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 functions.
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 1 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
then
satisfies Assumptions 1.1 and 1.2 and
also satisfies Assumptions 1.1 and 1.2.
Question 1.11. A firm buys inputs at levels
to produce a level of output
input prices
and
and
on competitive markets and uses them
Its technology is such that the minimum cost of producing
at
is given by the cost function
Page 2 of 10
where
and
are constant parameters.
(a) What parameter condition does the homogeneity of this cost function imply?
and
(b) Derive the conditional demand functions
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 represented 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 3 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 4 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 then
and
Substituting these into the constraint, we can solve for
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
Therefore,
and
let
and
We then have
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 5 of 10
Answer 1.4. From Figure 1.2, we see that the minimum point is
on the ratio of
Therefore, the cost is
or
or
depending
That is,
x2
ax1 + bx 2 = y
Isoquant
w1 x1 + w 2 x 2 = c
.
x1
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 problem. One way to do is to use a graph.
x2
ax1 = bx2
y
b
y = f ( x)
y/a
x1
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.
Page 6 of 10
Answer 1.7. The cost function and the expenditure function in consumer theory are mathematically the same.
Answer 1.8. (1) Since
is homogeneous of degree
is linearly homogeneous and since
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 7 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
Assumption 1.1 is satisfied. Since
Answer 1.10. 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 8 of 10
When the functions are differentiable, taking derivatives is often the easiest way to find monotonicity 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
Since the labor
it needs at least
amount of 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 9 of 10
Therefore, Assumptions 1.2 is satisfied.
(b) The problem is
The solution is:
Therefore, the supplies are:
Page 10 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
demand for good is defined as
∗
( , )
The income elasticity of
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 1 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
( ̅)
(a) Show that
̅ ( , )
∗
∗
of a utility-maximizing consumer.
̅ ( , )
(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
Suppose that the individual has an income
with wage rate
and food
of the form:
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 consumer 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 consumer demand for leisure?
Question 2.11. One popular functional form in empirical work for ordinary demand functions
∗
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 2 of 10
(a) Interpret
good is
∗
∗
and
in terms of elasticity, where the price elasticity of demand for
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 restrictions implied by this fact? If goods 1 and 2 are gross substitutes, are there additional
parameter restrictions?
of present
Question 2.12. A consumer has an intertemporal utility function
consumption
and future consumption
He takes as given the spot prices
can borrow and lend freely at an interest rate
units of the commodity in the present and
He
He has an initial endowment of
units of the commodity in the future.
(a) Find the utility-maximizing consumption bundle of the consumer, and compute his marginal 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
and
where is the input in period 1 and
is the output in period 2. (NOTE:
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 3 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
preferences satisfy local nonsatiation. But for two points
and
but
It is easy to see that the
and
we have
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 4 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 5 of 10
Answer 2.5. Since
is linearly homogeneous in
given that fact that
Answer 2.6. We can easily show that
Then,
ous of degree
in
is linearly homogeneous in
and
is homogene-
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 6 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
(d) For each
denotes the null space of
by (2), since by assumption
's.
( ̅)
one of the
̅
must be
nonnegative.
Page 7 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 8 of 10
Answer 2.11. (a) We have
∗
∗
(b) For any
∗
since
∗
is homogeneous of degree
we have
∗
∗
Therefore,
∗
Normality implies that
Similarly, using the 2nd equation, we also have
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 9 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 10 of 10
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
Denote
and a zero-mean random variable
define
by
Derive
show that the ex-
Question 3.3. For a quadratic utility function
