Universidad Carlos III de Madrid Economics of Information

Exercise List 3: Adverse Selection

Exercise 1. A good of two qualities, high (H) and low (L), trades in competitive
markets in which each seller has a single unit and each buyer wants to buy a single
unit. There are nH sellers with a unit of high quality whose opportunity cost is cH
euros, nL sellers with a unit of low quality whose opportunity cost is cL euros, and n
buyers who value a unit of high quality in uH euros and a unit of low quality in uL
euros. Assume that uH > cH > uL > cL .
(a) Suppose that quality is observable. Calculate the competitive prices for the
cases n > nH + nL and n < nH + nL : Discuss if these competitive equilibria generate
the maximum surplus. (If you …nd it helpful, assume uH = 10; cH = 7; uL = 5;
cL = 0; nH = 1, nL = 1 and n 2 f1; 3g:)
(b) Now suppose that quality is not observable and both qualities trade in the
same market. Also assume that n = nH + nL : Represent the supply and demand
schedules in the plane (q; p) and calculate the competitive equilibria of this market
when the expected value of a random unit,
nH nL
u(nH ; nL ) = uH + uL ;
nH + nL nH + nL

is greater than cH ; and when it is less than cH : (If you …nd it helpful, use the parameter
values suggested in part (a), and consider the cases nL = 1 and nL = 2.)

Exercise 2. Consider a market for used cars whose qualities, indexed by the sellers’
cost, are uniformly distributed in the interval [2; 6]. Buyers are risk-neutral and value
each quality 20% more than sellers. Naturally, each seller knows the quality of the
good he sells, but quality is not observable to buyers prior to purchase. Assume that
there are more buyers than sellers.
(a) Determine the market supply and the average quality of the cars o¤ered at
each price.
(b) Calculate the market equilibrium.

Exercise 3. Consider an insurance market in which all individuals have the same
initial wealth W = 1 and the same preferences, which are represented by the von
p
Neumann-Morgenstern utility function u(x) = x, where x is the individual’s dis-
posable income. Each individual faces the risk of having an accident resulting in
losing his wealth. For a fraction 2 (0; 1) of individuals the probability of having
this accident is pL = 1=2 whereas for the remaining fraction 1 this probability is
pH = 4=5. Insurance companies know this information, but at the time of signing a
policy do not observe whether the probability of having an accident for a particular
individual is pL or pH .
(a) If it is mandatory that policies o¤er full insurance, which policies will compa-
nies o¤er for each value of ? Which individuals would subscribe them?
(b) If companies are free to o¤er any policy, which policies will be o¤ered for each
value of ? (Here you need to identify a separating equilibrium.)

Exercise 4. Consider an economy with three types of workers whose abilities and
reservation utilities are (a1 ; a2 ; a3 ) = (1; 2; 3) and (u1 ; u2 ; u3 ) = (1=2; 1; 22=10), re-
spectively. The probability of each type is 1=3. A perfectly competitive …rm con-
siders hiring the workers, but the productivity is private information only known
to each worker. Once the workers are hired, the production function of the …rm is
y = a1 L1 + a2 L2 + a3 L3 , where Li is the amount of work done by a worker of type ai .
The product y is sold at the price of 1.
(a) Would the three types of workers accept an average salary?
(b) Can the …rm o¤er a salary that in equilibrium attracts only workers of types
1 and 2?

Exercise 5. In an insurance market there are two types of agents A and B, in equal
proportions. Both types of agents have the same initial wealth w = 1 and the same
p
preferences on money represented by the utility function u = x, where x is money.
However, their risk of loss is di¤erent. Agents of type A have a probability of loss of
0; 5, whereas the probability of loss for type B agents is 0; 2. The insurance companies
can distinguish the types of the agents but the agents do not know their types.
(a) Compute the competitive equilibrium in this market.
(b) The government regulates the market and forces the companies to o¤er a
unique full insurance policy and cannot reject any customer. Compute the new equi-
librium and explain why this would not be an equilibrium if there were no regulation.
(c) Discuss the e¢ ciency in the situation (a) and (b).

Exercise 6. The genetic advances allow the technology that determines the risk
of acquiring a certain disease through a test. The government is considering the
possibility of allowing the insurance companies to make the above test, before they
o¤er the insurance policy to the buyers. What would you advise to the government?
Assume that the cause of the disease is only genetic, so it is not determined by the
habits of the consumer.

Exercise 7. A monopolist who produces with marginal cost 1 faces two consumers

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with demands q1 = 8 p1 and q2 = 10 p2 , respectively. The monopolist cannot
distinguish the demand of each consumer.
(a) Compute the price that the monopolist would choose and the pro…ts it would
obtain if it cannot discriminate consumers in any way.
(b) Suppose now that the monopolist can discriminate and it is thinking on a
second degree price discrimination. This consists in a two-part tari¤ (Q; p) in which
consumers pay a one-time access fee Q for the right to buy a product, and a per-unit
price p for each unit they consume. In this manner, the monopolist obtains revenues
from the access fee and from the sales of the good. Suppose the monopolist sets the
tari¤ so that both consumers pay it. That is, it sets the access fee Q to be less or equal
than the surplus of consumer 1 at the price p. Write the pro…ts of the monopolist
as a function of p and compute the optimal tari¤ for the monopolist and its pro…ts
under the optimal tari¤. If the monopolist is not interested in attracting consumer 1,
what is the optimal two-part tari¤? Compare the pro…ts of this case with the pro…ts
in part (b) above. Compute the surplus of each consumer.
The monopolist is considering the possibility of a menu of two-part tari¤s (Q1 ; p1 ) and
(Q2 ; p2 ). These tari¤s are designed so that the consumers with demand q1 = 8 p1
choose (Q1 ; p1 ) and the consumers with demand q2 = 10 p2 choose (Q2 ; p2 ).
(c) What are the conditions under which the consumers accept to purchase the
two-part tari¤s?
(d) What are the conditions under which each type of consumers prefer the two-
part tari¤ addressed to them?
(e) Write the optimization problem for the monopolist if it wants to …nd the opti-
mal menu of two-part tari¤s (Q1 ; p1 ) and (Q2 ; p2 ) that satis…es the above constraints
and solve this problem.
(f) Compute the total surplus in the above equilibrium and compare it with the
surplus obtained in part (b) of problem 1.

Exercise 8. Exercises 1, 2, 3, 4 and 6 in chapter 4 of the textbook by Macho and