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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: π 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 ≥ π 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 first-order condition for the above problem gives us the quantity q i which maximizes i ’s profit: dπ 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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This note was uploaded on 01/14/2012 for the course ECON 201 taught by Professor Witte during the Spring '08 term at Northwestern.
- Spring '08