ be the consumer’s demand for good The income elasticity of

demand for good is defined as ∗( , )

Show that, if all income elasticities are constant

and equal, they must all be one.

Question 2.4. Show that the cross-price effects for ordinary demand are symmetric iff all

goods have the same income elasticity: ∗( , ) ∗( , )

Question 2.5. A consumer has expenditure function / What is the value

of ?

Question 2.6. Suppose the consumer’s utility function is homogeneous of degree 1. Show

that the consumer’s demand functions have constant income elasticity equals 1.

Page 2 of 10

Question 2.7. Use the envelope theorem to show that the Lagrange multiplier associated

with the budget constraint is the marginal utility of income; that is, ( , )

Question 2.8. Suppose that the consumer's demand function for good has constant income

elasticity Show that the demand function can be written as ∗ ∗

Question 2.9. Consider the substitution matrix ̅ ( , )

of a utility-maximizing consumer.

(a) Show that ( ̅) ̅ ( , )

(b) Conclude that the substitution matrix is singular and that the price vector lies in its null

space.

(c) Show that this implies that there is some entry in each row and column of the substitution

matrix that is nonnegative.

Question 2.10. An individual has a utility function for leisure and food of the form:

Suppose that the individual has an income with wage rate and price of food

(a) Derive the individual's compensated demand functions for food and leisure.

(b) Verify Shephard's lemma and Roy's identity for this individual's demand functions.

(c) Suppose that there is an increase in the price of food. Divide the total effect on the con-

sumer demand for leisure into income and substitution effects.

(d) Is there a price of food at which a further rise in the price will lead to a decrease in con-

sumer demand for leisure?

Question 2.11. One popular functional form in empirical work for ordinary demand func-

tions ∗ and ∗ is the double logarithmic demand system: ∗∗where is the income and the price vector. The parameters are unknown

and are to be estimated.

Page 3 of 10

(a) Interpret and in terms of elasticity, where the price elasticity of demand for

good is ∗ ∗ and the income elasticity of demand for good is ∗ ∗

(b) Show that in order that the above demand functions can be interpreted as having been

derived from utility maximizing behavior, the following parameter restrictions must be

imposed:

If good 1 is a normal good and is not a Giffen good, are there additional parameter re-

strictions implied by this fact? If goods 1 and 2 are gross substitutes, are there additional

parameter restrictions?

Question 2.12. A consumer has an intertemporal utility function of present

consumption and future consumption He takes as given the spot prices He

can borrow and lend freely at an interest rate He has an initial endowment of

units of the commodity in the present and units of the commodity in the future.

(a) Find the utility-maximizing consumption bundle of the consumer, and compute his mar-

ginal rate of substitution between present and future consumption.

(b) What is the effect of a change in the interest rate on savings?

(c) Suppose, in addition to his endowment, the consumer owns a firm with a production

function where is the input in period 1 and is the output in period 2. (NOTE:

and are in the units of the commodity in period 1; and are in the units of the

commodity in period 2.) Determine the level at which the consumer will operate the firm

and the utility-maximizing consumption bundle he attains.

(d) Demonstrate that Fisher's Separation Theorem holds by showing that the problem can be

decomposed into two separate problems: a maximization of profits; and a maximization of

utility subject to a wealth constraint.

Page 4 of 10

Answer Set 2

Answer 2.1. Since in any neighborhood of we can always find a point such that and

strong monotonicity thus implies local nonsatiation.

Suppose the preferences are defined by It is easy to see that the

preferences satisfy local nonsatiation. But for two points and we have

and but That is, the preferences don’t satisfy strong monotonicity.

Answer 2.2.

(a) The consumer’s problem is

Let The FOC’s

imply Substituting this into the budget constraint will immediately give us

∗By symmetry, we also have

∗(b) Substituting the consumer’s demands into the utility function will give us

(c) Let i.e.

which immediately gives us the expenditure function:

(d) Substituting for in the consumer’s demand functions we get

Page 5 of 10

By symmetry,

Answer 2.3. Using the adding-up condition ∗we can take derivative w.r.t. on both sides of the equation to get: ∗implying ∗ ∗ ∗ ∗If then ∗that is,

Answer 2.4. By Shephard’s lemma,

By Slutsky equation, ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗where is the income elasticity of demand for good Similarly, ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗By (1) and the fact that then ∗ ∗

Page 6 of 10

Answer 2.5. Since is linearly homogeneous in

Answer 2.6. We can easily show that given that fact that

Then, is linearly homogeneous in and is homogene-

ous of degree in By Roy’s identity, we then have

∗ ∗Taking the derivative w.r.t. , we then have ∗ ∗Setting we then have

∗ ∗

Answer 2.7. The problem is

The Lagrange function for this problem is

We have

, Then by the Envelop Theorem,

Answer 2.8. Given

∗ ∗for all we have ∗∗ ∗Thus,

∗ ∗ ∗

Page 7 of 10

Therefore,

∗ ∗

Answer 2.9. (a) We have

By taking derivative w.r.t. on both sides of above equation, we have

(b) Part (a) implies

(3)

where

is the substitution matrix, and

By the assumption that (3) implies that must be singular. By the FOC

we then have

This means that

where denotes the null space of 's.

(d) For each by (2), since by assumption ( ̅)

one of the ̅

must be

nonnegative.

Page 8 of 10

Answer 2.10. (a) We have

implying

implying

(b) Taking derivatives w.r.t. the prices,

Therefore, the Shephard's Lemma is verified. From utility maximization, we can find the

consumer demand functions:

∗ ∗From the expenditure function,

implying

Therefore, the Roy's Identity is verified.

(c) We have

∗ ∗

(e) We see that the two effects cancel out, and thus the total effect ∗ is zero. That is,

changes in the price of food will not affect the demand for leisure.

Page 9 of 10

Answer 2.11. (a) We have ∗∗

(b) For any since ∗ is homogeneous of degree we have ∗ ∗∗

Therefore, Similarly, using the 2nd equation, we also have

Normality implies that ∗

We hence have

∗ ∗ ∗Since good 1 is not a Giffen good,

∗ We hence have

∗ ∗ ∗If good 1 is a substitute for good 2, then

∗ We hence have

∗ ∗ ∗If good 2 is a substitute for good 1, then

∗ We hence have

∗ ∗ ∗

Answer 2.12. (a) The consumer's problem is

The marginal rate of substitution between present and future consumption is

This should be equal to the price ratio at the optimal consumption levels. That is, √

Thus, ∗ and hence ∗ from the budget constraint.

(b) By (a),

∗

Page 10 of 10

Then, by the budget constraint,

∗which implies that ∗ decreases as increases, and hence savings ∗ increases as in-

creases. This is what we would expect in reality.

(c) The consumer's problem is

where This problem can be reduced to the following problem by eliminating and

using the two restrictions:

, We then have ∗ and ∗ Then, ∗ and ∗

(d) The profit maximization problem is

which gives solution ∗ ∗ The problem of utility maximization subject to

wealth constraint is

which gives solution ∗ ∗ Since the solutions in (d) and (c) are the same, Fisher's

Separation Theorem is verified.

Page 1 of 8

Problem Set 3

Micro Analysis, S. Wang

Question 3.1. Suppose that an expected utility function has constant absolute risk aversion ( )( ) What must the form of the utility function be?

Question 3.2. Given any constant and a zero-mean random variable define by

Denote Derive

Question 3.3. For a quadratic utility function show that the ex-

pected utility of a random payoff is a function of the mean and variance of

Question 3.4. A sports fan’s preferences can be represented by an expected utility. He has

subjective probability that the Lions will win their next football game and probability

that they will not win. He chooses to bet on the Lions so that if the Lions win, he wins

and if the Lions lose he loses The fan's initial wealth is

(a) How can we determine his subjective odds by observing his optimal bet ∗

(b) Under what condition does an increase in lead to a higher bet ∗

Question 3.5. Suppose that a consumer has a differentiable expected utility function for

money with The consumer is offered a bet with probability of winning and

probability of losing Show that, if is small enough, the consumer will always take the bet.

Page 2 of 8

Question 3.6. Let individual A have an expected utility function and let individual B

have an expected utility function where is income. Let be a monotonic in-

creasing, strictly concave function, and suppose that That is, is a concave

monotonic transformation of

(a) Show that individual A is more risk-averse than individual B in the sense of the absolute

risk aversion.

(b) Let be a random variable with Define “risk premiums” and by

Here is initial wealth. If show that

(c) Interpret the risk premium in words.

Question 3.7. For Exercise 3.4, when the probability of winning goes up, do you expect

the amount ∗ that a person is willing to gamble to go up? Prove your claim.

Question 3.8. Suppose a farmer is deciding to use fertilizer or not. But there is uncertainty

about the rain, which will also help the crops. Suppose that the farmer's choices consist of two

lotteries:

Suppose that the farmer is an expected utility maximizer and has monotonic preferences.

What would the farmer choose if he were (i) risk loving? (ii) risk neutral? (iii) risk averse?

