Mr. X and Ms. Y have entered into the following contractual arrangement with D. M. Kreps. Between now and the end of next week, each will install a certain amount of capacity for the manufacture of square based tri-prong slot poiuyts. Then Kreps will assume control of the capacity provided to him and produce and sell poiuyts. Demand for poiuyts is given by the demand function D = 10-P. (To keep matters simple, assume that production takes place for a single period.) The marginal cost of production of poiuyts is zero up to the total level of capacity that Mr. X and Ms. Y install. That is, if Mr. X installs capacity kx and Ms. Y installs capacity ky, then Kreps can produce up to k, + ky poiuyts and sell them, at zero additional cost. Kreps can, however, produce less if he so chooses. The contract signed by X, Y, and Kreps calls for Kreps to act in the best interests of X and Y. (He receives a smallf'ixed fee for doing so, which you may ignore.) So that there is no ambigUity about what are the best interests of X and Y, the contract calls for Kreps to set output so that gross profits (= revenue) are maximized. (Kreps, who knows a lot about linear demand systems and zero marginal cost production functions, can be counted on to do this.) These gross profits are split between X and Y in proportion to the capacities they build. That is, if kx = 2ky, then X receives two-thirds of the gross profits and Y receives one-third. Capacity is costly for X and Y. Precisely, there is a constant marginal cost of 2 per unit of installed capacity. Hence X's net from the entire arrangement, if he installs kx, Y installs ky, and Kreps produces Q ≤ kx + ky, is and similarly for Y. (a) Suppose that X and Y collude in terms of the capacities they install. That is, kx and ky are set to maximize the sum of the net profits of X and Y from this arrangement. (Assume, for the sake of definiteness, that kx = ky) What capacities will X and Y install? (b) (Good luck!) Unhappily, X and Y cannot (or do not) collude. They install capacity levels kx and ky "Cournot" style, choosing their capacity levels simultaneously and independently. What is (or are) the equilibrium (or equilibria) in this instance? (c) If demand is P A - Q and the marginal cost of capacity is r, can you find any values of A and r such that the total amount of capacity installed in an equilibrium exceeds A?
Mr. X and Ms. Y have entered into the following contractual arrangement with D. M. Kreps. Between now and the end of next week, each will install a certain amount of capacity for the manufacture of square based tri-prong slot poiuyts. Then Kreps will assume control of the capacity provided to him and produce and sell poiuyts.
The contract signed by X, Y, and Kreps calls for Kreps to act in the best interests of X and Y. (He receives a smallf'ixed fee for doing so, which you may ignore.) So that there is no ambigUity about what are the best interests of X and Y, the contract calls for Kreps to set output so that gross profits (= revenue) are maximized. (Kreps, who knows a lot about linear demand systems and zero marginal cost production functions, can be counted on to do this.) These gross profits are split between X and Y in proportion to the capacities they build. That is, if kx = 2ky, then X receives two-thirds of the gross profits and Y receives one-third.
Capacity is costly for X and Y. Precisely, there is a constant marginal cost of 2 per unit of installed capacity. Hence X's net from the entire arrangement, if he installs kx, Y installs ky, and Kreps produces Q ≤ kx + ky, is
and similarly for Y.
(a) Suppose that X and Y collude in terms of the capacities they install. That is, kx and ky are set to maximize the sum of the net profits of X and Y from this arrangement. (Assume, for the sake of definiteness, that kx = ky) What capacities will X and Y install?
(b) (Good luck!) Unhappily, X and Y cannot (or do not) collude. They install capacity levels kx and ky "Cournot" style, choosing their capacity levels simultaneously and independently. What is (or are) the equilibrium (or equilibria) in this instance?
(c) If demand is P A - Q and the marginal cost of capacity is r, can you find any values of A and r such that the total amount of capacity installed in an equilibrium exceeds A?
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