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a tariff rate of about 20 percent. Knight, F. H. Risk, Uncertainty and Proﬁt. Boston: Houghton Mifﬂin, 1921, chaps. 5 and 6. Classic treatment of the role of economic events in motivating industry behavior in the long run. 17The seventeenth-century French ﬁnance minister Jean-Baptiste Colbert captured the essence of the problem with his memorable statement that ‘‘the art of taxation consists in so plucking the goose as to obtain the largest possible amount of feathers with the smallest amount of hissing.’’ 18See F. Ramsey, ‘‘A Contribution to the Theory of Taxation,’’ Economic Journal (March 1927): 47–61. 452 Part 5: Competitive Markets Marshall, A. Principles of Economics, 8th ed. New York: Crowell-Collier and Macmillan, 1920, book 5, chaps. 1, 2, and 3. Salanie, B. The Economics of Taxation. Cambridge, MA: MIT Press, 2003. Classic development of the supply–demand mechanism. Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. New York: Oxford University Press, 1995, chap. 10. Provides a compact analysis at a high level of theoretical precision. There is a good discussion of situations where competitive markets may not reach an equilibrium. Reynolds, L. G. ‘‘Cut-Throat Competition.’’ American Economic Review 30 (December 1940): 736–47. Critique of the notion that there can be ‘‘too much’’ competition in an industry. Robinson, J. ‘‘What Is Perfect Competition?’’ Quarterly Journal of Economics 49 (1934): 104–20. Critical discussion of the perfectly competitive assumptions. This provides a compact study of many issues in taxation. Describes a few simple models of incidence and develops some general equilibrium models of taxation. Stigler, G. J. plated.’’ Journal of Political Economy 65 (1957): 1–17. ‘‘Perfect Competition, Historically Contem- Fascinating discussion of the historical development of the competitive model. Varian, H. R. Microeconomic Analysis, 3rd ed. New York: W. W. Norton, 1992, chap. 13.
Terse but instructive coverage of many of the topics in this chapter. The importance of entry is stressed, although the precise nature of the long-run supply curve is a bit obscure. DEMAND AGGREGATION AND ESTIMATION EXTENSIONS In Chapters 4–6 we showed that the assumption of utility maximization implies individual demand functions: several properties for • • • • the functions are continuous; the functions are homogeneous of degree 0 in all prices and income; income-compensated substitution effects are negative; and cross-price substitution effects are symmetric. In this extension we will examine the extent to which these properties would be expected to hold for aggregated market demand functions and what, if any, restrictions should be placed on such functions. In addition, we illustrate some other issues that arise in estimating these aggregate functions and some results from such estimates. E12.1 Continuity The continuity of individual demand functions clearly implies the continuity of market demand functions. But there are situations in which market demand functions may be continuous, whereas individual functions are not. Consider the case where goods—such as an automobile—must be bought in large, discrete units. Here individual demand is discontinuous, but the aggregated demands of many people are (nearly) continuous. E12.2 Homogeneity and income aggregation Because each individual’s demand function is homogeneous of degree 0 in all prices and income, market demand functions are also homogeneous of degree 0 in all prices and individual incomes. However, market demand functions are not necessarily homogeneous of degree 0 in all prices and total income. To see when demand might depend just on total income, suppose individual i’s demand for X is given by xi ¼ ai(P Þ þ b(P)yi, 1, n, i ¼ (i) where P is the vector of all market prices, ai(P) is a set of individual-speciﬁc price effects, and b(P) is a marginal propensity-to-spend function that is the same across all individuals (although the value of this parameter may depend on market prices). In this case the market demand functions will depend on P and on total income: n yi: (ii) g ¼ 1 i X ¼ This shows that market demand reﬂects the behavior of a single ‘‘typical’’ consumer. Gorman (1959) shows that this is the most general form of
demand function that can represent such a typical consumer. E12.3 Cross-equation constraints Suppose a typical individual buys k items and that expenditures on each are given by pjxj ¼ k 1 i X ¼ aijpi þ bjy, j ¼ 1, k: (iii) If expenditures on these k items exhaust total income, that is, k 1 j X ¼ pjxj ¼ y, then summing over all goods shows that and that k 1 j X ¼ aij ¼ 0 for all i k 1 j X ¼ 1 bj ¼ (iv) (v) (vi) for each person. This implies that researchers are generally not able to estimate expenditure functions for k goods independently. Rather, some account must be taken of relationships between the expenditure functions for different goods. E12.4 Econometric practice The degree to which these theoretical concerns are reﬂected in the actual practices of econometricians varies widely. At the least sophisticated level, an equation similar to Equation iii might be estimated directly using ordinary least squares (OLS) with little attention to the ways in which the assumptions might be violated. Various elasticities could be calculated 454 Part 5: Competitive Markets TABLE 12.3 REPRESENTATIVE PRICE AND INCOME ELASTICITIES OF DEMAND Price Elasticity Income Elasticity Food Medical services Housing Rental Owner occupied Electricity Automobiles Gasoline Beer Wine Marijuana Cigarettes Abortions Transatlantic air travel Imports Money 0.21 & 0.18 & 0.18 1.20 & & 1.14 & 1.20 & 0.55 & 0.26 & 0.88 & 1.50 & 0.35 & 0.81 & 1.30 & 0.58 & 0.40 & 0.28 þ 0.22 þ 1.00 þ 1.20 þ 0.61 þ 3.00 þ 1.60 þ 0.38 þ 0.97 þ 0.00 0.50 þ 0.79 þ 1.40 þ 2.73 þ 1.00 þ Note: Price elasticity refers to interest rate elasticity. SOURCES: Food: H. Wold and L. Jureen, Demand Analysis (New York: John Wiley & Sons, 1953): 203. Medical services: income elasticity from R. Andersen and L. Benham, ‘‘Factors Affecting the Relationship between Family Income and Medical Care Consumption,’
’ in Herbert Klarman, Ed., Empirical Studies in Health Economics (Baltimore: Johns Hopkins University Press, 1970); price elasticity from W. C. Manning et al., ‘‘Health Insurance and the Demand for Medical Care: Evidence from a Randomized Experiment,’’ American Economic Review (June 1987): 251–77. Housing: income elasticities from F. de Leeuw, ‘‘The Demand for Housing,’’ Review for Economics and Statistics (February1971); price elasticities from H. S. Houthakker and L. D. Taylor, Consumer Demand in the United States (Cambridge, MA: Harvard University Press, 1970): 166–67. Electricity: R. F. Halvorsen, ‘‘Residential Demand for Electricity,’’ unpublished Ph.D. dissertation, Harvard University, December 1972. Automobiles: Gregory C. Chow, Demand for Automobiles in the United States (Amsterdam: North Holland, 1957). Gasoline: C. Dahl, ‘‘Gasoline Demand Survey,’’ Energy Journal 7 (1986): 67–82. Beer and wine: J. A. Johnson, E. H. Oksanen, M. R. Veall, and D. Fritz, ‘‘Short-Run and LongRun Elasticities for Canadian Consumption of Alcoholic Beverages,’’ Review of Economics and Statistics (February 1992): 64–74. Marijuana: T. C. Misket and F. Vakil, ‘‘Some Estimate of Price and Expenditure Elasticities among UCLA Students,’’ Review of Economics and Statistics (November 1972): 474–75. Cigarettes: F. Chalemaker, ‘‘Rational Addictive Behavior and Cigarette Smoking,’’ Journal of Political Economy (August 1991): 722–42. Abortions: M. H. Medoff, ‘‘An Economic Analysis of the Demand for Abortions,’’ Economic Inquiry (April 1988): 253–59. Transatlantic air travel: J. M. Cigliano, ‘‘Price and Income Elasticities for Airline Travel,’’ Business Economics (September 1980): 17–21. Imports: M. D. Chinn, ‘‘Beware of Econometricians Bearing Estimates,’’
both quantity and quality demanded. But it does suggest why the automobile industry is so sensitive to the business cycle. References Gorman, W. M. ‘‘Separable Utility and Aggregation.’’ Econo- metrica (November 1959): 469–81. Theil, H. Principles of Econometrics. New York: John Wiley & Sons, 1971, pp. 326–46. ———. Theory and Measurement of Consumer Demand, vol. 1. Amsterdam: North Holland, 1975, chaps. 5 and 6. This page intentionally left blank CHAPTER THIRTEEN General Equilibrium and Welfare The partial equilibrium models of perfect competition that were introduced in Chapter 12 are clearly inadequate for describing all the effects that occur when changes in one market have repercussions in other markets. Therefore, they are also inadequate for making general welfare statements about how well market economies perform. Instead, what is needed is an economic model that permits us to view many markets simultaneously. In this chapter we will develop a few simple versions of such models. The Extensions to the chapter show how general equilibrium models are applied to the real world. Perfectly Competitive Price System The model we will develop in this chapter is primarily an elaboration of the supply– demand mechanism presented in Chapter 12. Here we will assume that all markets are of the type described in that chapter and refer to such a set of markets as a perfectly competitive price system. The assumption is that there is some large number of homogeneous goods in this simple economy. Included in this list of goods are not only consumption items but also factors of production. Each of these goods has an equilibrium price, established by the action of supply and demand.1 At this set of prices, every market is cleared in the sense that suppliers are willing to supply the quantity that is demanded and consumers will demand the quantity that is supplied. We also assume that there are no transaction or transportation charges and that both individuals and ﬁrms have perfect knowledge of prevailing market prices. The law of one price Because we assume zero transaction cost and perfect information, each good obeys the law of one price: A homogeneous good trades at the same price no matter who buys it or which ﬁrm sells it. If one good traded at two different prices, demanders would rush to buy the good where it was cheaper, and ﬁrms would try to sell all their output where the good was more expensive. These actions in themselves would tend to equalize
the price of the good. In the perfectly competitive market, each good must have only one price. This is why we may speak unambiguously of the price of a good. 1One aspect of this market interaction should be made clear from the outset. The perfectly competitive market determines only relative (not absolute) prices. In this chapter, we speak only of relative prices. It makes no difference whether the prices of apples and oranges are $.10 and $.20, respectively, or $10 and $20. The important point in either case is that two apples can be exchanged for one orange in the market. The absolute level of prices is determined mainly by monetary factors—a topic usually covered in macroeconomics. 457 458 Part 5: Competitive Markets Behavioral assumptions The perfectly competitive model assumes that people and ﬁrms react to prices in speciﬁc ways. 1. There are assumed to be a large number of people buying any one good. Each person takes all prices as given and adjusts his or her behavior to maximize utility, given the prices and his or her budget constraint. People may also be suppliers of productive services (e.g., labor), and in such decisions they also regard prices as given.2 2. There are assumed to be a large number of ﬁrms producing each good, and each ﬁrm produces only a small share of the output of any one good. In making input and output choices, ﬁrms are assumed to operate to maximize proﬁts. The ﬁrms treat all prices as given when making these proﬁt-maximizing decisions. These various assumptions should be familiar because we have been making them throughout this book. Our purpose here is to show how an entire economic system operates when all markets work in this way. A Graphical Model of General Equilibrium with Two Goods We begin our analysis with a graphical model of general equilibrium involving only two goods, which we will call x and y. This model will prove useful because it incorporates many of the features of far more complex general equilibrium representations of the economy. General equilibrium demand Ultimately, demand patterns in an economy are determined by individuals’ preferences. For our simple model we will assume that all individuals have identical preferences, which can be represented by an indifference curve map3 deﬁned over quantities of the two goods, x and y. The beneﬁt of this approach for our purposes is that this indifference curve map (which is
identical to the ones used in Chapters 3–6) shows how individuals rank consumption bundles containing both goods. These rankings are precisely what we mean by ‘‘demand’’ in a general equilibrium context. Of course, we cannot illustrate which bundles of commodities will be chosen until we know the budget constraints that demanders face. Because incomes are generated as individuals supply labor, capital, and other resources to the production process, we must delay any detailed illustration until we have examined the forces of production and supply in our model. General equilibrium supply Developing a notion of general equilibrium supply in this two-good model is a somewhat more complex process than describing the demand side of the market because we have two goods simultaneously. Our not thus far illustrated production and supply of 2Hence, unlike our partial equilibrium models, incomes are endogenously determined in general equilibrium models. 3There are some technical problems in using a single indifference curve map to represent the preferences of an entire community of individuals. In this case the marginal rate of substitution (i.e., the slope of the community indifference curve) will depend on how the available goods are distributed among individuals: The increase in total y required to compensate for a one-unit reduction in x will depend on which speciﬁc individual(s) the x is taken from. Although we will not discuss this issue in detail here, it has been widely examined in the international trade literature. Chapter 13: General Equilibrium and Welfare 459 approach is to use the familiar production possibility curve (see Chapter 1) for this purpose. By detailing the way in which this curve is constructed, we can illustrate, in a simple context, the ways in which markets for outputs and inputs are related. Edgeworth box diagram for production Construction of the production possibility curve for two outputs (x and y) begins with the assumption that there are ﬁxed amounts of capital and labor inputs that must be allocated to the production of the two goods. The possible allocations of these inputs can be illustrated with an Edgeworth box diagram with dimensions given by the total amounts of capital and labor available. In Figure 13.1, the length of the box represents total labor-hours, and the height of the box represents total capital-hours. The lower left corner of the box represents the ‘‘origin’’ for measuring capital and labor devoted to production of good x. The upper right corner of the box represents the origin for resources devoted to y. Using these conventions, any point
in the box can be regarded as a fully employed allocation of the available resources between goods x and y. Point A, for example, represents an allocation in which the indicated number of labor hours are devoted to x production together with a speciﬁed number of hours of capital. Production of good y uses whatever labor and capital are ‘‘left over.’’ Point A in Figure 13.1, for example, also shows the exact amount of labor and capital used in the production of good y. Any other point in the box has a similar interpretation. Thus, the Edgeworth box shows every possible way the existing capital and labor might be used to produce x and y. FIGURE 13.1 Construction of an Edgeworth Box Diagram for Production The dimensions of this diagram are given by the total quantities of labor and capital available. Quantities of these resources devoted to x production are measured from origin Ox; quantities devoted to y are measured from Oy. Any point in the box represents a fully employed allocation of the available resources to the two goods. Labor for x Labor in y production Labor for y O y Capital in y production Capital in x production O x Labor in x production A Total labor 460 Part 5: Competitive Markets Efﬁcient allocations Many of the allocations shown in Figure 13.1 are technically inefﬁcient in that it is possible to produce both more x and more y by shifting capital and labor around a bit. In our model we assume that competitive markets will not exhibit such inefﬁcient input choices (for reasons we will explore in more detail later in the chapter). Hence we wish to discover the efﬁcient allocations in Figure 13.1 because these illustrate the production outcomes in this model. To do so, we introduce isoquant maps for good x (using Ox as the origin) and good y (using Oy as the origin), as shown in Figure 13.2. In this ﬁgure it is clear that the arbitrarily chosen allocation A is inefﬁcient. By reallocating capital and labor, one can produce both more x than x2 and more y than y2. The efﬁcient allocations in Figure 13.2 are those such as P1, P2, P3, and P4, where the isoquants are tangent to one another. At any other points in the box diagram, the two goods’ isoquants will intersect,
and we can show inefﬁciency as we did for point A. At the points of tangency, however, this kind of unambiguous improvement cannot be made. In going from P2 to P3, for example, more x is being produced, but at the cost of less y being produced; therefore, P3 is not ‘‘more efﬁcient’’ than P2—both of the points are efﬁcient. Tangency of the isoquants for good x and good y implies that their slopes are equal. That is, the RTS of capital for labor is equal in x and y production. Later we will show how competitive input markets will lead ﬁrms to make such efﬁcient input choices. Therefore, the curve joining Ox and Oy that includes all these points of tangency shows all the efﬁcient allocations of capital and labor. Points off this curve are inefﬁcient in that unambiguous increases in output can be obtained by reshufﬂing inputs between the two goods. Points on the curve OxOy are all efﬁcient allocations, however, because more x can be produced only by cutting back on y production and vice versa. FIGURE 13.2 Edgeworth Box Diagram of Efﬁciency in Production This diagram adds production isoquants for x and y to Figure 13.1. It then shows technically efﬁcient ways to allocate the ﬁxed amounts of k and l between the production of the two outputs. The line joining Ox and Oy is the locus of these efﬁcient points. Along this line, the RTS (of l for k) in the production of good x is equal to the RTS in the production of y. O y y1 P3 y2 P2 x1 Total l Total k y4 y3 P1 O x P4 x3 A x4 x2 Chapter 13: General Equilibrium and Welfare 461 FIGURE 13.3 Production Possibility Frontier The production possibility frontier shows the alternative combinations of x and y that can be efﬁciently produced by a ﬁrm with ﬁxed resources. The curve can be derived from Figure 13.2 by varying inputs between the production of x and y while maintaining the conditions for efﬁciency.