Question 3.9. What axiom is violated by

Question 3.10. Show that the following two utility functions — one is a monotonic transfor-

mation of the other — imply the same preferences with certainty consumption bundles, but

not with uncertainty consumption bundles:

Page 3 of 8

Question 3.11. For the insurance problem:

where is the loss, is the probability of the bad event, is the price of

insurance, is initial wealth, and

(a) If the insurance market is not competitive and the insurance company makes a positive

expected profit: will the consumer demand full-insurance ∗ under-

insurance ∗ or over-insurance ∗ Show your answer.

(b) Show the above solution on a diagram.

Page 4 of 8

Answer Set 3

Answer 3.1. We have

where is some constant. Then

where and are some constants. Therefore, for ( )( ) if and only if there

are two constants and such that

Answer 3.2. By definition,

(A)

By Taylor's expansion,

and

Equalizing above two formulae immediately implies an approximated solution of

Answer 3.3. We have

Answer 3.4. (a) The individual problem is

Page 5 of 8

The first-order condition implies that

∗ ∗ (1)

implying ∗∗By knowing ∗ and can then be determined using above equation.

(b) By taking the derivative w.r.t on the FOC (1), we get

∗ ∗ ∗∗ ∗ (2)

By (2), for a risk averse person with increasing utility function we have ∗

Answer 3.5. We need to show that

(3)

when is small for a differentiable utility function with ( may not be concave). By

Taylor expansion, there are and such that

Therefore, (3) is true if and only if

Letting we have and and then

Therefore, when is small, (3) is true.

Answer 3.6. Since

if we have

Page 6 of 8

(b) We know that if is a convex function, then1 By definition,

Since is concave, is convex. Therefore,

Assuming is strictly increasing, then

(d) The risk premium is the maximum amount of money that an expect utility maximizer

is willing to pay to avoid risk.

Answer 3.7. For a risk averse person with increasing utility function, the answer is Yes. The

first-order condition is ∗ ∗By taking the derivative w.r.t. on above equation, we get ∗ ∗ ∗∗ ∗Of course, for a risk loving person with increasing utility function, the opposite is true.

Answer 3.8. If he is risk loving, then

Since by monotonicity this farmer will choose “fertilizer.” If he is

risk neutral, then he only cares about the expected income. Since

this farmer will still choose “fertilizer.” If he is risk averse, then

1For those who want to know, let 1 2 be a partition of the value space of the random

variable and the probability of Then, by the continuity and convexity of we

have

→ → →

Page 7 of 8

this farmer's choice will depend on his particular preferences. From the given information, we

don't know what this farmer will choose.

Note that by comparing the two distribution functions, the two lotteries don't dominate

each other by FOSD or SOSD. Thus, stochastic dominance cannot help determine the prefer-

ences.

Answer 3.9. If RCLA were not violated, then

which would immediately imply a contradiction. Therefore, RCLA must has been violated.

Answer 3.10. Let Then Since is a strictly increasing

function, and are equivalent over certainty consumption bundles. But for uncertainty

consumption bundles:

we have

Hence, and are not equivalent over uncertainty consumption bundles.

Answer 3.11.

(a) At the optimal point ∗ ∗ ∗∗The expected profit is Then, Thus,

Then, ∗ ∗ or ∗ ∗ i.e., ∗ ∗ It implies ∗ that is, we

have under-insurance.

(b) When in Example 1.12, we have shown that the solution must be on the ∘ line.

When the budget line is flatter, and the tangent point must be below the ∘ line. That

is, the individual is under-insured.

Page 8 of 8

I

I

1

2

w

w-l

slope=

slope=1-pp

45o

..

1-ππ

.

Figure 5.1. Insurance in a non-competitive market

1/8

Problem Set 4

Micro Analysis, S. Wang

Question 4.1. There are two consumers A and B with utility functions and endowments:

Calculate the GE price(s) and allocation(s).

Question 4.2 (PhD). We have agents with identical strictly concave utility functions. There

is some initial bundle of goods Show that equal division is a Pareto efficient allocation.

Question 4.3 (PhD). We have two agents with indirect utility functions

and initial endowments

Calculate the GE prices.

Question 4.4 (PhD). Suppose that we have two consumers and with identical utility

functions

Suppose that the total available amount of good 1 is and the total available amount of good 2

is i.e., Draw an Edgeworth box to illustrate the strongly Pareto optimal and the

(weakly) Pareto optimal sets.

Question 4.5. Consider a two-consumer, two-good economy. Both consumers have the same

Cobb-Douglas utility functions:

2/8

There is one unit of each good available. Calculate the set of Pareto efficient allocations and

illustrate it in an Edgeworth box.

Question 4.6. Consider an economy with two firms and two consumers. Denote as the

number of guns, as the amount of butter, and as the amount of oil. The utility functions for

consumers are . .Each consumer initially owns units of oil: Consumer 1 owns firm 1 which has

production function consumer 2 owns firm 2 which has production function

Find the general equilibrium.

Question 4.7. Suppose that there are one consumer, one firm, and one good The firm is

owned by the consumer. The consumer has an endowment of unit of time for working and

enjoying leisure, and has utility function for good and leisure

time The firm inputs amount of labor to produce amount of good. Find the GE.

Question 4.8. Suppose that the economy is the same as in Question 4.7 except that the firm

has production function Find the GE.

Question 4.9. There are two goods and with prices and respectively, and two indi-

viduals and with and

(a) Derive the contract curve. Suppose Draw it in an Edgeworth box.

(b) Derive the GE price ratio(s)

Question 4.10. There are two goods and and two individuals

with and

(a) Find all the Pareto optimal allocations. Are they strongly Pareto optimal?

(b) Find all the GE price ratio(s)

3/8

Answer Set 4

Answer 4.1. Individual A’s utility function is equivalent to Let

and Then the income is and the demands are:

For individual B, by its utility function, we know that the demands must satisfy Then

by budget constraint the demands are:

In equilibrium, the total supply of good 1 must be equal to the total demand for good 1:

Therefore, ∗ and the allocation is

∗ ∗ ∗ ∗

Answer 4.2. Denote If is not Pareto optimal, then there is

another allocation such that

(1)

and

By (1), Then, by concavity of

By (2), Then above inequality implies This is a

contradiction. Therefore, allocation must be Pareto optimal.

Answer 4.3. Let and Then the incomes are By Roy's Identity,

In equilibrium, the total supply of good 1 must be equal to the total demand of good 1:

4/8

Therefore, the equilibrium price ratio is:

∗

Answer 4.4. In the following charts, the left chart indicates the Edgeworth box and the indif-

ference curves. The right chart indicates the Pareto optimal points.

A

B

Bu

Au

A

BF

E D

C

Figure 1. Pareto Optimal Points

As indicated by the right chart, the set of weakly P.O. points consists of five intervals AC,

CD, DE, EF, and FB:

the set of strong P.O. points consists of only two points C and F:

Answer 4.5. By Proposition 1.27, the following equation defines the set of P.O. points:

Feasibility requires

Let and Then above two equations imply

Therefore,

5/8

This set is the diagonal line in the following diagram.

1

2

y=xP.O.

y

x

Figure 2. P.O. Allocations

Answer 4.6. Denote price of guns price of butter price of oil (we can arbitrarily choose one of prices. We can do that because of the homogeneity of demand functions). The two consumers are: . .. .

Firm 1’s problem:

It implies

Note that the only possible equilibrium is when Zero-profit argument is not accurate

here.

Firm 2’s problem:

It implies

Consumer 1’s problem:

, . .Its solution is

6/8

Consumer 2’s problem:

, . .The solution is

Market clearing conditions:

Because of Walras Law, we only need two of these three conditions to determine the equilibri-

um. They imply that ∗ and ∗ Therefore, the equilibrium is:

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

Answer 4.7. Firm's problem:

, The solution is

The only possible equilibrium is when We thus only consider

Consumer's problem:

, gives solution

Market clearing conditions:

Because of Walras Law, we only need to use one of conditions to determine the equilibrium.

The first condition implies that ∗ Then, ∗ ∗ and ∗ implies ∗

Therefore, the equilibrium is:

7/8

∗ ∗ ∗

Answer 4.8. We can arbitrarily set Firm’s problem:

, gives

Consumer’s problem:

, gives solution

Market clearing conditions:

Because of Walras Law, we only need to use one of conditions to determine the equilibrium. It

implies that ∗ Therefore, the equilibrium is:

∗ ∗ ∗

Answer 4.9. (a) We have

which gives the contract curve as goes from to We have

8/8

y

xy

ω

A

B

.x

Au

Bu

.

Contract curve

Figure 3. Contract Curve and Equilibrium

(b) We have

Equilibrium condition

implies

which can be solved to get the equilibrium price ratio

Answer 4.10. (a) The contract curve is the diagonal line in the chart. The points on the con-

tract curve are strongly P.O.

.

x

y

1

2

contract line

W

1u.

2u

Figure 4. Contract Curve and Equilibria

(b) The set of equilibria is That is, all the possible values of are

equilibria.

Page 1 of 1

Problem Set 5

Micro Analysis, S. Wang

There are no exercises for Chapter 5.

