Ch12 - Monopolistic Competition and Oligopoly

B suppose a second firm enters the market let q 1 be

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b. Suppose a second firm enters the market.  Let Q 1  be the output of the first firm and  Q 2  be the output of the second.  Market demand is now given by Q 1  + Q 2  = 53 - P. Assuming that this second firm has the same costs as the first, write the  profits of each firm as functions of Q 1  and Q 2 . When the second firm enters, price can be written as a function of the output of two  firms:  P  = 53 -  Q 1  -  Q 2 .  We may write the profit functions for the two firms: π 1 = PQ 1 - C Q 1 ( 29 = 53 - Q 1 - Q 2 ( 29 Q 1 - 5 Q 1 ,  or  π 1 1 1 2 1 2 1 53 5 = - - - Q Q Q Q Q and π 2 = PQ 2 - C Q 2 ( 29 = 53 - Q 1 - Q 2 ( 29 Q 2 - 5 Q 2 ,  or  π 2 2 2 2 1 2 2 53 5 = - - - Q Q Q Q Q . c. Suppose (as in the Cournot model) that each firm chooses its profit-maximizing  level of output on the assumption that its competitor’s output is fixed.  Find each  firm’s “reaction curve” (i.e., the rule that gives its desired output in terms of its  competitor’s output). Under the Cournot assumption, Firm 1 treats the output of Firm 2 as a constant in  its maximization of profits.  Therefore, Firm 1 chooses  Q 1  to maximize  π 1  in  with  Q being treated as a constant.  The change in  π 1  with respect to a change in  Q 1  is ∂π 1 1 1 2 1 2 53 2 5 0 24 2 Q Q Q Q Q = - - - = = - , .  or  197
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Chapter  12:  Monopolistic Competition and Oligopoly This   equation   is   the   reaction   function   for   Firm   1,   which   generates   the   profit-  maximizing  level of output,  given  the  constant  output  of Firm  2.   Because  the  problem is symmetric, the reaction function for Firm 2 is Q Q 2 1 24 2 = - . d. Calculate the Cournot equilibrium (i.e., the values of Q 1   and Q 2   for which both  firms are doing as well as they can given their competitors’ output).  What are the  resulting market price and profits of each firm? To find the level of output for each firm that would result in a stationary equilibrium,  we   solve   for   the   values   of   Q 1   and   Q 2   that   satisfy   both   reaction   functions   by  substituting the reaction function for Firm 2 into the one for Firm 1: By symmetry,  Q 2  = 16. To determine the price, substitute  Q 1  and  Q 2  into the demand equation: = 53 - 16 - 16 = $21. Profits are given by π i  =  PQ i  -  C ( Q i ) =  π i  = (21)(16) - (5)(16) = $256. Total profits in the industry are  π 1  +  π 2  = $256 +$256 = $512. *e. Suppose there are   N   firms in the industry, all with the same constant marginal  cost, MC = 5.  Find the Cournot equilibrium.  How much will each firm produce,  what will be the market price, and how much profit will each firm earn?   Also,  show that as  N
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