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Unformatted text preview: Consider a product, which is made by n ≥ 1 companies. Its unit production cost is c . Let q denote the total quantity produced. The unit selling price, p , is then given by: p = M q, q ≤ M = 0 , q > M What is the total quantity produced, the price charged and the companies’ profit • when they compete? • when they cooperate? • n = 1 (Monopoly): If the firm produces q units, then p = M q . The firm’s profit Π = ( p c ) q = ( M q c ) q The firm seeks to maximize its profit. So, d Π dq = M c 2 q = 0 ⇒ ˜ q = ( M c ) / 2 Price ˜ p = M + c 2 The firm’s profit = ( M c ) 2 / 4. • n = 2 (Duopoly): – Competition: Nash equilibrium. Each firm independently chooses its production level q i . Let q = q 1 + q 2 be the total quantity produced. Price p = M q . The payoff to each firm is the profit it gets. So, firm i ’s ( i = 1 , 2) payoff is given by: Π i ( q 1 ,q 2 ) = ( p c ) q i We want to find a NE (˜ q 1 , ˜ q 2 ). q 1 being firm 1’s best reply to firm 2’s choice of q 2 ⇒ ∂ Π 1 ( q 1 ,q 2 ) ∂q 1 = 0 ∂ ( M q 1 q 2 c ) q 1 ∂q 1 = 0 2 q 1 = M q 2 c Similarly, q 2 being a best response to q 1 ⇒ ∂ Π 2 ( q 1 ,q 2 ) ∂q 2 = 0 2 q 2 = M q 1 c A NE occurs where the reaction curves of the 2 players cross, i.e., 2˜ q 1 + ˜ q 2 = M c ˜ q 1 + 2˜ q 2 = M c Solving for ˜ q 1 , ˜ q 2 , ˜ q 1 = ˜ q 2 = ( M c ) / 3 Price ˜ p = M ˜ q 1 ˜ q 2 = ( M + 2 c ) / 3 The firms’ profits are ( M c ) 2 / 9 – Cooperation: If firm 1 produces...
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This note was uploaded on 09/17/2007 for the course ORIE 435 taught by Professor Todd,m. during the Fall '06 term at Cornell.
 Fall '06
 TODD,M.
 Game Theory, Alice, best reply, production level qi

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