(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.
(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:
