Dr. Donna Feir
Economics 313
Problem Set 6: Solutions
Asymmetric Information
1. (Adverse Selection and Moral Hazard) Suppose that a company offers “grade insurance” that works
as follows. For each course in which you get a grade below a C, the company pays you money.
Before offering the insurance policy for sale, the company looks over the transcripts of university
students and finds that, on average, 30% of all grades given are below a C.
a. Explain why the insurance company would be incorrect in assuming that it would only have
to pay claims on about 30% of its policies. Explain how adverse selection and moral hazard
would each affect this market
Grade insurance could create both a moral hazard and adverse selection problem. First, the
only students that would by grade insurance would be the students most likely to get a C. If
these types of students have a probability of getting a C that is 50%, then the probability the
insurance company would have to pay out is much here. Second, even if all students had the
same chances of getting a C, compensating students financially for their C would make it less
painful to get a C and therefore make them work less hard, increasing the probably they get
Cs.
Assume there are two types of students. Ones that care about grades and ones that don’t. The
ones that care about grades will only get a C with 10% probability. A C reduces their utility by
$1000 dollars. The ones that don’t care about grades will get a C with 40% probability. A C
reduces their utility by $500 dollars. Half of the students care about grades and half don’t.
b. How much is each student willing to pay as a premium to the insurance company to be fully
insured if they are risk neutral?
Students that care about grades premium: 𝟎. 𝟏 ∗ 𝟏𝟎𝟎𝟎 = 𝟏𝟎𝟎
Students that don’t care about grades premium: 𝟎. 𝟒 × 𝟓𝟎𝟎 = 𝟐𝟎𝟎
c. If a firm could observe each type of student, what would be the fair premium for each type?
What is the fair premium from the firm’s perspective if they can’t tell types of students
apart?
The fair premium for each type is given above. If their firm cannot tell the students apart,
it means there is a 50 percent probability they have students who care about grades who
they have to pay out 10 percent of the time and a 50 percent probability they have
students who do not care and they have to pay out 40 percent of the time. This implies
the insurance firm faces an average risk of pay out of 𝟎. 𝟏 × 𝟎. 𝟓𝟎 + 𝟎. 𝟒 × 𝟎. 𝟓𝟎 =
𝟎. 𝟎𝟓 + 𝟎. 𝟐 = 𝟎. 𝟐𝟓. In other words, the probability of the bad state from the perspective
of the insurance firm is 25 %. In addition, average damages if the bad state happens are
𝟎. 𝟓 × 𝟏𝟎𝟎𝟎 + 𝟎. 𝟓 × 𝟓𝟎𝟎 = 𝟓𝟎𝟎 + 𝟐𝟓𝟎 = 𝟕𝟓𝟎. Thus the fair premium from the firms
Dr. Donna Feir
Economics 313
perspective is 𝟎. 𝟐𝟓 × 𝟕𝟓𝟎 = 𝟏𝟖𝟕. 𝟓. Note that this is not the only way you could have
calculated the fair premium for the firm. It would depend on how much you assumed the
firm would pay out in the event of a C. In the problems we did in class, we assume the
“safe” and “unsafe” type had the same loss should the bad even happened and just that
the probability of the bad event differed.
d. What if the firm could not differentiate between these types of students and offer two
policies they can choose from. One with the fair premium for the student who cares under
which the firm would pay-out $1000 dollars in the event of a C and a fair premium for the
student who doesn’t and a pay-out of $500. Is this a profitable strategy for the firm?
No this would not be a profitable strategy for the firm because the students that don’t
care would have an incentive to pretend they do care and purchase the insurance meant
for the students that would actually care.