The negative of the slope of the production possibility curve is called the rate of product transformation (RPT). Quantity of y O x y4 y3 y2 y1 P1 P2 A P3 P4 x1 x2 x3 x4 O y Quantity of x Production possibility frontier The efﬁciency locus in Figure 13.2 shows the maximum output of y that can be produced for any preassigned output of x. We can use this information to construct a production possibility frontier, which shows the alternative outputs of x and y that can be produced with the ﬁxed capital and labor inputs. In Figure 13.3 the OxOy locus has been taken from Figure 13.2 and transferred onto a graph with x and y outputs on the axes. At Ox, for example, no resources are devoted to x production; consequently, y output is as large as is possible with the existing resources. Similarly, at Oy, the output of x is as large as possible. The other points on the production possibility frontier (say, P1, P2, P3, and P4) are derived from the efﬁciency locus in an identical way. Hence we have derived the following deﬁnition Production possibility frontier. The production possibility frontier shows the alternative combinations of two outputs that can be produced with ﬁxed quantities of inputs if those inputs are employed efﬁciently. Rate of product transformation The slope of the production possibility frontier shows how x output can be substituted for y output when total resources are held constant. For example, for points near Ox on 462 Part 5: Competitive Markets 1/4; this the production possibility frontier, the slope is a small negative number—say, implies that, by reducing y output by 1 unit, x output could be increased by 4. Near Oy, on the other hand, the slope is a large negative number (say, 5), implying that y output must be reduced by 5 units to permit the production of one more x. The slope of the production possibility frontier clearly shows the possibilities that exist for trading y for x in production. The negative of this slope is called the rate of product transformation (RPT).!! Rate of product transformation. The rate of product transformation (RPT) between two outputs is the negative of the slope of the production possibility frontier for those outputs. Mathematically, RPT of x for y ð Þ ¼
!½ slope of production possibility frontier dy dx ð along OxOyÞ, ¼! & (13:1) The RPT records how x can be technically traded for y while continuing to keep the available productive inputs efﬁciently employed. Shape of the production possibility frontier The production possibility frontier illustrated in Figure 13.3 exhibits an increasing RPT. For output levels near Ox, relatively little y must be sacriﬁced to obtain one more x (–dy/dx is small). Near Oy, on the other hand, additional x may be obtained only by substantial reductions in y output (–dy/dx is large). In this section we will show why this concave shape might be expected to characterize most production situations. A ﬁrst step in that analysis is to recognize that RPT is equal to the ratio of the marginal cost of x (MCx) to the marginal cost of y (MCy). Intuitively, this result is obvious. Suppose, for example, that x and y are produced only with labor. If it takes two labor hours to produce one more x, we might say that MCx is equal to 2. Similarly, if it takes only one labor hour to produce an extra y, then MCy is equal to 1. But in this situation it is clear that the RPT is 2: two y must be forgone to provide enough labor so that x may be increased by one unit. Hence the RPT is equal to the ratio of the marginal costs of the two goods. More formally, suppose that the costs (say, in terms of the ‘‘disutility’’ experienced by factor suppliers) of any output combination are denoted by C(x, y). Along the production possibility frontier, C(x, y) will be constant because the inputs are in ﬁxed supply. If we call this constant level of costs C, we can write C 0. It is this implicit function that underlies the production possibility frontier. Applying the results from Chapter 2 for such a function yields: x, y Þ! ¼ C ð RPT dy dx jC ¼ x Cx Cy ¼! MCx MCy : (13:2) To demonstrate reasons why the RPT might be expected to increase for clockwise movements along the production possibility frontier, we can proceed by showing why the ratio of MCx to MCy should increase as x output expands and
y output contracts. We ﬁrst present two relatively simple arguments that apply only to special cases; then we turn to a more sophisticated general argument. Diminishing returns The most common rationale offered for the concave shape of the production possibility frontier is the assumption that both goods are produced under conditions of diminishing returns. Hence increasing the output of good x will raise its marginal cost, whereas Chapter 13: General Equilibrium and Welfare 463 decreasing the output of y will reduce its marginal cost. Equation 13.2 then shows that the RPT will increase for movements along the production possibility frontier from Ox to Oy. A problem with this explanation, of course, is that it applies only to cases in which both goods exhibit diminishing returns to scale, and that assumption is at variance with the theoretical reasons for preferring the assumption of constant or even increasing returns to scale as mentioned elsewhere in this book. Specialized inputs If some inputs were ‘‘more suited’’ for x production than for y production (and vice versa), the concave shape of the production frontier also could be explained. In that case, increases in x output would require drawing progressively less suitable inputs into the production of that good. Therefore, marginal costs of x would increase. Marginal costs for y, on the other hand, would decrease because smaller output levels for y would permit the use of only those inputs most suited for y production. Such an argument might apply, for example, to a farmer with a variety of types of land under cultivation in different crops. In trying to increase the production of any one crop, the farmer would be forced to grow it on increasingly unsuitable parcels of land. Although this type of specialized input assumption has considerable importance in explaining a variety of real-world phenomena, it is nonetheless at variance with our general assumption of homogeneous factors of production. Hence it cannot serve as a fundamental explanation for concavity. Differing factor intensities Even if inputs are homogeneous and production functions exhibit constant returns to scale, the production possibility frontier will be concave if goods x and y use inputs in different proportions.4 In the production box diagram of Figure 13.2, for example, good x is capital intensive relative to good y. That is, at every point along the OxOy contract curve, the ratio of k to l in x production exceeds the ratio of k to l in y production: The bowed curve OxOy is always above the main diagonal of the Edgeworth box. If, on the other hand,
good y had been relatively capital intensive, the OxOy contract curve would have been bowed downward below the diagonal. Although a formal proof that unequal factor intensities result in a concave production possibility frontier will not be presented here, it is possible to suggest intuitively why that occurs. Consider any two points on the frontier OxOy in Figure 13.3—say, P1 (with coordinates x1, y4) and P3 (with coordinates x3, y2). One way of producing an output combination ‘‘between’’ P1 and P3 would be to produce the combination x1 þ 2 x3, y4 þ 2 y2 : Because of the constant returns-to-scale assumption, that combination would be feasible and would fully use both factors of production. The combination would lie at the midpoint of a straight-line chord joining points P1 and P3. Although such a point is feasible, it is not efﬁcient, as can be seen by examining points P1 and P3 in the box diagram of Figure 13.2. Because of the bowed nature of the contract curve, production at a point midway between P1 and P3 would be off the contract curve: Producing at a point such as P2 would provide more of both goods. Therefore, the production possibility frontier in Figure 13.3 must ‘‘bulge out’’ beyond the straight line P1P3. Because such a proof could be constructed for any two points on OxOy, we have shown that the frontier is concave; that is, the RPT increases as the output of good X increases. When production is reallocated in a northeast 4If, in addition to homogeneous factors and constant returns to scale, each good also used k and l in the same proportions under optimal allocations, then the production possibility frontier would be a straight line. 464 Part 5: Competitive Markets direction along the OxOy contract curve (in Figure 13.3), the capital–labor ratio decreases in the production of both x and y. Because good x is capital intensive, this change increases MCx. On the other hand, because good y is labor intensive, MCy decreases. Hence the relative marginal cost of x (as represented by the RPT) increases. Opportunity cost and supply The production possibility curve demonstrates that there are many possible efﬁcient combinations of the two goods and that producing more of one good necessitates cutting
back on the production of some other good. This is precisely what economists mean by the term opportunity cost. The cost of producing more x can be most readily measured by the reduction in y output that this entails. Therefore, the cost of one more unit of x is best measured as the RPT (of x for y) at the prevailing point on the production possibility frontier. The fact that this cost increases as more x is produced represents the formulation of supply in a general equilibrium context. EXAMPLE 13.1 Concavity of the Production Possibility Frontier In this example we look at two characteristics of production functions that may cause the production possibility frontier to be concave. Diminishing returns. Suppose that the production of both x and y depends only on labor input and that the production functions for these goods are lxÞ ¼ ð lyÞ ¼ f ð Hence production of each of these goods exhibits diminishing returns to scale. If total labor supply is limited by x ¼ y ¼ (13:3) f l 0:5 x, l 0:5 y : then simple substitution shows that the production possibility frontier is given by lx þ ly ¼ 100, x2 y2 þ ¼ 100 for x, y 0: ( (13:4) (13:5) In this case, the frontier is a quarter-circle and is concave. The RPT can now be computed directly from the equation for the production possibility frontier (written in implicit form as f x, y 100 0): x2 y2 ð Þ ¼ þ! ¼ RPT dy dx ¼!ð! fx fyÞ ¼ 2x 2y ¼ x y, ¼! (13:6) and this slope increases as x output increases. A numerical illustration of concavity starts by noting that the points (10, 0) and (0, 10) both lie on the frontier. A straight line joining these two points would also include the point (5, 5), but that point lies below the frontier. If equal 50p, which yields amounts of labor are devoted to both goods, then production is x more of both goods than the midpoint. ¼ ¼ y ﬃﬃﬃﬃﬃ Factor intensity. To show how differing factor intensities yield a concave production possibility frontier, suppose that the two goods are produced under constant returns to scale but with different Cobb–Dou
glas production functions: x y f g ¼ ¼ k, l ð k, l ð Þ ¼ Þ ¼ k0:5 x l 0:5 x, k0:25 l 0:75 y y : (13:7) Chapter 13: General Equilibrium and Welfare 465 Suppose also that total capital and labor are constrained by It is easy to show that kx þ ky ¼ 100, lx þ ly ¼ 100: RTSx ¼ kx lx ¼ jx, RTSy ¼ 3ky ly ¼ 3jy, (13:8) (13:9) RTSy or ki/li. Being located on the production possibility frontier requires RTSx ¼ where ki ¼ 3ky. That is, no matter how total resources are allocated to production, being on the kx ¼ production possibility frontier requires that x be the capital-intensive good (because, in some sense, capital is more productive in x production than in y production). The capital–labor ratios in the production of the two goods are also constrained by the available resources: ky ly ¼ kx lx þ kx þ lx þ ly)—that is, a is the share of total labor devoted to x production. Using the 3ky, we can ﬁnd the input ratios of the two goods in terms of the overall 100 100 ¼ ajx þ ð ky lx þ jy ¼ Þ (13:10) ly ¼ ly þ! 1, a 1 lx/(lx þ where a ¼ condition that kx ¼ allocation of labor: jy ¼ 1, 2a jx ¼ 1 : 2a (13:11) 1 þ 3 þ Now we are in a position to phrase the production possibility frontier in terms of the share of labor devoted to x production: j0:5 x lx ¼ ¼ j0:5 x a 100 ð Þ ¼ 100a " 3 1 2a # þ 0:5, j0:25 y ly ¼ ¼ j0:25 y a 1 ð! Þð 100 Þ ¼ 1 100 ð! a Þ x y (13:12) 0:25 : 1 1 " þ 2a # We could push this algebra even further to eliminate a from these two equations to get an explicit functional form for the
and the set of indifference curves represents individuals’ preferences for these goods. First, consider the price ratio px/py. At this price ratio, ﬁrms will choose to produce the output combination x1, y1. Proﬁtmaximizing ﬁrms will choose the more proﬁtable point on PP. At x1, y1 the ratio of the two goods’ prices (px/py) is equal to the ratio of the goods’ marginal costs (the RPT); thus, proﬁts are maximized there. On the other hand, given this budget constraint (line C),5 individuals will demand x01, y01. Consequently, with these prices, there is an excess demand for good x (individuals demand more than is being produced) but an excess supply of good y. The workings of the marketplace will cause px to increase and py to decrease. The price ratio px/py will increase; the price line will take on a steeper slope. Firms will respond to these price changes by moving clockwise along the production possibility frontier; that is, they will increase their production of good x and decrease their production of good y. Similarly, individuals will respond to the changing prices by substituting y for x in their consumption choices. These actions of both ﬁrms and individuals serve to eliminate the excess demand for x and the excess supply of y as market prices change. 5It is important to recognize why the budget constraint has this location. Because px and py are given, the value of total producpy Æ y1. This is the value of ‘‘GDP’’ in the simple economy pictured in Figure 13.4. It is also, therefore, the total tion is px Æ x1 þ income accruing to people in society. Society’s budget constraint therefore passes through x1, y1 and has a slope of –px/py. This is precisely the budget constraint labeled C in the ﬁgure. Chapter 13: General Equilibrium and Welfare 467 Equilibrium is reached at x), y) with a price ratio of p)x=p)y. With this price ratio,6 supply and demand are equilibrated for both good x and good y. Given px and py, ﬁrms will produce x) and y) in maximizing their proﬁts. Similarly
, with a budget constraint given by C ), individuals will demand x) and y). The operation of the price system has cleared the markets for both x and y simultaneously. Therefore, this ﬁgure provides a ‘‘general equilibrium’’ view of the supply–demand process for two markets working together. For this reason we will make considerable use of this ﬁgure in our subsequent analysis. Comparative Statics Analysis As in our partial equilibrium analysis, the equilibrium price ratio p)x=p)y illustrated in Figure 13.4 will tend to persist until either preferences or production technologies change. This competitively determined price ratio reﬂects these two basic economic forces. If preferences were to shift, say, toward good x, then px/py would increase and a new equilibrium would be established by a clockwise move along the production possibility curve. More x and less y would be produced to meet these changed preferences. Similarly, technical progress in the production of good x would shift the production possibility curve outward, as illustrated in Figure 13.5. This would tend to decrease the relative price of x and increase the quantity of x consumed (assuming x is a normal good). In the ﬁgure the quantity of y FIGURE 13.5 Effects of Technical Progress in x Production Technical advances that lower marginal costs of x production will shift the production possibility frontier. This will generally create income and substitution effects that cause the quantity of x produced to increase (assuming x is a normal good). Effects on the production of y are ambiguous because income and substitution effects work in opposite directions. Quantity of y y1 y0 E1 E0 U1 U0 x0 x1 Quantity of x 6Notice again that competitive markets determine only equilibrium relative prices. Determination of the absolute price level requires the introduction of money into this barter model. 468 Part 5: Competitive Markets consumed also increases as a result of the income effect arising from the technical advance; however, a slightly different drawing of the ﬁgure could have reversed that result if the substitution effect had been dominant. Example 13.2 looks at a few such effects. EXAMPLE 13.2 Comparative Statics in a General Equilibrium Model To explore how general equilibrium models work, let’s start with a simple example based on the production possibility frontier in Example 13.1. In that example we assumed that production of both goods was characterized by decreasing returns x and also that total labor available was given
by lx þ 100. The resulting production possibility frontier was given by ly ¼ x2 x/y. To complete this model we assume that the typical individual’s 100, and RPT ¼ utility function is given by U(x, y) x0.5y0.5, so the demand functions for the two goods are and y l0:5 x l0:5 y y2 ¼ ¼ þ ¼ ¼ x y x px, py, I ð ¼ Þ ¼ y px, py, I ð ¼ Þ ¼ 0:5I px 0:5I py, : (13:14) Base-case equilibrium. Proﬁt maximization by ﬁrms requires that px/py ¼ x/y, and utility-maximizing demand requires that px/py ¼ ¼ x/y ¼ shows that RPT y/x. Thus, equilibrium requires that y. Inserting this result into the equation for the production possibility frontier MCx/MCy ¼ y/x, or x ¼ x) y) ¼ ¼ 50p ﬃﬃﬃﬃﬃ ¼ 7:07 and px py ¼ 1: (13:15) This is the equilibrium for our base case with this model. that The budget constraint. The budget constraint faces individuals is not especially transparent in this illustration; therefore, it may be useful to discuss it explicitly. To bring some degree of absolute pricing into the model, let’s consider all prices in terms of the wage rate, w. Because total labor supply is 100, it follows that total labor income is 100w. However, because of the diminishing returns assumed for production, each ﬁrm also earns proﬁts. For ﬁrm x, say, 50p. Therefore, the the total cost function is C(w, x) proﬁts for ﬁrm x are px ¼ 50w. A similar computation shows that proﬁts for ﬁrm y are also given by 50w. Because general equilibrium models must obey the national income identity, we assume that consumers are also shareholders in the two ﬁrms and treat these proﬁts also as part of their spendable incomes. Hence total consumer income is wx2
, so px ¼ MCx ¼ wx2 (px – wx)x ¼ ¼ wlx ¼ ¼ ¼ (px – ACx)x 2wx 2w ﬃﬃﬃﬃﬃ ¼ total income ¼ ¼ labor income + profits 50w 100w ð Þ ¼ þ 2 200w: This income will just permit consumers to spend 100w on each good by buying price of 2w 50p, so the model is internally consistent. (13:16) 50p units at a ﬃﬃﬃﬃﬃ A shift in supply. There are only two ways in which this base-case equilibrium can be disturbed: (1) by changes in ‘‘supply’’—that is, by changes in the underlying technology of this economy; or (2) by changes in ‘‘demand’’—that is, by changes in preferences. Let’s ﬁrst consider changes in technology. Suppose that there is technical improvement in x production so that the production function is x 100, and RPT. Now the production possibility frontier is given by x2/4 x/4y. Proceeding as before to ﬁnd the equilibrium in this model: 2l 0:5 x y2 þ ¼ ¼ ﬃﬃﬃﬃﬃ ¼ Chapter 13: General Equilibrium and Welfare 469 px py ¼ px py ¼ x 4y ð y x ð supply, Þ demand, Þ so x2 ¼ 4y2 and the equilibrium is 2 50p, x) ¼ y) ¼ 50p and ﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃ px py ¼ 1 2 : (13:17) (13:18) Technical improvements in x production have caused its relative price to decrease and the consumption of this good to increase. As in many examples with Cobb–Douglas utility, the income and substitution effects of this price decrease on y demand are precisely offsetting. Technical improvements clearly make consumers better off, however. Whereas utility was previously given 0:5 50p by U x, y ð Þ 0:5 50p Þ
ð ﬃﬃﬃﬃﬃ 50p 2 ¼ ð 10. Technical change has increased consumer welfare substantially. 7:07, now it has increased to U A shift in demand. If consumer preferences were to switch to favor good y as U(x, y) x0.1y0.9, then demand functions would be given by x equilibrium would require px/py ¼ to arrive at an overall equilibrium, we have ¼ 0.9I/py, and demand 0.1I/px and y y/9x. Returning to the original production possibility frontier x0:5y0:5 50p Þ ¼ 2p ¼ x, y ð x0:5y0:5 ¼ ¼ Þ ¼ ﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃ ¼ ¼ ¼ ﬃﬃﬃ * px py ¼ px py ¼ x y ð y 9x ð supply, Þ demand, Þ so 9x2 ¼ y2 and the equilibrium is given by 10p, x) ¼ y) ¼ 3 10p and ﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃ px py ¼ 1 3 (13:19) (13:20) Hence the decrease in demand for x has signiﬁcantly reduced its relative price. Observe that in this case, however, we cannot make a welfare comparison to the previous cases because the utility function has changed. QUERY: What are the budget constraints in these two alternative scenarios? How is income distributed between wages and proﬁts in each case? Explain the differences intuitively. General Equilibrium Modeling and Factor Prices This simple general equilibrium model reinforces Marshall’s observations about the importance of both supply and demand forces in the price determination process. By providing an explicit connection between the markets for all goods, the general equilibrium model makes it possible to examine more complex questions about market relationships than is possible by looking at only one market at a time. General equilibrium modeling also permits an examination of the connections between goods and factor markets; we can illustrate that