Page 1 of 8

Problem Set 6

Micro Analysis, S. Wang

Question 6.1. You have just been asked to run a company that has two factories producing

the same good and sells its output in a perfectly competitive market. The production function

for each factory is:

Initially, the capital stocks in the two factories are respectively and The

wage rate for labor is and the rental rate for capital is In the short run, the capital stock

for each factory is fixed, and only labor can be varied. In long run, both capital and labor can

be varied.

(a) Find the short-run total cost function for each factory.

(b) Find the company’s short-run supply function of output and demand functions for labor.

(c) Find the long-run total cost function for each factory and the long-run supply curve of the

company.

(d) If all companies in the industry are identical to your company, what is the long-run indus-

try equilibrium price?

(e) Let Suppose the cost of labor services increases from to per unit. What is

the new long-run industry equilibrium price? Can you determine whether the quantity of

capital used in the long run will increase or decrease as a result of the increase in the wage

rate from to ?

Question 6.2. Suppose that two identical firms are operating at the cooperative solution and

that each firm believes that if it adjusts its output the other firm will adjust its output to keep

its market share equal to What kind of industry structure does this imply?

Question 6.3. Consider an industry with two firms, each having marginal costs equal to zero.

The industry demand is

where is total output.

Page 2 of 8

(a) What is the competitive equilibrium output?

(b) If each firm behaves as a Cournot competitor, what is firm 1’s optimal output given firm 2’s

output?

(c) Calculate the Cournot equilibrium output for each firm.

(d) Calculate the cooperative output for the industry.

(e) If firm 1 behaves as a follower and firm 2 behaves as a leader, calculate the Stackelberg

equilibrium output of each firm.

Question 6.4. Consider a Cournot industry in which the firms’ outputs are denoted by

aggregate output is denoted by the industry demand curve is denoted by

and the cost function of each firm is given by For simplicity, assume

Suppose that each firm is required to pay a specific tax of on output.

(a) Devise the first-order conditions for firm

(b) Show that the industry output and price only depend on the sum of tax rates

(c) Consider a change in each firm’s tax rate that does not change the tax burden on the indus-

try. Letting denote the change in firm ’s tax rate, we require that Assum-

ing that no firm leaves the industry, calculate the change in firm ’s equilibrium output

[Hint: use the equations from the derivations of (a) and (b)].

Question 6.5. (Entry Cost in a Bertrand Model). Consider an industry with an entry cost. Let

where and are two constants. Find the equilibrium solution for the following two-

stage game.

Stage 1. All potential firms simultaneously decide to be in or out. If a firm decides to be in, it

pays a setup cost

Stage 2. All firms that have entered play a Bertrand game.

Question 6.6. Verify the socially optimal number of firms to be ( ) // in Section

6.9 of the book.

Page 3 of 8

Answer Set 6

Answer 6.1. (a) For each factory with capital stock

Therefore, the short-run cost functions are

(b) The firm cares about the total profit from its two factories. The objective of firm is

therefore to maximize the total profit:

, The FOCs give us the well-known equality:

We have and Then and imply that

and Thus, and Therefore, the short-run supply function

is:

The labor demands for the factories are:

Therefore, the labor demand is

(c) The cost for each factory is

, The Lagrange function is

implying

The total cost is then

Page 4 of 8

From the profit function we immediately find the long-run

supply function:

That is, the long-run industry supply curve is horizontal. In this case, the equilibrium output is

determined by demand (which is not given).

(d) In a competitive market, with a horizontal industry supply curve, the long-run equilib-

rium price must be whatever the industry demand curve is.

(e) The original long-run equilibrium price is and the new price is The to-

tal capital investment is

With an increase in and output is reduced. With going down and going up, the

change is in is ambiguous; it demands on the demand.

p

p

y

sy

D

..

Answer 6.2. Let be the market price of the good when the output is is the cost of

firm when its output is The two firms have the same cost function. The cartel maximizes

their total profit:

, The FOCs are

∗ ∗ ∗ ∗We look for a solution for which ∗ ∗ (the symmetric solution). Thus, the FOC becomes

Page 5 of 8

∗ ∗ ∗ ∗We can rewrite (2) as

∗ ∗where On the other hand, the Cournot output is determined by

∗ ∗ ∗ ∗

.

p

Y

B A

)(YMR

2Yc

æ ö÷ç¢ ÷ç ÷çè ø

D1( ) ( )2

MR Y p Y Y¢-

.C .

Figure 6.1. A market-share Cournot equilibrium

In the diagram, point is the ‘competitive solution,’ for which each firm takes the market

price as given; point is our solution, for which each firm acts upon a decreasing demand and

assume equal market share as the other’s reaction; point is the Cournot equilibrium.

From the diagram, we can conclude that

• The equilibrium output at is lower than the output at the ‘competitive solution’ and the

output at the Cournot equilibrium.

• The equilibrium price at is higher than the price at the ‘competitive solution’ and the

price at the Cournot equilibrium.

Answer 6.3. (a) For competitive output, firms take price as given in maximizing their own

profits:

which implies

∗

Page 6 of 8

That is, the firms’ supply curve is the horizontal line at So is the industry supply curve.

The equilibrium industry supply is thus ∗ and the equilibrium price is ∗

(b) Firm 1 maximizes his own profit, given any

which gives the FOC:

Firm 1’s reaction function is thus

(c) By symmetry, the outputs for the two firms should be the same in equilibrium. By the

reaction function in (b), we hence have which gives Therefore, the

Cournot equilibrium is

∗ ∗(d) Suppose the two firms collude. They form a monopoly and maximizes their total profit:

which gives the cartel output: ∗

(e) Firm 1 will behave as in (b), and reacts according to his reaction function

Firm 2 will take this into consideration when maximizing his own profit:

which implies ∗ Then, ∗

In summary, the competitive industry output is the highest, the Stackelberg industry out-

put is the second, the Cournot industry output is the third, and cartel output is the lowest.

Answer 6.4. (a) The profit maximization for firm is

The FOC is

(3)

(b) By summarizing (3) from to we have

(4)

Page 7 of 8

This equation determines the industry output which obviously depends on rather

than the individual tax rates ’s.

(c) Since the total output depends only on and the latter has no change, doesn’t

change for a tax change. Then, by (3), i.e.,

where is determined by (4).

Answer 6.5. This is from Example 12.E.2 on page 407 of MWG (1995). Once identical firms

are in the industry, they play a Bertrand game. As we know, if the result is the competi-

tive outcome, i.e., ∗ and the profit without including the entry cost is zero for all the

firms. This means that each firm loses in the long run. Knowing this, once one firm has

entered the industry, all other firms will stay out. Therefore, more intense competition in stage

2 results in a less competitive industry!

This single firm will be the monopoly and produces at the monopolist output re-

sulting the monopoly price The monopoly profit is

As long as a firm will enter and that is the only firm in the industry.

Answer 6.6. We have

where Then,

Page 8 of 8

implying /implying /

/ //

1/12

Problem Set 7

Micro Analysis, S. Wang

Try to do more problems in MWG (1995), Chapters 7–9.

Question 7.1 (Mixed-Strategy Nash Equilibrium) (PhD). A principal hires an agent to per-

form some service at a price (which is supposed to equal the cost of the service). The principal

and the agent have initial wealth and respectively. The principal can poten-

tially lose If the agent offers low quality, the probability of losing is if the

agent offers high quality, the probability of losing is The quality is unobservable to

the principal. The price of a low quality product is (paid to the agent) is and the price

of a high quality product is by the competitive market assumption, and are the

costs of producing the products (the agent bears the costs). The agent is required by regulation

to provide high-quality services, but he may cheat. After such a bad event happens, the princi-

pal can spend in an investigation; if the agent is found to have provided low-quality

services, the agent will have to pay for the loss to the principal. This game can be written in

the following normal form:

low quality, high quality,

investigate,

not to investigate,

where

Find the mixed-strategy Nash equilibria.

Question 7.2 (Pure-Strategy Nash Equilibrium) (PhD). Find the pure-strategy Nash equilib-

ria in the above exercise.

2/12

Question 7.3. For the following game, find the pure-strategy NEs. Show whether or not they

are trembling-hand perfect.

Player 2

1, 6 0, 5

1, 1 1, 2

Question 7.4 (PhD). For the following game (Mas-Colell et al. 1995, p.271), find all the pure-

strategy Nash equilibria.

.

312

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø

156

æ ö- ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø

201

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø

.

. .

544

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø

017

æ ö÷ç ÷ç ÷ç- ÷ç ÷ç ÷÷ç ÷çè ø

220

æ ö- ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø

1L 1R

l r 2L 2R

P1

P2P3

P3 P3

rrl l

Question 7.5. In the following game, explain why there are mixed-strategy NEs in which P1

mixes and arbitrarily and P2 chooses

o

11

æ ö- ÷ç ÷ç ÷÷ç-è ø

P1

. .

12

æ ö÷ç ÷ç ÷÷çè ø

10

æ ö- ÷ç ÷ç ÷÷çè ø

11

æ ö÷ç ÷ç ÷÷çè ø

P200

æ ö÷ç ÷ç ÷÷çè ø

H

1L 1M 1R

2L 2R 2R2L

11s 21s31s

1m 2m

3/12

Question 7.6. Consider the following game.

o

21

æ ö- ÷ç ÷ç ÷÷ç-è ø

P1

. .

