Econ501aL12 - Prot Maximization by a Competitive Firm...

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Pro¯t Maximization by a Competitive Firm Having derived the total cost function (either long run or short run), we can now solve for the pro¯t- maximizing output level, x ¤ .G i v e n x ¤ ,wecanthen compute the unconditional demand for inputs such as capital and labor. The pro¯t function is total revenue minus total cost, ¼ ( x )= TR ( x ) ¡ TC ( x ) : For an interior choice of x that maximizes pro¯t, set theder ivat iveequa ltozero : ( x ) dx = dTR ( x ) dx ¡ dTC ( x ) dx = MR ( x ) ¡ MC ( x )=0 : Thus, a pro¯t maximizing ¯rm (either competitive or one with market power) chooses x so that marginal revenue equals marginal cost.
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A perfectly competitive ¯rm cannot in°uence the price, p x . Therefore, marginal revenue is just the price, since one more unit receives additional revenues of p x . A perfectly competitive ¯rm's pro¯t function is then ¼ ( x )= p x x ¡ TC ( x ) ; and the condition for (interior) pro¯t maximization is p x = MC ( x ) : (1) To make sure that we have a pro¯t maximum (and not a minimum!), use the second-order condition d 2 ¼ dx 2 = ¡ dMC ( x ) dx < 0 ; which says that the marginal cost curve should be upward sloping (increasing in x).
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The interpretation of equation (1), p x = MC ( x ), is not that the ¯rm is choosing the price. Rather, the ¯rm takes the price as determined in the market, and chooses x at the point where marginal cost equals the price.
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This note was uploaded on 07/17/2008 for the course ECON 501.02 taught by Professor Yang during the Spring '08 term at Ohio State.

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Econ501aL12 - Prot Maximization by a Competitive Firm...

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