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
and if the Lions lose he loses
on the Lions so that if the Lions win, he wins
The fan's initial wealth is
(a) How can we determine his subjective odds
(b) Under what condition does an increase in
by observing his optimal bet
lead to a higher bet
∗
∗
Question 3.5. Suppose that a consumer has a differentiable expected utility function for
The consumer is offered a bet with probability of winning
and
money with
probability of losing
Show that, if is small enough, the consumer will always take the bet.
Page 1 of 8
Question 3.6. Let individual A have an expected utility function
and let individual B
have an expected utility function
be a monotonic in-
where
is income. Let
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
Here
is initial wealth. If
Define “risk premiums”
and
by
show that
(c) Interpret the risk premium in words.
Question 3.7. For Exercise 3.4, when the probability
the amount
∗
of winning
goes up, do you expect
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 transformation of the other — imply the same preferences with certainty consumption bundles, but
not with uncertainty consumption bundles:
Page 2 of 8
Question 3.11. For the insurance problem:
where
insurance,
is the loss,
is the probability of the bad event,
is initial wealth,
is the price of
and
(a) If the insurance market is not competitive and the insurance company makes a positive
expected profit:
insurance
∗
will the consumer demand full-insurance
or over-insurance
∗
∗
under-
Show your answer.
(b) Show the above solution on a diagram.
Page 3 of 8
Answer Set 3
Answer 3.1. We have
where
where
is some constant. Then
and
are two constants
are some constants. Therefore, for
and
( )
( )
if and only if there
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 4 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
∗
∗
∗
∗
By (2), for a risk averse person
(2)
∗
with increasing utility function
we have
∗
Answer 3.5. We need to show that
(3)
when is small for a differentiable utility function
Taylor expansion, there are
and
with
( may not be concave). By
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 5 of 8
(b) We know that if
Since
is concave,
Assuming
is a convex function, then1
By definition,
is convex. Therefore,
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
variable
those who want to know, let
and
1
be a partition of the value space of the random
2
the probability of
Then, by the continuity and convexity of
we
have
→
→
→
Page 6 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 preferences.
Answer 3.9. If RCLA were not violated, then
which would immediately imply a contradiction. Therefore, RCLA must has been violated.
Then
Answer 3.10. Let
function,
and
Since
is a strictly increasing
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,
∗
∗
or
Then,
∗
∗
i.e.,
∗
Thus,
∗
It implies
∗
that is, we
have under-insurance.
(b) When
When
in Example 1.12, we have shown that the solution must be on the
the budget line is flatter, and the tangent point must be below the
∘
∘
line.
line. That
is, the individual is under-insured.
Page 7 of 8
I2
1-π
slope= π
slope=
.
w-l
.
45o
1-p
p
.
w
I1
Figure 5.1. Insurance in a non-competitive market
Page 8 of 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
is
i.e.,
and the total available amount of good 2
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:
1/8
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
number of guns,
as the amount of butter, and
as the
as the amount of oil. The utility functions for
consumers are
.
.
Each consumer initially owns
production function
units of oil:
Consumer 1 owns firm 1 which has
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
owned by the consumer. The consumer has an endowment of
unit of time for working and
enjoying leisure, and has utility function
time
for good
The firm inputs amount of labor to produce
The firm is
and leisure
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
Find the GE.
has production function
Question 4.9. There are two goods
viduals
and
and
with prices
and
with
respectively, and two indiand
Draw it in an Edgeworth box.
(a) Derive the contract curve. Suppose
(b) Derive the GE price ratio(s)
Question 4.10. There are two goods
with
and
and two individuals
and
(a) Find all the Pareto optimal allocations. Are they strongly Pareto optimal?
(b) Find all the GE price ratio(s)
2/8
Answer Set 4
Let
Answer 4.1. Individual A’s utility function is equivalent to
and
Then the income is
and the demands are:
For individual B, by its utility function, we know that the demands must satisfy
by budget constraint
Then
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
contradiction. Therefore, allocation
Answer 4.3. Let
and
This is a
must be Pareto optimal.
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:
3/8
Therefore, the equilibrium price ratio
is:
∗
Answer 4.4. In the following charts, the left chart indicates the Edgeworth box and the indifference curves. The right chart indicates the Pareto optimal points.