with an important historical case. 470 Part 5: Competitive Markets The Corn Laws debate High tariffs on grain imports were imposed by the British government following the Napoleonic wars. Debate over the effects of these Corn Laws dominated the analytical efforts of economists between the years 1829 and 1845. A principal focus of the debate concerned the effect that elimination of the tariffs would have on factor prices—a question that continues to have relevance today, as we will see. The production possibility frontier in Figure 13.6 shows those combinations of grain (x) and manufactured goods (y) that could be produced by British factors of production. Assuming (somewhat contrary to actuality) that the Corn Laws completely prevented trade, market equilibrium would be at E with the domestic price ratio given by p)x=p)y. Removal of the tariffs would reduce this price ratio to p0x=p0y. Given that new ratio, Britain would produce combination A and consume combination B. Grain imports would amount to xB – xA, and these would be ﬁnanced by export of manufactured goods equal to yA – yB. Overall utility for the typical British consumer would be increased by the opening of trade. Therefore, use of the production possibility diagram demonstrates the implications that relaxing the tariffs would have for the production of both goods. Trade and factor prices By referring to the Edgeworth production box diagram (Figure 13.2) that lies behind the production possibility frontier (Figure 13.3), it is also possible to analyze the effect of FIGURE 13.6 Analysis of the Corn Laws Debate Reduction of tariff barriers on grain would cause production to be reallocated from point E to point A; consumption would be reallocated from E to B. If grain production is relatively capital intensive, the relative price of capital would decrease as a result of these reallocations. Output of manufactured goods (y) P yA yE yB * * Slope = −px /py A Slope = −p′x /p′y E B U2 U1 xA xE xBP Output of grain (x) Chapter 13: General Equilibrium and Welfare 471 tariff reductions on factor prices. The movement from point E to point A in Figure 13.6 is similar to a movement from P3 to P1 in Figure 13.2, where production of x is decreased and production of y is increased. This ﬁgure also records the reallocation of capital and labor made necessary by such a
move. If we assume that grain production is relatively capital intensive, then the movement from P3 to P1 causes the ratio of k to l to increase in both industries.7 This in turn will cause the relative price of capital to decrease (and the relative price of labor to increase). Hence we conclude that repeal of the Corn Laws would be harmful to capital owners (i.e., landlords) and helpful to laborers. It is not surprising that landed interests fought repeal of the laws. Political support for trade policies The possibility that trade policies may affect the relative incomes of various factors of production continues to exert a major inﬂuence on political debates about such policies. In the United States, for example, exports tend to be intensive in their use of skilled labor, whereas imports tend to be intensive in unskilled labor input. By analogy to our discussion of the Corn Laws, it might thus be expected that further movements toward free trade policies would result in increasing relative wages for skilled workers and in decreasing relative wages for unskilled workers. Therefore, it is not surprising that unions representing skilled workers (the machinists or aircraft workers) tend to favor free trade, whereas unions of unskilled workers (those in textiles, shoes, and related businesses) tend to oppose it.8 A Mathematical Model of Exchange Although the previous graphical model of general equilibrium with two goods is fairly instructive, it cannot reﬂect all the features of general equilibrium modeling with an arbitrary number of goods and productive inputs. In the remainder of this chapter we will illustrate how such a more general model can be constructed, and we will look at some of the insights that such a model can provide. For most of our presentation we will look only at a model of exchange—quantities of various goods already exist and are merely traded among individuals. In such a model there is no production. Later in the chapter we will look brieﬂy at how production can be incorporated into the general model we have constructed. Vector notation Most general equilibrium modeling is conducted using vector notation. This provides great ﬂexibility in specifying an arbitrary number of goods or individuals in the models. Consequently, this seems to be a good place to offer a brief introduction to such notation. A vector is simply an ordered array of variables (which each may take on speciﬁc values). Here we will usually adopt the convention that the vectors we use are column vectors. Hence we will write an n 1 column vector as: + 7In the Corn
Laws debate, attention centered on the factors of land and labor. 8The ﬁnding that the opening of trade will raise the relative price of the abundant factor is called the Stolper–Samuelson theorem after the economists who rigorously proved it in the 1950s. 472 Part 5: Competitive Markets 2 x ¼ x1 x2 : : : xn 13:21) 1 column + where each xi is a variable that can take on any value. If x and y are two n vectors, then the (vector) sum of them is deﬁned as x1 x2 : : : xn y1 y2 : : : yn x1 þ x2 þ : : : xn þ y1 y2 3 7 7 7 7 7 7 5 yn 6 6 6 6 6 6 4 : (13:22) Notice that this sum only is deﬁned if the two vectors are of equal length. In fact, checking the length of vectors is one good way of deciding whether one has written a meaningful vector equation. The (dot) product of two vectors is deﬁned as the sum of the component-by-component product of the elements in the two vectors. That is: xy x1y1 þ x2y2 þ * * * þ ¼ xnyn: (13:23) Notice again that this operation is only deﬁned if the vectors are of the same length. With these few concepts we are now ready to illustrate the general equilibrium model of exchange. Utility, initial endowments, and budget constraints In our model of exchange there are assumed to be n goods and m individuals. Each individual gains utility from the vector of goods he or she consumes ui(xi) where i 1... m. Individuals also possess initial endowments of the goods given by xi. Individuals are free to exchange their initial endowments with other individuals or to keep some or all the endowment for themselves. In their trading individuals are assumed to be price-takers—that is, they face a price vector (p) that speciﬁes the market price for each of the n goods. Each individual seeks to maximize utility and is bound by a budget constraint that requires that the total amount spent on consumption equals the total value of his or her endowment: ¼ pxi ¼ pxi: (13:24) Although this budget constraint has
a simple form, it may be worth contemplating it for a minute. The right side of Equation 13.24 is the market value of this individual’s endowment (sometimes referred to as his or her full income). He or she could ‘‘afford’’ to consume this endowment (and only this endowment) if he or she wished to be self-sufﬁcient. But the endowment can also be spent on some other consumption bundle (which, presumably, provides more utility). Because consuming items in one’s own endowment has an opportunity cost, the terms on the left of Equation 13.24 consider the costs of all items that enter into the ﬁnal consumption bundle, including endowment goods that are retained. Demand functions and homogeneity The utility maximization problem outlined in the previous section is identical to the one we studied in detail in Part 2 of this book. As we showed in Chapter 4, one outcome of Chapter 13: General Equilibrium and Welfare 473 this process is a set of n individual demand functions (one for each good) in which quantities demanded depend on all prices and income. Here we can denote these in vector form as xi. These demand functions are continuous, and, as we showed in ChapÞ ter 4, they are homogeneous of degree 0 in all prices and income. This latter property can be indicated in vector notation by p, pxi ð xi tp, tpxi ð Þ ¼ xi p, pxi ð Þ (13:25) for any t > 0. This property will be useful because it will permit us to adopt a convenient normalization scheme for prices, which, because it does not alter relative prices, leaves quantities demanded unchanged. Equilibrium and Walras’ law Equilibrium in this simple model of exchange requires that the total quantities of each good demanded be equal to the total endowment of each good available (remember, there is no production in this model). Because the model used is similar to the one originally developed by Leon Walras,9 this equilibrium concept is customarily attributed to him Walrasian equilibrium. Walrasian equilibrium is an allocation of resources and an associated price vector, p), such that m 1 i X ¼ xi p), p)xi ð Þ ¼ m xi, 1 i X ¼ (13:26) where the summation is taken over the m individuals in
this exchange economy. The n equations in Equation 13.26 state that in equilibrium demand equals supply in each market. This is the multimarket analog of the single market equilibria examined in the previous chapter. Because there are n prices to be determined, a simple counting of equations and unknowns might suggest that the existence of such a set of prices is guaranteed by the simultaneous equation solution procedures studied in elementary algebra. Such a supposition would be incorrect for two reasons. First, the algebraic theorem about simultaneous equation systems applies only to linear equations. Nothing suggests that the demand equations in this problem will be linear—in fact, most examples of demand equations we encountered in Part 2 were deﬁnitely nonlinear. A second problem with Equation 13.26 is that the equations are not independent of one another—they are related by what is known as Walras’ law. Because each individual in this exchange economy is bound by a budget constraint of the form given in Equation 13.24, we can sum over all individuals to obtain m m pxi ¼ 1 i X ¼ 1 i X ¼ pxi or m 1 i X ¼ xi p ð! xi Þ ¼ 0: (13:27) In words, Walras’ law states that the value of all quantities demanded must equal the value of all endowments. This result holds for any set of prices, not just for equilibrium 9The concept is named for the nineteenth century French/Swiss economist Leon Walras, who pioneered the development of general equilibrium models. Models of the type discussed in this chapter are often referred to as models of Walrasian equilibrium, primarily because of the price-taking assumptions inherent in them. 474 Part 5: Competitive Markets prices.10 The general lesson is that the logic of individual budget constraints necessarily creates a relationship among the prices in any economy. It is this connection that helps to ensure that a demand–supply equilibrium exists, as we now show. Existence of equilibrium in the exchange model The question of whether all markets can reach equilibrium together has fascinated economists for nearly 200 years. Although intuitive evidence from the real world suggests that this must indeed be possible (market prices do not tend to ﬂuctuate wildly from one day to the next), proving the result mathematically proved to be rather difﬁcult. Walras himself thought he had a good proof that relied on evidence from the market to adjust prices toward equilibrium. The price would
increase for any good for which demand exceeded supply and decrease when supply exceeded demand. Walras believed that if this process continued long enough, a full set of equilibrium prices would eventually be found. Unfortunately, the pure mathematics of Walras’ solution were difﬁcult to state, and ultimately there was no guarantee that a solution would be found. But Walras’ idea of adjusting prices toward equilibrium using market forces provided a starting point for the modern proofs, which were largely developed during the 1950s. A key aspect of the modern proofs of the existence of equilibrium prices is the choice of a good normalization rule. Homogeneity of demand functions makes it possible to use any absolute scale for prices, providing that relative prices are unaffected by this choice. Such an especially convenient scale is to normalize prices so that they sum to one. Consider an arbitrary set of n non-negative prices p1, p2... pn. We can normalize11 these to form a new set of prices p0i ¼ pi n : pk (13:28) 1 k ¼ P These new prices will have the properties that maintained: n 1 k ¼ P pk pk ¼ 1 and that relative price ratios are (13:29) p0k ¼ pi pj : p0i p0j ¼ pi= pj P $P Because this sort of mathematical process can always be done, we will assume, without loss of generality, that the price vectors we use (p) have all been normalized in this way. Therefore, proving the existence of equilibrium prices in our model of exchange amounts to showing that there will always exist a price vector p) that achieves equilibrium in all markets. That is, m 1 i X ¼ xi p), p)xi ð Þ ¼ xi or xi p), p)xi ð Þ! m 1 i X ¼ xi ¼ 0 or p) z ð Þ ¼ 0, (13:30) where we use z(p) as a shorthand way of recording the ‘‘excess demands’’ for goods at a particular set of prices. In equilibrium, excess demand is zero in all markets.12 10Walras’ law holds trivially for equilibrium prices as multiplication of Equation 13.26 by p shows. 11This is possible only if at least one of the prices is nonzero. Throughout our discussion we will assume that
not all equilibrium prices can be zero. 12Goods that are in excess supply at equilibrium will have a zero price. We will not be concerned with such ‘‘free goods’’ here. FIGURE 13.7 A Graphical Illustration of Brouwer’s Fixed Point Theorem Chapter 13: General Equilibrium and Welfare 475 Because any continuous function must cross the 45! line somewhere in the unit square, this function must have a point for which f (x)) x). This point is called a ﬁxed point. ¼ f (x) 1 f (x) Fixed point f (x) 45° 0 x 1 x Now consider the following way of implementing Walras’ idea that goods in excess demand should have their prices increased, whereas those in excess supply should have their prices reduced.13 Starting from any arbitrary set of prices, p0, we deﬁne a new set, p1, as f ð k z ð, p0Þ p1 ¼ p0Þ ¼ p0 þ where k is a positive constant. This function will be continuous (because demand functions are continuous), and it will map one set of normalized prices into another (because of our assumption that all prices are normalized). Hence it will meet the conditions of the Brouwer’s ﬁxed point theorem, which states that any continuous function from a closed compact set onto itself (in the present case, from the ‘‘unit simplex’’ onto itself) f (x). The theorem is illustrated for a single dimenwill have a ‘‘ﬁxed point’’ such that x sion in Figure 13.7. There, no matter what shape the function f(x) takes, as long as it is continuous, it must somewhere cross the 45! line and at that point x (13:31) f(x). ¼ If we let p) represent the ﬁxed point identiﬁed by Brouwer’s theorem for Equation ¼ 13.31, we have: p) f p) ð ¼ Þ ¼ p) p) k z ð Þ : þ (13:32) ¼ Hence at this point z(p)) 0; thus, p) is an equilibrium price vector. The proof that Walras sought is easily accomplished using an important
mathematical result developed a few years after his death. The elegance of the proof may obscure the fact that it uses a number of assumptions about economic behavior such as: (1) price-taking by all parties; (2) homogeneity of demand functions; (3) continuity of demand functions; and (4) presence of budget constraints and Walras’ law. All these play important roles in showing that a system of simple markets can indeed achieve a multimarket equilibrium. 13What follows is an extremely simpliﬁed version of the proof of the existence of equilibrium prices. In particular, problems of free goods and appropriate normalizations have been largely assumed away. For a mathematically correct proof, see, for example, G. Debreu, Theory of Value (New York: John Wiley & Sons, 1959). 476 Part 5: Competitive Markets First theorem of welfare economics Given that the forces of supply and demand can establish equilibrium prices in the general equilibrium model of exchange we have developed, it is natural to ask what are the welfare consequences of this ﬁnding. Adam Smith14 hypothesized that market forces provide an ‘‘invisible hand’’ that leads each market participant to ‘‘promote an end [social welfare] which was no part of his intention.’’ Modern welfare economics seeks to understand the extent to which Smith was correct. Perhaps the most important welfare result that can be derived from the exchange model is that the resulting Walrasian equilibrium is ‘‘efﬁcient’’ in the sense that it is not possible to devise some alternative allocation of resources in which at least some people are better off and no one is worse off. This deﬁnition of efﬁciency was originally developed by Italian economist Vilfredo Pareto in the early 1900s. Understanding the deﬁnition is easiest if we consider what an ‘‘inefﬁcient’’ allocation might be. The total quantities of goods included in initial endowments would be allocated inefﬁciently if it were possible, by shifting goods around among individuals, to make at least one person better off (i.e., receive a higher utility) and no one worse off. Clearly, if individuals’ preferences are to count, such a situation would be undesirable. Hence we have a formal deﬁnition Pareto ef�
�cient allocation. An allocation of the available goods in an exchange economy is efﬁcient if it is not possible to devise an alternative allocation in which at least one person is better off and no one is worse off. A proof that all Walrasian equilibria are Pareto efﬁcient proceeds indirectly. Suppose that p) generates a Walrasian equilibrium in which the quantity of goods consumed by each person is denoted by )xk. Now assume that there is some alternative Þ ¼ allocation of the available goods 0xk such that, for at least one person, say, 1... m person i, it is that case that 0xi is preferred to )xi. For this person, it must be the case that )0 xi > p) )xi (13:33) because otherwise this person would have bought the preferred bundle in the ﬁrst place. If all other individuals are to be equally well off under this new proposed allocation, it must be the case for them that p)0 xk p) )xk k 1... m, k i: (13:34) ¼ If the new bundle were less expensive, such individuals could not have been minimizing expenditures at p). Finally, to be feasible, the new allocation must obey the quantity constraints 6¼ ¼ m m 1 i X ¼ Multiplying Equation 13.35 by p)yields 0xi ¼ xi: 1 i X ¼ m p)0 xi m 1 i X ¼ p) xi, ¼ 1 i X ¼ 14Adam Smith, The Wealth of Nations (New York: Modern Library, 1937) p. 423. (13:35) (13:36) Chapter 13: General Equilibrium and Welfare 477 but Equations 13.33 and 13.34 together with Walras’ law applied to the original equilibrium imply that m 1 i X ¼ m m p)0 xi > p) )xi 1 i X ¼ p) xi: ¼ 1 i X ¼ (13:37) Hence we have a contradiction and must conclude that no such alternative allocation can exist. Therefore, we can summarize our analysis with the following deﬁnition First theorem of welfare economics. Every Walrasian equilibrium is Pareto efﬁcient. The signiﬁcance of this ‘‘
theorem’’ should not be overstated. The theorem does not say that every Walrasian equilibrium is in some sense socially desirable. Walrasian equilibria can, for example, exhibit vast inequalities among individuals arising in part from inequalities in their initial endowments (see the discussion in the next section). The theorem also assumes price-taking behavior and full information about prices—assumptions that need not hold in other models. Finally, the theorem does not consider possible effects of one individual’s consumption on another. In the presence of such externalities even a perfect competitive price system may not yield Pareto optimal results (see Chapter 19). Still, the theorem does show that Smith’s ‘‘invisible hand’’ conjecture has some validity. The simple markets in this exchange world can ﬁnd equilibrium prices, and at those equilibrium prices the resulting allocation of resources will be efﬁcient in the Pareto sense. Developing this proof is one of the key achievements of welfare economics. x xA and yB A graphic illustration of the ﬁrst theorem In Figure 13.8 we again use the Edgeworth box diagram, this time to illustrate an exchange economy. In this economy there are only two goods (x and y) and two individuals (A and B). The total dimensions of the Edgeworth box are determined by the total quantities of the two goods available (x and y). Goods allocated to individual A are recorded using 0A as an origin. Individual B gets those quantities of the two goods that are ‘‘left over’’ and can be measured using 0B as an origin. Individual A’s indifference curve map is drawn in the usual way, whereas individual B’s map is drawn from the perspective of 0B. Point E in the Edgeworth box represents the initial endowments of these two individuals. Individual A starts with xA and yA. Individual B starts with xB A for person A and U 2 ¼ The initial endowments provide a utility level of U 2 A (point B). Or we could increase person A’s utility to U 3 B for person B. These levels are clearly inefﬁcient in the Pareto sense. For example, we could, by reallocating the available goods,15 increase person B’s utility to U 3 B while holding person A’s utility constant
at U 2 A while keeping person B on the U 2 B indifference curve (point A). Allocations A and B are Pareto efﬁcient, however, because at these allocations it is not possible to make either person better off without making the other worse off. There are many other efﬁcient allocations in the Edgeworth box diagram. These are identiﬁed by the tangencies of the two individuals’ indifference curves. The set of all such efﬁcient points is shown by the line joining OA to OB. This line is sometimes called the ‘‘contract curve’’ because it represents all the Paretoefﬁcient contracts that might be reached by these two individuals. Notice, however, that (assuming that no individual would voluntarily opt for a contract that made him or her yA: ¼!! y 15This point UB xB, yB could in principle be subject to the constraint UA xA, yA ð Þ ð U 2 A. See Example 13.3. Þ ¼ found by solving the following constrained optimization problem: Maximize 478 Part 5: Competitive Markets FIGURE 13.8 The First Theorem of Welfare Economics With initial endowments at point E, individuals trade along the price line PP until they reach point E). This equilibrium is Pareto efﬁcient worse off ) only contracts between points B and A are viable with initial endowments given by point E. The line PP in Figure 13.8 shows the competitively established price ratio that is guaranteed by our earlier existence proof. The line passes through the initial endowments (E) and shows the terms at which these two individuals can trade away from these initial positions. Notice that such trading is beneﬁcial to both parties—that is, it allows them to get a higher utility level than is provided by their initial endowments. Such trading will continue until all such mutual beneﬁcial trades have been completed. That will occur at allocation E) on the contract curve. Because the individuals’ indifference curves are tangent at this point, no further trading would yield gains to both parties. Therefore, the competitive allocation E) meets the Pareto criterion for efﬁciency, as we showed mathematically earlier. Second theorem of welfare economics The ﬁrst theorem of welfare economics shows that a Walras
