978-0393919684 Chapter 8 Solution Manual Part 4

Solutions to Chapter 8 Exercises

UNSOLVED EXERCISES

U1. (a) Jack’s expected utility is 0.5 • √250,000 + 0.5 • √90,000 = 400.

(d) Under the agreement, the expected utility for each participant is 0.25 • √250,000 + 0.25 •

√170,000 + 0.25 • √170,000 + 0.25 • √90,000 ≈ 406.1552.

(f) The eight possible luck outcome triplets for Jack, Janet, and Chrissy are (good, good,

(g) Under the agreement, the expected utility for each participant is 0.125 • √250,000 + 3 •

(h) For Jack, Jill, or Chrissy to achieve a utility of 408.2744, he or she (or she) would need a

U2. (a) Month 1:

Willingness to accept

of sellers

Willingness to pay

of buyers

Month 2:

Willingness to accept

of sellers

Willingness to pay

of buyers

Month 3:

Willingness to accept

of sellers

Willingness to pay

of buyers

(b) Risk-neutral buyers of a citrus of unknown type are willing to pay up to 0.8 • (willingness

$10,000

$10,500

$11,000

$11,500

$12,000

$12,500

$13,000

$13,500

$14,000

1 2 3 4 5 6 7 8 9 10 11 12

Price

Month

willingness to accept of orange

sellers

willingness to pay of buyers of an

unknown Citrus type

(c) Yes, there is a market for oranges in month 3, since the willingness to pay of buyers for

(d) The market for oranges collapses in month 6, when the willingness to pay of the buyers

(e) Whether or not lemons depreciate for potential sellers of lemons does not affect the

(f) Depreciation of lemons among potential buyers of a citrus of unknown type does affect

the timing of the collapse of the market for oranges. If lemons never depreciate among buyers, their

U3. (a) To achieve separation, the incentive compatibility conditions are

(b) If the signal is not available, then every worker is treated like a random draw from the

U4. (a) In a separating equilibrium, St. Anford will set a threshold number of publications to

determine whether it will grant tenure. Brilliant professors will signal their type by reaching that

(b) One pooling equilibrium has both Brilliant and Good types not applying for tenure at St.

A second pooling equilibrium has both types applying for and being granted tenure. In this case N

(c) To achieve a separation of types both of the following inequalities must hold:

The first inequality implies that N < 50, while the second implies that N > 25. Any number of publications

N in the set (25, 50) will achieve a separation of professor types for St. Anford.

U5. (a) In terms of z, Figure 8.10 is shown in the table below:

Fordor

Regardless (II) Conditional (OI)

(b) When z = 0 (that is, when Tudor is definitely high cost), the game table is as follows:

Fordor

Regardless (II) Conditional (OI)

There is only one pure-strategy Nash equilibrium: (Honest, Regardless).

The equilibrium found in the previous figure is separating. The two types of Tudor charge

(c) When z = 1 (that is, when Tudor is definitely low cost), the game table is as shown below:

Fordor

Regardless (II) Conditional (OI)

There are two pure-strategy Nash equilibria: (Bluff, Conditional) and (Honest, Conditional). (Bluff,

Conditional) is a pooling equilibrium, since both Tudor types charge the same price in the first period.

(d) There are two possible pooling equilibria: (Bluff, Regardless) and (Bluff, Conditional).

Next consider the pooling equilibrium (Bluff, Conditional). Fordor will play Conditional if 0 ≥

(e) If z is too low, Fordor can be fairly confident that Tudor is high cost, and so Fordor will

U6. (a) The game tree for a risk-averse Tudor with a low per-unit cost of 10 follows:

(b) If the probability that Tudor is low cost is z = 0.4, the game table for the previous game

tree is as follows:

Fordor

Regardless (II) Conditional (OI)

Nature

Tudor’s

cost high

Prob. 0.6

Tudor

Price

low

Price

high

Fordor

In

Out

5 + √3 , 5

In

Out

√19 + √3 , 5

√19 + √3 , 0

(–15) • 0.4 + 5 • 0.6 = –3

The unique Nash equilibrium is (Bluff, Conditional). Since both Tudor types send the same signal (they

both set low prices in the first period), this is a pooling equilibrium.

(c) If z = 0.1, the payoffs in the game table become

Fordor

Regardless (II) Conditional (OI)

Here, the unique Nash equilibrium is (Honest, Regardless), which is a separating equilibrium.

(d) Tudor’s risk aversion will not affect the threshold level of z required for a pooling

U7. (a) When Tudor’s low per-unit cost is 6 and its probability of being low cost is z, the game

table is as follows:

Fordor

Regardless (II) Conditional (OI)

(b) When z = 0.1, the payoffs are as follows:

Fordor

Regardless (II) Conditional (OI)

The equilibrium is (Honest, Regardless). It is a separating equilibrium because the two types of Tudor

send different signals according to their type.

(c) When z = 0.2, the payoffs are as follows:

Fordor

Regardless (II) Conditional (OI)

Here the equilibrium is (Bluff, Conditional). It is a pooling equilibrium because both types of Tudor send

the same signal regardless of their type.

(d) When z = 0.3, the payoffs are as follows:

Fordor

Regardless (II) Conditional (OI)

The equilibrium is again the pooling equilibrium (Bluff, Conditional).

(e) In part (d) of Exercise U5, the lowest z for which a pooling equilibria could occur was