B F
B
E
D
A
C
uB
uA
A
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,
4/8
This set is the diagonal line in the following diagram.
2
y
P.O.
y=x
x
1
Figure 2. P.O. Allocations
price of guns
price of butter
price of
Answer 4.6. Denote
(we can arbitrarily choose one of prices. We can do that because of the homogeneity
oil
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
5/8
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 equilibrium. 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:
6/8
∗
∗
∗
Firm’s problem:
Answer 4.8. We can arbitrarily set
,
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
7/8
y
B
x
ω
.
.
uA
uB
Contract curve
x
A
y
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 contract curve are strongly P.O.
2
y
.
contract line
W
.
u1
u2
1
x
Figure 4. Contract Curve and Equilibria
(b) The set of equilibria is
That is, all the possible values of
are
equilibria.
8/8
Problem Set 5
Micro Analysis, S. Wang
There are no exercises for Chapter 5.
Page 1 of 1
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
wage rate for labor is
and the rental rate for capital is
and
The
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 industry 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 1 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
Suppose that each firm is required to pay a specific tax of
For simplicity, assume
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 industry. 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
6.9 of the book.
(
)
/
/
in Section
Page 2 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
and
Thus,
Then
and
and
imply that
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 3 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 equilibrium 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
change is in
and
output
is reduced. With
going down and
going up, the
is ambiguous; it demands on the demand.
p
.
p
.
ys
D
y
Answer 6.2. Let
be the market price of the good when the output is
firm when its output is
is the cost of
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 4 of 8
∗
∗
∗
∗
We can rewrite (2) as
∗
∗
where
On the other hand, the Cournot output is determined by
∗
∗
∗
∗
p
æY ö
c ¢ ç ÷÷
çè 2 ÷ø
.B
A
.C .
MR(Y )
D
Y
1
MR(Y ) - p¢(Y )Y
2
Figure 6.1. A market-share Cournot equilibrium
In the diagram, point
price as given; point
is the ‘competitive solution,’ for which each firm takes the market
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 5 of 8
That is, the firms’ supply curve is the horizontal line at
∗
The equilibrium industry supply is thus
So is the industry supply curve.
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 output 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 6 of 8
This equation determines the industry output
than the individual tax rates
rather
’s.
(c) Since the total output depends only on
change for a tax change. Then, by (3),
where
which obviously depends on
and the latter has no change,
doesn’t
i.e.,
is determined by (4).
Answer 6.5. This is from Example 12.E.2 on page 407 of MWG (1995). Once
are in the industry, they play a Bertrand game. As we know, if
tive outcome, i.e.,
∗
the result is the competi-
and the profit without including the entry cost
firms. This means that each firm loses
identical firms
is zero for all the
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
sulting the monopoly price
As long as
re-
The monopoly profit is
a firm will enter and that is the only firm in the industry.
Answer 6.6. We have
where
Then,
Page 7 of 8
implying
/
implying
/
/
/
/
Page 8 of 8
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 perform some service at a price (which is supposed to equal the cost of the service). The principal
and the agent have initial wealth
tially lose
and
respectively. The principal can poten-
If the agent offers low quality, the probability of losing is
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
of a high quality product is
if the
by the competitive market assumption,
and the price
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 principal 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 equilibria in the above exercise.
1/12
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 purestrategy Nash equilibria.
P1
L1
R1
.
.
P2
P3
l
æ2ö
çç ÷÷
çç0 ÷÷
çç ÷÷÷
çè1 ÷ø
L2
r
R2
.
.
P3
P3
æ- 1ö
çç ÷÷
çç 5÷÷
çç ÷÷÷
çè 6÷ø
l
æ3 ö
çç ÷÷
çç1 ÷÷
çç ÷÷÷
çè 2 ÷ø
r
l
æ5 ö
çç ÷÷
çç 4 ÷÷
çç ÷÷÷
çè 4 ÷ø
æ 0ö
çç ÷÷
çç- 1÷÷
çç ÷÷÷
çè 7 ÷ø
r
æ- 2 ö
çç ÷÷
çç 2 ÷÷
çç ÷÷÷
çè 0÷ø
Question 7.5. In the following game, explain why there are mixed-strategy NEs in which P1
mixes
and
arbitrarily and P2 chooses
o
s11 L1
æ 0ö
çç ÷÷
çè0÷ø
L2
æ-1ö
çç ÷÷
çè-1÷ø
P1
s 21 M 1
s31 R1
.
H
m1
R2
æ1 ö
çç ÷÷
çè 2÷ø
m2
L2
æ-1ö
çç ÷÷
èç 0 ÷ø
.
P2
R2
æ1ö
çç ÷÷
èç1÷ø
2/12
Question 7.6. Consider the following game.
P1
o
L1
R1
x
.
P1
R̂1
L̂1
æ 0ö
çç ÷÷
çè0÷ø
.m
m2
H
1
.
P2
L2
R2
L2
R2
æ-2ö
çç ÷÷
çè-1÷ø
æ1ö
çç ÷÷
çè-2÷ø
æ-1ö
çç ÷÷
èç 1 ÷ø
æ 2ö
çç ÷÷
èç 3÷ø
(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.
Firm E
o
In
Out
x1
0
 