ian equilibrium is Pareto efﬁcient, but the social welfare consequences of this result are limited because of the role played by initial endowments in the demonstration. The location of the Walrasian equilibrium at E) in Figure 13.8 was signiﬁcantly inﬂuenced by the designation of E as the starting point for trading. Points on the contract curve outside the range of AB are not attainable through voluntary transactions, even though these may in fact be more socially desirable than E) (perhaps because utilities are more equal). The second theorem of welfare economics addresses this issue. It states that for any Pareto optimal allocation of resources there exists a set of initial endowments and a related price vector such that this allocation is also a Walrasian equilibrium. Phrased another way, any Pareto optimal allocation of resources can also be a Walrasian equilibrium, providing that initial endowments are adjusted accordingly. Chapter 13: General Equilibrium and Welfare 479 FIGURE 13.9 The Second Theorem of Welfare Economics If allocation Q) is regarded as socially optimal, this allocation can be supported by any initial endowments on the price line P0P0. To move from E to, say, Q would require transfers of initial endowments. P' E* A Q* E O B Q P graphical proof of the second theorem should sufﬁce. Figure 13.9 repeats the key aspects of the exchange economy pictures in Figure 13.8. Given the initial endowments at point E, all voluntary Walrasian equilibrium must lie between points A and B on the contract curve. Suppose, however, that these allocations were thought to be undesirable— perhaps because they involve too much inequality of utility. Assume that the Pareto optimal allocation Q) is believed to be socially preferable, but it is not attainable from the initial endowments at point E. The second theorem states that one can draw a price line through Q) that is tangent to both individuals’ respective indifference curves. This line is denoted by P 0P 0 in Figure 13.9. Because the slope of this line shows potential trades these individuals are willing to make, any point on the line can serve as an initial endowment from which trades lead to Q). One such point is denoted by Q. If a benevolent government wished to ensure that Q) would emerge as a Walrasian equilibrium, it would have to transfer initial
endowments of the goods from E to Q (making person A better off and person B worse off in the process). EXAMPLE 13.3 A Two-Person Exchange Economy To illustrate these various principles, consider a simple two-person, economy. Suppose that total quantities of the goods are ﬁxed at x y utility takes the Cobb–Douglas form: ¼ two-good exchange 1,000. Person A’s ¼ UAð and person B’s preferences are given by: xA, yAÞ ¼ A y1=3 x2=3 A, UBð xB, yBÞ ¼ B y2=3 x1=3 B : (13:38) (13:39) 480 Part 5: Competitive Markets Notice that person A has a relative preference for good x and person B has a relative preference for good y. Hence you might expect that the Pareto-efﬁcient allocations in this model would have the property that person A would consume relatively more x and person B would consume relatively more y. To ﬁnd these allocations explicitly, we need to ﬁnd a way of dividing the available goods in such a way that the utility of person A is maximized for any preassigned utility level for person B. Setting up the Lagrangian expression for this problem, we have: UAð Substituting for the explicit utility functions assumed here yields xA, yAÞ ¼ xA, yAÞ þ UBð 1,000 k ½ Lð xA, 1,000! yAÞ! : U B&! xA, yAÞ ¼ and the ﬁrst-order conditions for a maximum are A þ 1,000 Lð ½ð k! A y1=3 x2=3 1=3 xAÞ 1,000 ð! 2=3 yAÞ, U B&! @ L @xA ¼ @ L @yA ¼ 2 3 1 3 1=3 2=3 yA xA" # xA yA" # k 3!! " 2k 3 " 1,000 1,000 yA xA!! 1,000 1,000 # xA yA #!! 2=3 0; ¼ 1=3 0: ¼ Moving the terms in l to the right and dividing the
top equation by the bottom gives or yA 2 xA" # 1 2 ¼ " 1,000 1,000 yA xA!! # xA 1,000! 4yA xA ¼ 1,000! : yA (13:40) (13:41) (13:42) (13:43) This equation allows us to identify all the Pareto optimal allocations in this exchange economy. For example, if we were to arbitrarily choose xA ¼ 500, Equation 13.43 would become 4yA 1,000 yA ¼! 1 so xB ¼ yA ¼ 200, yB ¼ 800: (13:44) This allocation is relatively favorable to person B. At this point on the contract curve UA ¼ 5002/32001/3 683. Notice that although the available quantity of x is divided evenly (by assumption), most of good y goes to person B as efﬁciency requires. 369, UB ¼ 5001/38002/3 ¼ ¼ Equilibrium price ratio. To calculate the equilibrium price ratio at this point on the contract curve, we need to know the two individuals’ marginal rates of substitution. For person A, and for person B MRS @UA=@xA @UA=@yA ¼ 2 yA xA ¼ 2 200 500 ¼ ¼ 0:8 MRS @UB=@xB @UB=@yB ¼ ¼ 0:5 yA xA ¼ 0:5 800 500 ¼ 0:8: (13:45) (13:46) 0.8. Hence the marginal rates of substitution are indeed equal (as they should be), and they imply a price ratio of px/py ¼ Initial endowments. Because this equilibrium price ratio will permit these individuals to trade 8 units of y for each 10 units of x, it is a simple matter to devise initial endowments optimum. Consider, consistent with endowment the 1, the value of person A’s initial 680. If px ¼ 350, yA ¼ xA ¼ endowment is 600. If he or she spends two thirds of this amount on good x, it is possible to Pareto 650, yB ¼ for 0.8, py ¼ this 320; xB ¼ example, Chapter 13: General Equilibrium and Welfare
481 ¼ purchase 500 units of good x and 200 units of good y. This would increase utility from UA ¼ 3502/3 3201/3 340 to 369. Similarly, the value of person B’s endowment is 1,200. If he or she spends one third of this on good x, 500 units can be bought. With the remaining two thirds of the value of the endowment being spent on good y, 800 units can be bought. In the process, B’s utility increases from 670 to 683. Thus, trading from the proposed initial endowment to the contract curve is indeed mutually beneﬁcial (as shown in Figure 13.8). QUERY: Why did starting with the assumption that good x would be divided equally on the contract curve result in a situation favoring person B throughout this problem? What point on the contract curve would provide equal utility to persons A and B? What would the price ratio of the two goods be at this point? Social welfare functions Figure 13.9 shows that there are many Pareto-efﬁcient allocations of the available goods in an exchange economy. We are assured by the second theorem of welfare economics that any of these can be supported by a Walrasian system of competitively determined prices, providing that initial endowments are adjusted accordingly. A major question for welfare economics is how (if at all) to develop criteria for choosing among all these allocations. In this section we look brieﬂy at one strand of this large topic—the study of social welfare functions. Simply put, a social welfare function is a hypothetical scheme for ranking potential allocations of resources based on the utility they provide to individuals. In mathematical terms: ½ ¼ SW x1 U1ð Social Welfare x2, U2ð Þ The ‘‘social planner’s’’ goal then is to choose allocations of goods among the m individuals in the economy in a way that maximizes SW. Of course, this exercise is a purely conceptual one—in reality there are no clearly articulated social welfare functions in any economy, and there are serious doubts about whether such a function could ever arise from some type of democratic process.16 Still, assuming the existence of such a function can help to illuminate many of the thorniest problems in welfare economics...., Umð (13:47) : Þ& xm, Þ A ﬁrst observation that might be made about the
social welfare function in Equation 13.47 is that any welfare maximum must also be Pareto efﬁcient. If we assume that every individual’s utility is to ‘‘count,’’ it seems clear that any allocation that permits further Pareto improvements (that make one person better off and no one else worse off) cannot be a welfare maximum. Hence achieving a welfare maximum is a problem in choosing among Pareto-efﬁcient allocations and their related Walrasian price systems. We can make further progress in examining the idea of social welfare maximization by considering the precise functional form that SW might take. Speciﬁcally, if we assume utility is measurable, using the CES form can be particularly instructive: SW U1, U2, ð..., UmÞ : R, (13:48) Because we have used this functional form many times before in this book, its properties should by now be familiar. Speciﬁcally, if R 1, the function becomes: ¼ U1 þ..., UmÞ ¼ U1, U2, ð U2 þ (13:49) Um: SW... þ 16The ‘‘impossibility’’ of developing a social welfare function from the underlying preferences of people in society was ﬁrst studied by K. Arrow in Social Choice and Individual Values, 2nd ed. (New York: Wiley, 1963). There is a large body of literature stemming from Arrow’s initial discovery. 482 Part 5: Competitive Markets Thus, utility is a simple sum of the utility of every person in the economy. Such a social welfare function is sometimes called a utilitarian function. With such a function, social welfare is judged by the aggregate sum of utility (or perhaps even income) with no regard for how utility (income) is distributed among the members of society. At the other extreme, consider the case R. In this case, social welfare has a ‘‘ﬁxed proportions’’ character and (as we have seen in many other applications),..., UmÞ ¼..., Um& U1, U2, U1, U2, Min SW ð ½ : (13:50) ¼!1 Therefore, this function focuses on the worse-off person in any allocation and chooses that allocation
for which this person has the highest utility. Such a social welfare function is called a maximin function. It was made popular by the philosopher John Rawls, who argued that if individuals did not know which position they would ultimately have in society (i.e., they operate under a ‘‘veil of ignorance’’), they would opt for this sort of social welfare function to guard against being the worse-off person.17 Our analysis in Chapter 7 suggests that people may not be this risk averse in choosing social arrangements. However, Rawls’ focus on the bottom of the utility distribution is probably a good antidote to thinking about social welfare in purely utilitarian terms. It is possible to explore many other potential functional forms for a hypothetical welfare function. Problem 13.14 looks at some connections between social welfare functions and the income distribution, for example. But such illustrations largely miss a crucial point if they focus only on an exchange economy. Because the quantities of goods in such an economy are ﬁxed, issues related to production incentives do not arise when evaluating social welfare alternatives. In actuality, however, any attempt to redistribute income (or utility) through taxes and transfers will necessarily affect production incentives and therefore affect the size of the Edgeworth box. Therefore, assessing social welfare will involve studying the trade-off between achieving distributional goals and maintaining levels of production. To examine such possibilities we must introduce production into our general equilibrium framework. A Mathematical Model of Production and Exchange Adding production to the model of exchange developed in the previous section is a relatively simple process. First, the notion of a ‘‘good’’ needs to be expanded to include factors of production. Therefore, we will assume that our list of n goods now includes inputs whose prices also will be determined within the general equilibrium model. Some inputs for one ﬁrm in a general equilibrium model are produced by other ﬁrms. Some of these goods may also be consumed by individuals (cars are used by both ﬁrms and ﬁnal consumers), and some of these may be used only as intermediate goods (steel sheets are used only to make cars and are not bought by consumers). Other inputs may be part of individuals’ initial endowments. Most importantly, this is the way labor supply is treated in general equilibrium models. Individuals are endowed with a certain number of potential labor hours. They may sell these to ﬁrms by taking jobs at competitively
determined wages, or they may choose to consume the hours themselves in the form of ‘‘leisure,’’ In making such choices we continue to assume that individuals maximize utility.18 We will assume that there are r ﬁrms involved in production. Each of these ﬁrms is bound by a production function that describes the physical constraints on the ways the 17J. Rawls, A Theory of Justice (Cambridge, MA: Harvard University Press, 1971). 18A detailed study of labor supply theory is presented in Chapter 16. Chapter 13: General Equilibrium and Welfare 483 ﬁrm can turn inputs into outputs. By convention, outputs of the ﬁrm take a positive sign, whereas inputs take a negative sign. Using this convention, each ﬁrm’s production plan can 1... r), which contains both positive and be described by an n negative entries. The only vectors that the ﬁrm may consider are those that are feasible given the current state of technology. Sometimes it is convenient to assume each ﬁrm produces only one output. But that is not necessary for a more general treatment of production. 1 column vector, y j( j + ¼ Firms are assumed to maximize proﬁts. Production functions are assumed to be sufﬁciently convex to ensure a unique proﬁt maximum for any set of output and input prices. This rules out both increasing returns to scale technologies and constant returns because neither yields a unique maxima. Many general equilibrium models can handle such possibilities, but there is no need to introduce such complexities here. Given these assumptions, the proﬁts for any ﬁrm can be written as: py j if pjð p Þ ¼ < 0: p 0 if pjð Þ pjð y j (13:51) and Þ ( ¼ p 0 Hence this model has a ‘‘long run’’ orientation in which ﬁrms that lose money (at a particular price conﬁguration) hire no inputs and produce no output. Notice how the convention that outputs have a positive sign and inputs a negative sign makes it possible to phrase proﬁts in a compact way.19 Budget constraints and Walras’ law In an exchange model, individuals’ purchasing power is determined by the values of their initial endowments. Once
ﬁrms are introduced, we must also consider the income stream that may ﬂow from ownership of these ﬁrms. To do so, we adopt the simplifying assumption that each individual owns a predeﬁned share, si ð of all ﬁrms. That is, each person owns an ‘‘index fund’’ that can claim a proportionate share of all ﬁrms’ proﬁts. We can now rewrite each individual’s budget constraint (from Equation 13.24) as: of the proﬁts si ¼ where 1 i ¼ P 1 Þ m pxi si ¼ r 1 j X ¼ pyj pxi þ i ¼ 1... m: (13:52) Of course, if all ﬁrms were in long-run equilibrium in perfectly competitive industries, all proﬁts would be zero and the budget constraint in Equation 13.52 would revert to that in Equation 13.24. But allowing for long-term proﬁts does not greatly complicate our model; therefore, we might as well consider the possibility. As in the exchange model, the existence of these m budget constraints implies a constraint of the prices that are possible—a generalization of Walras’ law. Summing the budget constraints in Equation 13.52 over all individuals yields: and letting x Walras’ law xi m p xi yj xi, (13:53) 1 i X ¼ xi provides a simple statement of P p px ð Þ ¼ P px: py p ð Þ þ (13:54) 19As we saw in Chapter 11, proﬁt functions are homogeneous of degree 1 in all prices. Hence both output supply functions and input demand functions are homogeneous of degree 0 in all prices because they are derivatives of the proﬁt function. 484 Part 5: Competitive Markets Notice again that Walras’ law holds for any set of prices because it is based on individuals’ budget constraints. Walrasian equilibrium As before, we deﬁne a Walrasian equilibrium price vector (p)) as a set of prices at which demand equals supply in all markets simultaneously. In mathematical terms this means that: x p) y p) x: (13:55) �
Speciﬁcation of any general equilibrium model begins by deﬁning the number of goods to be included in the model. These ‘‘goods’’ include not only consumption goods but also intermediate goods that are used in the production of other goods (e.g., capital equipment), productive inputs such as labor or natural resources, and goods that are to be produced by the government (public goods). The goal of the model is then to solve for equilibrium prices for all these goods and to study how these prices change when conditions change. Some of the goods in a general equilibrium model are produced by ﬁrms. The technology of this production must be speciﬁed by production functions. The most common such speciﬁcation is to use the types of CES production functions that we studied in Chapters 9 and 10 because these can yield some important insights about the ways in which inputs are substituted in the face of changing prices. In general, ﬁrms are assumed to maximize their proﬁts given their production functions and given the input and output prices they face. Demand is speciﬁed in general equilibrium models by deﬁning utility functions for various types of households. Utility is treated as a function both of goods that are consumed and of inputs that are not supplied to the marketplace (e.g., available labor that is not supplied to the market is consumed as leisure). Households are assumed to maximize utility. Their incomes are determined by the amounts of inputs they ‘‘sell’’ in the market and by the net result of any taxes they pay or transfers they receive. Finally, a full general equilibrium model must specify how the government operates. If there are taxes in the model, how those taxes are to be spent on transfers or on public goods (which provide utility to consumers) must be modeled. If government borrowing is allowed, the bond market must be explicitly modeled. In short, the model must fully specify the ﬂow of both sources and uses of income that characterize the economy being modeled. Solving general equilibrium models Once technology (supply) and preferences (demand) have been speciﬁed, a general equilibrium model must be solved for equilibrium prices and quantities. The proof earlier in this chapter shows that such a model will generally have such a solution, but actually ﬁnding that solution can sometimes be difﬁcult—especially when the number of
goods and households is large. General equilibrium models are usually solved on computers via modiﬁcations of an algorithm originally developed by Herbert Scarf in the 1970s.22 This algorithm (or more modern versions of it) searches for market equilibria by mimicking the way markets work. That is, an initial solution is speciﬁed and then prices are raised in markets with excess demand and lowered in markets with excess supply until an equilibrium is found in which all excess demands are zero. Sometimes multiple equilibria will occur, but usually economic models have sufﬁcient curvature in the underlying production and utility functions that the equilibrium found by the Scarf algorithm will be unique. 21For more detail on the issues discussed here, see W. Nicholson and F. Westhoff, ‘‘General Equilibrium Models: Improving the Microeconomics Classroom,’’ Journal of Economic Education (Summer 2009): 297–314. 22Herbert Scarf with Terje Hansen, On the Computation of Economic Equilibria (New Haven, CT: Yale University Press, 1973). Chapter 13: General Equilibrium and Welfare 487 Economic insights from general equilibrium models General equilibrium models provide a number of insights about how economies operate that cannot be obtained from the types of partial equilibrium models studied in Chapter 12. Some of the most important of these are: • All prices are endogenous in economic models. The exogenous elements of models are preferences and productive technologies. • All ﬁrms and productive inputs are owned by households. All income ultimately accrues to households. • Any model with a government sector is incomplete if it does not specify how tax receipts are used. • The ‘‘bottom line’’ in any policy evaluation is the utility of households. Firms and governments are only intermediaries in getting to this ﬁnal accounting. • All taxes distort economic decisions along some dimension. The welfare costs of such distortions must always be weighed against the beneﬁts of such taxes (in terms of public good production or equity-enhancing transfers). Some of these insights are illustrated in the next two examples. In later chapters we will return to general equilibrium modeling whenever such a perspective seems necessary to gain a more complete understanding of the topic being covered. EXAMPLE 13.4 A Simple General Equilibrium Model Let’s look at a simple general equilibrium model with only two households, two consumer goods (x and y), and two
inputs (capital k and labor l). Each household has an ‘‘endowment’’ of capital and labor that it can choose to retain or sell in the market. These endowments are denoted by k1, l1 and k2, l2, respectively. Households obtain utility from the amounts of the consumer goods they purchase and from the amount of labor they do not sell into the market (i.e., leisure li! li). The households have simple Cobb–Douglas utility functions: ¼ U 1 ¼ 1 y0:3 x0:5 1 ð l1! 0:2, l1Þ U 2 ¼ 2 y0:4 x0:4 2 ð l2! 0:2: l2Þ (13:57) Hence household 1 has a relatively greater preference for good x than does household 2. Notice that capital does not enter into these utility functions directly. Consequently, each household will provide its entire endowment of capital to the marketplace. Households will retain some labor, however, because leisure provides utility directly. Production of goods x and y is characterized by simple Cobb–Douglas technologies: k0:2 x l 0:8 x, x ¼ k0:8 y l 0:2 y : y ¼ (13:58) Thus, in this example, production of x is relatively labor intensive, whereas production of y is relatively capital intensive. To complete this model we must specify initial endowments of capital and labor. Here we assume that k1 ¼ 40, l1 ¼ 24 and k2 ¼ 10, l2 ¼ 24: (13:59) Although the households have equal labor endowments (i.e., 24 ‘‘hours’’), household 1 has signiﬁcantly more capital than does household 2. Base-case simulation. Equations 13.57–13.59 specify our complete general equilibrium model in the absence of a government. A solution to this model will consist of four equilibrium prices (for x, y, k, and l ) at which households maximize utility and ﬁrms maximize proﬁts.23 23Because ﬁrms’ production functions are characterized by constant returns to scale, in equilibrium each earns zero proﬁts; therefore, there is no need to specify ﬁrm ownership in this model. 488 Part 5: Competitive
