6001: ECON Solution to Assignment 9
(1) Suppose you have 1 dollar to invest in a project. The outcome of the
project depends on the state of the world. There are two states of
the world: 1 and 2 and they are equally likely. In state 1, the project
returns 4, 2 or nothing. Each event has an equal chance. In state 2,
the return is zero for sure. Before making the investment, you can
consult a nancial analyst. This expert does not know the true state
of the world but gets imperfect signal about the states: in particular,
there are two signals {s1 , s2 }. In state 1, signal s1 is received with
probability 3/4 while signal s2 is received with probability 1/4. On
the other hand, in state 2, signal s1 is received with probability 1/4
while signal s2 is received with probability 3/4. If you pay the analyst,
you can get to observe the signal she receives. If you are risk-neutral,
what is the maximum amount that you will be willing to pay for the
analysts service?
First note that by Bayes rule 1 the conditional probability that the
state is 1 given signal s1 is given by P (state is 1|s1 ) which equals probability that state is 1 and the signal s1 is received divided by the probability of receiving signal s1 . The probability of receiving signal s1
equals 1/2.3/4 + 1/2.1/4 = 1/2 while in state 1, s1 is received with
probability 3/4. Thus,
P (state is 1|s1 ) =
1/2.3/4
= 3/4
1/2
Similarly given that the signal is s2 , one can show that the probability
that the state is 1 is 1/4. Clearly, if you observe s2 , you are not going
to invest. If however, signal is s1 , the expected return is 3/4.21 = 1/2
and thus you will invest.
Hence ex-ante, your payo to visiting the expert is
1/2[3/4.1] + 1/2[1/4(1)] = 1/4
This is the maximum F you are willing to pay.
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Bayes Rule: Conditional on the event B , the probability of event A is given by
P rob(A|B ) =
P rob(A and B )
P rob(B )
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(2) In a market for used cars, there are two types of cars: good or bad. A
buyer values a good car at 15, 000 and a bad car at 1, 000. Moreover,
75% of all the present car owners also have the same valuation. The
remaining 25% of them however are leaving town and they will sell
their cars as long as they get 1,000, irrespective of their quality. The
buyers can not distinguish between qualities but knows that 50% of
the used cars are good. Find out the maximum price at which a used
car can be sold. What is the average quality of cars that get sold at
this price?
It is easy to check that P must be less than 15, 000. Now for any
P > 1000, all bad quality sellers will sell and 1/4 of the good quality
sellers will sell. Thus the probability that a car oered for sale is good
is given by 1/5 (use Bayes rule one again.) Thus the maximum price
that a buyer is willing to pay is 1/5(15, 00) + 4/5(1000) = 3, 800.
(3) An entrepreneur needs 1 dollar to invest in a project. An entrepreneur
can be of two types: safe or risky. The safe projects return 1.5 for the
dollar invested while the risky project returns 4 with probability 1/3
and 0 with probability 2/3. A potential lender knows that 50% of the
borrowers have safe projects. The borrowers are protected by limited
liability, (i.e, if the return is zero, the borrower does not have to pay.)
The loan market is competitive and the risk free interest rate is zero
and thus a potential lender will lend as long as he gets back his money.
(a) If the market could observe the identity of the entrepreneur,
what will be the credit market outcome, i.e, which group of entrepreneurs will obtain loan and at what interest rate?
Risky types have a net return of 1/3(4) 1 > 0 and the safe
type also has a strictly positive net return. When there is full
information, the safe types will receive the loan at zero rate of
interest while the risky types have to pay an interest r such that
1+r
3 = 1, or that r = 2.
(b) If the market could not observe the identity of the entrepreneur,
what will be equilibrium interest rate? Does the entrepreneur
with the safe project gets to invest in this equilibrium?
The safe type will pay a maximum interest of r = .5 while risky
types will pay a maximum interest of r = 3. Now if r .5, the
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expected payo to a lender is 1/2(1+ r)+1/2[1/3(1+ r)] = 2(1+r) .
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At r = .5, this payo equals 1. Thus, there is one equilibrium
with r = .5 where both types of borrowers get the loan.