2
Firm E
Large
Niche
Small
Niche
.
Small
Niche
 Firm E' s Payoff 


 Firm I' s Payoff 
.
 − 6


 − 6
.
HI
Large
Niche
 − 1
 
 1
Small
Niche
Firm I
Large
Niche
 1  − 3 
   
 − 1  − 3 
3/12
Question 7.8. For the following game, find all the pure-strategy NE, all the SPNEs and all the
BEs.
Firm E
o
Out
In
σ 1E
1 − σ 1E
z
0
 
2
.
Fight
σ 2E
Fight
σI
π E 
 
π I 
HI
.
m
 − 3
 
−1
Firm E
Accom
1 − σ 2E
.
Firm I
Accom
1−σ I
Fight
Accom
1−σ I
σI
− 2


 − 1
 1


 − 2
1- m
 3
 
 1
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?
P1
o
γ1
B
T
γ2
.
.
P2
P2
δ1
4
 
2
D
U δ2
1
 
1
δ1
 5
 
 1
D
U
δ2
2
 
2
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.
4/12
Question 7.10 (PhD). One problem with a BE is that it may not be trembling-hand perfect.
Consider the following game.
P1
o
L1
R1
.
.μ
μ2
1
L2
æ1 ö
çç ÷÷
èç2÷ø
R2
L2
æ0 ö
çç ÷÷
çè2÷ø
æ 0ö
çç ÷÷
çè1 ÷ø
P2
R2
æ3ö
çç ÷÷
çè3÷ø
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.
5/12
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
6/12
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 purestrategy 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:
indifferent from
probability
and
and
In
given
player 1 is
However, if player 2 may make some mistakes by taking
no matter how small
is, player 1 will be strictly prefer
is not a trembling-hand NE, while
to
with
Thus,
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,
the information set in the middle is reached, and
is the action if
is the action if the information set on the
right is reached. Player 3 has eight strategies:
The normal form is
P1 plays
P2:
P3
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)
7/12
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,
P1 does,
and
are indifferent to P1. On the other hand, whatever
is always better to P2.
Answer 7.6.
(a) P2 has one information set
and
where
an action at
containing two nodes. P1 has two information sets
contains the initial node. Denote P1’s strategies as
and
is an action at
where
is
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
8/12
(c) Let us find BEs. For P2,
es
at node
Since
iff
Then, since choosing
or
means a payoff of
P1 chooses
If so, P1 choosat the beginning.
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
In this case, consistency is required and it implies
at the initial node.
which can be satisfied. Hence, there
is another BE:
∗
Further, if
∗
P2 is indifferent between
mixed strategy
with
iff
∗
and
that is, P2’s strategy can be any
Then, at node
P1’s preference would be
However, this is completely impossible. We in fact always
have
Hence, P1 will always choose
iff
Since
, i.e.,
at
Then, P1’s preference at the initial point is
. Hence, if
P1’s strategy is
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
ble. Hence, there is no BE in this case. If
case, if P1 takes
If P1 takes
P1 is indifferent between
This is impossiand
. In this
with a positive probability, consistency is required and it cannot be satisfied.
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 conclude that BEs may not be SPNEs.
Answer 7.7. In the proper subgame with the normal form:
Firm I
Small,
Firm E:
Small,
Large,
The equilibrium
Large,
-6, -6
(-1, 1)
(1, -1)
-3, -3
is to make firm E indifferent between his two strategies:
9/12
∗
implying
∗
∗
Since the game is symmetric, we also have
∗
We also have
∗
∗
Then, the expected payoff is
The game is reduced to:
Firm E
o
Out
In
 − 199 
 19 
− 9 
0
 
2
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
strategies as
and
where
where
is an action at
contains the initial node. Denote firm E’s
and
is an action at
We can then
find the normal form:
Firm E
Firm I:
<out, fight>
<out, accom>
<in, fight>
<in, accom>
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.
10/12
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.
be a typical P2’s strate-
Answer 7.9. (a) There are two information sets for P2. Let
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
P1:
<D, D>
<D, U>
<U, D>
<U, U>
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:
P1
o
γ1
B
.
δ1
4
 
2
D
P2
T
γ2
.
H2
U δ2
δ1
 5
 
 1
1
 
1
D
U
δ2
2
 
2
The normal form is
P2
P1:
D
U
B
4, 2
1, 1
T
5, 1
(2, 2)
11/12
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.
12/12
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
is to name a price
and
of the partnership, respectively. Partner 1
and then partner 2 chooses either to buy partner 1’s share for
his share to partner 1 for
Assume that partner 1’s value of the firm is
the whole firm and zero otherwise; and partner 2’s value of the firm is
or to sell
if she owns
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
(the uniform distribution on
and
independently follow
) 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
knows her own valuation
(and hence
that the seller’s valuation follows
makes a single offer
with known parameter
but the buyer doesn’t know
The seller
The buyer knows
(the uniform distribution on
). The buyer
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 legislative process in which a feasible policy is
the Congress is
where
The status quo is
and the ideal policy for
The ideal policy for the president is
which is private
information of the president. The Congress only knows that follows
proposes a policy
and the president either signs or vetoes. Given a policy
Congress and the president are respectively
BNE
∗
and verify
∗
The Congress
the payoffs of the
and
Find the
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
1
in the first period and the Congress proposes
based on a belief
which
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 1 of 6
is the probability that the president has type when message
may take a pure strategy
or a mixed strategy
is observed. The president
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 2 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
case, the firm goes to partner 1 iff
and it goes to partner 2 iff
In this
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 3 of 6
∗
Therefore, there is no trade if
there is a trade if
and there may or may not be a
trade if
the president will sign the proposal. Hence,
Answer 8.3. If and only if
the Congress’s problem is
(
)
(
(
)
)
(
)
We have
(
(
If
)
(
)
)(
)
then
Then, the FOC for
is
or
The left-hand side is positive. But, since
sible. Hence, we must have
With
the right-hand side is negative. It is impos-
∗
we have
The FOC for is
implying
implying
implying
implying
Page 4 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
Congress then responds with proposal
is
with message
The game is drawn below. In the figure, player P
is the president and player C is the Congress. Let
payoff under policy
The
and
be type
president’s
be the payoff of the Congress under policy
Nature
tL
.
tH
1- d
d
.
P
P
L
p
L
R
.
.
C 1- q
1
C -p
q C
C
R
aL
aH
aL
aH
aL
aH
aL
aH
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 president plays a pure strategy
the density of type is
In the second step, when the Congress sees
and its proposal
it guesses that
is a solution of the following problem:
(1)
( , )
( , )
( , )
( , )
Page 5 of 6
In the first step, knowing the Congress’s proposal
the president considers how to send a
message. His problem is
(2)
Let
This implies
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 mixedstrategy 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
such that
∗
∗
if there is a
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 6 of 6
Problem Set 9
Micro Analysis, S. Wang
Question 11.1. We have two agents with identical strictly convex preferences and equal endowments. 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
ଵ
ଶ
and preferences) and they have strict monotonic and
strict convex preferences. Show that the initial endowment point
1 Strict
convexity of preferences means that:
and
must be in the core.1
for
Page 1 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.
2
y
2
y
2
y
.
core
core
W
core
.W
1
1
x
(a)
x
(b)
.W
1
x
(c)
Figure 1. The Core
(b) No. Let
ଵ
with
ଵ
and
ଵ
ଶ
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
blocks
That is,
is not in the core. Then, there is a feasible allocation
஺
and
(or
஻
and
஺
஻
஺
஻
that
By the strictly convexity of
the preferences,
By the feasibility, however,
monotonicity. Therefore,
஺
஻
ଵ
ଵ
ଶ ஺
ଶ
஻
i.e.,
ଵ
ଵ
ଶ ஺
ଶ
஻
This contradicts with strict
must be in the core.
Page 2 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
஺
ଵ
where
௝
௝
஺
஻
is
஻
ଵ
is the demand of individual
஺
ଶ
in good
஻
ଶ
Obviously, the feasible allocation
satisfies the above two conditions, and is thus Pareto optimal.
is thus in the core.
Page 3 of 3
Problem Set 10
Micro Analysis, S. Wang
Question 10.1 (Akerlof). In the Akerlof model, we now suppose that the buyers can be guaranteed a minimum quality of the car by inspection and test drive. Specifically, instead of the
for used cars in the market, suppose that all the cars have a minimum
minimum quality
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 information. The insurance company offers a price
tion
for an insurance policy that pays a compensa-
if an accident happens. Let
and
(a) Compute the demand functions
(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 information. Suppose that insurance companies offer price-quantity contracts. There are two types
of agents with type
the probability
initial wealth
or
The initial wealth for all agents is
of losing an amount
the same possible loss
An agent with type has
when the bad event happens. All agents have the same
and the same utility function
of income
Let
1/12
(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 equilibrium 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
(c) For any
and type H chooses
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.
2/12
(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 information, suppose now that there are three types of agents (high risk
risk ), rather than two types. For the convenience of explanation, let
ence curve for type and
medium risk
and low
-curve be an indiffer-
-line be the break-even line when only type buys the insurance
scheme, where
or
insurance scheme,
-line be the break-even line when types
and let
-line be the break-even line when all types join the
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
-curve and its break-even line
is, the steeper its indifference curve
-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 outcome 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 solution?
3/12
Answer Set 10
the car quality is uniformAnswer 10.1. Since the seller will still sell her car for a price
ly distributed along interval
Thus, the average quality of cars on the market is
Since there is demand if
quality
any car can be sold for
This means that any car with
or less will be traded in the market, i.e., the seller with car quality
So, there is a market, and the market is
to find a buyer and trade the car at a price
for cars with quality in the range
will be able
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. However, 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
4/12
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 individual 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
through
on Figure 8.1 where
is
The indifference curve going
is
i.e.,
(4)
5/12
I2
pooling
πP
πH
πL
45° line
.
A
uH
.
C
.
D
.C
*
H
*
L
.o
uL
I1
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
(c) The indifference curve going through
and
is
6/12
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
the population share of type L is less than
Therefore, when
there exists a separating equilibrium, defined
by
7/12
y
y = λ2 − 0.95λ + 0.16
λ
0.73
0.22
Figure 2
When
Notice that we should ignore the situation with
line will cut the indifference curve
the pooling line
-curve, but the cutting is below the initial point
-
which
will not upset the separating equilibrium. See the figure below.
I2
πL
uL
uH
πH
45° line
A
.B
.E
..
o
πP
I1
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.,
8/12
(9)
iff
For type H, he will choose
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
9/12
(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 equilibrium in which type L chooses
choose
and type H chooses
The conditions for type L to
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
there is a separating equilibrium in which type L chooses
Answer 10.6. (a) In the full-information solution,
That is, under (18),
and type H chooses
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?
10/12
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, a person with
the employers are not quite sure which type a person is. For
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 latter means that the Cho-Kreps’ (1987) intuitive criterion and RS (1976) screening mechanism cannot even guarantee that a signaling solution is better than the no-signaling solution.
(b) In all the models mentioned above, education does not contribute to productivity. If individuals 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 highproductivity 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
come is the set of three contracts
the
-line and the
((that passes through
-curve,
and
in Figure 1. A separating out-
in Figure 2, where
is the intersection point of
is the intersection point of the
-line and the
-curve
is the intersection point of the
-line and the
-curve
(that passes through
11/12
πL
I2
π LMH
πL
I2
πM
π LMH
45° line
πH
B
.
.
.o
45° line
uM
uH
πH
C
π M uL
F
.
uM
uL
..
A
B
C
.o
I1
I1
Figure 2. The Separating Equilibrium
Figure 1. No Pooling Equilibrium
(b) There is no pooling equilibrium. A pooling outcome such as point
easily destroyed by another contract such as point
in Figure 1 can be
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
curve,
cannot be a separating equilibrium. If the
-line cuts the
-line cuts the
-
-curve,
cannot be a separating equilibrium; in this case, one can design a contract that is
below
the
-line and attracts types
-curve (point
and
If the intersection point of the
is on the left of the
-line,
cannot be a separating equilib-
rium; in this case, one can design a contract that is below
and
but not
 The
 The
Therefore, the conditions for
-line doesn’t cut the
-line doesn’t cut the
 The intersection point
-curve and
-line and attracts types
to be a separating equilibrium are:
-curve.
-curve.
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 right of the
-line is always on the left of the
-line and the
-line).
Answer 10.9. You try out first. I will show you the solution later in Chapter 11.
12/12
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
product for the seller of type
is
The cost of investment for the seller is
buyer’s value from the product of quality
ment
Hence, the buyer can offer a deal
surplus is
of the
is
The
The buyer can observe and verify investto the seller. Given a contract
and the buyer’s surplus is
the seller’s
Given the seller’s reservation
value , suppose the buyer’s deal ensures an ex ante IR condition to induce the seller’s acceptence. 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
known only to the worker himself, and the firm only knows the distribution of
density function is
for
where
With education level
is the wage and
where the
the worker’s utility function is
is the worker’s private cost of education.
The revenue function is
Hence, the firm’s payoff is
offers allocation scheme
to the worker. The IR constraint is:
where
is
The firm
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
1/6
where
is the quality of a product,
buyer to the seller. Quantity
is the quantity traded, and
and payment
is the payment from the
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
tion function on
i.e.,
(a) Find the optimal solution
be the uniform distribu-
for
∗
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.
2/6
Answer Set 11
Answer 11.1. The direct mechanism is: the seller of type
buyer pays the price
reports his type
for the product and demands investment
and then the
for the production of
the product. By the revelation principle, the buyer can confine her search for an optimal contract 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
for the seller is
Then, the FOC for reporting
the value function
is
(1)
Taking a derivative on the truth reporting condition
The SOC is
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
3/6
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)
and
Answer 11.3. We have
with
(a) Equation (9.73) in the book becomes
implying
∗∗
We have
∗∗
Thus,
∗∗
∗∗
is decreasing when
need
Thus, if
and
∗∗
is increasing when
is decreasing around a point
Since
we
by the first argument following
(9.77) in the book, we have
∗∗
around the point
creasing if
for
Thus, since
∗∗
is not decreasing if
By the requirement
where
(5)
cannot be strictly de-
must be constant on
Let
By (9.78) and (9.79) in the book, we have
∗∗
implying
4/6
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
decreasing on
∗∗
implying
is constant. That is,
for
must be strictly
In summary,
∗∗
where
(b)
is determined by (6).
∗∗
is decreasing when
and
∗∗
is increasing when
By this knowledge, we can now draw the picture for
Also,
∗∗
is convex.
∗∗
x
x ** (θ )
x (θ )
b
Figure 1.
1
2
1
∗∗
θ
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
5/6
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
actual payment. We have
If
since
∗
∗
but the term
when
∗
and
reduces the
∗
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.
6/6
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
each agent has probability
dollars. During the period,
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
reasons, assume
in any state. For technical
and
(a) Find the simplest Pareto equilibrium solution. It is the equilibrium solution for the ArrowDebreu 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 probability 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
government can observe
Assume the
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 probability of having an accident by taking some level of precaution
1 Consider
The level of precaution
this as a pension plan, for which the dead get nothing and what the dead have left is shared among
the living population.
Page 1 of 14
costs
By investing more
matter how high
an individual lowers his probability of loss. Assume that no
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 Section 1, let
The density function
mean
states that the output follows the exponential distribution with
and variance
(a) Show that the second-best solution is
∗
[Hint: assume
( , )
( , )
(b) Find the first-best solution
∗
for any
∗∗
and
∗∗
and verify this later].
[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 2 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 3 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
If
the individual's problem is
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 possiThe insurance
ble, but he is limited by the no-borrowing condition. The solution is
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 equilibrium 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 4 of 14
i.e.,
which yields
∗∗
To find the competitive equilibrium solution, we repeat the derivation in (b). The individual'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
The problem can thus be simplified to
as possible. Thus, with the budget,
The FOC is
The insurance company's profit is
Thus, the competitive solution
∗
Zero profit then implies
Thus,
is the solution of the following equation:
∗
∗
∗
We have
∗∗
We have
(
∗∗ )
if
By the concavity of
in
this implies
∗∗
means that in the competitive equilibrium, each individual will invest too much in
∗
This
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 5 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 6 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
Given price
such as
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
the solution must be
∗∗
Thus,
∗
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
shifting to the left as
increases. This line will be becoming steeper and at the same time
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 7 of 14
∗
the line being moved too much to the left. The optimal
tradeoff. The separating equilibrium is the pair
is the value that gives the optimal
of contracts.
I2
πL
π H*
πH
.
45° line
B
.E
uH
..o
w−L
w− L − z*
I1
w − z* 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
shifting to the left as
increases. This line will be becoming steeper and at the same time
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
tradeoff. The separating equilibrium is the pair
∗
is the value that gives the optimal
of contracts.
I2
π L*
πL
πH
w−L
w− L − z*
.
45° line
B
.E u
.. o
w − z* w
H
I1
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 prevention by the high-risk individuals. In (b), only the low-risk individuals are better off, and the
Page 8 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
The FOC for
is
for any
and verify this later.
which implies
(1)
Three conditions, the IC condition, the IR condition and (1), can determine the three parameters
and
∗
The IR condition implies
Page 9 of 14
implying
By the IC condition, we have
implying
implying
(3)
By (1),
Page 10 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 11 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
2 Using
we have
the Lagrange method, the FOC for
is
implying
implying
∗∗
Page 12 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 13 of 14
Thus,
∗
Thus,
∗
∗
Then,
∗
(b) The first best is determined by
implying
∗∗
∗∗
Obviously, the effort levels are higher in the first best.
Page 14 of 14