Markets Because any general equilibrium model can compute only relative prices, we are free to impose a price-normalization scheme. Here we assume that the prices will always sum to unity. That is, Solving24 for these prices yields px þ py þ pk þ pl ¼ 1: px ¼ 0:363, py ¼ 0:253, pk ¼ 0:136, pl ¼ 0:248: (13:60) (13:61) At these prices, total production of x is 23.7 and production of y is 25.1. The utility-maximizing choices for household 1 are x1 ¼ y1 ¼ for household 2, these choices are 15:7, 8:1; l1! l1 ¼ 24! 14:8 9:2, U 1 ¼ ¼ 13:5; (13:62) 8:1, 11:6; y2 ¼ x2 ¼ l2 ¼ Observe that household 1 consumes quite a bit of good x but provides less in labor supply than does household 2. This reﬂects the greater capital endowment of household 1 in this base-case simulation. We will return to this base case in several later simulations. 5:9, U 2 ¼ l2! (13:63) 8:75: 18:1 24 ¼! QUERY: How would you show that each household obeys its budget constraint in this simulation? Does the budgetary allocation of each household exhibit the budget shares that are implied by the form of its utility function? EXAMPLE 13.5 The Excess Burden of a Tax In Chapter 12 we showed that taxation may impose an excess burden in addition to the tax revenues collected because of the incentive effects of the tax. With a general equilibrium model we can show much more about this effect. Speciﬁcally, assume that the government in the economy of Example 13.4 imposes an ad valorem tax of 0.4 on good x. This introduces a wedge between what demanders pay for this good x (px) and what suppliers receive for the good (p0x ¼ (1 – t)px ¼ 0.6px). To complete the model we must specify what happens to the revenues generated by this tax. For simplicity we assume that these revenues are rebated to the households in a 50–50 split. In all other respects the economy remains as described in Example 13
.4. Solving for the new equilibrium prices in this model yields px = 0.472, py = 0.218, pk = 0.121, pl = 0.188. (13:64) At these prices, total production of x is 17.9, and total production of y is 28.8. Hence the allocation of resources has shifted signiﬁcantly toward y production. Even though the relative px/py ¼ 2.17) has increased signiﬁcantly price of x experienced by consumers ( from its value (of 1.43) in Example 13.4, the price ratio experienced by ﬁrms (0.6px/py ¼ 1.30) has decreased somewhat from this prior value. Therefore, one might expect, based on a partial equilibrium analysis, that consumers would demand less of good x and likewise that ﬁrms would similarly produce less of that good. Partial equilibrium analysis would not, however, allow us to predict the increased production of y (which comes about because the relative price 0.472/0.218 ¼ ¼ 24The computer program used to ﬁnd these solutions is accessible at www.amherst.edu/ CompEquApplet.html. - fwesthoff/compequ/FixedPoints Chapter 13: General Equilibrium and Welfare 489 of y has decreased for consumers but has increased for ﬁrms) nor the reduction in relative input prices (because there is less being produced overall). A more complete picture of all these effects can be obtained by looking at the ﬁnal equilibrium positions of the two households. The posttax allocation for household 1 is for household 2, x1 ¼ 11:6, y1 ¼ 15:2, l1! l1 ¼ 11:8, U1 ¼ 12:7; x2 ¼ 6.3, y2 ¼ 13.6, l2! l2 ¼ 7.9, U 2 ¼ 8.96. (13:65) (13:66) Hence imposition of the tax has made household 1 considerably worse off: utility decreases from 13.5 to 12.7. Household 2 is made slightly better off by this tax and transfer scheme, primarily because it receives a relatively large share of the tax proceeds that come mainly from household 1. Although total utility has decreased (as predicted by the simple partial equilibrium analysis of excess burden
), general equilibrium analysis gives a more complete picture of the distributional consequences of the tax. Notice also that the total amount of labor supplied decreases as a result of the tax: total leisure increases from 15.1 (hours) to 19.7. Therefore, imposition of a tax on good x has had a relatively substantial labor supply effect that is completely invisible in a partial equilibrium model. QUERY: Would it be possible to make both households better off (relative to Example 13.4) in this taxation scenario by changing how the tax revenues are redistributed? SUMMARY This chapter has provided a general exploration of Adam Smith’s conjectures about the efﬁciency properties of competitive markets. We began with a description of how to model many competitive markets simultaneously and then used that model to make a few statements about welfare. Some highlights of this chapter are listed here. • Preferences and production technologies provide the building blocks on which all general equilibrium models are based. One particularly simple version of such a model uses individual preferences for two goods together with a concave production possibility frontier for those two goods. • Competitive markets can establish equilibrium prices by making marginal adjustments in prices in response to information about the demand and supply for individual goods. Walras’ law ties markets together so that such a solution is assured (in most cases). • General equilibrium models can usually be solved by using computer algorithms. The resulting solutions yield many insights about the economy that are not obtainable from partial equilibrium analysis of single markets. • Competitive prices will result in a Pareto-efﬁcient allocation of resources. This is the ﬁrst theorem of welfare economics. • Factors that interfere with competitive markets’ abilities to achieve efﬁciency include (1) market power, (2) externalities, (3) existence of public goods, and (4) imperfect information. We explore all these issues in detail in later chapters. • Competitive markets need not yield equitable distributions of resources, especially when initial endowments are highly skewed. In theory, any desired distribution can be attained through competitive markets accompanied by appropriate transfers of initial endowments (the second theorem of welfare economics). But there are many practical problems in implementing such transfers. 490 Part 5: Competitive Markets PROBLEMS 13.1 Suppose the production possibility frontier for guns (x) and butter (y) is given by x 2 2y 2 þ ¼ 900. a. Graph this frontier. b. If individuals
always prefer consumption bundles in which y c. At the point described in part (b), what will be the RPT and hence what price ratio will cause production to take place at 2x, how much x and y will be produced? ¼ that point? (This slope should be approximated by considering small changes in x and y around the optimal point.) d. Show your solution on the ﬁgure from part (a). 13.2 Suppose two individuals (Smith and Jones) each have 10 hours of labor to devote to producing either ice cream (x) or chicken soup (y). Smith’s utility function is given by whereas Jones’ is given by x 0:3y 0:7, U S ¼ U J ¼ The individuals do not care whether they produce x or y, and the production function for each good is given by x0:5y0:5: where l is the total labor devoted to production of each good. a. What must the price ratio, px /py, be? b. Given this price ratio, how much x and y will Smith and Jones demand? Hint: Set the wage equal to 1 here. c. How should labor be allocated between x and y to satisfy the demand calculated in part (b)? 2l and y x ¼ 3l, ¼ 13.3 Consider an economy with just one technique available for the production of each good. Good Food Cloth Labor per unit output Land per unit output 1 2 1 1 a. Suppose land is unlimited but labor equals 100. Write and sketch the production possibility frontier. b. Suppose labor is unlimited but land equals 150. Write and sketch the production possibility frontier. c. Suppose labor equals 100 and land equals 150. Write and sketch the production possibility frontier. Hint: What are the intercepts of the production possibility frontier? When is land fully employed? Labor? Both? d. Explain why the production possibility frontier of part (c) is concave. e. Sketch the relative price of food as a function of its output in part (c). f. If consumers insist on trading 4 units of food for 5 units of cloth, what is the relative price of food? Why? g. Explain why production is exactly the same at a price ratio of pF /pC ¼ h. Suppose that capital is also required for producing food and clothing and that capital requirements per unit of food and per unit of clothing are 0.8 and 0.9,
respectively. There are 100 units of capital available. What is the production possibility curve in this case? Answer part (e) for this case. 1.1 as at pF /pC ¼ 1.9. 13.4 Suppose that Robinson Crusoe produces and consumes ﬁsh (F) and coconuts (C). Assume that, during a certain period, he has decided to work 200 hours and is indifferent as to whether he spends this time ﬁshing or gathering coconuts. Robinson’s production for ﬁsh is given by Chapter 13: General Equilibrium and Welfare 491 and for coconuts by F ¼ lF p ﬃﬃﬃﬃ C ¼ lC, where lF and lC are the number of hours spent ﬁshing or gathering coconuts. Consequently, p ﬃﬃﬃﬃ lC þ Robinson Crusoe’s utility for ﬁsh and coconuts is given by 200: lF ¼ a. If Robinson cannot trade with the rest of the world, how will he choose to allocate his labor? What will the optimal levels utility =. Cp F * ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ of F and C be? What will his utility be? What will be the RPT (of ﬁsh for coconuts)? b. Suppose now that trade is opened and Robinson can trade ﬁsh and coconuts at a price ratio of pF /pC ¼ 2/1. If Robinson continues to produce the quantities of F and C from part (a), what will he choose to consume once given the opportunity to trade? What will his new level of utility be? c. How would your answer to part (b) change if Robinson adjusts his production to take advantage of the world prices? d. Graph your results for parts (a), (b), and (c). 13.5 Smith and Jones are stranded on a desert island. Each has in his possession some slices of ham (H) and cheese (C). Smith is a choosy eater and will eat ham and cheese only in the ﬁxed proportions of 2 slices of cheese to 1 slice of ham. His utility function is given by US ¼ slices of ham and
200 slices of cheese. Jones is more ﬂexible in his dietary tastes and has a utility function given by UJ ¼ 3C. Total endowments are 100 min(H, C/2). 4H þ a. Draw the Edgeworth box diagram that represents the possibilities for exchange in this situation. What is the only exchange ratio that can prevail in any equilibrium? b. Suppose Smith initially had 40H and 80C. What would the equilibrium position be? c. Suppose Smith initially had 60H and 80C. What would the equilibrium position be? d. Suppose Smith (much the stronger of the two) decides not to play by the rules of the game. Then what could the ﬁnal equilibrium position be? 13.6 In the country of Ruritania there are two regions, A and B. Two goods (x and y) are produced in both regions. Production functions for region A are given by q here lx and ly are the quantities of labor devoted to x and y production, respectively. Total labor available in region A is 100 units; that is, xA ¼ yA ¼ p lx, ; ﬃﬃﬃﬃ ly ﬃﬃﬃﬃ Using a similar notation for region B, production functions are given by lx + ly ¼ 100. There are also 100 units of labor available in region B: 1 xB ¼ 2 1 p yB ¼ 2 lx, ﬃﬃﬃﬃ ly ﬃﬃﬃﬃ : q lx + ly ¼ 100. 492 Part 5: Competitive Markets a. Calculate the production possibility curves for regions A and B. b. What condition must hold if production in Ruritania is to be allocated efﬁciently between regions A and B (assuming labor cannot move from one region to the other)? c. Calculate the production possibility curve for Ruritania (again assuming labor is immobile between regions). How much total y can Ruritania produce if total x output is 12? Hint: A graphical analysis may be of some help here. 13.7 Use the computer algorithm discussed in footnote 24 to examine the consequences of the following changes to the model in Example 13.4. For each change, describe the ﬁnal results of the modeling and offer some
intuition about why the results worked as they did. a. Change the preferences of household 1 to U 1 ¼ b. Reverse the production functions in Equation 13.58 so that x becomes the capital-intensive good. c. Increase the importance of leisure in each household’s utility function. 1 (l1! 1 y0:2 x0:6 l1)0:2: Analytical Problems 13.8 Tax equivalence theorem Use the computer algorithm discussed in footnote 24 to show that a uniform ad valorem tax of both goods yields the same equilibrium as does a uniform tax on both inputs that collects the same revenue. Note: This tax equivalence theorem from the theory of public ﬁnance shows that taxation may be done on either the output or input sides of the economy with identical results. 13.9 Returns to scale and the production possibility frontier The purpose of this problem is to examine the relationships among returns to scale, factor intensity, and the shape of the production possibility frontier. Suppose there are ﬁxed supplies of capital and labor to be allocated between the production of good x and good y. The production functions for x and y are given (respectively) by where the parameters a, b, g, and d will take on different values throughout this problem. Using either intuition, a computer, or a formal mathematical approach, derive the production possibility frontier for x and y x ¼ kal b and y kgl d, ¼ in the following cases. a. a b. a c. a d. a e. a f/2, g 1/2, g g d ¼ 0.6, g 0.7, g ¼ ¼ ¼ ¼ ¼ ¼ 1/2. 1/3, d d ¼ 2/3. 0.2, d 0.6, d 2/3. ¼ 2/3. 1.0. 0.8. ¼ ¼ Do increasing returns to scale always lead to a convex production possibility frontier? Explain. 13.10 The trade theorems The construction of the production possibility curve shown in Figures 13.2 and 13.3 can be used to illustrate three important ‘‘theorems’’ in international trade theory. To get started, notice in Figure 13.2 that the efﬁciency line Ox,Oy is bowed above the main diagonal of the Edgeworth box. This shows that the production
of good x is always ‘‘capital intensive’’ relative to the x > k production of good y. That is, when production is efﬁcient, y no matter how much of the goods are produced. Demonstration of the trade theorems assumes that the price ratio, p px/py, is determined in international markets—the % & ¼ domestic economy must adjust to this ratio (in trade jargon, the country under examination is assumed to be ‘‘a small country in a large world’’). % & k l l a. Factor price equalization theorem: Use Figure 13.4 to show how the international price ratio, p, determines the point in the Edgeworth box at which domestic production will take place. Show how this determines the factor price ratio, w/v. If production functions are the same throughout the world, what will this imply about relative factor prices throughout the world? Chapter 13: General Equilibrium and Welfare 493 b. Stolper–Samuelson theorem: An increase in p will cause the production to move clockwise along the production possibility frontier—x production will increase and y production will decrease. Use the Edgeworth box diagram to show that such a move will decrease k/l in the production of both goods. Explain why this will cause w/v to decrease. What are the implications of this for the opening of trade relations (which typically increases the price of the good produced intensively with a country’s most abundant input). c. Rybczynski theorem: Suppose again that p is set by external markets and does not change. Show that an increase in k will increase the output of x (the capital-intensive good) and reduce the output of y (the labor-intensive good). 13.11 An example of Walras’ law Suppose there are only three goods (x1, x2, x3) in an economy and that the excess demand functions for x2 and x3 are given by ED2 ¼! ED3 ¼! 3p2 p1 þ 4p2 p1! 2p3 p1! 2p3 p1! 1, 2: a. Show that these functions are homogeneous of degree 0 in p1, p2, and p3. b. Use Walras’ law to show that, if ED2 ¼ 0, then ED1 must also be 0. Can you also use Walras’
law to calculate ED1? c. Solve this system of equations for the equilibrium relative prices p2 /p1 and p3 /p1. What is the equilibrium value for p3 /p2? ED3 ¼ ¼ 13.12 Productive efﬁciency with calculus In Example 13.3 we showed how a Pareto efﬁciency exchange equilibrium can be described as the solution to a constrained maximum problem. In this problem we provide a similar illustration for an economy involving production. Suppose that there is only one person in a two-good economy and that his or her utility function is given by U(x, y). Suppose also that this economy’s production possibility frontier can be written in implicit form as T(x, y) 0. a. What is the constrained optimization problem that this economy will seek to solve if it wishes to make the best use of its available resources? b. What are the ﬁrst-order conditions for a maximum in this situation? c. How would the efﬁcient situation described in part (b) be brought about by a perfectly competitive system in which this individual maximizes utility and the ﬁrms underlying the production possibility frontier maximize proﬁts. d. Under what situations might the ﬁrst-order conditions described in part (b) not yield a utility maximum? 13.13 Initial endowments, equilibrium prices, and the ﬁrst theorem of welfare economics In Example 13.3 we computed an efﬁcient allocation of the available goods and then found the price ratio consistent with this allocation. That then allowed us to ﬁnd initial endowments that would support this equilibrium. In that way the example demonstrates the second theorem of welfare economics. We can use the same approach to illustrate the ﬁrst theorem. Assume again that the utility functions for persons A and B are those given in the example. a. For each individual, show how his or her demand for x and y depends on the relative prices of these two goods and on the 1 and let p represent the price of x (relative initial endowment that each person has. To simplify the notation here, set py ¼ to that of y). Hence the value of, say, A’s initial endowment can be written as pxA þ b. Use the equilibrium conditions that total quantity demanded of goods x and y must
equal the total quantities of these two goods available (assumed to be 1,000 units each) to solve for the equilibrium price ratio as a function of the initial endowments of the goods held by each person (remember that initial endowments must also total 1,000 for each good). yA. c. For the case xA ¼ d. Describe in general terms how changes in the initial endowments would affect the resulting equilibrium prices in this 500, compute the resulting market equilibrium and show that it is Pareto efﬁcient. yA ¼ model. Illustrate your conclusions with a few numerical examples. 13.14 Social welfare functions and income taxation The relationship between social welfare functions and the optimal distribution of individual tax burdens is a complex one in welfare economics. In this problem, we look at a few elements of this theory. Throughout we assume that there are m individuals in the economy and that each individual is characterized by a skill level, ai, which indicates his or her ability to earn income. Without loss of generality suppose also that individuals are ordered by increasing ability. Pretax income itself is 494 Part 5: Competitive Markets I(ai, ci). Suppose also that determined by skill level and effort, ci, which may or may not be sensitive to taxation. That is, Ii ¼ 0: Finally, the government wishes to choose a schedule of, w0 > 0, w00 < 0, w the utility cost of effort is given by w 0 ð income taxes and transfers, T(I), which maximizes social welfare subject to a government budget constraint satisfying m c Þ ð Þ ¼ T IiÞ ¼ ð R (where R is the amount needed to ﬁnance public goods). 1 i ¼ P a. Suppose that each individual’s income is unaffected by effort and that each person’s utility is given by ui ¼ ui[Ii – T(Ii) – c(c)]. Show that maximization of a CES social welfare function requires perfect equality of income no matter what the precise form of that function. (Note: for some individuals T(Ii) may be negative.) b. Suppose now that individuals’ incomes are affected by effort. Show that the results of part (a) still hold if the government based income taxation on ai rather than on Ii. c. In general show that if income taxation is based on
observed income, this will affect the level of effort individuals under- take. d. Characterization of the optimal tax structure when income is affected by effort is difﬁcult and often counterintuitive. Diamond25 shows that the optimal marginal rate schedule may be U-shaped, with the highest rates for both low- and highincome people. He shows that the optimal top rate marginal rate is given by 1 1 eL,wÞð kiÞ ð þ! 1 eL,wÞð 1 2eL,w þ ð! þ where ki(0 1) is the top income person’s relative weight in the social welfare function and eL,w is the elasticity of labor supply with respect to the after-tax wage rate. Try a few simulations of possible values for these two parameters, and describe what the top marginal rate should be. Give an intuitive discussion of these results. ImaxÞ ¼ ki, kiÞ T 0, ð, SUGGESTIONS FOR FURTHER READING Arrow, K. J., and F. H. Hahn. General Competitive Analysis. Amsterdam: North-Holland, 1978, chaps. 1, 2, and 4. Harberger, A. ‘‘The Incidence of the Corporate Income Tax.’’ Journal of Political Economy (January/February 1962): 215–40. Sophisticated mathematical treatment of general equilibrium analysis. Each chapter has a good literary introduction. Nice use of a two-sector general equilibrium model to examine the ﬁnal burden of a tax on capital. Debreu, G. Theory of Value. New York: John Wiley & Sons, 1959. Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. Oxford, UK: Oxford University Press, 1995. Basic reference; difﬁcult mathematics. Does have a good introductory chapter on the mathematical tools used. Debreu, G. ‘‘Existence of Competitive Equilibrium.’’ In K. J. Arrow and M. D. Intriligator, Eds., Handbook of Mathematical Economics, vol. 2. Amsterdam: North-Holland, 1982, pp. 697–743. Fairly difﬁcult survey of existence proofs based on ﬁxed point theorems. Contains a comprehensive set of references
