Solutions to second problem set

Solutions to second problem set
1. The used car supply in a given town consists of 10,000 cars. The quality
of these cars range from 5,000 to 15,000 with exactly one car being worth
each pound amount between these two …gures. Used car owners are always
willing to sell their cars for what they are worth. Demanders of used cars
have no way of telling the quality of a particular car. Their demand
depends on the average quality of the cars in the market (Q) and on the
price P of the cars themselves, according to the equation:
D = 1:5Q
P
(a) If demanders base their estimate of Q on the entire used car market,
what will its value be and what will be the equilibrium price of used
cars?
AVERAGE QUALITY IS 10000. DEMAND = 15000 - P. EQUATING SUPPLY AND DEMAND ONE FINDS P=5000
(b) In the equilibrium described in part a, what will be the average quality of used cars actually traded in the market?
ONLY THE CAR WORTH 5000 WILL BE TRADED
(c) If demanders revise their estimate of Q on the basis of the average
quality of cars actually traded, what will be the new equilibrium price
of used cars? What is the average quality of cars traded now?
NEW DEMAND IS = 7500 -P. EQUATING SYPPLY AND DEMAND ONE FINDS NEGATIVE PRICE
(d) Is there a market equilibrium in this situation at which the actual
value of Q is consistent with supply and demand equilibrium at a
positive price and quantity?
NO. ASYMMETRIC INFORMATION HAS RESULTED IN MARKET BREAKDOWN.
2. There are many entrepreneurs and each has one project that requires an
upfront investment of 100. There are two types of projects. The “safe”
type of project yields 106 with probability 1 (net return equal to 6). The
“risky”project yields either 100 or 110 (a 0 or 10 percent net return), each
with probability one half. Entrepreneurs have no funds of their own, and
must borrow the entire investment cost of 100 from a bank. Assume that
banks are willing to make loans as long as their expected (mean) return
is at least 4 percent; entrepreneurs, on the other hand, will borrow if and
only if they have some chance of making strictly positive pro…ts.
(a) Suppose banks can tell which type of project each entrepreneur has
prior to making loans. Show that it is an equilibrium for banks to
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charge 4 percent interest to safe entrepreneurs and 8 percent interest to risky entrepreneurs. (By ‘equilibrium’ you need to show two
things: (1) banks are willing to lend because they are able to make an
expected return of 4 percent on their loans, and (2) entrepreneurs are
willing to borrow, because they have some chance of making positive
pro…ts).
RISKY ENTREPRENEURS ARE HAPPY LO BORROW BECAUSE
THEY WILL PAY BACK ONLY IN CASE OF SUCCESS. THE
BANK’S EXPECTED RETURN FROM A RISKY ENTREPRENEUR IS:
1
1
0+ 8=4
2
2
HENCE BANK IS HAPPY TO LEND
(b) Now suppose that banks cannot distinguish safe entrepreneurs from
risky entrepreneurs. Assume that the probability of funding a safe
(risky) entrepreneur is 50%.
i. What will be bank’s mean return be if it o¤ers a 4 percent interest
rate to all borrowers? Why isn’t this an equilibrium as de…ned
in part (a)?
BANK’S EXPECTED RETURN:
1 1
1
1
4 + ( 0 + 4) = 3
2
2 2
2
BANK WILL NOT LEND.
ii. Now suppose the bank charges 6 percent to all borrowers. What
would the mean return to the bank be if both types of entrepreneurs were willing to borrow at this rate? Will risky entrepreneurs actually be willing to borrow at this rate? Will safe
entrepreneurs actually be willing to borrow at this rate? What
will the mean return to the bank from charging 6 percent actually be, given the types of entrepreneurs who will actually want
to borrow at this rate? Is this an equilibrium as de…ned in part
(a)?
BANK’S EXPECTED RETURN IF EVERYONE BORROWS:
1 1
1
1
6 + ( 0 + 6) = 4:5
2
2 2
2
ONLY RISKY ENTREPRENEURS WILL BE WILLING TO
BORROW. HENCE ACTUAL MEAN RETURN IS:
1
1
0+ 6=3
2
2
BANK NOT HAPPY TO LEND
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iii. Suppose the bank charges 8 percent to all borrowers. Which type
of entrepreneur will be willing to borrow at this rate? What will
the bank’s mean return be, given the types of entrepreneurs who
will want to borrow at this rate? Is this an equilibrium?
ONLY RISKY ENTREPRENEURS. BANK HAPPY TO LEND
BECAUSE EXPECTED RETURN IS 4. THIS IS AN EQUILIBRIUM.
iv. Explain why the equilibrium in (iii) is ine¢ cient from society’s
viewpoint.
ADVERSE SELECTION PROBLEM DUE TO ASYMMETRIC
INFORMATION
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Answer to question 3:
4
5
Answer to question 4:
6
7
8