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ps9_sol_fall11 - Department of Economics Columbia...

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Department of Economics W3211 Columbia University Fall 2011 SO LUTIONS T O Probl e m S e t 9 GRADED PART ONL Y Int e rm e diat e Mi c ro ec onomi c s Prof . S e yhan Erd e n Arkona c 1 . Suppose the demand for pizza in a small isolated town is p = 10 - Q. The only two firms, A and B, behave as Cournot duopolists. Each has a cost function TC = 2 + Q. (a) Determine the Cournot equilibrium. (b) Determine the equilibrium quantities of each if firm A is the Stackelberg leader. ( c ) If the government wants to subsidize firm A to raise its output to that of a Stackelberg leader, how large should the subsidy be? Answer: (a) Firm A's profit is π = [10 - (q A + q B )] q A - 2 - q A . Maximizing with respect to its own output yields q A = 4.5 - q B /2. Similarly, firm B's best response is q B = 4.5 - q A /2. The equilibrium occurs when both firms produce 3 units. Price is 10 - 3 - 3 = 4. (b) Firm B's profit is π = [10 - (q A + q B )] q B - 2 - q B . Maximizing with respect to its own output yields firm B's best response q B = 4.5 - q A /2. Knowing this, firm A substitutes t his into demand and maximizes its profit. π = [10 - q A - (4.5 - q A /2)] q A - 2 - q A . Maximizing with respect to q A yields q A = 4.5. Firm B responds by producing 2.25. ( c ) Firm B's p rofit is π = [10 - (q A + q B )] q B - 2 - q B . Maximizing with respect to its own output yields firm B's best response q B = 4.5 - q A /2. Knowing this, firm A substitutes this into demand and maximizes its profit. π = [10 - q A - (4.5 - q A /2)] q A - 2 - q A . Maximizing with respect to q A yields q A = 4.5. Firm B responds by producing 2.25. The subsidy (S) must make firm A choose 4.5 units while acting as a Cournot firm. Firm A's reaction function as a Cournot firm is q A = [10 - 1 + S - q B ]/2. Setting q A = 4.5 and q B = 2.25 yields a subsidy of 2.25 per unit. 2 . Explain why the intersection of the best-response functions is the Cournot equilibrium. Answer: On a best-response function, a firm selects the profit-maximizing level of output given the level of output it believes the other firm will produce. At the intersection of the best-response functions, what one firm believes the other firm will produce actually is the amount the other firm produces. Thus, neither firm has an incentive to change the level of output. 3 . Draw a graph that shows the effect on the equilibrium quantities if the government subsidizes one firm in a Cournot duopoly with a per-unit subsidy. Assume that the best-response functions are linear. Explain the new equilibrium quantities.
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Answer: See the above figure. The subsidized firm, B, produces more output and the non-subsidized firm, A, produces less. The subsidized firm faces a lower after-subsidy marginal cost, so the profit-
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