.
Question 1
.
Consider a pure exchange economy with two consumers (A and B), two goods (1 and
2), and endowment point E = (A e, B e) (so that A e is the initial endowment of consumer A
and B e is the initial endowment of consumer B). Suppose the two consumers have identical
preferences represented by the utility function u. You may assume that u is smooth, strictly
increasing in each of its arguments, and strictly concave.
1. Show that X = (
A e+ B e
2
,
A e+ B e
2
) is a Pareto efficient allocation.
2. Can the allocation X be supported as a market equilibrium? If so, what prices would
support this allocation as a market equilibrium?
3. Are there other Pareto efficient allocations in this economy?
.
Answers to Question 1
.
A
B
1. Let z = e+2 e . Observe that for any h 6= 0:
u(z + h) + u(z − h) = 2
1
1
1
1
u(z + h) + u(z − h) < 2u( (z + h) + (z − h)) = 2u(z)
2
2
2
2
(1)
Alternatively, develop an argument based on the observation that at allocation X the two
consumers have identical MRS.
2. Yes. Follows from the Second Welfare Theorem. Prices would have to be proportional to
∂u
∂u
(z), ∂x
(z) .
∂x1
2
3. Yes, for example one consumer could receive the entire supply of each good.
.
Question 2
.
Consider a consumer living for 2 time periods only, today and tomorrow. Tomorrow may
be Bad (B), Mediocre (M), or Good (G). The consumer has no income today, and contingent
income tomorrow. If tomorrow is Bad then his income is 0, but his income is 400 NOK in the
Mediocre state and 800 NOK in the Good state. There are two goods in this world, apples
1
and oranges. The price of an orange is 1 NOK in all states of the world, and that of an apple
is 2 NOK in all states of the world. The consumer derives no utility from consuming today,
but derives positive utility from consumption tomorrow. Specifically, we suppose that the
consumer has Cobb-Douglas preferences such that, if xw
i represents consumption of good i in
state of the world w ∈ {B, M, G}, then:
u(x) =
Y
xw
i
(2)
i,w
The asset structure of the economy is given by A, where


2 0 2


A= 2 2 0 
0 2 2
(3)
Each tradable asset is traded today at the price of 1 NOK.
You are also being told that

A−1

1
1 −1
1

=  −1 1
1 
4
1 −1 1
(4)
1. What is the price of portfolio ϕ0 , where


200


ϕ0 =  −200 
0
(5)
2. Suppose that the consumer purchases portfolio ϕ0 today. How many NOK will the
consumer dispose of tomorrow, taking into account his own income?
3. Let b denote the bond of this economy. Express b. What is the interest rate in this
economy?
4. What probability does the market assign to each state of the world occurring?
5. Now suppose that the consumer wishes to maximizes his utility. What portfolio should
he purchase in this case?
2
.
Answers to Question 2
.
1. Price is q.ϕ0 = 0.
 
 
0
1
 
 
2. Aϕ0 + y = Aϕ0 + 400  1  = 400  1 .
2
1
 
 
1
1
 

1
3. Ab =  1 , from which b = 4  1 . Then q.b =
1
3
4
=
1
,
1+r
from which r = 13 .
1

 
1
1


1
1
1 
t
4. Use q = Aπ to retrieve π = 4  1 . Use then π = 1+r α to obtain α = 3  1 .
1
1
5. Let M denote the present value of lifetime income, so that M = π.y = 300. Optimal
consumption involves present value expenditure of M/3 = 100 in each future state of the
world. Expressed at spot values the expenditure
is thus
 400 in each future state of the world.
1
 
The optimal portfolio thus solves Aϕ + y = 400  1 . By 2. we thus obtain ϕ = ϕ0 .

1
3
Problem 1.
Consider the setup in the Akerlof “lemons” model. There are two groups of traders for cars: B and S
with (group) incomes YB and YS, respectively. Incomes include earnings from sale of cars. As in the
Akerlof model, consider each group to be acting as an individual player in the car-exchange market.
There are two goods which B and S care about: cars and M, where xi is the quality of car i, M is the
other “consumption good” and n is the number of cars bought.
Group B has the following utility function:
UB =M + ∑𝑛𝑖=1 2xi
Similarly, US =M + ∑𝑛𝑖=1 xi applies to group S.
Assume that group S has N cars with quality x where x is uniformly distributed on the interval [0, 3].
Group B has 0 cars to begin with.
Normalize the price of M to 1 and ignore any indivisibilities (for cars). Proceed by taking the price of
a car to be p.
(i) Write down the demand and supply functions of cars for each group.
(ii) Now calculate the equilibrium price and corresponding quantity of cars traded.
(iii) How is this different from the full information first-best outcome?
Problem 2.
Consider a risk-neutral principal and a risk-neutral agent in an asymmetric information setup.
The principal cares about the level of output q. The realization of q is stochastic: either high (qH) or
low (qL) where qL< qH.
The agent may perform a task requiring effort which is not observable by anyone other than the agent.
Exerting effort is costly for the agent: call this cost C (which is a fixed positive number). If the agent
undertakes effort, the probability that q=qH is π1. If the agent does not undertake effort, the probability
that q=qH is π0, where π0< π1. Assume that the agent has a reservation utility of 0. The principal must
design an incentive contract for the agent. Assume that C is low enough, so that the expected output
when effort is exerted less the cost of effort C exceeds expected output when no effort is exerted.
(i) Write down the principal’s (constrained optimization) problem.
(ii) Is moral hazard an issue here, in the sense of deviation from the first-best? Illustrate by
constructing a set of optimal transfers.
(iii) Now suppose, there is limited liability in the sense that transfers to the agent cannot be negative.
Can first-best be achieved under this additional constraint?
Solution to Problem 1.
Let μ denote the average quality of cars supplied when the price is p.
For group S, the demand is YS /p when μ>p and 0 for the reverse inequality. The supply is given by
pN/3 exploiting the fact that quality distribution is uniform on the interval [0, 3]. Thus the average
quality is simply p/3 when the price is p.
For group B the demand is YB /p when 2μ>p and 0 for the reverse inequality. This group has no cars to
supply.
So, total demand (supply) for any level of price is obtained by adding up the demand (supply) of the
two groups. Given that the expected quality is p/3 when the price is p, leads to no trade at any positive
price. This is clearly not first-best as both groups could benefit from trading at any price between 1
and 2.
Solution to Problem 2.
The principal’s problem is to choose transfers tL and tH in order to:
max
π1(qH - tH) + (1- π1) (qL – tL)
subject to the incentive and participation constraints (IC and PC) of the risk-neutral agent.
IC: π1 tH + (1- π1) tL – C ≥ π0 tH + (1- π0) tL.
PC: π1 tH + (1- π1) tL – C ≥ 0.
Given the nature of the constraints, they bind at the optima. Hence, we get the following:
tL= - C ( π0)/( π1- π0) which is negative and tH= C (1- π0)/( π1- π0) which is positive.
It is straightforward to see that the expected payment by the principal is simply C and such a scheme
of transfers does indeed implement the first-best level of effort; so moral hazard has no bite here.
With limited liability in the sense of non-negative transfers, we are in the world of second-best.