Solutions to Chapter 8 Exercises
SOLVED EXERCISES
S1. (a) Your neighbor has a sure income of $100,000. In addition, under the insurance
contract, he will receive x when you have a good year and pay you $60,000 when you have a bad
Rounding down to $55,009.88 would make your neighbor very slightly prefer not
entering the contract, so the minimum x that your neighbor will agree to is $55,009.89.
(b) Here we are looking for the level of x where you are indifferent between getting
insurance (where you pay x in a good year and receive $60,000 in a bad year) and not getting
insurance. That is, we’re looking for the x for which your expected utility with the insurance is
equal to your expected utility without the insurance:
Solutions to Chapter 8 Solved Exercises
S2. (a) To achieve separation, t must be such that
The minimum wait time that achieves separation is the smallest moment longer than 40
minutes.
(b) The expected benefit has changed for the college students, so t must now satisfy
The minimum wait time that achieves separation is now a shade more than 20 minutes.
The partial identification of college students reduces the minimum wait time required to achieve
separation. When the charity has more information about the patrons—even if only partial
information—this allows it to distinguish between the two types by means of a less stringent test.
S3. (a) Buyers expect a random Citrus to be an orange with probability f = 0.6 and a
lemon with probability 0.4. Risk-neutral buyers are then willing to pay up to
(b) The willingness to accept (WTA) of owners of oranges is $12,500. Since the
(c) If f = 0.2, risk-neutral buyers will be willing to pay up to
Solutions to Chapter 8 Solved Exercises
(d) There will not be a market for oranges when f = 0.2. The probability of a random
(e) The minimum f such that oranges are sold is the f that satisfies
(f) One way to look at it is to redo part (e) in the abstract:
Increasing WTPorange increases the size of the denominator of the fraction, so fthreshold
In an intuitive sense, the more highly buyers value oranges or lemons (or both), the more
willing they will be to take a greater gamble on a random Citrus. That is, they’ll be willing to pay
S4. A competent electrician’s payoff after obtaining the signal (certification) is √100 – C; in
the absence of the signal, this electrician’s payoff is √25. The competent electrician will signal as
Solutions to Chapter 8 Solved Exercises
S5. (a) The game table in terms of z follows:
Fordor
Regardless (II) Conditional (OI)
(b) When z = 0 (that is, when Tudor is definitely high cost), the game table, with best
responses underlined, follows:
Fordor
Regardless (II) Conditional (OI)
There are thus two pure-strategy Nash equilibria: (Honest, Regardless) and (Honest,
Conditional). Tudor will always signal that it is high cost (which it is), and Fordor will always
enter.
(c) When z = 1 (that is, when Tudor is definitely low cost), the game table, with best
responses underlined, follows:
Fordor
Regardless (II) Conditional (OI)
Solutions to Chapter 8 Solved Exercises
There are thus two pure-strategy Nash equilibria: (Bluff, Conditional) and (Honest,
Conditional). Tudor will always signal that it is low cost (which it is), and Fordor will never enter.
(d) When z is strictly between 0 and 1, the best responses are
Fordor
Regardless (II) Conditional (OI)
When z is strictly between 0 and 1 there is thus a unique pure-strategy Nash equilibrium:
(Honest, Conditional). Since a low-cost Tudor will always send a different signal from that of a
high-cost Tudor, this is a separating equilibrium.
S6. (a) The extensive form with a risk-averse Tudor (with square-root utility) follows:
Solutions to Chapter 8 Solved Exercises
(b) The equivalent game table for the tree in part (a), with z = 0.4, follows:
Fordor
Regardless (II) Conditional (OI)
102 0.4 + 5 0.6 = 8.7,
–29 0.4 + 5 0.6 = –8.6
5 0.6 = 3
The unique pure-strategy Nash equilibrium of the game is (Honest, Conditional). Since
the low-cost Tudor and the high-cost Tudor send different signals in equilibrium, this game has a
separating equilibrium.
(c) The equivalent game table for the tree in part (a), with z = 0.1, follows:
Solutions to Chapter 8 Solved Exercises
Nature
prob. z
Tudor’s
cost high
prob. 1 – z
Tudor
Price
low
Price
high
Fordor
Fordor
In
Out
(25 + 3), 45 – 40
In
Out
(0 + 3), 45 – 40
(0 + 25), 0
The unique pure-strategy Nash equilibrium of the game is also (Honest, Conditional), so
like the game in part (b), it also has a separating equilibrium.
S7. (a) The extensive form of this game, with the low per-unit cost equal to 6, is shown
below:
Solutions to Chapter 8 Solved Exercises
prob. z
Tudor
Price
high
(b) The normal form of the game follows:
Fordor
Regardless (II) Conditional (OI)
180 0.4 + 30 0.6 = 90,
(c) The unique pure-strategy Nash equilibrium of this game is (Bluff, Conditional).
This is a pooling equilibrium because both a low-cost Tudor and a high-cost Tudor send the same
signal: setting a low price.
S8. (a) The extensive form of this game is shown below:
Solutions to Chapter 8 Solved Exercises