PS4-09Ans - University of California, Davis Department of...

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University of California, Davis Department of Agricultural and Resource Economics ARE 100B Winter 200 9 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. "# œœ 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) Cournot competition. Firm 1's profit is [100 4(q q )]q 5q , and its reaction curve comes from C "" # " " œ ` ` œ  ß C " # " / q 0 or 100 8q 4q 5=0. Solving for q the reaction curve is RC : q 11.875 0.5q . " Firm 2's profit is [100 4(q q )]q 10q , and its reaction curve comes from C #" # # # ` ` œ œ ß C ## " # # / q 0 or 100 4q 8q 10 0. RC : q 11.25 0.5q . # The Cournot equilibrium is the quantities q q that solve the two reaction curves; substituting RC -8 # into RC , " q 11.875 0.5(11.25 0.5q ) so q (11.875 0.5 11.25)/.75 8.33. From RC q 11.25 0.5 8.33 7.083. __ _ # œ ß œ œ Market price is P 100 4(8.33 7.083) 38.33. Firm 1's profit is (p AC )q _ œ œ C " ¸ œ †¸ (38.33 5) 8.33 277.64. Firm 2 profit is (p AC )q (38.33 10) 7.083 200.67. _ _ C # (b) 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 " " " 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 " = q 11.25 0.5 (12.5) 5. # œ Market price is P 100 4(12.5 5) 30. Firm 1's profit is (p AC )q œ  œ C " 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. (c) 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 # # # Ð 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 # = q 11.875 0.5 (10.625) 6.5625.
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PS4-09Ans - University of California, Davis Department of...

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