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Unformatted text preview: Solutions to Problem Set 4 1 Cournot Competition with More than 2 Firms (a) The profit of firm i is the difference between its total revenue and total cost, and is therefore a function of the quantities set by all of the firms: i ( q 1 ,...,q n ) = q i P ( Q ) 10 q i = q i (100 Q 10) = q i (90 q 1 q n ) (b) Firm i s best response to the other firms quantities is the solution to its profit maximization problem, taking as given the quantities of the other firms. The profit maximization problem is: max q i i ( q 1 ,...,q n ) max q i q i (90 q 1 q 2 ... q n ) max q i 90 q i q 1 q i ... q 2 i ... q n q i The firstorder condition for the above problem gives us the quantity q i which maximizes i s profit: d i dq i = (90 q 1 ... 2 q i ... q n ) = 0 2 q i = 90 q 1 ... q i 1 q i +1 ... q n q i = 90 q 1 ... q i 1 q i +1 ... q n 2 This is the optimal quantity for firm i given the quantities of the other firms....
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 Spring '08
 Witte
 Macroeconomics

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