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Unformatted text preview: COLLEGE OF HUMAN ECOLOGY CORNELL UNIVERSITY ITHACA, NEW YORK Department of Policy Analysis and Management PAM 200 Intermediate Microeconomics A. Sinan ¨ Un¨ur – Fall 2007 Duopoly The earliest treatment of duopoly , a market where two firms produce a single homogenous product, was by French economist Cournot in his ground-breaking work entitled Researches into the Mathematical Principles of the Theory of Wealth published in 1838. Remark : If you are at all curious about the subject-matter, and you are not averse to taking a few simple derivatives, I heartily recommend this little book. An oligopolistic market is one that is comprised of very few firms selling a homogenous product. Duopoly is a special case of oligopoly where there are two firms in market. For our analysis, we will assume that the demand side of the market is described by a linear demand curve of the form: P = a- bQ T , where a is the vertical intercept, and b is the slope of the demand curve. In the equation above, Q T is the total quantity produced by the two firms in question. Our purpose is to figure out how much each firm will produce, and by implication, the price and quantity exchanged in this market in equilibrium. For simplicity, we will also assume that both firms have the same technology, and their costs function given by TC ( Q i ) = cQ i , i = 1 , 2 . That is, each firm’s marginal cost function is constant: MC ( Q i ) = c , i = 1 , 2 . 1 Reaction Functions Each firm tries to maximize its profits taking as given the production level of the other firm . First, consider Firm 1. It will choose a production level, Q 1 , taking as given Firm 2’s production level, say Q 2 . Given that Firm 2 is already producing Q 2 units, demand curve faced by Firm 1 is given by: P = a- b ( Q 1 + Q 2 ) = ( a- bQ 2 )- bQ 1 . Firm 1’s marginal revenue in this case is given by: MR 1 ( Q 1 ; Q 2 ) = ( a- bQ 2 )- 2 bQ 1 . The solution to Firm 1’s profit maximization problem will be found by setting marginal revenue equal to marginal cost in the usual manner: MR 1 ( Q 1 ; Q 2 ) = MC 1 ( Q 1 ) ( a- bQ 2 )- 2 bQ 1 = c ⇒ Q 1 = a- c 2 b- Q 2 2 . You can think of Q 2 as the production level Firm 1 expects Firm 2 to choose. Clearly, cor- responding to each production level Firm 2 might choose, Firm 1’s will have a different profit maximizing output choice. The equation above characterizes Firm 1’s best response (its profit maximizing reaction ) to any possible choice of output level by Firm 2....
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