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Unformatted text preview: Chapter 9 Monopoly As you will recall from intermediate micro, monopoly is the situation where there is a single seller of a good. Because of this, it has the power to set both the price and quantity of the good that will be sold. We begin our study of monopoly by considering the price that the monopolist should charge. 1 9.1 Simple Monopoly Pricing The object of the f rm is to maximize pro f t. However, the price that the monopolist charges a f ects the quantity it sells. The relationship between the quantity sold and the price charged is governed by the (aggregate) demand curve q ( p ) . Note, in order to focus on the relationship between q and p , we suppress the wealth arguments in the aggregate demand function. We can thus state the monopolists problem as follows: max p pq ( p ) c ( q ( p )) . Note, however, that there is a one-to-one correspondence between the price charged and the quantity the monopolist sells. Thus we can rewrite the problem in terms of quantity sold instead of the price charged. Let p ( q ) be the inverse demand function. That is, p ( q ( p )) = p. The f rms pro f t maximization problem can then be written as max q p ( q ) q c ( q ) . It turns out that it is usually easier to look at the problem in terms of setting quantity and letting price be determined by the market. For this reason, we will use the quantity-setting approach. 1 References: Tirole, Chapter 1; MWG, Chapter 12; Bulow, Durable-Goods Monopolists, JPE 90(2) 314-332. 233 Nolan Miller Notes on Microeconomic Theory: Chapter 9 ver: Aug. 2006 A D Q Q P P B Figure 9.1: The Monopolists Marginal Revenue In order for the solution to be unique, we need the objective function to be strictly concave (i.e. d 2 dq 2 < 0) . The second derivative of pro f t with respect to q is given by d 2 dq 2 ( p ( q ) q c ( q )) = p 00 ( q ) q + 2 p ( q ) c 00 ( q ) . If cost is strictly convex, c 00 ( q ) > , and since demand slopes downward, p ( q ) < . Hence the second and third terms are negative. Because of this, we dont need inverse demand to be concave. However, it cant be too convex. Generally speaking, well just assume that the objective function is concave without making additional assumptions on p () . Actually, to make sure the maximizing quantity is f nite, we need to assume that eventually costs get large enough relative to demand. This will always be satis f ed if, for example, the demand and marginal cost curves cross. The objective function is maximized by looking at the f rst derivative. At the optimal quantity, q , p ( q ) q + p ( q ) = c ( q ) On the left-hand side of the expression is the marginal revenue of increasing output a little bit. This has two parts - the additional revenue due to selling one more unit, p ( q ) (area B in Figure 9.1), and the decrease in revenue due to the fact that the f rm receives a lower price on all units it sells (area A in Figure 9.1). Hence the monopolists optimal quantity is where marginal revenue is equal to marginal cost, and price is de f ned by the demand curve p ( q ) ....
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