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Unformatted text preview: Econ 100A December 12, 2008 NonComprehensive Review of Market Structure Michael Schihl Note the caveat utilitor . 1 Profit Maximization All firms, by assumption, maximize profits. One can express profit as π i ( q i ) = TR i ( q i ) TC i ( q i ) (1) = p ( Q i ,q i ) q i TC i ( q i ) (2) Q i represents the output of all firms except firm i . When we optimize, i.e., find the extrema of, a function, we typically set the first derivative equal to zero. Here is the example from the beginning of the semester (a typical calculus problem). Suppose you have 20 meters of fence from which to make a rectangular fence. We know that the perimeter is the length of fence and that there are two sides, call them w for width and l for length. We can write this as 2 w + 2 l = 20, then, w = 10 l . Now, area can be written as A = wl = (10 l ) l = 10 l l 2 . As suggested, take the first derivative and set it equal to zero. d dl A = 0 (3) d dl [10 l l 2 ] = 0 (4) 10 2 l = 0 (5) 2 l = 10 (6) l = 5 (7) So we find that the way to maximize the area of a rectangular fence is to set the length, l , to 5, which makes the width, w , equal to 5. The maximized area is A max = 5 × 5 = 25. (I think the way to maximize the area is really to make a circular fence, but who makes circular fences?) Returning to the problem of maximizing profit, we’ll take the derivative of profit and set it equal to zero. d dq i π i ( q i ) = 0 (8) d dq i [ p ( Q i ,q i ) q i TC i ( q i )] = 0 (9) dp ( Q i ,q i ) dq i q i + p ( Q i ,q i )  {z } MR i ( q i ) MC i ( q i ) = 0 (10) MR i ( q i ) MC i ( q i ) = 0 (11) MR i ( q i ) = MC i ( q i ) (12) This gives us the familiar result that marginal revenue equals marginal cost. Note that, for a perfectly competitive firm, p ( Q i ,q i ) is a constant determined by the market, thus, its derivative is zero. So marginal revenue for a firm in a perfectly competitive market is just price. MR i ( q i ) = dp ( Q i ,q i ) dq i q i + p ( Q i ,q i ) (13) = 0 × q i + p ( Q i ,q i ) since dp ( Q i ,q i ) dq i = 0 for a firm in a perfectly competitive market (14) = p since p ( Q i ,q i ) is a constant in a perfectly competitive market (15) Now let’s look at something a bit more specific, in particular, linear demand functions. Let’s make another simplifying assumption; let’s suppose that firms have constant marginal costs. Say firm i has constant marginal cost 1 Market Inverse Demand Perfect Competition p ( Q i ,q i ) = k , a constant Monopoly p ( Q i ,q i ) = p (0 ,q i ) = p ( q i ), because there is only one firm (by definition) Cournot Duopoly p ( Q i ,q i ) = p ( q i ,q j ), because there are only two firms (by definition) Cournot Oligopoly p ( Q i ,q i ) = p ( q 1 ,q 2 ,...,q N ), because there are N firms Table 1: Inverse Demand Functions for Various Markets Market Inverse Demand Monopoly p ( q i ) = a bq i Cournot Duopoly p ( q i ,q j ) = a b ( q i + q j ) Cournot Oligopoly p ( q 1 ,q 2 ,...,q N ) = a b ( q 1 + q 2 + ···...
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This note was uploaded on 11/13/2009 for the course ECON 181 taught by Professor Kasa during the Spring '07 term at Berkeley.
 Spring '07
 Kasa

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