. Ginsburgh, V., and M. Keyzer. The Structure of Applied General Equilibrium Models. Cambridge, MA: MIT Press, 1997. Detailed discussions of the problems in implementing computable general equilibrium models. Some useful references to the empirical literature. Part Four is devoted to general equilibrium analysis. Chapters 17 (existence) and 18 (connections to game theory) are especially useful. Chapters 19 and 20 pursue several of the topics in the Extensions to this chapter. Salanie, B. Microeconomic Models of Market Failure. Cambridge, MA: MIT Press, 2000. Nice summary of the theorems of welfare economics along with detailed analyses of externalities, public goods, and imperfect competition. Sen, A. K. Collective Choice and Social Welfare. San Francisco: Holden-Day, 1970, chaps. 1 and 2. Basic reference on social choice theory. Early chapters have a good discussion of the meaning and limitations of the Pareto efﬁciency concept. 25P. Diamond ‘‘Optimal income taxation: An example with a U-shaped pattern of optimal marginal tax rates’’ American Economic Review, March 1998, pages 83–93 COMPUTABLE GENERAL EQUILIBRIUM MODELS EXTENSIONS As discussed brieﬂy in Chapter 13, recent improvements in computer technology have made it feasible to develop computable general equilibrium (CGE) models of considerable detail. These may involve literally hundreds of industries and individuals, each with somewhat different technologies or preferences. The general methodology used with these models is to assume various forms for production and utility functions, and then choose particular parameters of those functions based on empirical evidence. Numerical general equilibrium solutions are then generated by the models and compared with real-world data. After ‘‘calibrating’’ the models to reﬂect reality, various policy elements in the models are varied as a way of providing general equilibrium estimates of those policy changes. In this extension we brieﬂy review a few of these types of applications. the overall impact of E13.1 Trade models One of the ﬁrst uses for applied general equilibrium models was to the study of the impact of trade barriers. Because much of the debate over the effects of such barriers (or of their reduction) focuses on impacts on real wages, such general equilibrium models are especially appropriate for the task. Two unusual features tend to characterize such models. First, because the models often have an explicit
focus on domestic versus foreign production of speciﬁc goods, it is necessary to introduce a large degree of product differentiation into individuals’ utility functions. That is, ‘‘U.S. textiles’’ are treated as being different from ‘‘Mexican textiles’’ even though, in most trade theories, textiles might be treated as homogeneous goods. Modelers have found they must allow for only limited substitutability among such goods if their models are to replicate actual trade patterns. A second feature of CGE models of trade is the interest in incorporating increasing returns-to-scale technologies into their production sectors. This permits the models to capture one of the primary advantages of trade for smaller economies. Unfortunately, introduction of the increasing returns-to-scale assumption also requires that the models depart from perfectly competitive, price-taking assumptions. Often some type of markup pricing, together with Cournot-type imperfect competition (see Chapter 15), is used for this purpose. North American free trade Some of the most extensive CGE modeling efforts have been devoted to analyzing the impact of the North American Free Trade Agreement (NAFTA). Virtually all these models ﬁnd that the agreement offered welfare gains to all the countries involved. Gains for Mexico accrued primarily because of reduced U.S. trade barriers on Mexican textiles and steel. Gains to Canada came primarily from an increased ability to beneﬁt from economies of scale in certain key industries. Brown (1992) surveys a number of CGE models of North American free trade and concludes that gains on the order of 2–3 percent of GDP might be experienced by both countries. For the United States, gains from NAFTA might be considerably smaller; but even in this case, signiﬁcant welfare gains were found to be associated with the increased competitiveness of domestic markets. E13.2 Tax and transfer models A second major use of CGE models is to evaluate potential changes in a nation’s tax and transfer policies. For these applications, considerable care must be taken in modeling the factor supply side of the models. For example, at the margin, the effects of rates of income taxation (either positive or negative) can have important labor supply effects that only a general tax/ equilibrium approach can model properly. Similarly, transfer policy can also affect savings and investment decisions, and for these too it may be necessary to adopt more detailed modeling procedures (e.g., differentiating individuals by age
to examine effects of retirement programs). The Dutch MIMIC model Probably the most elaborate tax/transfer CGE model is that developed by the Dutch Central Planning Bureau—the Micro Macro Model to Analyze the Institutional Context (MIMIC). This model puts emphasis on social welfare programs and on some of the problems they seek to ameliorate (most notably unemployment, which is missing from many other CGE models). Gelauff and Graaﬂund (1994) summarize the main features of the MIMIC model. They also use it to analyze such policy proposals as the 1990s tax reform in the Netherlands and potential changes to the generous unemployment and disability beneﬁts in that country. 496 Part 5: Competitive Markets E13.3 Environmental models CGE models are also appropriate for understanding the ways in which environmental policies may affect the economy. In such applications, the production of pollutants is considered as a major side effect of the other economic activities in the model. By specifying environmental goals in terms of a given reduction in these pollutants, it is possible to use these models to study the economic costs of various strategies for achieving these goals. One advantage of the CGE approach is to provide some evidence on the impact of environmental policies on income distribution—a topic largely omitted from more narrow, industry-based modeling efforts. Assessing CO2 reduction strategies Concern over the possibility that CO2 emissions in various energy-using activities may be contributing to global warming has led to a number of plans for reducing these emissions. Because the repercussions of such reductions may be widespread and varied, CGE modeling is one of the preferred assessment methods. Perhaps the most elaborate such model is that developed by the Organisation for Economic Co-operation and Development (OECD)—the General Equilibrium Environmental (GREEN) model. The basic structure of this model is described by Burniaux, Nicoletti, and Oliviera-Martins (1992). The model has been used to simulate various policy options that might be adopted by European nations to reduce CO2 emissions, such as institution of a carbon tax or increasingly stringent emissions regulations for automobiles and power plants. In general, these simulations suggest that economic costs of these policies would be relatively modest given the level of restrictions currently anticipated. But most of the policies would have adverse distributional effects that may require further attention through government transfer policy. E13.4 Regional and urban models A ﬁnal way in which CGE models can be used is to examine economic issues that have important spatial dimensions. Construction
of such models requires careful attention to issues of transportation costs for goods and moving costs associated with labor mobility because particular interest is focused on where transactions occur. Incorporation of these costs into CGE models is in many ways equivalent to adding extra levels of product differentiation because these affect the relative prices of otherwise homogeneous goods. Calculation of equilibria in regional markets can be especially sensitive to how transport costs are speciﬁed. Changing government procurement CGE regional models have been widely used to examine the local impact of major changes in government spending policies. For example, Hoffmann, Robinson, and Subramanian (1996) use a CGE model to evaluate the regional impact of reduced defense expenditures on the California economy. They ﬁnd that the size of the effects depends importantly on the assumed costs of migration for skilled workers. A similar ﬁnding is reported by Bernat and Hanson (1995), who examine possible reductions in U.S. price-support payments to farms. Although such reductions would offer overall efﬁciency gains to the economy, they could have signiﬁcant negative impacts on rural areas. References Bernat, G. A., and K. Hanson. ‘‘Regional Impacts of Farm Programs: A Top-Down CGE Analysis.’’ Review of Regional Studies (Winter 1995): 331–50. Brown, D. K. ‘‘The Impact of North American Free Trade Area: Applied General Equilibrium Models.’’ In N. Lus-tig, B. P. Bosworth, and R. Z. Lawrence, Eds., North American Free Trade: Assessing the Impact. Washington, DC: Brookings Institution, 1992, pp. 26–68. Burniaux, J. M., G. Nicoletti, and J. Oliviera-Martins. ‘‘GREEN: A Global Model for Quantifying the Costs of Policies to Curb CO2 Emissions.’’ OECD Economic Studies (Winter 1992): 49–92. Gelauff, G. M. M., and J. J. Graaﬂund. Modeling Welfare State Reform. Amsterdam: North Holland, 1994. Hoffmann, S., S. Robinson, and S. Subramanian. ‘‘The Role of Defense Cuts in the California Recession: Computable General Equilibrium Models and Interstate Fair Mobility.’’ Journal of Regional
Science (November 1996): 571–95. This page intentionally left blank Market Power P A R T SIX Chapter 14 Monopoly Chapter 15 Imperfect Competition In this part we examine the consequences of relaxing the assumption that ﬁrms are price-takers. When ﬁrms have some power to set prices, they will no longer treat them as ﬁxed parameters in their decisions but will instead treat price setting as one part of the proﬁt-maximization process. Usually this will mean prices no longer accurately reﬂect marginal costs and the efﬁciency theorems that apply to competitive markets no longer hold. Chapter 14 looks at the relatively simple case where there is only a single monopoly supplier of a good. This supplier can choose to operate at any point on the demand curve for its product that it ﬁnds most proﬁtable. Its activities are constrained only by this demand curve, not by the behavior of rival producers. As we shall see, this offers the ﬁrm a number of avenues for increasing proﬁts, such as using novel pricing schemes or adapting the characteristics of its product. Although such decisions will indeed provide more proﬁts for the monopoly, in general they will also result in welfare losses for consumers (relative to perfect competition). In Chapter 15 we consider markets with few producers. Models of such markets are considerably more complicated than are markets of monopoly (or of perfect competition, for that matter) because the demand curve faced by any one ﬁrm will depend in an important way on what its rivals choose to do. Studying the possibilities will usually require game-theoretic ideas to capture accurately the strategic possibilities involved. Hence you should review the basic game theory material in Chapter 8 before plunging into Chapter 15, whose general conclusion is that outcomes in markets with few ﬁrms will depend crucially on the details of how the ‘‘game’’ is played. In many cases the same sort of inefﬁciencies that occur in monopoly markets appear in imperfectly competitive markets as well. 499 This page intentionally left blank C H A P T E R FOURTEEN Monopoly A monopoly is a single ﬁrm that serves an entire market. This single ﬁrm faces the market demand curve for its output. Using its knowledge of this demand curve, the monopoly makes a decision on how much to
produce. Unlike the perfectly competitive ﬁrm’s output decision (which has no effect on market price), the monopoly’s output decision will, in fact, determine the good’s price. In this sense, monopoly markets and markets characterized by perfect competition are polar-opposite cases Monopoly. A monopoly is a single supplier to a market. This ﬁrm may choose to produce at any point on the market demand curve. At times it is more convenient to treat monopolies as having the power to set prices. Technically, a monopoly can choose that point on the market demand curve at which it prefers to operate. It may choose either market price or quantity, but not both. In this chapter we will usually assume that monopolies choose the quantity of output that maximizes proﬁts and then settle for the market price that the chosen output level yields. It would be a simple matter to rephrase the discussion in terms of price setting, and in some places we shall do so. Barriers to Entry The reason a monopoly exists is that other ﬁrms ﬁnd it unproﬁtable or impossible to enter the market. Therefore, barriers to entry are the source of all monopoly power. If other ﬁrms could enter a market, then the ﬁrm would, by deﬁnition, no longer be a monopoly. There are two general types of barriers to entry: technical barriers and legal barriers. Technical barriers to entry A primary technical barrier is that the production of the good in question may exhibit decreasing marginal (and average) costs over a wide range of output levels. The technology of production is such that relatively large-scale ﬁrms are low-cost producers. In this situation (which is sometimes referred to as natural monopoly), one ﬁrm may ﬁnd it proﬁtable to drive others out of the industry by cutting prices. Similarly, once a monopoly has been established, entry will be difﬁcult because any new ﬁrm must produce at relatively low levels of output and therefore at relatively high average costs. It is important to stress that the range of declining costs need only be ‘‘large’’ relative to the market in questhe tion. Declining costs on some absolute scale are not necessary. For example, 501 502 Part 6: Market Power production and delivery of concrete does not exhibit declining marginal costs over a
broad range of output when compared with the total U.S. market. However, in any particular small town, declining marginal costs may permit a monopoly to be established. The high costs of transportation in this industry tend to isolate one market from another. Another technical basis of monopoly is special knowledge of a low-cost productive technique. The monopoly has an incentive to keep its technology secret; but unless this technology is protected by a patent (see next paragraph), this may be extremely difﬁcult. Ownership of unique resources—such as mineral deposits or land locations, or the possession of unique managerial talents—may also be a lasting basis for maintaining a monopoly. Legal barriers to entry Many pure monopolies are created as a matter of law rather than as a matter of economic conditions. One important example of a government-granted monopoly position is the legal protection of a product by a patent or copyright. Prescription drugs, computer chips, and Disney animated movies are examples of proﬁtable products that are shielded (for a time) from direct competition by potential imitators. Because the basic technology for these products is uniquely assigned to one ﬁrm, a monopoly position is established. The defense made of such a governmentally granted monopoly is that the patent and copyright system makes innovation more proﬁtable and therefore acts as an incentive. Whether the beneﬁts of such innovative behavior exceed the costs of having monopolies is an open question that has been much debated by economists. A second example of a legally created monopoly is the awarding of an exclusive franchise to serve a market. These franchises are awarded in cases of public utility (gas and electric) service, communications services, the post ofﬁce, some television and radio station markets, and a variety of other situations. Usually the restriction of entry is combined with a regulatory cap on the price the franchised monopolist is allowed to charge. The argument usually put forward in favor of creating these franchised monopolies is that the industry in question is a natural monopoly: average cost is diminishing over a broad range of output levels, and minimum average cost can be achieved only by organizing the industry as a monopoly. The public utility and communications industries are often considered good examples. Certainly, that does appear to be the case for local electricity and telephone service where a given network probably exhibits declining average cost up to the point of universal coverage. But recent deregulation in telephone services and electricity generation show that, even for these industries, the natural monopoly rationale may not be
all-inclusive. In other cases, franchises may be based largely on political rationales. This seems to be true for the postal service in the United States and for a number of nationalized industries (airlines, radio and television, banking) in other countries. Creation of barriers to entry Although some barriers to entry may be independent of the monopolist’s own activities, other barriers may result directly from those activities. For example, ﬁrms may develop unique products or technologies and take extraordinary steps to keep these from being copied by competitors. Or ﬁrms may buy up unique resources to prevent potential entry. The De Beers cartel, for example, controls a large fraction of the world’s diamond mines. Finally, a would-be monopolist may enlist government aid in devising barriers to entry. It may lobby for legislation that restricts new entrants to ‘‘maintain an orderly market’’ or for health and safety regulations that raise potential entrants’ costs. Because the monopolist has both special knowledge of its business and signiﬁcant incentives to pursue these goals, it may have considerable success in creating such barriers to entry. FIGURE 14.1 Proﬁt Maximization and Price Determination for a Monopoly Chapter 14: Monopoly 503 The attempt by a monopolist to erect barriers to entry may involve real resource costs. Maintaining secrecy, buying unique resources, and engaging in political lobbying are all costly activities. A full analysis of monopoly should involve not only questions of cost minimization and output choice (as under perfect competition) but also an analysis of proﬁt-maximizing creation of entry barriers. However, we will not provide a detailed investigation of such questions here.1 Instead, we will take the presence of a single supplier on the market, and this single ﬁrm’s cost function, as given. Proﬁt Maximization and Output Choice To maximize proﬁts, a monopoly will choose to produce that output level for which marginal revenue is equal to marginal cost. Because the monopoly, in contrast to a perfectly competitive ﬁrm, faces a negatively sloped market demand curve, marginal revenue will be less than the market price. To sell an additional unit, the monopoly must lower its price on all units to be sold if it is to generate the extra demand necessary to absorb this marginal unit. The proﬁt-maximizing output level for
a ﬁrm is then the level Q! in Figure 14.1. At that level, marginal revenue is equal to marginal costs, and proﬁts are maximized. A proﬁt-maximizing monopolist produces that quantity for which marginal revenue is equal to marginal cost. In the diagram this quantity is given by Q!, which will yield a price of P! in the market. Monopoly proﬁts can be read as the rectangle of P!EAC. Price, costs D P* C MC E A AC D MR Q* Output per period 1For a simple treatment, see R. A. Posner, ‘‘The Social Costs of Monopoly and Regulation,’’ Journal of Political Economy 83 (August 1975): 807–27. 504 Part 6: Market Power Given the monopoly’s decision to produce Q!, the demand curve D indicates that a market price of P! will prevail. This is the price that demanders as a group are willing to pay for the output of the monopoly. In the market, an equilibrium price–quantity combination of P!, Q! will be observed. Assuming P! > AC, this output level will be proﬁtable, and the monopolist will have no incentive to alter output levels unless demand or cost conditions change. Hence we have the following principle Monopolist’s output. A monopolist will choose to produce that output for which marginal revenue equals marginal cost. Because the monopolist faces a downward-sloping demand curve, market price will exceed marginal revenue and the ﬁrm’s marginal cost at this output level. The inverse elasticity rule, again In Chapter 11 we showed that the assumption of proﬁt maximization implies that the gap between a price of a ﬁrm’s output and its marginal cost is inversely related to the price elasticity of the demand curve faced by the ﬁrm. Applying Equation 11.14 to the case of monopoly yields P MC " P 1 eQ, P, ¼ " (14:1) where now we use the elasticity of demand for the entire market (eQ, P) because the monopoly is the sole supplier of the good in question. This observation leads to two general conclusions about monopoly pricing. First, a monopoly will choose to operate only in regions in which the market demand curve is elastic (eQ, P < 1). If demand
were inelastic, then marginal revenue would be negative and thus could not be equated to marginal cost (which presumably is positive). Equation 14.1 also shows that eQ, P > 1 implies an (implausible) negative marginal cost. " " ¼ A second implication of Equation 14.1 is that the ﬁrm’s ‘‘markup’’ over marginal cost (measured as a fraction of price) depends inversely on the elasticity of market demand. For example, if eQ, P ¼ " 2MC, whereas if eQ, P ¼ 2, then Equation 14.1 shows that P 1.11MC. Notice also that if the elasticity of demand were constant along 10, then P " ¼ the entire demand curve, the proportional markup over marginal cost would remain unchanged in response to changes in input costs. Therefore, market price moves proportionally to marginal cost: Increases in marginal cost will prompt the monopoly to increase its price proportionally, and decreases in marginal cost will cause the monopoly to reduce its price proportionally. Even if elasticity is not constant along the demand curve, it seems clear from Figure 14.1 that increases in marginal cost will increase price (although not necessarily in the same proportion). As long as the demand curve facing the monopoly is downward sloping, upward shifts in MC will prompt the monopoly to reduce output and thereby obtain a higher price.2 We will examine all these relationships mathematically in Examples 14.1 and 14.2. Monopoly proﬁts Total proﬁts earned by the monopolist can be read directly from Figure 14.1. These are shown by the rectangle P!EAC and again represent the proﬁt per unit (price minus average cost) times the number of units sold. These proﬁts will be positive if market price exceeds average total cost. If P! < AC, however, then the monopolist can operate only at a long-term loss and will decline to serve the market. 2The comparative statics of a shift in the demand curve facing the monopolist are not so clear, however, and no unequivocal prediction about price can be made. For an analysis of this issue, see the discussion that follows and Problem 14.4. Chapter 14: Monopoly 505 Because (by assumption) no entry is possible into a monopoly market, the monopolist’s positive proﬁts can exist even
