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solutions_Chapter27

solutions_Chapter27 - Chapter 27 NAME Oligopoly...

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Chapter 27 NAME Oligopoly Introduction. In this chapter you will solve problems for firm and indus- try outcomes when the firms engage in Cournot competition, Stackelberg competition, and other sorts of oligopoly behavior. In Cournot competi- tion, each firm chooses its own output to maximize its profits given the output that it expects the other firm to produce. The industry price de- pends on the industry output, say, q A + q B , where A and B are the firms. To maximize profits, firm A sets its marginal revenue (which depends on the output of firm A and the expected output of firm B since the expected industry price depends on the sum of these outputs) equal to its marginal cost. Solving this equation for firm A’s output as a function of firm B’s expected output gives you one reaction function; analogous steps give you firm B’s reaction function. Solve these two equations simultaneously to get the Cournot equilibrium outputs of the two firms. Example: In Heifer’s Breath, Wisconsin, there are two bakers, Anderson and Carlson. Anderson’s bread tastes just like Carlson’s—nobody can tell the difference. Anderson has constant marginal costs of $1 per loaf of bread. Carlson has constant marginal costs of $2 per loaf. Fixed costs are zero for both of them. The inverse demand function for bread in Heifer’s Breath is p ( q ) = 6 . 01 q , where q is the total number of loaves sold per day. Let us find Anderson’s Cournot reaction function. If Carlson bakes q C loaves, then if Anderson bakes q A loaves, total output will be q A + q C and price will be 6 . 01( q A + q C ). For Anderson, the total cost of producing q A units of bread is just q A , so his profits are pq A q A = (6 . 01 q A . 01 q C ) q A q A = 6 q A . 01 q 2 A . 01 q C q A q A . Therefore if Carlson is going to bake q C units, then Anderson will choose q A to maximize 6 q A . 01 q 2 A . 01 q C q A q A . This expression is maximized when 6 . 02 q A . 01 q C = 1. (You can find this out either by setting A’s marginal revenue equal to his marginal cost or directly by setting the derivative of profits with respect to q A equal to zero.) Anderson’s reaction function, R A ( q C ) tells us Anderson’s best output if he knows that Carlson is going to bake q C . We solve from the previous equation to find R A ( q C ) = (5 . 01 q C ) /. 02 = 250 . 5 q C . We can find Carlson’s reaction function in the same way. If Carlson knows that Anderson is going to produce q A units, then Carlson’s profits will be p ( q A + q C ) 2 q C = (6 . 01 q A . 01 q C ) q C 2 q C = 6 q C . 01 q A q C . 01 q 2 C 2 q C . Carlson’s profits will be maximized if he chooses q C to satisfy the equation 6 . 01 q A . 02 q C = 2. Therefore Carlson’s reaction function is R C ( q A ) = (4 . 01 q A ) /. 02 = 200 . 5 q A .
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328 OLIGOPOLY (Ch. 27) Let us denote the Cournot equilibrium quantities by ¯ q A and ¯ q C . The Cournot equilibrium conditions are that ¯ q A = R A q C ) and ¯ q C = R C q A ).
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