12

æ ö÷ç ÷ç ÷÷ç-è ø

11

æ ö- ÷ç ÷ç ÷÷çè ø

23

æ ö÷ç ÷ç ÷÷çè ø

P200

æ ö÷ç ÷ç ÷÷çè ø

H

1L

1L̂

2L 2R 2R2L

1m 2m

.1R̂

1R

P1

x

(a) Find all pure-strategy NEs.

(b) Find all SPNEs.

(c) Find all BEs.

(d) Are all the BEs subgame perfect?

Question 7.7. Find all the mixed strategy SPNE in the following game.

o

.

−−

66

Firm E

Firm E

. .

−11

− 1

1

−−

33

Out In

SmallNiche

PayoffsI' Firm Payoffs E'Firm

Firm I

20

SmallNiche

SmallNiche

LargeNiche

LargeNiche

LargeNiche

IH

1x

4/12

Question 7.8. For the following game, find all the pure-strategy NE, all the SPNEs and all the

BEs.

o

.

−−

13

Firm E

Firm E

. .

− 2

1

−−

12

13

Out In

Fight Fight

Fight Accom

I

E

ππ

Firm I

20

Accom Accom

m

Iσ IσIσ−1 Iσ−1

E2σ

E1σ

E21 σ−

E11 σ−

IH

z

1 m-

Question 7.9 (PhD). A revised version of Exercise 9.C.7 in Mas-Colell et al. (1995, p.304)].

(a) For the following game, find all the pure-strategy NEs. Which one is a SPNE?

o

P2

24

..

P1

22

1δ 2δ

1γ 2γ

1δ 2δ

B T

D U D U

11

15

P2

Figure 7.1. NEs and SPNEs

(b) Now suppose that P2 cannot observe P1’s move. Draw the game tree, and find all the

mixed-strategy NEs.

5/12

Question 7.10 (PhD). One problem with a BE is that it may not be trembling-hand perfect.

Consider the following game.

o

..

12

æ ö÷ç ÷ç ÷÷çè ø33

æ ö÷ç ÷ç ÷÷çè ø

02

æ ö÷ç ÷ç ÷÷çè ø

01

æ ö÷ç ÷ç ÷÷çè ø

P1

P22μ1μ

1L 1R

2L 2R 2L 2R

Figure 7.2. Trembling-Hand Perfect Equilibrium

(a) Show that we have the following BE: ∗ ∗ ∗ ∗ ∗with payoff pair

(b) Show that this BE is a SE. Note that we already know in Example 7.10 that this strategy

profile ∗ ∗ is not trembling-hand perfect.

6/12

Answer Set 7

Answer 7.1. Assume that the principal can commit ex ante to investigate or not before a loss

occurs. In other words, the principal can only make up his mind on investigation before she

has suffered a loss. Before a loss occurs, the game box of surpluses is

low quality, high quality,

investigate,

not to investigate,

In each cell, the value on the left is the surplus of the principal and the value on the right is the

surplus of the agent.

The optimal choice of is to make the principal indifferent between investigation and no

investigation:

(1)

implying

implying

implying

The choice of is to make the agent indifferent between cheating and no cheating:

(2)

implying

7/12

Answer 7.2. By substituting the parameter values into the game box of surpluses, we have

cheat, not to cheat,

investigate,

not to investigate,

By Proposition 7.2 in the book, to find pure-strategy Nash equilibria, we can restrict to pure

strategies only. Thus, simply by inspecting each cell one by one, we know that there is no pure-

strategy Nash equilibrium.

Answer 7.3. This is a situation in which a player is indifferent from two alternative strategies,

one of which is the equilibrium strategy. This player has no incentive to deviate if other players

don’t make any mistakes. However, the situation changes if possible mistakes by other players

are taken into account. There two NEs: and In given player 1 is

indifferent from and However, if player 2 may make some mistakes by taking with

probability no matter how small is, player 1 will be strictly prefer to Thus,

is not a trembling-hand NE, while is.

Answer 7.4. The strategy sets for players 1 and 2 are simple:

There are three information sets for player 3. Denote a typical strategy of player 3 as

where is the action if the information set on the left is reached, is the action if

the information set in the middle is reached, and is the action if the information set on the

right is reached. Player 3 has eight strategies:

The normal form is

P1 plays P3

P2: 2,0,1 -1,5,6 2,0,1 -1,5,6 2,0,1 -1,5,6 2,0,1 -1,5,6

2,0,1 -1,5,6 2,0,1 (-1,5,6) 2,0,1 -1,5,6 2,0,1 (-1,5,6)

8/12

P1 plays P3

P2: 3,1,2 3,1,2 3,1,2 3,1,2 (5,4,4) (5,4,4) (5,4,4) (5,4,4)

0,-1,7 0,-1,7 -2,2,0 -2,2,0 0,-1,7 0,-1,7 -2,2,0 -2,2,0

All the pure strategy Nash equilibria are indicated in the boxes.

To find all the Nash equilibria, we can check each cell one by one. A cell cannot be a Nash

equilibrium if one of the players doesn’t stick to it. In each cell, we can first check to see if

player 3 will stick to his strategy, by which we can quickly eliminate many cells.

A sequentially rational NE must be an outcome from backward induction. Example 7.14 in

the book shows that backward induction only leads to one outcome:

which is one of the Nash equilibria.

Answer 7.5. Whatever P2 does, and are indifferent to P1. On the other hand, whatever

P1 does, is always better to P2.

Answer 7.6.

(a) P2 has one information set containing two nodes. P1 has two information sets

and where contains the initial node. Denote P1’s strategies as where is

an action at and is an action at The normal form, where the payoff profile in each cell

is (P2’s payoff, P1’s payoff), is:

P2\P1

(0, 0) (0, 0) -1, -2 1, -1

0, 0 0, 0 -2, 1 (3, 2)

The pure-strategy NEs are indicated in the above table.

(b) Since in the real subgame SG(x), there is only one NE in SG(x). Hence,

there is one SPNE, which is

9/12

(c) Let us find BEs. For P2, iff or If so, P1 choos-

es at node Then, since choosing means a payoff of P1 chooses at the beginning.

Since is not on the equilibrium path in this case, any belief is acceptable. Hence we have a

BE:

∗ ∗ ∗If then then P1 chooses at and then P1 choose at the initial node.

In this case, consistency is required and it implies which can be satisfied. Hence, there

is another BE: ∗ ∗ ∗Further, if P2 is indifferent between and that is, P2’s strategy can be any

mixed strategy with Then, at node P1’s preference would be

iff However, this is completely impossible. We in fact always

have Hence, P1 will always choose at Then, P1’s preference at the initial point is

iff , i.e., . Hence, if P1’s strategy is

Since is not on the equilibrium path, any belief is acceptable. Thus, we have a BE:

∗ ∗ ∗If is on the equilibrium path, by which consistency requires This is impossi-

ble. Hence, there is no BE in this case. If P1 is indifferent between and . In this

case, if P1 takes with a positive probability, consistency is required and it cannot be satisfied.

If P1 takes for sure, consistency on is not required and hence can be allowed.

Hence, we have another BE:

∗ ∗ ∗This BE4 can be combined with BE3.

(d) Since the BE1, BE3 and BE4 (the strategies of these BEs) are not the SPNE, we con-

clude that BEs may not be SPNEs.

Answer 7.7. In the proper subgame with the normal form:

Firm I

Small, Large,

Firm E: Small, -6, -6 (-1, 1)

Large, (1, -1) -3, -3

The equilibrium is to make firm E indifferent between his two strategies:

10/12

implying ∗ Since the game is symmetric, we also have ∗ Then, the expected payoff is ∗ ∗ ∗ We also have ∗ The game is reduced to:

o

Out In

−−

919

919

Firm E

20

Then, firm E will choose ‘out.’ Thus, the SPNE is

∗∗

Answer 7.8. Firm I has one information set containing two nodes. Based on this

information, firm I has two strategies:

Firm E has two information sets and where contains the initial node. Denote firm E’s

strategies as where is an action at and is an action at We can then

find the normal form:

Firm E

<out, fight> <out, accom> <in, fight> <in, accom>

Firm I: fight (2, 0) (2, 0) -1, -3 -1, -2

accom 2, 0 2, 0 -2, 1 (1, 3)

We can easily find the pure-strategy Nash equilibria, as indicated in the above box:

There is only one SPNE, which is NE3, i.e.,

One of BEs is

This example indicates that BE and SPNE don’t imply each other: BE eliminates two NEs, one

of which is SPNE; SPNE also eliminates two NEs, one of which is BE.

11/12

There are three BEs:

BE2 is the same as the SPNE.

This example indicates that BE and SPNE don’t imply each other: BE eliminates NE1;

SPNE eliminates NE1 and NE2, one of which is BE. In other examples, we also know that BE

sometimes eliminates SPNEs.

