Homework _2_2016_Solution - EconS425 Homework#2(Due on...

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1 Instructor: Ana Espinola EconS425 - Homework #2 (Due on February 8 th , 2016) 1. Exercise 4 and 5 from Chapter 5 in Shy (page 93) Solution Exercise #4: a) The monopoly’s profit function is: 2 1 2 1 1 2 2 1 2 ( , ) (100 / 2) (100 ) ( ) q q q q q q q q b) The two first order conditions are given by: 1 1 2 1 2 1 2 1 100 2( ) 0 100 2 2( ) 0 q q q q q q q q Solving for 1 M q and 2 M q yields that 1 25 M q and 2 12.5 M q . c) Substitute the profit maximizing quantity sold in market 1 and 2 into the market demand function yields: 1 1 100 / 2 87.5 M M p q and 2 2 100 87.5 M M p q . Hence the profit level of the discriminating monopoly is: 2 1 2 ( , ) 87.5 25 87.5 12.5 (25 12.5) 1875 M M q q d) Now that each plant sells only in one market, each plant’s cost function only depends on their own production and is given by: 2 1 1 1 ( ) TC q q and 2 2 2 2 ( ) TC q q . The profit functions are: 2 1 1 1 1 1 2 2 2 2 2 2 ( ) (100 / 2) ( ) (100 ) q q q q q q q q The two first order conditions are given by: 1 1 1 1 2 2 2 2 100 2 0 100 2 2 0 q q q q q q Solving for 1 M q and 2 M q yields that 1 100/ 3 M q and 2 25 M q . e) Substitute the profit maximizing quantity sold in market 1 and 2 into the market demand function yields: 1 1 100 / 2 250/ 3 M M p q and 2 2 100 75 M M p q . Hence the sum of profits of the two plants is:
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2 Instructor: Ana Espinola 2 2 1 1 2 2 250 100 100 ( ) ( ) 75 25 25 2917 3 3 3 M M q q f) This decomposition increase the monopoly’s profit since the technology exhibits DRS. Solution Exercise #5: By section 5.3, the discriminating monopoly will set quantities to satisfy MR 1 (q M 1 ) =MR 2 (q M 2 ). Hence, by Proposition 3.3: MR 1 (q 1 ) = p M 1 (1 +1/ 2)= p M 2 (1 +1/ 4)= MR2(q2). Thus, p M 1 = 1.5p M 2 . 2. Exercise 3 and 5 from Chapter 6 in Shy (pages 128-129) Solution of exercise 3 from Chapter 6: In the third period, firm 3 takes 1 q and 2 q as given and chooses 3 q to maximize its profit. The solution to this problem yields firm 3’s best -response function 3 1 2 ( , ) R q q which is given in (6.8). Hence, using (6.8) we have it that 1 2 3 1 2 1 2 1 120 ( , ) ( ) 2 2 2 2 q q a c c R q q q q b In the second period, firm 2 takes 1 q and 3 1 2 ( , ) R q q as given and chooses 2 q that solves 2 2 2 2 1 2 3 1 2 2 2 max [120 ( , )] q pq cq q q R q q q cq The first-order condition is given by 2 1 2 2 0 60 / 2 / 2 d q q c dq . Therefore, the best- response function of firm 2 is given by 1 2 1 120 ( ) 2 q c R q
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