To see this consider the incentive compatibility constraint of the student that doesn’t care
to purchase the insurance that is meant for them:
𝑬𝒙𝒑𝒆𝒄𝒕𝒆𝒅 𝑼𝒕𝒊𝒍𝒊𝒕𝒚 𝒇𝒓𝒐𝒎 𝒑𝒂𝒚𝒊𝒏𝒈 𝒕𝒉𝒆 𝒑𝒓𝒆𝒎𝒊𝒖𝒎 𝒇𝒐𝒓 𝒕𝒉𝒆 𝒔𝒕𝒖𝒅𝒆𝒏𝒕𝒔 𝒘𝒉𝒐 𝒄𝒂𝒓𝒆 𝒂𝒏𝒅 𝒈𝒆𝒕𝒕𝒊𝒏𝒈
𝒕𝒉𝒆𝒊𝒓 𝒓𝒂𝒕𝒆 𝒐𝒇 𝒄𝒐𝒎𝒑𝒆𝒏𝒔𝒂𝒕𝒊𝒐𝒏
< 𝑬𝒙𝒑𝒆𝒄𝒕𝒆𝒅 𝒖𝒕𝒊𝒍𝒊𝒕𝒚 𝒇𝒓𝒐𝒎 𝒑𝒂𝒚𝒊𝒏𝒈 𝒑𝒓𝒆𝒎𝒊𝒖𝒎𝒔 𝒇𝒐𝒓 𝒔𝒕𝒖𝒅𝒆𝒏𝒕𝒔 𝒘𝒉𝒐 𝒅𝒐𝒏′ 𝒕 𝒄𝒂𝒓𝒆 𝒂𝒏𝒅 𝒈𝒆𝒕𝒕𝒊𝒏𝒈
𝒕𝒉𝒂𝒕 𝒑𝒂𝒚 𝒐𝒖𝒕
𝟎. 𝟔(𝟎 − 𝟏𝟎𝟎) + 𝟎. 𝟒(𝟎 − 𝟓𝟎𝟎 − 𝟏𝟎𝟎 + 𝟏𝟎𝟎𝟎) < 𝟎. 𝟔𝟎(𝟎 − 𝟐𝟎𝟎) + 𝟎. 𝟒(𝟎 − 𝟓𝟎𝟎 + 𝟓𝟎𝟎 − 𝟐𝟎𝟎)
−𝟔𝟎 + 𝟏𝟔𝟎 < −𝟏𝟐𝟎 − 𝟖𝟎
𝟏𝟎𝟎 < −𝟐𝟎𝟎
Obviously, this inequality doesn’t hold so the incentive compatibility constrain doesn’t bind for the
student that doesn’t care. This will result in the firm having negative expected profits because
everyone would buy the insurance meant for the low risk type:
Profits if both types buy insurance for low risk type:
𝟎.𝟗𝟎+𝟎.𝟔
𝟎.𝟏+𝟎.𝟒
(𝟏𝟎𝟎) +
(−𝟏𝟎𝟎𝟎 +
𝟐
𝟐
𝟏𝟎𝟎) = 𝟕𝟓 −
𝟐𝟐𝟓 = −𝟏𝟓𝟎
2. (Moral Hazard) Ed Bull works in a china shop. If he is careful, there is a 50% chance that he will break
some china. If he is careless, the chance of breaking some china rises to 75%. If Ed breaks china, he will
lose his job and have no wealth; but if he avoids breaking any china, he will keep his job and have wealth
of W. Ed dislikes being careful and values being careless by E, a lump-sum of utility that is added to his
utility of wealth function, i.e u(x)+E is his utility if he is careless and u(x) is his utility if he is careful,
where x is the amount of wealth that he has. He has a strictly concave utility function of wealth.
Dr. Donna Feir
Economics 313
a. The insurance company cannot observe Ed’s actions. As a result, if any china is broken, the company
will not know whether Ed’s carelessness caused the accident. Denote P the premium and D the
deductible for Ed. Write down the incentive compatibility constraint and the participation
constraints for Ed and the insurance company to characterize the optimal insurance contract.
Assume a competitive insurance market and assume that in the absence of this contract Ed’s best
option would be i) to not buy insurance, ii) to buy full coverage from the insurance at the actuarially
fair premium when he is careless.
We need the following incentive compatibility constraint:
.5u(W-P)+.5u(0-P-D+W) .25[u(W-P)+E]+.75[u(0-P-D+W)+E]
Participation constraint must make Ed as least as well off as his best alternative. There are
two possibilities in this example: Best alternative could be i) without insurance contract:
.5u(W-P)+.5u(0-P-D+W)  .5u(W)+.5u(0)
or ii) with insurance at the actuarially fair premium when Ed is not careful: ¾W is the
premium that Ed would have to pay if the insurance charges a premium for full coverage at which
it breaks even under the assumption that Ed will be careless.
.5u(W-P)+.5u(0-P-D+W)  u(¼W)+E,
For the insurance, we have the following participation constraint when its next best
alternative is to break even:
.5P + .5(P+D-W) = 0.
b. Suppose W=50, E=.1, and u(x) = x½. Check whether an insurance contract with a premium P=20 and
D=10 would satisfy the constraints from a).
With P=20 and D=10, then we have a situation where the insurance breaks even, and Ed gets an
expected utility of 4.97 if he is careful, 4.82 if he is careless. Ed’s next best alternative if full
insurance is not possible is to buy no insurance and be careful, which yields a utility of 3.54.
3. (Adverse Selection) Suppose there are two types of drivers, the safe types and the unsafe types. All
drivers cause the same damage if they have an accident, but the safe types’ probability of getting into an
accident is lower than the probability of the unsafe types of getting into an accident. Also assume that
both types have the same amount of wealth in the good state and that the damage in an accident is
equal to half of their wealth. If the insurance market is perfectly competitive, explain why there will
never be a pooling equilibrium, even if both types would be better off buying insurance at the actuarially
fair premium based on the average risk of the population. Use diagrams in the state-contingent space to
make your argument.
In the first diagram you can see that both types would be better off in point A, than without
insurance. Point A lies to the right of both types’ indifference curves that go through the point where
they do not have insurance. Point A is the result of full coverage to both types of drivers at the
actuarially fair premium of the average risk
Dr. Donna Feir
Economics 313
wG
ICU without insurance policy A
A
wealth
ICS without insurance policy A
FIBLS
FIBLU
wealth/2
FIBLP
wB
Dr. Donna Feir
Economics 313
In the next graph we see that if one firm provides an insurance policy resulting in point A, another
firm can steal all the safe customers (because the policy results in a point that would make safe types
better off than in A, but unsafe types worse off) away from this firm and make a profit (because the
policy would lead to point B which is to the left of the FIBL for safe types). The argument we have
made is a very general one, and relies only on the fact that ICU is steeper than ICS through any point.
No matter how many people are of the unsafe type and how many people are of the safe type, as long
as there are people of both types, one policy cannot be sold to both types in equilibrium.
wG
ICU with insurance policy A
B
A
wealth
ICS with insurance policy A
FIBLS
FIBLU
wealth/2
FIBLP
wB
Dr. Donna Feir
Economics 313
4. (Contracts) Wayne Corp. needs to hire a salesperson to sell a new product it has developed. If the
salesperson works hard, there is a 90% chance that he will sell $100,000 worth of product and only a
10% chance that he will sell only $50,000 worth of product. If he shirks, there is only a 20% chance that
he will make sales of $100,000 and an 80% chance that he will make sales of $50,000.
a.
Assume that any salesperson can easily find a job that pays $20,000 and that requires no
effort. Bob, who is a salesperson, is considering working for Wayne Corp. Bob’s utility is of
the form U(w,e)= w-e, where w is the wage paid to Bob, and e is the cost of effort in terms
of dollars. A high level of effort for Bob is equivalent to a cost of $10,000, while shirking is
equivalent to cost of $0. If Bob’s actions are completely observable by Wayne Corp., what is
the optimal contract they should offer him? (Write down the participation and incentive
compatibility constraints and give an example of wH and wL that maximizes the profit of
Wayne Corp.)
If Bob’s effort can be completely observed, Wayne Corp. can structure the contract depending on
Bob’s level of effort. If we let wH be the wage offered if Bob is observed working hard and wS be the
wage offered if Bob is observed shirking, we know that for Bob to want to be hired and work hard, it
must be true that
PC: wH – 10,000 ≥ 20,000 and
IC: wH –10,000 ≥ wS.
Therefore, it must be true that wH ≥ 30,000 and wS ≤ 20,000.
We know that for Bob to want to be hired and shirk, it must be true that
PC: wS ≥20,000
and
IC: wS ≥ wH – 10,000.
Therefore, if must be true that wH ≤ 30,000 and wS ≥ 20,000.
Minimizing cost for the principal implies that wH = 30,000 if the principal wants to induce effort and if
the principal wants Bob to shirk wS =20,000.