Answer 7.9. (a) There are two information sets for P2. Let be a typical P2’s strate-

gy, where is an action taken at the left information set and is an action taken at the right

information set. The normal form of the game is

P2

<D, D> <D, U> <U, D> <U, U>

P1: B 4, 2 (4, 2) 1, 1 1, 1

T 5, 1 2, 2 5, 1 (2, 2)

There are two pure-strategy NEs: ∗ and ∗ The first one is a

SPNE.

(b) The game tree is:

o

P2

24

..

P1

22

1δ 2δ2H

1γ 2γ

1δ 2δ

B T

D U D U

11

15

The normal form is

P2

D U

P1: B 4, 2 1, 1

T 5, 1 (2, 2)

12/12

There is a pure-strategy NE: ∗ Since playing is a strictly dominant strategy for P1,

this NE is the NE.

Answer 7.10. It is simple. You do by yourself.

Page 1 of 6

Problem Set 8

Micro Analysis, S. Wang

Question 8.1 (Gibbons 1992, p.250, Exercise 4.10). It is a buyback solution to dissolve a

partnership. Partners 1 and 2 own shares and of the partnership, respectively. Partner 1

is to name a price and then partner 2 chooses either to buy partner 1’s share for or to sell

his share to partner 1 for Assume that partner 1’s value of the firm is if she owns

the whole firm and zero otherwise; and partner 2’s value of the firm is if he owns the whole

firm and zero otherwise. Suppose that each partner’s valuation is private information and the

other partner only knows the distribution only. Suppose and independently follow

(the uniform distribution on ) What is the BNE?

Question 8.2 (Gibbons 1992, p.250, Exercise 4.11). A buyer and a seller have valuations

and respectively. The buyer’s valuation is with known parameter The seller

knows her own valuation (and hence but the buyer doesn’t know The buyer knows

that the seller’s valuation follows (the uniform distribution on ). The buyer

makes a single offer which the seller either accepts or rejects. Find the BNE.

Question 8.3. (Gibbons 1992, p.253, the first part of Exercise 4.15) (PhD). Consider a legisla-

tive process in which a feasible policy is The status quo is and the ideal policy for

the Congress is where The ideal policy for the president is which is private

information of the president. The Congress only knows that follows The Congress

proposes a policy and the president either signs or vetoes. Given a policy the payoffs of the

Congress and the president are respectively and Find the

BNE ∗ and verify ∗ in equilibrium.

Question 8.4 (A cheap-talk game) (PhD).1 The basic game setup is the same as in Question

8.3. Now, suppose that the president can engage in rhetoric (send a cheap-talk message)

before the Congress proposes a policy. Consider a two-step PBE in which the president sends a

message in the first period and the Congress proposes based on a belief which

1 This is from Gibbons (1992, p.253, the second part of Exercise 4.15). Ignore this exercise if I didn’t cover

cheap-talk games in class.

Page 2 of 6

is the probability that the president has type when message is observed. The president

may take a pure strategy or a mixed strategy with and

(a) Define the PBE when the president takes a pure strategy.

(b) Define the PBE when the president takes a mixed strategy.

(c) Show that

In equilibrium, there are only two possible proposals and .

Derive the PBE and shows that

Page 3 of 6

Answer Set 8

Answer 8.1. Partner 1’s problem is

( ) ( )

The FOC is

implying ∗This ∗ is the BNE. In equilibrium, who owns the firm? Partner 1 owns the firm if ∗ ∗

or

otherwise partner 2 owns the firm.

Note that, in the above, we assume that partner 1 decides the price and partner 2 decides

whether to sell. If both partners have the right to decide whether to sell or buy, in order for

partner 1 to have the firm, partner 1 should be willing to buy (when

and partner 2 is willing to sell (when in order

for partner 2 to have the firm, partner 1 should be willing to sell (when

and partner 2 is willing to buy (when In this

case, the firm goes to partner 1 iff and it goes to partner 2 iff This

situation is complicated.

Answer 8.2. If and only if the seller will accept the price offer. Hence, the buyer’s

problem is

We have

Hence, the optimal pricing is

Page 4 of 6

∗Therefore, there is no trade if there is a trade if and there may or may not be a

trade if

Answer 8.3. If and only if the president will sign the proposal. Hence,

the Congress’s problem is

( ) ( ) ( ) ( )We have

( ) ( )( )( )

If then

Then, the FOC for is

or

The left-hand side is positive. But, since the right-hand side is negative. It is impos-

sible. Hence, we must have ∗

With we have

The FOC for is

implying

implying

implying

implying

Page 5 of 6

Hence,

∗We have ∗ iff

or

which is always true. Hence, we indeed have ∗ .

Answer 8.4. This problem is from Matthews (1989, QJE, 347-369). When the Congress sees

message it has the belief the probability of type is with message The

Congress then responds with proposal The game is drawn below. In the figure, player P

is the president and player C is the Congress. Let be type president’s

payoff under policy and be the payoff of the Congress under policy

..

. .

Lt Ht

L R L R

Nature

PP

C C

La Ha

C Cp q 1 p- 1 q-

d 1 d-

La Ha La Ha La Ha

P P P P P P P P

Figure 1. A Free-Talk Game

(a) Following Gibbons (1992), we first consider a pure-strategy BE. Suppose that the pres-

ident plays a pure strategy In the second step, when the Congress sees it guesses that

the density of type is and its proposal is a solution of the following problem:

( , ) ( , ) ( , ) ( , ) (1)

Page 6 of 6

In the first step, knowing the Congress’s proposal the president considers how to send a

message. His problem is

(2)

This implies Let be the true density function of the type. The equilibrium consisten-

cy condition requires that, if is a message that is sent in equilibrium, i.e., for some

then

( ) (3)

Under three conditions (1)–(3), we have a BE:

(b) Following Crawford-Sobel (1982) and Matthews (1989), we now consider a mixed-

strategy BE. Suppose that the president plays a pure strategy. In the second step, when the

Congress sees it guesses that the density of type is and its proposal is a solu-

tion of the following problem:

( , ) ( , ) ( , ) ( , ) (4)

In the first step, knowing the Congress’s proposal the president considers how to send

his message strategy. He applies a mixed strategy where, for a message ∗ if there is a

such that ∗ then

∗ (5)

Let be the true density function of the type. The equilibrium consistency condition re-

quires that, if is a message that is sent in equilibrium, i.e., for some 2 then

(6)

Under three conditions (4)–(6), we have a BE:

(c) The mixed-strategy BE is the same as that in Matthews (1989). Hence, the solution can

be found in Matthews (1989).

2 Following Matthews (1989), an alternative to this

Page 1 of 3

Problem Set 9 Micro Analysis, S. Wang

Question 11.1. We have two agents with identical strictly convex preferences and equal en-

dowments. Describe the core and illustrate it with an Edgeworth box.

Question 11.2. For a two-good two-agent economy,

(a) Explain graphically that the core depends on the initial endowments.

(b) Is it true that if the initial allocation is already in the core, then it is the only point in the

core? Explain.

(c) Try to suggest some mild conditions under which the statement in (b) is correct.

Question 11.3. In a two-agent two-good economy, suppose that the two agents are identical

(with the same endowment and preferences) and they have strict monotonic and

strict convex preferences. Show that the initial endowment point must be in the core.1

1 Strict convexity of preferences means that: and for

Page 2 of 3

Answer Set 9

Answer 11.1. Using Figure 1 done in the book, one can easily figure out the core to be the

initial endowment point. The core contains a unique point, which is the initial endowment

point.

Answer 11.2. (a) The dependency of the core on the initial endowment point is shown

clearly by the following diagram.

y

x1

.

(a)

core

W

2 y

x1 .(b)

core

W

2y

x1

.

(c)

core

W

2

Figure 1. The Core

(b) No. Let with and We see in the

above diagram (b) that all the points on the diagonal line are in the core.

(c) The weakest conditions are strict quasi-concavity and strict monotonicity for all the

utility functions.

Answer 11.3. There are two alternative ways to prove.

Proof 1: Suppose is not in the core. Then, there is a feasible allocation that

blocks That is, and (or and By the strictly convexity of

the preferences,

By the feasibility, however, i.e., This contradicts with strict

monotonicity. Therefore, must be in the core.

Page 3 of 3

Proof 2: Obviously, no single person would block the distribution We thus only

need to show that it is also Pareto optimal, i.e., the whole society won't block it either. By

Proposition 4.4, the Pareto optimality of is

where is the demand of individual in good Obviously, the feasible allocation

satisfies the above two conditions, and is thus Pareto optimal. is thus in the core.

1/12

Problem Set 10 Micro Analysis, S. Wang

Question 10.1 (Akerlof). In the Akerlof model, we now suppose that the buyers can be guar-

anteed a minimum quality of the car by inspection and test drive. Specifically, instead of the

minimum quality for used cars in the market, suppose that all the cars have a minimum

quality

(1) Will adverse selection disappear?

(2) Is it possible to have cars with a range of qualities to be traded in the market?

Question 10.2 (Akerlof). In the Akerlof model, what would be the result if we changed the

buyer’s utility to

That is, the buyer’s MU for a car is now instead of How will such an increase in desire

for a car change the results? Explain your conclusion intuitively.

Question 10.3 (RS Insurance). Consider the RS insurance model under complete infor-

mation. The insurance company offers a price for an insurance policy that pays a compensa-

tion if an accident happens. Let

(a) Compute the demand functions and

(b) Compute the slopes of demand ( )

and ( )

and interpret.

(c) Under what price would a person demands full insurance, i.e., ?

Question 10.4 (RS Insurance). Consider the RS insurance model under asymmetric infor-

mation. Suppose that insurance companies offer price-quantity contracts. There are two types

of agents with type or The initial wealth for all agents is An agent with type has

the probability of losing an amount when the bad event happens. All agents have the same

initial wealth the same possible loss and the same utility function of income Let

2/12

(a) Compute the marginal rates of substitution for the two types and explain their relative

magnitudes.

(b) Compute the separating equilibrium, assuming its existence.

(c) Determine the condition under which the separating equilibrium survives.

Question 10.5 (Spence). For the Spence Model in Example 8.1, suppose that the employers

hold the following belief:

• If a job applicant has education he is of type L for certain.

• If a job applicant has education he is of type H for certain.

• If a job applicant has education satisfying he is type L with probability and

is type H with probability

Given this belief, find the wage contract in a competitive labor market, and then find an equi-

librium for each of the following three cases. Let be the population share of type L.

(a) For find a pooling equilibrium in which both types choose

(b) For find a separating equilibrium in which type L chooses and type H chooses

(c) For any find a separating equilibrium in which type L chooses and type H

chooses

Question 10.6 (Spence). Efficiency analysis for the above problem.

(a) In comparison with the full-information solution, who is better off and who is worse off in

the pooling solution? Why?

(b) In comparison with the full-information solution, who is worse off in a separating solution?

Why?

(c) In Exercise 8.5 (c), why does type H want to choose a higher education when is

enough to distinguish themselves from type L?

Question 10.7 (Spence). In the Akerlof and RS insurance models, we learn that asymmetric

information can leads to inefficiency in a free market. In the Spence and RS labor models, we

discuss ways to improve efficiency by signaling and screening. Has this task been successfully

accomplished? The answer is No.

3/12

(a) Explain this using the results from Spence Model.

(b) Propose a potential revision to Spence Model to solve the problem.

Question 10.8 (RS Insurance) (PhD). For the RS insurance model under asymmetric infor-

mation, suppose now that there are three types of agents (high risk medium risk and low

risk ), rather than two types. For the convenience of explanation, let -curve be an indiffer-

ence curve for type and -line be the break-even line when only type buys the insurance

scheme, where or and let -line be the break-even line when all types join the

insurance scheme, -line be the break-even line when types and join the scheme, etc.

Use only the simple arguments in the RS insurance model to establish an equilibrium; ignore

the fancy arguments such as Riley’s reactive equilibrium concept and Cho-Kreps’ intuitive

criterion. [Hint: no need to write even a single equation; all discussions can be carried out

verbally using a few diagrams. The smaller a type’s risk is, the steeper its indifference curve

-curve and its break-even line -line.]

(a) Using a diagrammatic analysis, find a potential pooling outcome (pool all types) and a

potential separating outcome (separate all types) [Hint: point out the outcomes in figures;

no need to explain or prove. The results resemble the ones with two types.]

(b) Is an outcome in the above a sustainable equilibrium? If not, explain briefly (by one or two

sentences) and use a figure. If it is, write out (speculate) conditions under which an out-

come is an equilibrium; in this case, no need to explain or prove.

Question 10.9 (RS Insurance under Monopoly). Consider the RS insurance model under

asymmetric information. Instead of a competitive insurance market, assume that there is

single monopoly in the insurance market. What is this monopoly’s profit maximization solu-

tion?

4/12

Answer Set 10

Answer 10.1. Since the seller will still sell her car for a price the car quality is uniform-

ly distributed along interval Thus, the average quality of cars on the market is

Since there is demand if any car can be sold for This means that any car with

quality or less will be traded in the market, i.e., the seller with car quality will be able

to find a buyer and trade the car at a price So, there is a market, and the market is

for cars with quality in the range However, it is still a market for lemons since it is

only for low-quality cars.

In summary, there is a range of qualities in which cars with those qualities are sold. How-

ever, adverse selection still exists, since only low-quality cars are chosen by sellers to be on the

market.

Answer 10.2. For the case with asymmetric information, the decision rule for the buyer is

and for the seller is still By the decision rule, the average quality is still

Thus, any car can be sold and the buyer’s decision is to buy any car at the market price. The

intuition is this: the buyer is desperate for a car so that as long as the price and quality are not

too far apart, he will buy the car. Since all the used cars will be on the market, the mean is

Thus, the market price is With this price, the buyer will buy any car and the seller

is willing to sell her car.

Answer 10.3. (a) With the FOC becomes

(1)

The budget constraint is

(2)

The two equations (1) and (2) determine the two unknowns and The solution is

(3)

(b) The slopes of demand are

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and are respectively the demands for income in good and bad times. The signs of the

slopes can be interpreted as: if the price of insurance against the bad time is high, the individ-

ual will buy less insurance for the bad time but will try to enjoy more in the good time.

(c) By (3), we find that if and only if That is, only if the company be-

haves like a perfectly competitive firm, the individual will choose full insurance.

Answer 10.4. (a) The MRS is a typical person with probability is

Thus, the MRS for the two types are respectively

At each point we always have That is, since the slope of an indifference

curve is the MRS, the indifference curve for type L is always steeper than the indifference

curve for type H at any point. The intuition is clear; since MRS is an individual’s internal price

of the good time, type L values the good time highly since they are less likely to have a bad

time.

(b) The zero-profit line for type H is

i.e.,

Thus, the point on Figure 8.1 where is The indifference curve going

through is

i.e.,

(4)

6/12

. o

1I

2I

..Hu

Lu

A

pooling

Hπ

LπPπ

line°45

.D

*HC

*LC

.

Figure 1. Separating Equilibrium

The zero-profit line for type L is

i.e.,

(5)

Then, the point on Figure 1 is determined jointly by (4) and (5):

To solve this equation set, let and Then,

It implies

which gives

There are two possible values for As indicated by Figure 1, we should pick the lower value.

Thus,

Hence, the point is

The separating equilibrium is a set of contracts and

(c) The indifference curve going through is

7/12

i.e.,

(6)

The budget line for a pooling equilibrium is

where and is the population proportion of type L. Thus,

(7)

In order for the separating equivalent to be sustainable, we need to show that (6) and (7) don’t

intersect. In other words, we need to show that the following equation set has no solution:

Again, let and The equation set now becomes

which implies

implying

implying

As we know, an equation doesn’t to have a solution if and only if

For our problem, this condition is

i.e.,

i.e.,

(8)

The solutions of are

As indicated by the following chart, (8) holds if and only if Therefore, when

the population share of type L is less than there exists a separating equilibrium, defined

by

8/12

22.0 73.0 λ

16.095.02 +−= λλy

y

Figure 2

Notice that we should ignore the situation with When the pooling line -

line will cut the indifference curve -curve, but the cutting is below the initial point which

will not upset the separating equilibrium. See the figure below.

.o1I

2I

..

Hu

Lu

A

B

Hπ

Lπ

Pπ

line°45

E

.

Answer 10.5. With zero-profit, this belief implies the following pay scheme:

Workers decide to choose or or (no point to choose other levels).

(a) Let us try to find a pooling equilibrium. Consider a pooling equilibrium in which both

types choose For type L, he will choose iff

i.e.,

i.e.,

9/12

(9)

For type H, he will choose iff

i.e.,

i.e.,

(10)

Thus, if

(11)

then both (9) and (10) are satisfied. In this case, if the employers’ belief is correct. We

thus have a pooling equilibrium.

(b) Let us now find a separating equilibrium. We first try to find a separating equilibrium

in which type L chooses and type H chooses The conditions for type L to choose

are

i.e.,

i.e.,

(12)

The conditions for type H to choose are

i.e.,

i.e.,

(13)

Conditions (12) and (13) are satisfied if

10/12

(14)

Note that (14) implies In this separating equilibrium, the employers’ belief is correct if

Then, condition (14) becomes

(15)

That is, with under (15), there is a separating equilibrium in which type L chooses

and type H chooses

(c) Let us now find another separating equilibrium. We want to find a separating equilib-

rium in which type L chooses and type H chooses The conditions for type L to

choose are

i.e.,

i.e.,

(16)

The conditions for type H to choose are

i.e.,

i.e.,

(17)

Conditions (16) and (17) are satisfied if

(18)

In this separating equilibrium, the employers’ belief is correct for any That is, under (18),

there is a separating equilibrium in which type L chooses and type H chooses

Answer 10.6. (a) In the full-information solution, and Thus, in the pooling

solution, type L is better and type H is worse off. The reason is that in the pooling solution,

type H subsidies type L. Why then does type H choose so that they are pooled with type L?

11/12

The reason is that turns out to be too costly for type H to distinguish themselves from type

L.

(b) Type L is indifferent between a separating solution and the full-information solution.

