aps7 rev - Intermediate Microeconomics - Wissink BRIEF...

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1's reaction function 6 b-cn 2 a b 6 .85 a-cn 2's reaction function Intermediate Microeconomics - Wissink BRIEF ANSWERS TO PROBLEM SET #7 1. Recall: demand: p = 12 - 2X where X = x 1 + x 2 and total costs: c(x 1 ) = 1 + 5(x 1 ) 2 and c(x 2 ) = 2 + (x 2 ) 2 Let x 1 = a and let x 2 = b (for notational ease) a. For firm 1: profit=(12-2a-2b)(a) - 1 - 5a 2 . Set the derivative equal to zero and get: 12-2a-2b-2a-10a = 0 or 12-14a-2b=0 or 14a=12-2b so: a R =(6/7)-(1/7)b For firm 2: profit=(12-2a-2b)(b) - 2 - b 2 . Set the derivative equal to zero and get: 12-2a-2b-2b-2b = 0 or 12-2a-6b=0 or 6b=12-2a so: b R =2-(1/3)a b. Now find a and b by solving a R and b R simultaneously. That is, plug the equation for b R into where you see b in the equation for a R and then solve for a-cn. Once you get a-cn you just use that value back in the b R function to get the value of b-cn. a-cn = 3/5 b-cn = 9/5 X-cn = 12/5 p-cn = 36/5 Firm 1's profit = 38/25 Firm 2's profit = 193/25 Joint profit = 231/25 c. For the multi-plant monopoly: 1) get the marginal costs for each firm; 2) invert them and add
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aps7 rev - Intermediate Microeconomics - Wissink BRIEF...

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