# (Optional, requires calculus) You are Oceania’s Minister for Peace, and it is your job to purchase..

(Optional, requires calculus) You are Oceania’s Minister

for Peace, and it is your job to purchase war materials for your country. The

net benefit, measured in Oceanic dollars, from quantity Q of these materials is

2Q1 2 2 M, where M is the amount of money paid for the materials. There is just

one supplier—Baron Myerson’s Armaments (BMA). You do not know BMA’s cost of

production. Everyone knows that BMA’s cost per unit of output is constant, and

that it is equal to 0.10 with probability p 5 0.4 and equal to 0.16 with

probability 12 p. Call BMA “low cost” if its unit cost is 0.10 and “high cost”

if it is 0.16. Only BMA knows its true cost type with certainty. In the past,

your ministry has used two kinds of purchase contracts: cost plus and fixed

price. But cost-plus contracts create an incentive for BMA to overstate its

costs, and fixed-price contracts may compensate the firm more than is

necessary. You decide to offer a menu of two possibilities:

Contract 1: Supply us quantity Q1, and we will pay you

money M1.

Contract 2: Supply us quantity Q2, and we will pay you

money M2. The idea is to set Q1, M1, Q2, and M2 such that a low-cost BMA will

find contract 1 more profitable, and a high-cost BMA will find contract 2 more

profitable. If another contract is exactly as profitable, a low-cost BMA will

choose contract 1, and a high-cost BMA will choose contract 2. Further,

regardless of its cost, BMA will need to receive at least zero economic profit

in any contract it accepts.

(a) Write expressions for the profit of a low-cost BMA

and a high-cost BMA when it supplies quantity Q and is paid M.

(b) Write the incentive-compatibility constraints to

induce a low-cost BMA to select contract 1 and a high-cost BMA to select

contract 2.

(c) Give the participation constraints for each type of

BMA.

(d) Assuming that each of the BMA types chooses the

contract designed for it, write the expression for Oceania’s expected net

benefit.

Now your problem is to choose Q1, M1, Q2, and M2 to

maximize the expected net benefit found in part (d) subject to the incentive-compatibility

(IC) and participation constraints (PC). (e) Assume that Q1 . Q2, and further

assume that constraints IC1 and PC2 bind—that is, they will hold with

equalities instead of weak inequalities. Use these constraints to derive lower

bounds on your feasible choices of M1 and M2 in terms of Q1 and Q2. (f) Show

that when IC1 and PC2 bind, IC2 and PC1 are automatically satisfied. (g)

Substitute out for M1 and M2, using the expressions found in part (e) to

express your objective function in terms of Q1 and Q2. (h) Write the

first-order conditions for the maximization, and solve them for Q1 and Q2. (i)

Solve for M1 and M2. (j) What is Oceania’s expected net benefit from offering

this menu of contracts? (k) What general principles of screening are illustrated

in the menu of contracts you found?