Problem Set 6
Economics 703
Spring 2016
Due: Friday March 18
1. Consider the adverse selection model we looked at in class. Suppose that θ is distributed uniformly on [θ, θ̄] where 0 < θ < θ̄. Assume that r(θ) = aθ where 0 < a < 1.
What is the unique equilibrium price and average quality as a function of a, θ, and θ̄?
How do you know the equilibrium is unique? How does the price depend on a, θ, and θ̄ (is
it increasing, decreasing, or ambiguous as a function of the parameter)? What happens
if θ = 0? What if we assume θ is uniformly distributed between 0 and 1 and change r to
r(θ) = aθ − b where .5 + b < a < 1 and b > 0?
2. Suppose there are two types of entrepreneurs in the world who differ only in terms of
how likely their project is to succeed. More specifically, each entrepreneur has a project
which pays off S > 0 if it succeeds and 0 if it fails. Good entrepreneurs have a project that
has a probability of ph of success, while the projects of bad entrepreneurs have probability
p` of success where 0 < p` < ph < 1. A fraction λ ∈ (0, 1) of the entrepreneurs are good.
To do the project, the entrepreneur must get a loan of L > 0. If an entrepreneur’s project
succeeds, he gets S minus L and the interest on the loan. He cannot default on the loan
in this case. If his project fails, he defaults and gets D > 0, while the bank gets nothing.
If he doesn’t get a loan at all, his payoff is 0. Assume that S − L > D. Also, assume
p` S + (1 − p` )D − L > 0
and
λph + (1 − λ)p` <
ph L
.
ph S + (1 − ph )D
Banks are risk neutral and compete so that their expected profits are zero. If the
borrower borrows L and repays with interest, the bank gets L(1 + r). If the borrower
defaults, the bank gets 0. If the bank doesn’t make the loan, its payoff is L.
(a) Suppose the bank sees the type of the entrepreneur before setting the interest
rate. What interest rate will be offered to an entrepreneur? Under what conditions does
he take the loan?
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(b) Now suppose the bank cannot see the type of the entrepreneur. What is the
equilibrium outcome and how does it depend on λ?
3. 1. Suppose there are two types of workers, type I and type II. Half the workforce is
of each type. The productivity of a type I is $100 while the productivity of a type II is
$40. An employer cannot observe a worker’s type directly.
(a) Suppose the cost of a year of education is $35 to a type II and only $25 to a type I.
Find the separating and pooling equilibria.
(b) Now suppose that the cost of a year of education is $30 for each type. What is the
set of separating and pooling equilibria?
(c) Now suppose there are three types. Types I and II are as in (a), while type III has
productivity $30 and education costs $45. Now find the (fully) separating and (fully)
pooling equilibria.
4. Consider a variation on the Spence model where there are two types of jobs, A and
B, as well as two types of workers, tA and tB . The productivity of a worker depends on
the job he is assigned to, his type, and his education level. Specifically, the productivity
levels are
t√
tB
A
job A 4 e 0
job B 0
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where e is the worker’s level of education. As in the usual Spence model, assume there
are two employers who observe the worker’s education level but not his type and then
offer competing wages to try to hire the worker. Either employer can use the worker in
either job. Suppose the cost per unit of education for type tA workers is 1 while the cost
for the type tB workers is 3.
Find all separating equilibria with the property that the beliefs are always degenerate
(that is, one type is given probability 1, where the type given this probability can depend
on what has been observed).
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