PS4-10A - University of California Davis Department of...

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University of California, Davis Department of Agricultural and Resource Economics ARE 100B Winter 2010 Dr. Larson Problem Set 4 Answers 1. Two firms are considering producing a new product, tempered glass for the auto market. They will make the same product and face the same demand curve, given by P 100 4Q, where P is in dollars œ per pound of glass produced and Q is thousands of pounds produced per month. Firm 1 has marginal cost MC $5 per pound while firm 2 has MC $10 per pound. " # œ œ Under the following forms of market rivalry with Nash equilibrium, determine each firm's optimal quantity choice and profits in the market , as well as the resulting market price. (a) Firm 1 decides to be a Stackelberg leader, with 2 the follower. Firm 1 uses 2's reaction curve in making its production decision, so its profit is [100 4(q 11.25 0.5q )]q 5q C " " " " " œ  Ð Ñ Þ The profit maximizing output q comes from / q 0, which in this case is " = " " ` ` œ C 100 8q 45 4q 5=0, = = " " so q 50/4 12.5. With this output by 1, we know immediately from 2's reaction curve that " = œ œ q 11.25 0.5 (12.5) 5. # œ œ Market price is P 100 4(12.5 5) 30. Firm 1's profit is (p AC )q œ œ œ C " " " œ œ œ œ œ (30 5) 12.5 312.50. Firm 2 profit is (p AC )q (30 10) 5 100. C # # # Total profit for both firms is lower, but the leader has higher profit. (b) Firm 2 is a Stackelberg leader, with 1 the follower. Here, Firm 2 uses 1's reaction curve in making its production decision, so its profit is [100 4( 11.875 0.5q q ]q 10q C # # # # # œ Ð Ñ  Þ The profit maximizing output q comes from / q 0, which is " = # # ` ` œ C 100 47.5 4q 8q 10 0, œ = = # # so q 42.5/4 10.625. With this output by 2, 1's output is # = œ œ q 11.875 0.5 (10.625) 6.5625. " œ œ Market price is P 100 4(10.625 6.5625) 31.25 Firm 1's profit is (p AC )q œ œ œ C " " " œ ¸ œ œ (31.25 5) 6.5625 172.27. Firm 2 profit is (p AC )q (31.25 10) 10.625 C # # # ¸ 225.78. Here too, the Stackelberg leader (2) gets more than it would under Cournot competition, with total profits lower and the follower much worse off.
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