However, r if > 1/2, only risky types will apply for the loan and
then for the lender to break even we must have r = 2. This is
the second type of equilibrium.
(4) Consider a used car market. A car can be of good or bad quality.
A good quality car has 1/4 probability of a break down while this
probability is 4/5 for a bad quality car. A buyers utility from owning
the car is 100 if the car does not break down and it is zero otherwise. A
prospective seller knows the true quality of the cars while a buyer only
knows that 50% of the owners have good quality cars. The reservation
prices2 of 50% of the good car owners is sixty and it is twenty for
the remaining half. The bad quality car owners will sell at any non
negative price.
(a) Determine the maximum price at which the buyer is willing to
buy a car? At this maximum price, what is the proportion of the
bad quality cars that are oered for sale?
The utility from buying a good quality car is 1/4(0)+3/4.(100) =
75 and the utility from buying a bad quality car is 1/5(100) +
4/5(0) = 20.
If P 60, all sellers will sell. Then on average half of the cars
will be good. Thus the expected utility from buying a car is
1/2.(75) + 1/2(20) = 47.5. Since P 60, buyers will not buy at
such a price.
Now if 20 P < 60, all bad quality cars and half of the good
quality cars will be sold in the market. Thus, a buyer has 1/3
probability of getting a good quality car. His expected utility
then is 1/3(75) + 2/3(20) = 115/3. This is the maximum price
that a buyer is willing to pay. At this price, two thirds of the cars
in the market are bad.
(b) Does there exist a warranty scheme in which only the good quality
sellers will oer their cars for sale? Explain your answer.
Let W be the warranty amount and let P be the price. For all
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The minimum price at which the seller is willing to sell.
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good quality sellers to oer this scheme, we must have
P
W
60
4
(0.1)
while for bad quality sellers not to use it, we will also require that
P
4W
0
5
(0.2)
Finally, the scheme must give a positive net utility to the buyer.
The net utility of the buyer is
300 W
+
P 0
4
4
(0.3)
It is easy to check that there are schemes (W, P ) that satisfy all
three of these equations.
(c) Assume now that a seller can take his car to a mechanic (for
inspection) who will certify the quality of the car. Good quality
cars always pass the inspection test while the bad quality cars
pass this test with probability 1/2. If the mechanics fee is 20
dollars, is there an equilibrium where only good quality cars are
sold in the market?
Given any P , a bad quality seller will use the mechanics service
if P/2 20. If P 40, these sellers can stay out of the market.
Given that P 40, a good quality seller with reservation price
of 20 will use the mechanics service if and only if P 20 20.
Thus, the only possibility for an equilibrium is with P = 40.
Good quality sellers (whose reservation price equals 20) will sell
their cars and the bad quality sellers are happy to stay out the
market.
(5) There are two jobs A, B and two groups of workers. In Job A, Group 1
workers are worth 10,000 while Group 2 workers are worth only 1,000.
In job B, all workers are equally productive and are worth 5,000. Let p
be the proportion of Group 1 workers in the population. An employer
in job A can not distinguish between the workers. Show that if p less
than 4/9, then all workers will be employed in job B.
Let w be the wage paid in job A. Clearly, w 5000. Moreover,
p(10, 000) + (1 p)1, 000 w 5, 000. Thus, 9, 000p 5, 000, or
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p 5/9 for the rm to break even. Hence if p < 4/9, it will not be
possible to hire a worker in job A without making losses.
(6) Assume now that workers can spend 4, 000 to acquire a degree. The
probability of getting this degree however diers across these two
groups. In particular, this probability is 1 for group 1 workers and
it is only 1/4 for the second group. Does there exist an equilibrium in
which i) group 1 workers acquire the degree and get 10,000 in job A
and ii) group 2 workers take up job B?
The answer is yes. For this to happen, group 1 workers must pay 4,000
and acquire the degree rather than taking up a job in B . This is true
since 10, 000 4, 000 > 5, 000. Second, group 2 workers should not try
to get a degree, or, 1/4(10, 000) + 3/4(5000) 4000 5000. This is
also true.
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