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CHAPTER 18:
Asymmetric Information
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
18.7 This problem is a simple version of Akerlof’s lemons problem.
18.8 This problem has students work through a very simple model of a common-values
auction in which the winner’s curse arises.
18.9 Doctor–patient relationship. This problem works through a moral-hazard problem in
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
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
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,
E(Utility) = (0.5)(1,000/2) + (0.5)(400/2) – 100 = 250.
With a quarter share,
E(Utility) = (0.5)(1,000/4) + (0.5)(400/4) – 100 = 75.
18.2 a. The lawyer maximizes
2,
32
ll
b. The lawyer maximizes
2,
2
l
cl
(1 ).cc
c. The optimal contingency fee for the plaintiff is
*1 2,c
maximizing her surplus
(1 ).cc
Her surplus is 1/4 and the lawyers is 1/8.
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
case to a lawyer would have no incentive to exert effort for the case.
18.3 First solve for type
’s demand. Given linear price
,p
this type will choose
q
to
(2 ) .q pq
q
22
2
1 20 1 15 625( )
( ) ( ) .
2 2 2
pc
p c p c
p p p
The first-order condition with respect to
p
is
3
625(2 ) 0,
2
cp
p
implying
*2,pc
or
*10p
when
5.c
The monopolist’s expected profit at
*10p
is
15.625.
18.4 By Equation 18.51, the low type’s second-best quantity satisfies
** **
1 2 3 1
15 5 (20 15) ,
13
LL
qq
implying
**
15 5 10,
L
q
or
** 1.
L
q
This one-ounce cup is sold for
** (1) (15)(2 1) 30 cents.
LL
Tv
The high type’s cup has the same size as in the first best:
** 16.
H
q
The tariff is
** 20 2 16 2 1 15 2 1 150.
H
T
Compared to Example 18.4, the low type’s cup is distorted even further from the first
best because the cost of doing so has fallen since there are a smaller proportion of them.
The shop owner can then increase the high type’s tariff.
18.5 a. The premium satisfies
(0.5)(10,000) 5,000.p
b. The premium satisfies
(0.5)(5,000) 2,500.p
The individual’s utility is
0.5ln(20,000 2,500) 0.5ln(20,000 2,500 10,000 5,000)
9.6017
His utility from part (a) is
ln(20,000 5,000) 9.6158,
verifying that he prefers full to partial insurance.
c. The premium satisfies
(0.5)(7,000/ 2) 1,750.p
The individual’s utility from
partial insurance now is
0.5ln(20,000 1,750) 0.5ln(20,000 1,750 7,000 3,500)
9.7055,
so he now prefers partial to full insurance.
18.6 a. Full insurance for left handers involves a certain payout of $500. They would be
11
ln(100 ) ln(1,000 500) ln(1,000),
22
RH
p
implying
ln(100 ) ln( 500 1,000),
RH
p
in turn implying
*1,000 500 1,000 292.9.
RH
p
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:
ln(1,000 ) ln(1,000 500 ),
LH RH RH
p p x
The premium for right handers reduces them to their outside option of no
insurance:
11
ln(1,000 500 ) ln(1,000 )
22
11
ln(1,000 500) ln(1,000).
22
RH RH RH
p x p
Solving for
,
RH
x
(2)
(1,500 ).
1,000
RH RH
RH
RH
pp
xp
The insurer’s profit is
10( ) 100 .
2
LH
LH LH RH x
p x p
*500,
LH
x
c. The competitive equilibrium under full information results in full, fair insurance
**
500,
LH LH
px
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
equivalent to no insurance. On the other hand, any fairly priced contract attracting
2.
RH RH
px
18.7 a.
(1/ 2)(10,000) (1/ 2)(2,000) $6,000
.
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
1 1 1
Pr(other sees ) 0 Pr(other sees ) 1
2 2 2
1 1 1 1 1
2 2 2 2 2
1.
8
LH
conditional on
.H
To see this payoff, if the other sees
,L
the object is certainly
worthless. If the other sees
,H
he bids according to the specified strategies, and
the end up in a tie. One of them is randomly selected as the winner, so with
12
1 1 2 1 2.
Analytical Problems
18.9 Doctor–patient relationship
m
( ) ,
d d p
U I U
dm
I p m
pp
UU
1.
pd
mm
pp
Um U
pp
U x U x
m
( , ),
p c m
U m I p m
and an implied marginal rate of substitution of
.
p
m
p
Um
p
Ux
Graphically, the fully informed patient chooses a point of tangency A between his
indifference curve and the budget constraint. The doctor chooses a point B, which we
m
x
Up
A
B
Ic
Ic/pm
18.10 Increasing competition in an auction
a. Bidder 1 maximizes
1 2 1 1
Pr max( , , ) ( ),
n
b b b v b
Maximizing with respect to
1
b
yields
11
( 1) / .b v n n
Expected revenue is
()
( )( 1) / .
n
E v n n
This equals
( 1)/( 1)nn
using the formula (Equation 18.71)
for the expected value of the maximum order statistic
()
.
n
v
b. Expected revenue equals the expected value of the second highest bid. Since
ii
bv
c. Yes.
d. Bids converge to valuations in the first-price auction as the number of bidders
grows large. Bids remain at valuations in a second-price auction. In both cases,
which is negative for
1.n
Further,
2
21
lim 0.
2
n
n
n
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.
18.12 Nudging consumers into adverse selection
a. Red-car owners are fully insured in equilibrium, leading to certain “endgame”
wealth
100,000 5,000 95,000,
which is their certainty equivalent. Gray-car
owners’ certainty equivalent satisfies
ln( ) 0.15ln(80,000 453 3,021) 0.85ln(100,000 453),
G
CE
96,793
G
CE
11
(0.1)(95,000) (0.9) 96,614 95,000 95,807,
22
less than in part (a). Thus, a nudge to reduce the behavioral bias increases welfare.
(0.16)(20,000) 3,200.
all consumers a certainty equivalent of
100,000 3,200 96,800,
greater than in