Type H is worse off in a separating solution. The reason is that type H is forced to spend on

education in order to distinguish themselves from type L. The possibility of disguised type L

forces type H to spend on a signal.

(c) The reason is that for the employers are not quite sure which type a person is. For

the employers, a person with still have a chance of for being of type L. If is not too high,

type H finds that it is worthwhile to completely convince the employers of their type.

Answer 10.7.

(a) In Spence (1973), the pooling equilibria are inferior to the no-signaling solution. That is,

the existence of a signal (the education level) makes everyone worse off. How about the

separating equilibria, where the signal has a private value? Even in this case, when is

large enough, the no-signaling solution is better than a separating equilibrium. Even the

most efficient separating equilibrium can be inferior to the no-signaling solution. The lat-

ter means that the Cho-Kreps’ (1987) intuitive criterion and RS (1976) screening mecha-

nism cannot even guarantee that a signaling solution is better than the no-signaling solu-

tion.

(b) In all the models mentioned above, education does not contribute to productivity. If indi-

viduals can gain a productivity increase while obtaining the signal, it is likely that we

would have a signaling solution that is better than the no-signaling solution, especially

when the benefit of the productivity increase can fully cover the cost of education for high-

productivity individuals. (This claim is by my intuition; it is not yet rigorously proven).

Answer 10.8.

(a) A pooling outcome is a point on -line, such as point in Figure 1. A separating out-

come is the set of three contracts in Figure 2, where is the intersection point of

the -line and the -curve, is the intersection point of the -line and the -curve

((that passes through and is the intersection point of the -line and the -curve

(that passes through

12/12

. o

1I

2I

.

Lu

Hπ

Lπ

LMHπ line°45Mπ

Mu.C

B

Figure 1. No Pooling Equilibrium

. o

1I

2I

.

Lu

Hπ

Lπ

LMHπ line°45Mπ

Mu

.C

B

.A

Hu

F

Figure 2. The Separating Equilibrium

(b) There is no pooling equilibrium. A pooling outcome such as point in Figure 1 can be

easily destroyed by another contract such as point in Figure 1. However, there may exist

a separating equilibrium. What are the conditions that can guarantee the three contracts

in Figure 2 are a separating equilibrium? Obviously, if the -line cuts the -

curve, cannot be a separating equilibrium. If the -line cuts the -curve,

cannot be a separating equilibrium; in this case, one can design a contract that is

below -line and attracts types and If the intersection point of the -curve and

the -curve (point is on the left of the -line, cannot be a separating equilib-

rium; in this case, one can design a contract that is below -line and attracts types

and but not Therefore, the conditions for to be a separating equilibrium are:

The -line doesn’t cut the -curve.

The -line doesn’t cut the -curve.

The intersection point is on the right of the -line.

(My answer may not be completely right; there may be other conditions needed for the

separating equilibrium. Notice that the -line is always on the left of the -line and the

-line is always on the right of the -line).

Answer 10.9. You try out first. I will show you the solution later in Chapter 11.

1/6

Problem Set 11 Micro Analysis, S. Wang

Question 11.1. Suppose that a seller is selling a product to a buyer. The seller has type

which the buyer does not know. The buyer only knows that the type has density function

on The buyer knows that the relationship between investment and quality of the

product for the seller of type is The cost of investment for the seller is The

buyer’s value from the product of quality is The buyer can observe and verify invest-

ment Hence, the buyer can offer a deal to the seller. Given a contract the seller’s

surplus is and the buyer’s surplus is Given the seller’s reservation

value , suppose the buyer’s deal ensures an ex ante IR condition to induce the seller’s ac-

ceptence. Setup the buyer’s problem.

Question 11.2. For the Spence labor problem, instead of private companies that compete to

maximize expected profits, assume there is a state-owned company that maximizes expected

social welfare, where social welfare is the direct sum of the firm’s payoff and the worker’s

payoff (without weights).

Specifically, the state-owned company hires a worker from the labor market. The worker

has a continuum of possible types in where Assume that type is

known only to the worker himself, and the firm only knows the distribution of where the

density function is for With education level the worker’s utility function is

where is the wage and is the worker’s private cost of education.

The revenue function is Hence, the firm’s payoff is The firm

offers allocation scheme to the worker. The IR constraint is:

where is the worker’s reservation value. Setup the firm’s problem.

Question 11.3 (PhD). For the buyer-seller model with quasi-linear utility in Section 6.4, let

2/6

where is the quality of a product, is the quantity traded, and is the payment from the

buyer to the seller. Quantity and payment are observable, but quality is not observable to

the buyer. Given a payment, higher quality and higher quantity yield higher satisfaction for the

buyer but costs more for the seller. Let the distribution function be the uniform distribu-

tion function on i.e., for

(a) Find the optimal solution ∗ under asymmetric information and ∗∗ under complete

information using the direct mechanism that maximizes the buyer’s expected utility.

(b) Draw a figure for ∗ and ∗∗

Question 11.4 (PhD). Prove the Revenue Equivalence Theorem. Hint: verify that the seller’s

expected revenue ∈ℕ is dependent on and

only.

Question 11.5 (PhD). For the optimal auction in Section 2, assume two symmetric bidders

with and for both and where is large enough so that

Show that the transfer scheme is based on the second price.

3/6

Answer Set 11

Answer 11.1. The direct mechanism is: the seller of type reports his type and then the

buyer pays the price for the product and demands investment for the production of

the product. By the revelation principle, the buyer can confine her search for an optimal con-

tract to the set of incentive-compatible allocation schemes Hence, the

buyer’s problem is

(⋅) (⋅), (⋅)We now spell out the IC and IR conditions. Given an offer the value function

for the seller is Then, the FOC for reporting is

(1)

The SOC is Taking a derivative on the truth reporting condition

yields Then, the SOC becomes:

(2)

Also, given the reservation value , the ex ante IR condition is

Hence, the buyer’s problem is

( ), ( )(3)

Answer 11.2. The worker’s problem of reporting his type is

The FOC and SOC are

Hence, to induce true telling, we need

4/6

By take derivative w.r.t. on the FOC, the SOC becomes This is satisfied if

For type , social welfare is

Then, the social welfare maximum problem is

(⋅), (⋅)(4)

Answer 11.3. We have and with

(a) Equation (9.73) in the book becomes

implying

∗∗We have ∗∗ ∗∗Thus, ∗∗ is decreasing when and ∗∗ is increasing when Since we

need Thus, if is decreasing around a point by the first argument following

(9.77) in the book, we have

∗∗ (5)

around the point Thus, since ∗∗ is not decreasing if cannot be strictly de-

creasing if By the requirement must be constant on Let

for where By (9.78) and (9.79) in the book, we have

∗∗implying

5/6

implying

By substituting the second equation into the first one, we have an equation for

The solution is In since ∗∗ is strictly decreasing, by the condition (5), it is

impossible to have an open internal on which is constant. That is, must be strictly

decreasing on implying ∗∗ for In summary,

∗∗where is determined by (6).

(b) ∗∗ is decreasing when and ∗∗ is increasing when Also, ∗∗ is convex.

By this knowledge, we can now draw the picture for ∗∗

θ121b

)(** θx

)(θx

x

Figure 1. ∗∗ and

Answer 11.4. By the revelation principle, we know that any social choice function that is

implementable by a Bayesian Nash equilibrium must be incentive compatible. We can thus

restrict ourselves to incentive compatible social choice functions only.

The seller’s expected revenue is ∈ By the condition

Proposition 9.3, we have

6/6

Moreover, by integration by parts,

where is the distribution function of Thus,

∈Therefore, the revenue is

∈ ∈ ∈By inspection of the above formula, we see that any two Bayesian incentive compatible social

choice functions that generate the same functions and the same value

must imply the same expected revenue for the seller.

Answer 11.5. The optimal transfer scheme is

∗ ∗ ∗ ∗Here, the term ∗ says that the winner pays but the term ∗ reduces the

actual payment. We have If since ∗ when and ∗ when

we have

∗implying ∗ That is, the transfer scheme is based on the second price. Thus, the

optimal solution is the second-price sealed-bid auction.

Page 1 of 14

Problem Set 12 Micro Analysis, S. Wang

Question 10.1 (Insurance) (PhD). This exercise is from Helpman–Laffont (1975). Consider

an economy with one period and one good. The initial income is dollars. During the period,

each agent has probability of having an accident that results in a loss of dollars. The risk of

each individual is independent of others. An agent's utility function is state-dependent and is

defined by

Each agent is restricted to have no borrowing, i.e., income in any state. For technical

reasons, assume and

(a) Find the simplest Pareto equilibrium solution. It is the equilibrium solution for the Arrow-

Debreu world under complete markets.

(b) There is a competitive insurance company that offers a contract that makes a payment

when there is no accident, but no payment when there is an accident.1 Each agent can buy

any amount of insurance for a constant price (i.e., the insurance premium is ). Find

the competitive equilibrium. Is this solution a Pareto optimum?

