Ch12 - Monopolistic Competition and Oligopoly

Substituting for q 2 in the reaction function for

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Substituting for  Q 2  in the reaction function for Texas Air, Q 1  = 30 - 0.5(30 - 0.5 Q 1 ) = 20. By symmetry,  Q 2  = 20.  Industry output,  Q T , is  Q 1  plus  Q 2 , or Q T   = 20 + 20 = 40. Substituting industry output into the demand equation, we find  P  = 60.  Substituting  Q 1 Q 2 , and  P  into the profit function, we find π 1  =  π 2  = 60(20) -20 2  - (20)(20) = $400  for both firms in Cournot-Nash equilibrium. b. What would be the equilibrium quantity if Texas Air had constant marginal and  average costs of $25, and American had constant marginal and average costs of  $40? By solving for the reaction functions under this new cost structure, we find that profit  for Texas Air is equal to π 1 1 1 2 1 2 1 1 1 2 1 2 100 25 75 = - - - = - - Q Q Q Q Q Q Q Q Q . The change in profit with respect to  Q 1  is = - - π 1 1 1 2 75 2 Q Q Q . Set the derivative to zero, and solving for  Q 1  in terms of  Q 2 , Q 1  = 37.5 - 0.5 Q 2 . This is Texas Air’s reaction function.  Since American has the same cost structure as  in 8 .a ., American’s reaction function is the same as before: Q 2  = 30 - 0.5 Q 1 . To determine  Q 1 , substitute for  Q 2  in the reaction function for Texas Air and solve for  Q 1 : Q 1  = 37.5 - (0.5)(30 - 0.5 Q 1 ) = 30. 205
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Chapter  12:  Monopolistic Competition and Oligopoly Texas Air finds it profitable to increase output in response to a decline in its cost  structure. To determine  Q 2 , substitute for  Q 1  in the reaction function for American: Q 2  = 30 - (0.5)(37.5 - 0.5 Q 2 ) = 15. American has cut back slightly in its output in response to the increase in output by  Texas Air. Total quantity,  Q T ,  is  Q 1  +  Q 2 , or Q T  = 30 + 15 = 45. Compared to 8 a , the equilibrium quantity has risen slightly. c. Assuming that both firms have the original cost function, C(q) = 40q, how much  should Texas Air be willing to invest to lower its marginal cost from $40 to $25,  assuming that American will not follow suit?   How much should American be  willing to spend to reduce its marginal cost to $25, assuming that Texas Air will  have marginal costs of $25 regardless of American’s actions? Recall that profits for both firms were $400 under the original cost structure.  With  constant average and marginal costs of 25, Texas Air’s profits will be  (55)(30) - (25)(30) = $900. The difference in profit is $500.  Therefore, Texas Air should be willing to invest up to  $500 to  lower costs  from 40 to 25 per unit (assuming American does not follow suit).
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