(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:

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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?