# (Optional) Revisit the situation in which Oceania is procuring arms from BMA. (See Exercise S11.)…

(Optional) Revisit the situation in which Oceania is
procuring arms from BMA. (See Exercise S11.) Now consider the case in which BMA
has three possible cost types: c1, c2, and c3, where c3 . c2 . c1. BMA has cost
c1 with probability p1, cost c2 with probability p2, and cost c3 with
probability p3, where p1 1 p2 1 p3 5 1. In
what follows, we will say that BMA is of type i if its cost is ci, for i 5 1,
2, 3. You offer a menu of three possibilities: “Supply us quantity Qi, and we
will pay you Mi,” for i 5 1, 2, and 3. Assume that more than one contract is
equally profitable, so that a BMA of type i will choose contract i. To meet the
participation constraint, contract i should give BMA of type i nonnegative
profit.

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(Optional) Revisit the situation in which Oceania is procuring arms from BMA. (See Exercise S11.)…
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(a) Write an expression for the profit of type-i BMA when
it supplies quantity Q and is paid M.

(b) Give the participation constraints for each BMA type.

(c) Write the six incentive-compatibility constraints.
That is, for each type i give separate expressions that state that the profit
that BMA receives under contract i is greater than or equal to the profit it
would receive under the other two contracts.

(d) Write down the expression for Oceania’s expected net
benefit, B. This is the objective function (what you want to maximize). Now
your problem is to choose the three Qi and the three Mi to maximize expected
net benefit, subject to the incentive-compatibility (IC) and participation
constraints (PC).

(e) Begin with just three constraints: the IC constraint
for type 2 to prefer contract 2 over contract 3, the IC constraint for type 1
to prefer contract 1 over contract 2, and the participation constraint for type
3. Assume that Q1 . Q2 . Q3. Use these constraints to derive lower bounds on
your feasible choices of M1, M2, M3 in terms of c1, c2, and c3 and Q1, Q2, and
Q3. (Note that two or more of the cs and Qs may appear in the expression for
the lower bound for each of the Ms.)

(f) Prove that these three constraints—the two ICs and
one PC in part (e)—will be binding at the optimum.

(g) Now prove that when the three constraints in part (e)
are binding, the other six constraints (the remaining four ICs and two PCs) are
automatically satisfied.

(h) Substitute out for the Mi to express your objective
function in terms of the three Qi only.

(i) Write the first-order conditions for the
maximization, and solve for each of the Qi. That is, take the three partial
derivatives, set them equal to zero, and solve for Qi.

(j) Show that the assumption made above, Q1 . Q2 . Q3,
will be true at the optimum if: