Ch12 - Monopolistic Competition and Oligopoly

Ww profits will be π p 1 q 1 c 1 300 3 q 1 3 q 2 q 1

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WW profits will be: Π = P 1 q 1 - C 1 = (300 - 3 q 1 - 3 q 2 ) q 1 - (30 q 1 +1.5 q 1 2 ) P = 270 q 1 - 4.5 q 1 2 - 3 q 1 q 2 P = 270 q 1 - 4.5 q 1 2 - 3 q 1 (30 - 1 3 q 1 ) P =180 q 1 - 3.5 q 1 2 . Profit maximization implies: ∂Π q 1 = 180 - 7 q 1 = 0. This results in q 1 =25.7 and q 2 =21.4.  The equilibrium price and profits will then be: P = 200 - 2(q 1  + q 2 ) = 200 - 2(25.7 + 21.4) = $158.57 π 1  = (158.57) (25.7) - (30) (25.7) – 1.5*25.7 2  = $2313.51 π 2  = (158.57) (21.4)  - (30) (21.4) – 1.5*21.4 2  = $2064.46. WW is able to benefit from its first mover advantage by committing to a high level of  output.  Since firm 2 moves after firm 1 has selected its output, firm 2 can only react  to the output decision of firm 1.  If firm 1 produces its Cournot output as a leader,  212
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Chapter  12:  Monopolistic Competition and Oligopoly firm 2 produces its Cournot output as a follower.  Hence, firm 1 cannot do worse as a  leader than it does in the Cournot game.   When firm 1 produces more, firm 2  produces less, raising firm 1’s profits. *11.  Two firms compete by choosing price.  Their demand functions are Q 1  = 20 - P 1  + P 2           and          Q 2  = 20 + P 1  - P 2 where P 1  and P 2  are the prices charged by each firm respectively and Q 1  and Q 2  are the  resulting demands.  Note that the demand for each good depends only on the difference in  prices; if the two firms colluded and set the same price, they could make that price as  high as they want, and earn infinite profits.  Marginal costs are zero. a. Suppose the two firms set their prices at the  same time .  Find the resulting Nash  equilibrium.  What price will each firm charge, how much will it sell, and what will  its profit be?  (Hint: Maximize the profit of each firm with respect to its price.) To determine the Nash equilibrium, we first calculate the reaction function for each  firm, then solve for price.  With zero marginal cost, profit for Firm 1 is: π 1 = P 1 Q 1 = P 1 20 - P 1 + P 2 ( 29 = 20 P 1 - P 1 2 + P 2 P 1 . The marginal revenue is the slope of the total revenue function (here it is the slope of  the profit function because total cost is equal to zero): MR 1  = 20 - 2 P 1  +  P 2 . At the profit-maximizing price,  MR 1  = 0.  Therefore, P P 1 2 20 2 = + . This is Firm 1’s reaction function.  Because Firm 2 is symmetric to Firm 1, its reaction  function is  P P 2 1 20 2 = + .   Substituting Firm 2’s reaction function into that of Firm 1: 1 1 1 20 20 2 2 10 5 4 P P P . = + + = + + = 520 By symmetry,  P 2  = $20. To determine the quantity produced by each firm, substitute   P 1   and   P 2   into the  demand functions: Q 1  = 20 - 20 + 20 = 20  and       Q 2  = 20 + 20 - 20 = 20.
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