c 2009 Je rey A. Miron

Transcription

1 Lecture 22: Factor Markets c 2009 Je rey A. Miron Outline 1. Introduction 2. Monopoly in the Output Market 3. Monopsony 4. Upstream and Downstream Monopolies 1 Introduction The analysis in earlier lectures examined factor demands for a rm facing competitive output and input markets. In some interesting cases, however, the assumption of a competitive market (for either the output or the input) is not realistic. This lecture considers what happens to factor demands when the assumption of competition does not hold in one or both markets. Some of the results here are not enormously important in practice, but they are nice reviews of earlier material and clean applications of the tools already developed. 2 Monopoly in the Output Market When choosing how much of a factor to hire, a rm always makes the same calculation: it compares the marginal revenue from using a bit more of this factor to the marginal cost. This general rule takes di erent speci c forms depending on the circumstances. We assume now that the rm is a monopolist and uses one factor of production. We can therefore write the production function as 1

2 y = f(x) and the revenue the rm receives is R(y) = p(y)y Suppose the rm considers increasing x a little bit. Then the e ect on = p(f(x))f 0 (x) + f(x)p 0 (f(x))f 0 (x) = [p(y) + yp 0 (y)] f 0 (x) = p(y) 1 + yp0 (y) f 0 (x) p(y) = p(y) MP x This can also be = MR y MP x That is, the e ect of an increase in x on revenue is the e ect of the MP of x times the e ect of increasing x on the rm s MR. The overall expression is known as the marginal revenue product. The M RP is a generalization of the competitive case. elasticity of demand is in nite and the formula collapses to Under = pmp x since for a competitive rm marginal revenue equals price. In this case, therefore, the MRP is just equal to the value marginal product, where that value is determined by the market price. 2

3 It is useful to think about the how the value MP compares to the revenue MP. The MRP must always be less than the value MP since an increase in x by a monopolist means an increase in the amount of y, and this causes a decrease in p. This result ts with fact that a monopolist wants to produce less than a rm acting competitively, so the monopolist also purchases less of the input. The derivation so far shows the bene t from hiring more x. To determine equilibrium amount of x employed, the monopolist must compare the bene t to the cost. Assuming a competitive factor market, the price of the input is just!: The monopolist therefore chooses x to satisfy MRP (x) =! We can illustrate this graphically as below: 3

4 Graph: Factor Demand by a Monopolist FACTOR PRICE w MRP pmp xm xc FACTOR DEMAND The graph also shows the factor demand curve for a competitive rm, which must lie to the right of the demand curve of the monopolist. 3 Monopsony The kind of market power that receives most of the attention in economics, and more generally, is when only a single seller of a product exists in the market place. Another interesting possibility arises when only a single buyer of a product exists. This situation is known as monopsony. A standard example is the company town, meaning a location with a single employer who provides the only source of employment for residents of that location. We can model this situation in a straightforward way using our standard tools. For simplicity, assume the buyer produces output that is sold in a competitive market. As above, suppose the rm produces output using a single factor according to 4

5 y = f(x) This rm, however, dominates the factor market and recognizes that the amount of x it demands will in uence the price. In particular, if the rm hires only a few units of x, it can do so at a relatively low rate, but if it tries to hire more x, it will face a higher and higher price of x: We formalize this by assuming the rm faces an (inverse) supply curve for x of the form!(x) This says that if the rm want to hire x units of the factor, it must pay the price!(x). A rm in a competitive factor market faces a at supply curve; under monopsony the rm faces an upward sloping supply curve. Stated di erently, a rm in a competitive market is a price taker; a monopsonist is a price maker. The pro t maximization problem faced by the monopsonist is max pf(x)!(x)x and the solution is the x that solves the FOC: pf 0 (x) =!(x) + x! 0 (x) In words, this says that the marginal revenue from adding a bit more x should equal its marginal cost. If the factor market were competitive, the marginal cost of hiring a bit more x would just be!. For the monopsonist, however, the marginal cost must re ect the fact that hiring more x raises the price he must pay. This means the monopsonist hires less than a rm that does not have monopsony power. A di erent way of presenting this is as follows: MP x =! where is the elasticity of the supply curve for x. If the supply curve is perfectly elastic, so that is in nite, then the FOC reduces to the case of a rm facing a competitive factor market. Graphically, we can illustrate this as follows: 5