in the long run. For this reason, some authors refer to the proﬁts that a monopoly earns in the long run as monopoly rents. These proﬁts can be regarded as a return to that factor that forms the basis of the monopoly (e.g., a patent, a favorable location, or a dynamic entrepreneur); hence another possible owner might be willing to pay that amount in rent for the right to the monopoly. The potential for proﬁts is the reason why some ﬁrms pay other ﬁrms for the right to use a patent and why concessioners at sporting events (and on some highways) are willing to pay for the right to the concession. To the extent that monopoly rights are given away at less than their true market value (as in radio and television licensing), the wealth of the recipients of those rights is increased. Although a monopoly may earn positive proﬁts in the long run,3 the size of such proﬁts will depend on the relationship between the monopolist’s average costs and the demand for its product. Figure 14.2 illustrates two situations in which the demand, marginal revenue, and marginal cost curves are rather similar. As Equation 14.1 suggests, the price-marginal cost markup is about the same in these two cases. But average costs in Figure 14.2a are considerably lower than in Figure 14.2b. Although the proﬁt-maximizing decisions are similar in the two cases, the level of proﬁts ends up being different. In Figure 14.2a the monopolist’s price (P!) exceeds the average cost of producing Q! (labeled AC!) by a large extent, and sigAC! and the monopoly earns niﬁcant proﬁts are obtained. In Figure 14.2b, however, P! zero economic proﬁts, the largest amount possible in this case. Hence large proﬁts from a monopoly are not inevitable, and the actual extent of economic proﬁts may not always be a good guide to the signiﬁcance of monopolistic inﬂuences in a market. ¼ FIGURE 14.2 Monopoly Proﬁts Depend on the Relationship between the Demand and Average Cost Curves Both monopolies in this ﬁgure are equally ‘‘strong’’
if by this we mean they produce similar divergences between market price and marginal cost. However, because of the location of the demand and average cost curves, it turns out that the monopoly in (a) earns high proﬁts, whereas that in (b) earns no proﬁts. Consequently, the size of proﬁts is not a measure of the strength of a monopoly. Price D P* C* Price D P* = AC* D MC AC MR MC AC D MR Q* Quantity per period Q* Quantity per period (a) Monopoly with large proﬁts (b) Zero-proﬁt monopoly 3As in the competitive case, the proﬁt-maximizing monopolist would be willing to produce at a loss in the short run as long as market price exceeds average variable cost. 506 Part 6: Market Power There is no monopoly supply curve In the theory of perfectly competitive markets presented in Part 4, it was possible to speak of an industry supply curve. We constructed the long-run supply curve by allowing the market demand curve to shift and observing the supply curve that was traced out by the series of equilibrium price–quantity combinations. This type of construction is not possible for monopolistic markets. With a ﬁxed market demand curve, the supply ‘‘curve’’ for a monopoly will be only one point—namely, that price–quantity combination for which MR MC. If the demand curve should shift, then the marginal revenue curve would also shift, and a new proﬁt-maximizing output would be chosen. However, connecting the resulting series of equilibrium points on the market demand curves would have little meaning. This locus might have a strange shape, depending on how the market demand curve’s elasticity (and its associated MR curve) changes as the curve is shifted. In this sense the monopoly ﬁrm has no well-deﬁned ‘‘supply curve.’’ Each demand curve is a unique proﬁt-maximizing opportunity for a monopolist. ¼ EXAMPLE 14.1 Calculating Monopoly Output Suppose the market for Olympic-quality Frisbees (Q, measured in Frisbees bought per year) has a linear demand curve of the form or 2,000 Q ¼ " 20P 100 P ¼ " Q 20, and let
. It is convenient to think of a monopoly as arising from the ‘‘capture’’ of such a competitive industry and to treat the individual ﬁrms that constituted the competitive industry as now 4Notice that when c curve. ¼ 0, we have P ¼ a/2. That is, price should be halfway between zero and the price intercept of the demand 508 Part 6: Market Power being single plants in the monopolist’s empire. A prototype case would be John D. Rockefeller’s purchase of most of the U.S. petroleum reﬁneries in the late nineteenth century and his decision to operate them as part of the Standard Oil trust. We can then compare the performance of this monopoly with the performance of the previously competitive industry to arrive at a statement about the welfare consequences of monopoly. A graphical analysis Figure 14.3 provides a graphical analysis of the welfare effects of monopoly. If this market were competitive, output would be Qc—that is, production would occur where price is equal to long-run average and marginal cost. Under a simple single-price monopoly, output would be Qm because this is the level of production for which marginal revenue is equal to marginal cost. The restriction in output from Qc to Qm represents the misallocation brought about through monopolization. The total value of resources released by this output restriction is shown in Figure 14.3 as area FEQcQm. Essentially, the monopoly closes down some of the plants that were operating in the competitive case. These transferred inputs can be productively used elsewhere; thus, area FEQcQm is not a social loss. FIGURE 14.3 Allocational and Distributional Effects of Monopoly Monopolization of this previously competitive market would cause output to be reduced from Qc to Qm. Productive inputs worth FEQcQm are reallocated to the production of other goods. Consumer surplus equal to PmBCPc is transferred into monopoly proﬁts. Deadweight loss is given by BEF. Price A Pm Pc G MC Transfer from consumers to firm B E Deadweight loss C F D Value of transferred inputs MR Qm Qc Quantity per period Chapter 14: Monopoly 509 The restriction in output from Qc to Qm involves a total loss in consumer surplus of PmBEPc. Part of this loss, PmBCPc, is transferred to the monopoly as increased
proﬁt. Another part of the consumers’ loss, BEC, is not transferred to anyone but is a pure deadweight loss in the market. A second source of deadweight loss is given by area CEF. This is an area of lost producer surplus that does not get transferred to another source.5 The total deadweight loss from both sources is area BEF, sometimes called the deadweight loss triangle because of its roughly triangular shape. The gain PmBCPc in monopoly proﬁt from an increased price more than compensates for its loss of producer surplus CEF from the output reduction so that, overall, the monopolist ﬁnds reducing output from Qc to Qm to be proﬁtable. To illustrate the nature of this deadweight loss, consider Example 14.1, in which we calculated an equilibrium price of $75 and a marginal cost of $50. This gap between price and marginal cost is an indication of the efﬁciency-improving trades that are forgone through monopolization. Undoubtedly, there is a would-be buyer who is willing to pay, say, $60 for an Olympic Frisbee but not $75. A price of $60 would more than cover all the resource costs involved in Frisbee production, but the presence of the monopoly prevents such a mutually beneﬁcial transaction between Frisbee users and the providers of Frisbee-making resources. For this reason, the monopoly equilibrium is not Pareto optimal—an alternative allocation of resources would make all parties better off. Economists have made many attempts to estimate the overall cost of these deadweight losses in actual monopoly situations. Most of these estimates are rather small when viewed in the context of the whole economy.6 Allocational losses are larger, however, for some narrowly deﬁned industries. EXAMPLE 14.3 Welfare Losses and Elasticity The allocational effects of monopoly can be characterized fairly completely in the case of constant marginal costs and a constant price elasticity demand curve. To do so, assume again that constant marginal (and average) costs for a monopolist are given by c and that the demand curve has a constant elasticity form of ¼ where e is the price elasticity of demand (e < market will be Q P e, (14:13) 1). We know the competitive price in this " and the monopoly price is given by Pc ¼ c Pm
.5. What fraction of consumer surplus is lost through monopolization? 2? ¼ " ¼ " Monopoly, Product Quality, and Durability The market power enjoyed by a monopoly may be exercised along dimensions other than the market price of its product. If the monopoly has some leeway in the type, quality, or diversity of the goods it produces, then it would not be surprising for the ﬁrm’s decisions to differ from those that might prevail under a competitive organization of the market. Whether a monopoly will produce higher-quality or lower-quality goods than would be produced under competition is unclear, however. It all depends on the ﬁrm’s costs and the nature of consumer demand. Chapter 14: Monopoly 511 A formal treatment of quality Suppose consumers’ willingness to pay for quality (X) is given by the inverse demand function P(Q, X), where @P @Q < 0 and @P @X > 0: If the costs of producing Q and X are given by C(Q, X), the monopoly will choose Q and X to maximize " The ﬁrst-order conditions for a maximum are ¼ p P Q, X ð Q Þ C Q, X ð Þ : @p @Q ¼ Q, X P ð Þ þ Q @P @Q " CQ ¼ 0, @p @X ¼ Q @P @X " CX ¼ 0: (14:22) (14:23) (14:24) The ﬁrst of these equations repeats the usual rule that marginal revenue equals marginal cost for output decisions. The second equation states that, when Q is appropriately set, the monopoly should choose that level of quality for which the marginal revenue attainable from increasing the quality of its output by one unit is equal to the marginal cost of making such an increase. As might have been expected, the assumption of proﬁt maximization requires the monopolist to proceed to the margin of proﬁtability along all the dimensions it can. Notice, in particular, that the marginal demander’s valuation of quality per unit is multiplied by the monopolist’s output level when determining the proﬁt-maximizing choice. The level of product quality chosen under competitive conditions will also be the one that maximizes net social welfare: SW Q! ¼ 0 ð P Q, X ð �
� dQ C ð " Q, X, Þ (14:25) where Q! is the output level determined through the competitive process of marginal cost pricing, given X. Differentiation of Equation 14.25 with respect to X yields the ﬁrst-order condition for a maximum: @SW @X ¼ Q! 0 ð Q, X PXð Þ dQ CX ¼ " 0: (14:26) The monopolist’s choice of quality in Equation 14.24 targets the marginal consumer. The monopolist cares about the marginal consumer’s valuation of quality because increasing the attractiveness of the product to the marginal consumer is how it increases sales. The perfectly competitive market ends up providing a quality level in Equation 14.26, maximizing total consumer surplus (the total after subtracting the cost of providing that quality level), which is the same as the quality level that maximizes consumer surplus for the average consumer.7 Therefore, even if a monopoly and a perfectly competitive industry choose the same output level, they might opt for differing quality levels because each is 7The average marginal valuation (AV) of product quality is given by AV Q! ¼ 0 ð Q, X PX ð Þ dQ=Q: Hence Q 14.24. ’ AV ¼ Cx is the quality rule adopted to maximize net welfare under perfect competition. Compare this with Equation 512 Part 6: Market Power concerned with a different margin in its decision making. Only by knowing the speciﬁcs of the problem is it possible to predict the direction of these differences. For an example, see Problem 14.9; more detail on the theory of product quality and monopoly is provided in Problem 14.11. The durability of goods Much of the research on the effect of monopolization on quality has focused on durable goods. These are goods such as automobiles, houses, or refrigerators that provide services to their owners over several periods rather than being completely consumed soon after they are bought. The element of time that enters into the theory of durable goods leads to many interesting problems and paradoxes. Initial interest in the topic started with the question of whether monopolies would produce goods that lasted as long as similar goods produced under perfect competition. The intuitive notion that monopolies would ‘‘underproduce’’ durability (just as they choose an output below the competitive level) was soon shown to be incorrect by the Australian economist Peter Swan in
the early 1970s.8 Swan’s insight was to view the demand for durable goods as the demand for a ﬂow of services (i.e., automobile transportation) over several periods. He argued that both a monopoly and a competitive market would seek to minimize the cost of providing this ﬂow to consumers. The monopoly would, of course, choose an output level that restricted the ﬂow of services to maximize proﬁts, but—assuming constant returns to scale in production—there is no reason that durability per se would be affected by market structure. This result is sometimes referred to as Swan’s independence assumption. Output decisions can be treated independently from decisions about product durability. Subsequent research on the Swan result has focused on showing how it can be undermined by different assumptions about the nature of a particular durable good or by relaxing the implicit assumption that all demanders are the same. For example, the result depends critically on how durable goods deteriorate. The simplest type of deterioration is illustrated by a durable good, such as a lightbulb, that provides a constant stream of services until it becomes worthless. With this type of good, Equations 14.24 and 14.26 are identical, so Swan’s independence result holds. Even when goods deteriorate smoothly, the independence result continues to hold if a constant ﬂow of services can be maintained by simply replacing what has been used—this requires that new goods and old goods be perfect substitutes and inﬁnitely divisible. Outdoor house paint may, more or less, meet this requirement. On the other hand, most goods clearly do not. It is just not possible to replace a run-down refrigerator with, say, half of a new one. Once such more complex forms of deterioration are considered, Swan’s result may not hold because we can no longer fall back on the notion of providing a given ﬂow of services at minimal cost over time. In these more complex cases, however, it is not always the case that a monopoly will produce less durability than will a competitive market—it all depends on the nature of the demand for durability. Time inconsistency and heterogeneous demand Focusing on the service ﬂow from durable goods provides important insights on durability, but it does leave an important question unanswered—when should the monopoly produce the actual durable goods needed to provide the desired service ﬂow? Suppose, for example, that a lightbulb monopoly decides that its
proﬁt-maximizing output decision is to supply the services provided by 1 million 60-watt bulbs. If the ﬁrm decides to 8P. L. Swan, ‘‘Durability of Consumption Goods,’’ American Economic Review (December 1970): 884–94. Chapter 14: Monopoly 513 produce 1 million bulbs in the ﬁrst period, what is it to do in the second period (say, before any of the original bulbs burn out)? Because the monopoly chooses a point on the service demand curve where P > MC, it has a clear incentive to produce more bulbs in the second period by cutting price a bit. But consumers can anticipate this, so they may reduce their ﬁrst-period demand, waiting for a bargain. Hence the monopoly’s proﬁtmaximizing plan will unravel. Ronald Coase was the ﬁrst economist to note this ‘‘time inconsistency’’ that arises when a monopoly produces a durable good.9 Coase argued that its presence would severely undercut potential monopoly power—in the limit, competitive pricing is the only outcome that can prevail in the durable goods case. Only if the monopoly can succeed in making a credible commitment not to produce more in the second period can it succeed in its plan to achieve monopoly proﬁts on the service ﬂow from durable goods. Recent modeling of the durable goods question has examined how a monopolist’s choices are affected in situations where there are different types of demanders.10 In such cases, questions about the optimal choice of durability and about credible commitments become even more complicated. The monopolist must not only settle on an optimal scheme for each category of buyers, it must also ensure that the scheme intended for (say) type-1 demanders is not also attractive to type-2 demanders. Studying these sorts of models would take us too far aﬁeld, but some illustrations of how such ‘‘incentive compatibility constraints’’ work are provided in the Extensions to this chapter and in Chapter 18. Price Discrimination In some circumstances a monopoly may be able to increase proﬁts by departing from a single-price policy for its output. The possibility of selling identical goods at different prices is called price discrimination.11 Price discrimination. A monopoly engages in price discrimination if it is able to sell otherwise identical units of output at different prices.
Examples of price discrimination include senior citizen discounts for restaurant meals (which could instead be viewed as a price premium for younger customers), coffee sold at a lower price per ounce when bought in larger cup sizes, and different (net) tuition charged to different college students after subtracting their more or less generous ﬁnancial aid awards. A ‘‘nonexample’’ of price discrimination might be higher auto insurance premiums charged to younger drivers. It might be clearer to think of the insurance policies sold to younger and older drivers as being different products to the extent that younger drivers are riskier and result in many more claims having to be paid. Whether a price discrimination strategy is feasible depends crucially on the inability of buyers of the good to practice arbitrage. In the absence of transactions or information costs, the ‘‘law of one price’’ implies that a homogeneous good must sell everywhere for the same price. Consequently, price discrimination schemes are doomed to failure because demanders who can buy from the monopoly at lower prices will be more attractive sources 9R. Coase, ‘‘Durability and Monopoly,’’ Journal of Law and Economics (April 1972): 143–49. 10For a summary, see M. Waldman, ‘‘Durable Goods Theory for Real World Markets,’’ Journal of Economic Perspectives (Winter 2003): 131–54. 11A monopoly may also be able to sell differentiated products at differential price–cost margins. Here, however, we treat price discrimination only for a monopoly that produces a single homogeneous product. Price discrimination is an issue in other imperfectly competitive markets besides monopoly but is easiest to study in the simple case of a single ﬁrm. 514 Part 6: Market Power of the good—for those who must pay high prices—than is the monopoly itself. Proﬁtseeking middlemen will destroy any discriminatory pricing scheme. However, when resale is costly or can be prevented entirely, then price discrimination becomes possible. First-degree or perfect price discrimination If each buyer can be separately identiﬁed by a monopolist, then it may be possible to charge each the maximum price he or she would willingly pay for the good. This strategy of perfect (or ﬁrst-degree) price discrimination would then extract all available consumer surplus, leaving demanders as a group indifferent between buying the monopolist’s good or doing without it. The strategy
is illustrated in Figure 14.4. The ﬁgure assumes that buyers are arranged in descending order of willingness to pay. The ﬁrst buyer is willing to pay up to P1 for Q1 units of output; therefore, the monopolist charges P1 and obtains total revenues of P1Q1, as indicated by the lightly shaded rectangle. A second buyer is willing to pay up to P2 Q1) for Q2 " from this buyer. Notice that this strategy cannot succeed unless the second buyer is unable to resell the output he or she buys at P2 to the ﬁrst buyer (who pays P1 > P2). Q1 units of output; therefore, the monopolist obtains total revenue of P2(Q2 " The monopolist will proceed in this way up to the marginal buyer, the last buyer who is willing to pay at least the good’s marginal cost (labeled MC in Figure 14.4). Hence total quantity produced will be Q!. Total revenues collected will be given by the area DEQ!0. All consumer surplus has been extracted by the monopolist, and there is no deadweight loss in this situation. (Compare Figures 14.3 and 14.4.) Therefore, the allocation of resources under perfect price discrimination is efﬁcient, although it does entail a large transfer from consumer surplus into monopoly proﬁts. FIGURE 14.4 Perfect Price Discrimination Under perfect price discrimination, the monopoly charges a different price to each buyer. It sells Q1 units at P1, Q2 " will be DEQ!0. Q1 units at P2, and so forth. In this case the ﬁrm will produce Q!, and total revenues Price D P1 P2 MC E D 0 Q1 Q2 Q* Quantity per period Chapter 14: Monopoly 515 EXAMPLE 14.4 First-Degree Price Discrimination Consider again the Frisbee monopolist in Example 14.1. Because there are relatively few highquality Frisbees sold, the monopolist may ﬁnd it possible to discriminate perfectly among a few world-class ﬂippers. In this case, it will choose to produce that quantity for which the marginal buyer pays exactly the marginal cost of a Frisbee: 100 P ¼ " Q 20 ¼ MC ¼ 0:1Q: Hence and, at the margin, price and marginal cost are given by Q!