Now look at Wayne Corp.’s expected profit. If Bob works hard, the expected profit is 0.9*100,000 +
0.1* 50,000 – 30,000= 65,000. If Bob shirks, the expected profit is 0.2*100,000 + 0.8*50,000 – 20,000 =
40,000. Therefore, the optimal contract for Wayne Corp. to offer Bob is one that specifies w H =
$30,000 and for example wS = $0. Such a contract would insure that Bob would want to join the firm
and work hard.
b.
Now assume that Bob’s actions cannot be observed by Wayne Corp. Write down the
participation and incentive compatibility constraints for wG and wB to (i) induce effort and
(ii) to induce shirking. Illustrate the optimal contracts for each effort level in a diagram.
Which is the effort level that will maximize Wayne Corp.’s expected profits?
Dr. Donna Feir
Economics 313
Now, since Wayne Corp. can’t observe Bob’s effort level, it can only structure the contract on the
basis of whether or not a good outcome is observed. Let wG be the wage offered if $100,000 in sales is
observed and wB be the wage offered if $50,000 in sales is observed. For Bob to want to join Wayne
Corp. and work hard, the wage scheme must satisfy
PC: 0.9 wG + 0.1 wB – 10,000 ≥ 20,000
and
IC: 0.9 wG + 0.1 wB – 10,000 ≥ 0.2 wG + 0.8 wB
In order to draw the diagram, from PC we find w G ≥ 30,000/.9 –(1/9) wB and from IC we
get wG ≥ 100,000/7 + wB.
In the picture below the thick line shows where the optimal combinations of wG and wB lie that induce
effort and minimize costs for the firm given that it wants Bob to join and work with high effort. This
also shows that the expected payments to Bob must be equal to 30,000 in order to want him to work
for the firm, because on this line from PC we know that 0.9 wG + 0.1 wB = 30,000.
Dr. Donna Feir
Economics 313
Wage with good
outcome
IC and PC satisfied
33,333
$
30,000
28,571
PC satisfied
IC satisfied
20,000
14,286
10,000
80,000
160,000
240,000
300,000
Wage with bad
outcome
For Bob to want to join Wayne Corp. and shirk, the wage scheme must satisfy
PC: 0.2 wG + 0.8 wB ≥ 20,000
and
IC: 0.2 wG + 0.8 wB ≥ 0.9 wG + 0.1 wB – 10,000
In order to draw the diagram, from PC we find wG ≥ 20,000/.2 – 4 wB and from IC we get wG 
100,000/7 + wB.
In the picture below the thick line shows where the optimal combinations of wG and wB lie that induce
shirking and minimize costs for the firm given that it wants Bob to join and shirk. This also shows that
the expected payments to Bob must be equal to 20,000 in order to want him to work for the firm,
because on this line from PC we know that 0.2 wG + 0.8 wB = 20,000.
Dr. Donna Feir
Economics 313
Wage with good
outcome
PC satisfied
IC and PC satisfied
100,000
14,286
IC satisfied
7,000
14,000
21,000
25,000
Wage with bad
outcome
We now need to determine whether Wayne Corp. is better off inducing effort or shirking.
We already know the expected wage payments to Bob to induce effort are 30,000 and to induce
shirking 20,000.
We now calculate expected profits for both effort levels:
High effort: .9*100,000 + .1*50,000 – 30,000 = 65,000.
Shirking:
.2*100,000 + .8*50,000 – 20,000 = 40,000.
It is expected profits maximizing to induce high effort.
c.
Now assume that Bob’s disutility from working hard is $40,000. Write down the
participation and incentive compatibility constraints for wG and wB to (i) induce effort and
(ii) to induce shirking. Illustrate the optimal contracts for each effort level in a diagram.
Which is the effort level that will maximize Wayne Corp.’s expected profits?
Now in order to for Bob to work hard the constraints that must be satisfied are
PC: 0.9 wG + 0.1 wB – 40,000 ≥ 20,000
and
IC: 0.9 wG + 0.1 wB – 40,000 ≥ 0.2 wG + 0.8 wB
Dr. Donna Feir
Economics 313
In order to draw the diagram, from PC we find wG ≥ 200,000/3 –(1/9) wB and from IC we get wG ≥
400,000/7 + wB
In the picture below the thick line shows where the optimal combinations of wG and wB lie that induce
high effort and minimize costs for the firm given that it wants Bob to join and work with effort. This
also shows that the expected payments to Bob must be equal to 60,000 in order to want him to work
for the firm, because on this line from PC we know that 0.9 wG + 0.1 wB = 60,000.
