Section 11 Notes - Consider a product which is made by n 1...

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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?
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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.
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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
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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
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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
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Cooperation: If firm 1 produces q 1 and firm 2 pro- duces q 2 , the total quantity produced = q = q
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