(c) Reconsider the problem in (b), but now suppose that the agent can influence the probabil-

ity by spending dollars. Let the probability of having an accident be First find

the equilibrium solution for the Arrow-Debreu world with complete markets. Also find the

competitive equilibrium solution for which the insurance company cannot observe and

show that it is not a Pareto optimum. Explain why.

(d) Consider a tax scheme that levies a proportional tax on and redistribute the tax revenue

to those who do not have an accident using a uniform lump-sum transfer Assume the

government can observe Can this tax scheme restore the Pareto optimum? Are there any

other ways to restore the Pareto optimum?

Question 10.2 (Insurance) (PhD). Reconsider the competitive insurance industry in Chapter

4 (the RS model). There are two types of individuals. The individuals know their own types but

the company cannot observe the types. Assume now that the individuals can affect their prob-

ability of having an accident by taking some level of precaution The level of precaution

1 Consider this as a pension plan, for which the dead get nothing and what the dead have left is shared among

the living population.

Page 2 of 14

costs By investing more an individual lowers his probability of loss. Assume that no

matter how high is, the probability of loss for the high type is always higher than that for the

low type.

(a) Find an equilibrium (if one exists) under these circumstances. [That is, find one policy or a

pair of policies such that, when each individual chooses the policy (and the level of in the

case of a high-risk individual) that is the best for him, no firm can increase its profit by

dropping a policy or by offering a different one.] Clearly indicate the equilibrium policies

in a diagram and state the level of chosen in equilibrium.

(b) Now suppose that the low-risk individuals, rather than the high-risk individuals, can

choose a level of that affects their probability of loss. Assume that even if this

probability is lower than the probability of loss for the high risk individuals. What can you

say about the value of chosen in an equilibrium in this case? Given the value of chosen,

illustrate in a diagram the policies offered in an equilibrium (if it exists).

(c) Is there a welfare improvement for individuals with accident prevention?

Question 10.3 (The Standard Agency Model) (PhD). For the standard agency model in Sec-

tion 1, let

The density function states that the output follows the exponential distribution with

mean and variance

(a) Show that the second-best solution is

∗ ∗[Hint: assume

( , )( , ) for any and verify this later].

(b) Find the first-best solution ∗∗ and ∗∗ [Hint: equation has a numerical

solution of

Question 10.4. For the sharing contract in the case of double moral hazard and double risk

neutrality, consider the following parametric case:

Page 3 of 14

where is a random variable with and

(a) Derive the second-best solution.

(b) Derive the first-best solution. Do we have larger efforts in the first best?

Page 4 of 14

Answer Set 12

Answer 10.1. (a) There is a proportion of agents who have an accident. The total income is

thus

The egalitarian Pareto optimum yields an ante identical income to all those who can profit

from it. Thus, by dividing this income among those who don't have an accident, we obtain the

egalitarian Pareto optimum at which each of those who doesn't have an accident receives

and each of those who has an accident receives nothing

This solution is also the complete-market solution in the Arrow-Debreu world.

(b) The individual's income is

Given price the individual's problem is

If we have i.e., there is no demand and thus no profit. So, we must have in

which case there is a demand for insurance and the individual wants to buy as much as possi-

ble, but he is limited by the no-borrowing condition. The solution is The insurance

company's profit is Zero profit then implies Thus,

∗The incomes are

This solution is the same as the complete-market solution in (a). Thus, the competitive equi-

librium is a Pareto optimum.

(c) To find the equilibrium solution for the Arrow-Debreu world with complete markets,

we repeat the derivation in (a). The total income is thus

At the egalitarian Pareto optimum, each of those who doesn't have an accident receives ( )( ) and each of those who has an accident receives nothing A typical indi-

vidual solves the following problem

Page 5 of 14

i.e.,

which yields ∗∗To find the competitive equilibrium solution, we repeat the derivation in (b). The individ-

ual's income is

Given price the individual's problem is

, We must have otherwise there would be no demand for insurance. Without the budget

limit, as long as the individual would buy as much as possible. Thus, with the budget,

The problem can thus be simplified to

The FOC is

The insurance company's profit is Zero profit then implies Thus,

Thus, the competitive solution ∗ is the solution of the following equation:

∗ ∗∗We have

∗∗We have

( ∗∗) if By the concavity of in this implies ∗∗ ∗ This

means that in the competitive equilibrium, each individual will invest too much in and thus

the competitive equilibrium cannot be a Pareto optimum.

Collective waste occurs because each agent tries to protect himself against an accident.

E.g., each agent buys his own fire engine when it would be better for the society to provide one

fire engine for all. The marginal private gain from spending is larger than the marginal social

benefit at the social optimum.

Page 6 of 14

(d) Given price and the individual's problem is

, Again, without the budget limit, the individual would buy as much as possible. Thus, ( )

The problem can thus be simplified to

The FOC is

Zero profit implies Thus,

We also have

Thus,

The government will then solve the social welfare maximization problem:

, The Lagrangian is

The FOC are

Thus, and ∗Therefore, the tax scheme restores the competitive equilibrium to Pareto optimality.

We can then solve for the optimal tax rate:

∗ ∗ ∗∗ ∗∗

Page 7 of 14

Notice that ∗ is the same as ( ∗∗)

Are there any other ways to restore the Pareto optimum? Yes, there are obviously other

ways. For example, the government can impose a restriction limiting the use of such as

Given price the individual's problem is

,

Given the individual would like to buy as much as possible. Thus, The problem

becomes

As shown in (c), if there is no restriction, the individual will want more than ∗∗ Thus,

the solution must be ∗

Answer 10.2. Given a price of insurance for an individual with probability of accident,

his problem is

where is the initial wealth, is the potential loss of wealth and is the expenditure on acci-

dent prevention. The individual maximizes his expected utility subject to his budget line.

Zero profit for the insurance companies implies that must equal the probability of

accident for those who bought the policy. Thus, the break-even line is the budget line with

We can write the budget line as

which means that the budget line goes through the point and has a slope

(a) The break-even line for high-risk individuals is

where decreases as increases. This line will be becoming steeper and at the same time

shifting to the left as increases. The high-risk individuals may improve welfare if the break-

even line becomes steeper; however, if has increased too much, the break-even line will be

moved too much to the left. That is, there is a tradeoff between a steeper break-even line and

Page 8 of 14

the line being moved too much to the left. The optimal ∗ is the value that gives the optimal

tradeoff. The separating equilibrium is the pair of contracts.

. o

1I

2I

..

HuBHπ

Lπ line°45

E

Lw − .*zLw −−

*zw −

*Hπ

w Figure 1. Separating equilibrium with a precaution spending by high-risk individuals

(b) The break-even line for low-risk individuals is

where decreases as increases. This line will be becoming steeper and at the same time

shifting to the left as increases. The low-risk individuals may improve welfare if the break-

even line becomes steeper; however, if has increased too much, the break-even line will be

moved too much to the left. That is, there is a tradeoff between a steeper break-even line and

the line being moved too much to the left. The optimal ∗ is the value that gives the optimal

tradeoff. The separating equilibrium is the pair of contracts.

. o

1I

2I

..Hu

BHπ

Lπ line°45

E

Lw − .*zLw −−

*zw −

*Lπ

w

Figure 2. Separating equilibrium with a precaution spending by low-risk individuals

(c) In (a), both the high-risk and low-risk individual are better off with accident preven-

tion by the high-risk individuals. In (b), only the low-risk individuals are better off, and the

Page 9 of 14

high-risk individuals are indifferent. Notice that the low-risk individuals have been better off

otherwise they would have chosen ∗

Answer 10.3. (a) We have

The IC condition is:

The IR condition is

Let and be the Lagrange multipliers. Then, the Lagrangian is

The Hamiltonian for is

We have

The first-order condition for the Hamiltonian implies the Euler equation:

Together with the limited liability condition, the optimal contract is

∗We will assume that for any and verify this later.

The FOC for is which implies

(1)

Three conditions, the IC condition, the IR condition and (1), can determine the three parame-

ters and ∗ The IR condition implies

Page 10 of 14

implying

By the IC condition, we have

implying

implying

(3)

By (1),

Page 11 of 14

implying

In summary, we have

Then,

implying

Since we have

implying

∗Then,

∗The contract is

∗By (2) and (3), we have

We need which is obviously satisfied.

Page 12 of 14

(b) With a verifiable the principal's problem

∗∗ ∈ , ∈The first-best solution corresponds the case with From the expression of we im-

mediately find ∗∗By the IR constraint, we have

implying

Then, the objective function becomes

The FOC is

implying2

∗∗where satisfies

Since we have

2 Using the Lagrange method, the FOC for is

implying

implying ∗∗

Page 13 of 14

The numerical solution is Then, the solution is

∗∗ ∗∗ ∗∗

Answer 10.4. (a) We have

By Proposition 10.2, we have

∗ ∗ ∗ ∗We now solve for ∗ ∗ from

, ∈

(4)

With the specific functions, (4) becomes

, ∈or

, ∈The Lagrange function is

The FOCs are

∗And,

implying

Page 14 of 14

Thus,

∗Thus,

∗ ∗Then,

∗(b) The first best is determined by

implying ∗∗ ∗∗Obviously, the effort levels are higher in the first best.