Wage with good
outcome
66,667
IC and PC satisfied
57,143
PC satisfied
IC satisfied
150,000
300,000 450,000
600,000
Wage with bad
outcome
For Bob to want to join Wayne Corp. and shirk, the wage scheme must satisfy
PC: 0.2 wG + 0.8 wB ≥ 20,000
and
IC: 0.2 wG + 0.8 wB ≥ 0.9 wG + 0.1 wB – 40,000
In order to draw the diagram, from PC we find wG ≥ 100,000 – 4 wB and from IC we get wG
 400,000/7 + wB.
Dr. Donna Feir
Economics 313
In the picture below the thick line shows where the optimal combinations of wG and wB lie that induce
shirking and minimize costs for the firm given that it wants Bob to join and shirk. This also shows that
the expected payments to Bob must be equal to 20,000 in order to want him to work for the firm,
Wage with good
outcome
PC satisfied
IC and PC satisfied
100,000
57,143
IC satisfied
7,000
14,000
21,000
25,000
Wage with bad
outcome
because on this line from PC we know that 0.2 wG + 0.8 wB = 20,000.
We now need to determine whether Wayne Corp. is better off inducing effort or shirking.
We already know the expected wage payments to Bob to induce effort are 60,000 and to induce
shirking 20,000.
We now calculate expected profits for both effort levels:
High effort: .9*100,000 + .1*50,000 – 60,000 = 35,000.
Shirking:
.2*100,000 + .8*50,000 – 20,000 = 40,000.
It is expected profits maximizing to induce shirking.
d.
If Bob’s disutility of high effort remains at $10,000 but his outside option increases to
$30,000. Write down the participation and incentive compatibility constraints for wG and wB
to (i) induce effort and (ii) to induce shirking. Illustrate the optimal contracts for each effort
Dr. Donna Feir
Economics 313
level in a diagram. Which is the effort level that will maximize Wayne Corp.’s expected
profits?
Now in order for Bob to work hard the constraints that must be satisfied are
PC: 0.9 wG + 0.1 wB – 10,000 ≥ 30,000
and
IC: 0.9 wG + 0.1 wB – 10,000 ≥ 0.2wG + 0.8 wB
In order to draw the diagram, from PC we find wG ≥ 400,000/9 –(1/9) wB and from IC we get wG ≥
100,000/7 + wB
In the picture below the thick line shows where the optimal combinations of wG and wB lie that induce
high effort and minimize costs for the firm given that it wants Bob to join and work with high effort.
This also shows that the expected payments to Bob must be equal to 40,000 in order to want him to
work for the firm, because on this line from PC we know that 0.9 wG + 0.1 wB = 40,000.
Wage with good
outcome
44,444
IC and PC satisfied
30,000
20,000
PC satisfied
IC satisfied
14,286
10,000
100,000
200,000
300,000
400,000
For Bob to want to join Wayne Corp. and shirk, the wage scheme must satisfy
Dr. Donna Feir
Economics 313
PC: 0.2 wG + 0.8 wB ≥ 30,000
and
IC: 0.2 wG + 0.8 wB ≥ 0.9 wG + 0.1 wB – 10,000
In order to draw the diagram, from PC we find wG ≥ 150,000 – 4 wB and from IC we get wG  100,000/7
+ wB.
In the picture below the thick line shows where the optimal combinations of wG and wB lie that induce
shirking and minimize costs for the firm given that it wants Bob to join and shirk. This also shows that
the expected payments to Bob must be equal to 30,000 in order to want him to work for the firm,
because on this line from PC we know that 0.2 wG + 0.8 wB = 30,000.
Wage with good
outcome
PC satisfied
IC and PC satisfied
150,000
14,286
IC satisfied
10,000
20,000
30,000
37,500
Wage with bad
outcome
We now need to determine whether Wayne Corp. is better off inducing high effort or shirking.
We already know the expected wage payments to Bob to induce effort are 40,000 and to induce
shirking 30,000.
Dr. Donna Feir
Economics 313
We now calculate expected profits for both effort levels:
High effort: .9*100,000 + .1*50,000 – 40,000 = 55,000.
Shirking:
.2*100,000 + .8*50,000 – 30,000 = 30,000.
It is expected profits maximizing to induce high effort.