6 Graph: Monopsony w MR = MC MC = a + 2bL w(l) = a + bl (inverse supply) w* a MR = pmp_l L* LABOR The downward sloping line is the rm s demand for the input (here labeled as labor). It is downward sloping, as discussed in earlier lectures, because of the assumption of diminishing marginal product. The upward sloping line to the right is the supply curve for labor facing the rm. It is assumed here to be everywhere upward sloping; it takes a higher and higher wage rate to get more labor supplied. (Note that labor supply curves, as discussed in earlier lectures, do not have to slope upward, but we ignore that here for simplicity.) The upward sloping line to the left is the marginal cost of hiring an additional unit of labor. It rises more steeply than the labor supply curve for a reason analogous to why a marginal revenue curve is steeper than a demand curve. Given the assumption of one wage rate, a rm trying to hire more labor must not only pay a wage to that additional unit of labor; it must also pay a higher wage to all previously hired units of labor. The monopsonist therefore hires less labor, and pays a lower wage, than would occur if the input market were competitive. If it were, the equilibrium wage and amount of labor would be determined by the intersection of the demand and supply of labor. 6

7 4 Upstream and Downstream Monopolies We have considered two possible deviations from the base case of competitive input and output markets. Many other possible market structures are possible, but one variation is of particular interest. Consider a monopolist whose output is used as a factor of production by another monopolist. For example, a logging rm in a remote area might be the only supplier of logs, and a paper manufacturer might be the only purchaser of logs and a monopoly seller of paper. Suppose that the rst monopolist, whom we label the upstream monopolist (UM), produces output x at a constant MC of c. This UM sells its output, x, to a downstream monopolist (DM), at price k. The DM then uses x to produce a good y according to the production function y = f(x). This output is then sold by the UM in a market with inverse demand curve p(y). To keep the algebra simple, we assume p(y) = a by and that the production function is just y = x We assume the DM has no costs except buying x at price k. is The way to solve the problem is to solve the DM s problem rst. This problem and the FOC is max y p(y)y ky = [a by]y ky a 2by = k 7

8 or y = a k 2b Since the DM demands one unit of x for each unit of y produced, this expression also determines the demand function for x, x = a k 2b This tells us the relation between the factor price, k, and the amount of the factor the DM will demand. Now consider the problem faced by the U M. This rm presumably understands the DM s problem and can determine how much x it will sell if it sets price k. That is, it knows that the demand function for its output is identical to the UM s factor demand for x. The UM therefore solves max k xk(x) c(x) and for simplicity we assume the UM has no costs. its MR equal to MC: So, the FOC for the UM sets and imposing xk 0 (x) + k(x) = c k(x) = a 2bx means that the FOC reduces to a 4bx = c Thus, the optimal amount of x produced by the UM is just x = a c 4b Since one unit of input produces one unit of output, we also know that 8

9 y = a c 4b These are the solutions under the assumption that the two monopolists act independently. It is useful to compare this outcome to the one that would occur if the nal output were produced by a single, integrated monopolist. Suppose the U M and DM merged. This means a single rm faces the inverse demand function p = a by and maximizes pro ts by setting set MR = MC : a 2by = c This yields a pro t-maximizing output of y = a c 2b Thus, the integrated monopolist produces twice as must output as the non-integrated monopolist (and sells at a lower price). We can see this graphically as follows: 9

10 Graph: Upstream and Downstream Monopoly PRICE MR_U MC MR_D D ym yi QUANTITY The DM faces the inverse demand curve p(y) and sets price using the associated MR curve. The UM takes this MR curve as its demand curve and then derives its own MR curve. The UM monopolist then sets a price where its MR curve intersects MC; so this cost gets marked up once by the UM. The DM then takes this marked up price as its cost and equates it to its own MR curve to determine its price, i.e., it marks up the cost yet again. This is known as a double markup, or, more commonly, as double marginalization. It has huge implications for antitrust policy toward vertical integration. For decades antitrust policy was hostile to vertical integration (and to vertical restrictions more generally), assuming they were used to extend monopoly power. This example shows that vertical integration can make consumers better o. 10