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Most of the problems in this chapter focus on different applications of the principal–agent model.
Additional problems are provided on auctions and the lemons problem.
Problem 18.5 requires the solution to a complicated maximization problem that has to be
solved using numerical methods similar to Example 18.5 in the text. An Excel spreadsheet with
the solution method is provided on the textbook’s website. Other mathematical software can also
be used of course.
Comments on Problems
18.1 This problem studies the moral-hazard model in the context of shareholders inducing
effort from a manager using various contractual forms (profit sharing, bonuses, buyouts).
18.2 This problem applies the moral-hazard model to the relationship between a client (in the
role of principal) and a lawyer (in the role of agent).
18.3 This problem computes the optimal linear (i.e., per-unit) price for coffee to compare to
the optimal nonlinear tariff computed in Example 18.4. As a first step, the problem
requires students to convert representations of consumer utility functions into demand
functions.
18.4 This problem provides students with further practice on computing optimal nonlinear
tariffs by slightly changing the numbers used in Example 18.4.
18.5 This problem, similar to Example 18.2, provides students with further practice on moral
hazard in insurance.
18.6 This problem, similar to Example 18.5, provides students with further practice on adverse
selection in insurance. The tongue-in-cheek application, involving a higher accident rate
for left-handers, has an interesting history in the medical literature. Early studies
estimated a much higher mortality rate for left-handers by comparing mean ages at death
for left- and right-handers. Later studies suggested that the difference in mean death age
was an artifact of parents more recently abandoning the practice of training left-handed
children to be right-handed. The later parts of the problem treat competitive insurance as
in Example 18.6 in the text.
18.7 This problem is a simple version of Akerlof’s lemons problem.
CHAPTER 18:
Asymmetric Information
Chapter 18: Asymmetric Information
205
18.8 This problem has students work through a very simple model of a common-values
auction in which the winner’s curse arises.
Analytical Problems
18.9 Doctor–patient relationship. This problem works through a moral-hazard problem in
which the patient is the principal and the doctor is the agent. The problem is more
difficult than a standard one because the students are asked to work with general
functional forms.
18.10 Increasing competition in an auction. This problem provides students with practice
working through the calculation of optimal strategies in an auction by repeating the
analysis from the chapter except with n bidders rather than just two. The problem
establishes an interesting proposition about the benefits of competition (here, that
increasing competition in an auction increases the seller’s revenue).
18.11 Team effort. This problem works through the logic of Holmstrom’s famous “Moral
Hazard in Teams” article, showing that incentives are diluted in large teams. The problem
suggests that employee stock ownership plans may not provide good incentives to work
hard and so probably have other rationales.
Behavioral Problem
18.12 Nudging consumers into adverse selection. This problem shows that nudges that
improve individual consumers’ choices can exacerbate adverse selection and reviews a
recent empirical article by Benjamin Handel demonstrating this point for health
insurance.
Solutions
18.1 a. With a half share,
Chapter 18: Asymmetric Information
206
b. The most she would pay equals (0.5)(1,000) + (0.5)(400) – 100 = 600.
d.
18.2 a. The lawyer maximizes
b. The lawyer maximizes
d. With a 100% contingency fee, the lawyer chooses
*1l=
and earns a surplus of
1/2, which the plaintiff can extract initially by selling the case to him. This
Chapter 18: Asymmetric Information
207
18.3 First solve for type
’s demand. Given linear price
,p
this type will choose
q
to
18.4 By Equation 18.51, the low type’s second-best quantity satisfies
Chapter 18: Asymmetric Information
208
18.6 a. Full insurance for left handers involves a certain payout of $500. They would be
unwilling to pay more than the $500 they receive. So full insurance for them is
equivalent to no insurance. The premium for full insurance for right handers
b. Left handers are fully insured. Hence,
500.
LH
x=
The premium just satisfies
incentive compatibility, making them indifferent between their contract and the
one for right handers:
Chapter 18: Asymmetric Information
209
One can try to use calculus to find the maximum of this function, but perhaps a
simpler route is just to look at its graph.
c. The competitive equilibrium under full information results in full, fair insurance
d. The separating contract for the risky type involves full insurance. We saw that the
full insurance contract, if priced fairly as required under competition, was
Chapter 18: Asymmetric Information
210
b. If sellers value good cars at $8,000, they would not be willing to sell even at the
18.8 a. If a buyer’s signal is
,L
the object is certainly worth 0. If the buyer’s signal is
,H
b. If the buyers’ strategy is to bid 0 conditional on
L
and 1/2 conditional on
,H
a
buyer earns 0 conditional on
L
and
Analytical Problems
18.9 Doctor–patient relationship
Chapter 18: Asymmetric Information
211
A fully informed patient would choose
m
to maximize
( , ),
p c m
U m I p m−
yielding first-
order condition
18.10 Increasing competition in an auction
a. Bidder 1 maximizes
x
Up
Chapter 18: Asymmetric Information
212
b. Expected revenue equals the expected value of the second highest bid. Since
d. Bids converge to valuations in the first-price auction as the number of bidders
18.11 Team effort
a. Partner
obtains a
1n
share of the revenues. He chooses
e
to maximize
b. If worker
i
gets a 100% share, he chooses
i
e
to maximize
Chapter 18: Asymmetric Information
213
c. We have
d. The analysis suggests it is unlikely that the stock plan provides incentives in a
rational model. There may be psychological effects on employee morale.
Behavioral Problem
18.12 Nudging consumers into adverse selection
a. Red-car owners are fully insured in equilibrium, leading to certain “endgame”
b. Fixing the separating contracts from the previous part, there are four certainty
equivalents to be averaged together, depending on the type of owner and whether
Chapter 18: Asymmetric Information
214
c. There exists a competitive equilibrium with a pooling contract offering full