¼ 666 ¼ Now we can compute total revenues by integration: ¼ P MC 66:6: Q! P Q Þ ð 0 ð 55,511: R ¼ ¼ dQ ¼ 100Q! Q2 40 " 666 Q ¼ " Q ¼ 0 Total costs are total proﬁts are given by C Q ð Þ ¼ 0:05Q2 R p ¼ " þ C 10,000 ¼ 32,178; 23,333, ¼ (14:27) (14:28) (14:29) (14:30) (14:31) which represents a substantial increase over the single-price policy examined in Example 14.1 (which yielded 15,000). QUERY: What is the maximum price any Frisbee buyer pays in this case? Use this to obtain a geometric deﬁnition of proﬁts. Third-degree price discrimination through market separation First-degree price discrimination poses a considerable information burden for the monopoly—it must know the demand function for each potential buyer. A less stringent requirement would be to assume the monopoly can separate its buyers into relatively few identiﬁable markets (such as ‘‘rural–urban,’’ ‘‘domestic–foreign,’’ or ‘‘prime-time–offprime’’) and pursue a separate monopoly pricing policy in each market. Knowledge of the price elasticities of demand in these markets is sufﬁcient to pursue such a policy. The monopoly then sets a price in each market according to the inverse elasticity rule. Assuming that marginal cost is the same in all markets, the result is a pricing policy in which 1 ei Pi 1! þ Pj 1! þ ¼ " 1 ej " (14:32) or þ þ where Pi and Pj are the prices charged in markets i and j, which have price elasticities of demand given by ei and ej. An immediate consequence of this pricing policy is that the (14:33), Pi Pj ¼ 1 ð 1 ð 1=ejÞ 1=eiÞ 516 Part 6: Market Power FIGURE 14.5 Separated Markets Raise the Possibility of Third-Degree Price Discrimination If two markets are separate, then a monopolist can maximize proﬁts by selling its
product at different prices in the two markets. This would entail choosing that output for which MC MR in each of the markets. The diagram shows that the market with a less elastic demand curve will be charged the higher price by the price discriminator. ¼ Price P1 P2 MC D1 MR1 D2 MR2 Quantity in market 1 Q1 0 Q2 Quantity in market 2 proﬁt-maximizing price will be higher in markets in which demand is less elastic. If, for example, ei ¼ " 4/3—prices will be one third higher in market i, the less elastic market. 3, then Equation 14.33 shows that Pi/Pj ¼ 2 and ej ¼ " Figure 14.5 illustrates this result for two markets that the monopoly can serve at constant marginal cost (MC). Demand is less elastic in market 1 than in market 2; thus, the gap between price and marginal revenue is larger in the former market. Proﬁt maximization requires that the ﬁrm produce Q!1 in market 1 and Q!2 in market 2, resulting in a higher price in the less elastic market. As long as arbitrage between the two markets can be prevented, this price difference can persist. The two-price discriminatory policy is clearly more proﬁtable for the monopoly than a single-price policy would be because the ﬁrm can always opt for the latter policy should market conditions warrant. The welfare consequences of third-degree price discrimination are, in principle, ambiguous. Relative to a single-price policy, the discriminating policy requires raising the price in the less elastic market and reducing it in the more elastic one. Hence the changes have an offsetting effect on total allocational losses. A more complete analysis suggests the intuitively plausible conclusion that the multiple-price policy will be allocationally superior to a single-price policy only in situations in which total output is increased through discrimination. Example 14.5 illustrates a simple case of linear demand curves in which a single-price policy does result in greater allocational losses.12 12For a detailed discussion, see R. Schmalensee, ‘‘Output and Welfare Implications of Monopolistic Third-Degree Price Discrimination,’’ American Economic Review (March 1981): 242–47. See also Problem 14.13. Chapter 14: Monopoly 517 EXAMPLE 14.5 Third-Degree Price
Discrimination Suppose that a monopoly producer of widgets has a constant marginal cost of c products in two separated markets whose inverse demand functions are ¼ 6 and sells its 24 Q1 P1 ¼ and P2 ¼ Notice that consumers in market 1 are more eager to buy than are consumers in market 2 in the sense that the former are willing to pay more for any given quantity. Using the results for linear demand curves from Example 14.2 shows that the proﬁt-maximizing price–quantity combinations in these two markets are: 0:5Q2: (14:34) 12 " " 24 6 þ 2 ¼ P!1 ¼ 99. We can With this pricing strategy, proﬁts are p " compute the deadweight losses in the two markets by recognizing that the competitive output (with P 6) in market 1 is 18 and in market 2 is 12: 15, Q!1 ¼ (15 9, Q!2 ¼ 6) Æ 6 81 9, P!2 ¼ 6) Æ 9 (14:35) MC 18 (9 ¼ ¼ þ þ ¼ " 6: 12 6 þ 2 ¼ ¼ ¼ DW2 6 DW ¼ ¼ ¼ DW1 þ 0:5 P!1 " ð 40:5 9 þ ¼ 18 Þð " 49:5: 9 Þ þ 0:5 P!2 " ð 6 12 Þð 6 Þ " (14:36) A single-price policy. In this case, constraining the monopoly to charge a single price would reduce welfare. Under a single-price policy, the monopoly would simply cease serving market 2 because it can maximize proﬁts by charging a price of 15, and at that price no widgets will be bought in market 2 (because the maximum willingness to pay is 12). Therefore, total deadweight loss in this situation is increased from its level in Equation 14.36 because total potential consumer surplus in market 2 is now lost: DW1 þ DW2 ¼ (14:37) 12 ð 76:5: 40:5 40:5 DW Þ ¼ 0:5 36 12 ¼ þ " þ ¼ " Þð 6 0 This illustrates a situation where third-degree price discrimination is welfare improving over a single-price policy—when the discriminatory policy permits ‘‘smaller’’ markets to be served. Whether
such a situation is common is an important policy question (e.g., consider the case of U.S. pharmaceutical manufacturers charging higher prices at home than abroad). QUERY: Suppose these markets were no longer separated. How would you construct the market demand in this situation? Would the monopolist’s proﬁt-maximizing single price still be 15? Second-Degree Price Discrimination Through Price Schedules The examples of price discrimination examined in the previous section require the monopoly to separate demanders into a number of categories and then choose a proﬁtmaximizing price for each such category. An alternative approach would be for the monopoly to choose a (possibly rather complex) price schedule that provides incentives for demanders to separate themselves depending on how much they wish to buy. Such schemes include quantity discounts, minimum purchase requirements or ‘‘cover’’ charges, and tie-in sales. These plans would be adopted by a monopoly if they yielded greater proﬁts than would a single-price policy, after accounting for any possible costs of 518 Part 6: Market Power implementing the price schedule. Because the schedules will result in demanders paying different prices for identical goods, this form of (second-degree) price discrimination is feasible only when there are no arbitrage possibilities. Here we look at one simple case. The Extensions to this chapter and portions of Chapter 18 look at other aspects of second-degree price discrimination. Two-part tariffs One form of pricing schedule that has been extensively studied is a linear two-part tariff, under which demanders must pay a ﬁxed fee for the right to consume a good and a uniform price for each unit consumed. The prototype case, ﬁrst studied by Walter Oi, is an amusement park (perhaps Disneyland) that sets a basic entry fee coupled with a stated marginal price for each amusement used.13 Mathematically, this scheme can be represented by the tariff any demander must pay to purchase q units of a good: T q ð Þ ¼ a þ pq, (14:38) where a is the ﬁxed fee and p is the marginal price to be paid. The monopolist’s goal then is to choose a and p to maximize proﬁts, given the demand for this product. Because the average price paid by any demander is given by T q ¼ a q þ p, p ¼ (14:39
) this tariff is feasible only when those who pay low average prices (those for whom q is large) cannot resell the good to those who must pay high average prices (those for whom q is small). One approach described by Oi for establishing the parameters of this linear tariff would be for the ﬁrm to set the marginal price, p, equal to MC and then set a to extract the maximum consumer surplus from a given set of buyers. One might imagine buyers being arrayed according to willingness to pay. The choice of p MC would then maximize consumer surplus for this group, and a could be set equal to the surplus enjoyed by the least eager buyer. He or she would then be indifferent about buying the good, but all other buyers would experience net gains from the purchase. ¼ ¼ This feasible tariff might not be the most proﬁtable, however. Consider the effects on proﬁts of a small increase in p above MC. This would result in no net change in the profits earned from the least willing buyer. Quantity demanded would drop slightly at the margin where p MC, and some of what had previously been consumer surplus (and therefore part of the ﬁxed fee, a) would be converted into variable proﬁts because now p > MC. For all other demanders, proﬁts would be increased by the price increase. Although each will pay a bit less in ﬁxed charges, proﬁts per unit bought will increase to a greater extent.14 In some cases it is possible to make an explicit calculation of the optimal two-part tariff. Example 14.6 provides an illustration. More generally, however, optimal schedules will depend on a variety of contingencies. Some of the possibilities are examined in the Extensions to this chapter. 13W. Y. Oi, ‘‘A Disneyland Dilemma: Two-Part Tariffs for a Mickey Mouse Monopoly,’’ Quarterly Journal of Economics (February 1971): 77–90. Interestingly, the Disney empire once used a two-part tariff but abandoned it because the costs of administering the payment schemes for individual rides became too high. Like other amusement parks, Disney moved to a single-admissions price policy (which still provided them with ample opportunities for price discrimination, especially with the multiple parks at Disney World). 14This follows because qi(MC) > q1(MC), where qi(MC) is the quantity demanded when p
will require natural monopolies to operate at a loss. Natural monopolies, by deﬁnition, exhibit decreasing average costs over a broad range of output levels. The cost curves for such a ﬁrm might look like those shown in Figure 14.6. In the absence of regulation, the monopoly would produce output level QA and receive a price of PA for its product. Proﬁts in this situation are given by the rectangle PAABC. A regulatory agency might instead set a price of PR for the monopoly. At this price, QR is demanded, and the marginal cost of producing this output level is also PR. Consequently, marginal cost pricing has been achieved. Unfortunately, because of the negative slope of the ﬁrm’s average cost curve, the price PR ( marginal cost) decreases below average costs. With this regulated price, the monopoly must operate at a loss of GFEPR. Because no ﬁrm can operate indeﬁnitely at a loss, this poses a dilemma for the regulatory agency: Either it must abandon its goal of marginal cost pricing, or the government must subsidize the monopoly forever. ¼ Two-tier pricing systems One way out of the marginal cost pricing dilemma is the implementation of a multiprice system. Under such a system the monopoly is permitted to charge some users a high price while maintaining a low price for marginal users. In this way the demanders paying the high price in effect subsidize the losses of the low-price customers. Such a pricing scheme is shown in Figure 14.7. Here the regulatory commission has decided that some users will pay a relatively high price, P1. At this price, Q1 is demanded. Other users Chapter 14: Monopoly 521 FIGURE 14.7 Two-Tier Pricing Schedule By charging a high price (P1) to some users and a low price (P2) to others, it may be possible for a regulatory commission to (1) enforce marginal cost pricing and (2) create a situation where the proﬁts from one class of user (P1DBA) subsidize the losses of the other class (BFEC). Price P1 A P2 D B C Q1 F E D Q2 AC MC Quantity per period (presumably those who would not buy the good at the P1 price) are offered a lower price, Q1. Consequently, a total outP2. This lower price generates additional demand of Q2 " put
of Q2 is produced at an average cost of A. With this pricing system, the proﬁts on the sales to high-price demanders (given by the rectangle P1DBA) balance the losses incurred on the low-priced sales (BFEC). Furthermore, for the ‘‘marginal user,’’ the marginal cost pricing rule is being followed: It is the ‘‘intramarginal’’ user who subsidizes the ﬁrm so it does not operate at a loss. Although in practice it may not be so simple to establish pricing schemes that maintain marginal cost pricing and cover operating costs, many regulatory commissions do use price schedules that intentionally discriminate against some users (e.g., businesses) to the advantage of others (consumers). Rate of return regulation Another approach followed in many regulatory situations is to permit the monopoly to charge a price above marginal cost that is sufﬁcient to earn a ‘‘fair’’ rate of return on investment. Much analytical effort is then devoted to deﬁning the ‘‘fair’’ rate concept and to developing ways in which it might be measured. From an economic point of view, some of the most interesting questions about this procedure concern how the regulatory activity affects the ﬁrm’s input choices. If, for example, the rate of return allowed to ﬁrms exceeds what owners might obtain on investment under competitive circumstances, there will be an incentive to use relatively more capital input than would truly minimize costs. And if regulators delay in making rate decisions, this may give ﬁrms cost-minimizing 522 Part 6: Market Power incentives that would not otherwise exist. We will now brieﬂy examine a formal model of such possibilities.16 A formal model Suppose a regulated utility has a production function of the form This ﬁrm’s actual rate of return on capital is then deﬁned as q k, l f ð : Þ ¼ pf k, l Þ " ð k wl, s ¼ (14:43) (14:44) where p is the price of the ﬁrm’s output (which depends on q) and w is the wage rate for labor input. If s is constrained by regulation to be equal to (say) s, then the ﬁrm’s
problem is to maximize proﬁts ¼ subject to this regulatory constraint. The Lagrangian for this problem is Þ " " p pf k, l ð wl vk + pf k, l ð Þ " ¼ wl vk k wl ½ þ sk pf : k, l ð Þ) " þ " (14:45) (14:46) Notice that if l maximizing ﬁrm. If l ¼ 0, regulation is ineffective and the monopoly behaves like any proﬁt- 1, Equation 14.46 reduces to ¼ + s v k, Þ " which, assuming s > v (which it must be if the ﬁrm is not to earn less than the prevailing rate of return on capital elsewhere), means this monopoly will hire inﬁnite amounts of capital—an implausible result. Hence 0 < l < 1. The ﬁrst-order conditions for a maximum are ¼ ð (14:47) @+ @l ¼ @+ @k ¼ @+ @k ¼ pfl " pfk " wl sk " þ pf " 0, pf1Þ ¼ " 0, pfkÞ ¼ k, l ð Þ ¼ 0: (14:48) The ﬁrst of these conditions implies that the regulated monopoly will hire additional labor input up to the point at which pfl ¼ w—a result that holds for any proﬁt-maximizing ﬁrm. For capital input, however, the second condition implies that or 1 ð " pfk ¼ k Þ v " ks pfk ¼ v 1 ks " " Because s > v and l < 1, Equation 14.50 implies pfk < v: Þ : (14:49) (14:50) (14:51) 16This model is based on H. Averch and L. L. Johnson, Economic Review (December 1962): 1052–69. ‘‘Behavior of the Firm under Regulatory Constraint,’’ American Chapter 14: Monopoly 523 The ﬁrm will hire more capital (and achieve a lower marginal productivity of capital) than it would under unregulated conditions. Therefore, ‘‘overcapitalization’’
may be a regulatory-induced misallocation of resources for some utilities. Although we shall not do so here, it is possible to examine other regulatory questions using this general analytical framework. Dynamic Views Of Monopoly The static view that monopolistic practices distort the allocation of resources provides the principal economic rationale for favoring antimonopoly policies. Not all economists believe that the static analysis should be deﬁnitive, however. Some authors, most notably J. A. Schumpeter, have stressed the beneﬁcial role that monopoly proﬁts can play in the process of economic development.17 These authors place considerable emphasis on innovation and the ability of particular types of ﬁrms to achieve technical advances. In this context the proﬁts that monopolistic ﬁrms earn provide funds that can be invested in research and development. Whereas perfectly competitive ﬁrms must be content with a normal return on invested capital, monopolies have ‘‘surplus’’ funds with which to undertake the risky process of research. More important, perhaps, the possibility of attaining a monopolistic position—or the desire to maintain such a position—provides an important incentive to keep one step ahead of potential competitors. Innovations in new products and cost-saving production techniques may be integrally related to the possibility of monopolization. Without such a monopolistic position, the full beneﬁts of innovation could not be obtained by the innovating ﬁrm. Schumpeter stresses the point that the monopolization of a market may make it less costly for a ﬁrm to plan its activities. Being the only source of supply for a product eliminates many of the contingencies that a ﬁrm in a competitive market must face. For example, a monopoly may not have to spend as much on selling expenses (e.g., advertising, brand identiﬁcation, and visiting retailers) as would be the case in a more competitive industry. Similarly, a monopoly may know more about the speciﬁc demand curve for its product and may more readily adapt to changing demand conditions. Of course, whether any of these purported beneﬁts of monopolies outweigh their allocational and distributional disadvantages is an empirical question. Issues of innovation and cost savings cannot be answered by recourse to a priori arguments; detailed investigation of real-world markets is a necessity. SUMMARY In this chapter we have examined
models of markets in which there is only a single monopoly supplier. Unlike the competitive case investigated in Part 4, monopoly ﬁrms do not exhibit price-taking behavior. Instead, the monopolist can choose the price–quantity combination on the market demand curve that is most proﬁtable. A number of consequences then follow from this market power. • The most proﬁtable level of output for the monopolist is the one for which marginal revenue is equal to marginal cost. At this output level, price will exceed marginal cost. The proﬁtability of the monopolist will depend on the relationship between price and average cost. • Relative to perfect competition, monopoly involves a loss of consumer surplus for demanders. Some of this is transferred into monopoly proﬁts, whereas some of the loss in consumer supply represents a deadweight loss of overall economic welfare. 17See, for example, J. A. Schumpeter, Capitalism, Socialism and Democracy, 3rd ed. (New York: Harper & Row, 1950), especially chap. 8. 524 Part 6: Market Power • Monopolists may opt for different levels of quality than would perfectly competitive ﬁrms. Durable goods monopolists may be constrained by markets for used goods. • A monopoly may be able to increase its proﬁts further through price discrimination—that is, charging different prices to different categories of buyers. The ability the monopoly to practice price discrimination of depends on its ability to prevent arbitrage among buyers. • Governments often choose to regulate natural monopolies (ﬁrms with diminishing average costs over a broad range of output levels). The type of regulatory mechanisms adopted can affect the behavior of the regulated ﬁrm. PROBLEMS 14.1 A monopolist can produce at constant average and marginal costs of AC given by Q 53 P. ¼ " MC ¼ ¼ 5. The ﬁrm faces a market demand curve a. Calculate the proﬁt-maximizing price–quantity combination for the monopolist. Also calculate the monopolist’s proﬁts. b. What output level would be produced by this industry under perfect competition (where price c. Calculate the consumer surplus obtained by consumers in case (b). Show that this exceeds the sum of the monopolist’s proﬁts and the consumer surplus received in case (a). What is the value of the �
�‘deadweight loss’’ from monopolization? marginal cost)? ¼ 14.2 A monopolist faces a market demand curve given by a. If the monopolist can produce at constant average and marginal costs of AC Q 70 p: " ¼ MC 6, what output level will the monop- ¼ ¼ olist choose to maximize proﬁts? What is the price at this output level? What are the monopolist’s proﬁts? b. Assume instead that the monopolist has a cost structure where total costs are described by C Q ð Þ ¼ 0:25Q2 5Q þ " 300: With the monopolist facing the same market demand and marginal revenue, what price–quantity combination will be chosen now to maximize proﬁts? What will proﬁts be? c. Assume now that a third cost structure explains the monopolist’s position, with total costs given by C Q ð Þ ¼ 0:0133Q3 5Q þ " 250: Again, calculate the monopolist’s price–quantity combination that maximizes proﬁts. What will proﬁt be? Hint: Set MC MR as usual and use the quadratic formula to solve the second-order equation for Q. ¼ d. Graph the market demand curve, the MR curve, and the three marginal cost curves from parts (a), (b), and (c). Notice that the monopolist’s proﬁt-making ability is constrained by (1) the market demand curve (along with its associated MR curve) and (2) the cost structure underlying production. 14.3 A single ﬁrm monopolizes the entire market for widgets and can produce at constant average and marginal costs of Originally, the ﬁrm faces a market demand curve given by AC MC 10: ¼ ¼ 60 Q ¼ " P: a. Calculate the proﬁt-maximizing price–quantity combination for the ﬁrm. What are the ﬁrm’s proﬁts? b. Now assume that the market demand curve shifts outward (becoming steeper) and is given by 45 Q ¼ " 0:5P: Chapter 14: Monopoly 525 What is the ﬁrm’s proﬁt-maximizing
price–quantity combination now? What are the ﬁrm’s proﬁts? c. Instead of the assumptions of part (b), assume that the market demand curve shifts outward (becoming ﬂatter) and is given by What is the ﬁrm’s proﬁt-maximizing price–quantity combination now? What are the ﬁrm’s proﬁts? d. Graph the three different situations of parts (a), (b), and (c). Using your results, explain why there is no real supply curve for a monopoly. 100 Q ¼ " 2P: 14.4 Suppose the market for Hula Hoops is monopolized by a single ﬁrm. a. Draw the initial equilibrium for such a market. b. Now suppose the demand for Hula Hoops shifts outward slightly. Show that, in general (contrary to the competitive case), it will not be possible to predict the effect of this shift in demand on the market price of Hula Hoops. c. Consider three possible ways in which the price elasticity of demand might change as the demand curve shifts: It might increase, it might decrease, or it might stay the same. Consider also that marginal costs for the monopolist might be increasing, decreasing, or constant in the range where MR MC. Consequently, there are nine different combinations of ¼ types of demand shifts and marginal cost slope conﬁgurations. Analyze each of these to determine for which it is possible to make a deﬁnite prediction about the effect of the shift in demand on the price of Hula Hoops. 14.5 Suppose a monopoly market has a demand function in which quantity demanded depends not only on market price (P) but also on the amount of advertising the ﬁrm does (A, measured in dollars). The speciﬁc form of this function is The monopolistic ﬁrm’s cost function is given by Q 20 ¼ ð P 1 Þð þ " 0:1A " 0:01A2 : Þ a. Suppose there is no advertising (A yield? What will be the monopoly’s proﬁts? ¼ C ¼ 10Q 15 A: þ þ 0). What output will the proﬁt-maximizing ﬁrm choose? What market price will this b.
Now let the ﬁrm also choose its optimal level of advertising expenditure. In this situation, what output level will be chosen? What price will this yield? What will the level of advertising be? What are the ﬁrm’s proﬁts in this case? Hint: This can be worked out most easily by assuming the monopoly chooses the proﬁt-maximizing price rather than quantity. 14.6 Suppose a monopoly can produce any level of output it wishes at a constant marginal (and average) cost of $5 per unit. Assume the monopoly sells its goods in two different markets separated by some distance. The demand curve in the ﬁrst market is given by and the demand curve in the second market is given by Q1 ¼ 55 " P1, Q2 ¼ 70 " 2P2: a. If the monopolist can maintain the separation between the two markets, what level of output should be produced in each market, and what price will prevail in each market? What are total proﬁts in this situation? b. How would your answer change if it costs demanders only $5 to transport goods between the two markets? What would be the monopolist’s new proﬁt level in this situation? c. How would your answer change if transportation costs were zero and then the ﬁrm was forced to follow a single-price policy? d. Now assume the two different markets 1 and 2 are just two individual consumers. Suppose the ﬁrm could adopt a linear two-part tariff under which marginal prices charged to the two consumers must be equal but their lump-sum entry fees might vary. What pricing policy should the ﬁrm follow? 526 Part 6: Market Power 14.7 Suppose a perfectly competitive industry can produce widgets at a constant marginal cost of $10 per unit. Monopolized marginal costs increase to $12 per unit because $2 per unit must be paid to lobbyists to retain the widget producers’ favored position. Suppose the market demand for widgets is given by QD ¼ 1,000 50P: " a. Calculate the perfectly competitive and monopoly outputs and prices. b. Calculate the total loss of consumer surplus from monopolization of widget production. c. Graph your results and explain how they differ from the usual analysis. 14.8 Suppose the government wishes to combat the undesirable allocational effects of a monopoly through the use of a subsidy.
a. Why would a lump-sum subsidy not achieve the government’s goal? b. Use a graphical proof to show how a per-unit-of-output subsidy might achieve the government’s goal. c. Suppose the government wants its subsidy to maximize the difference between the total value of the good to consumers and the good’s total cost. Show that, to achieve this goal, the government should set where t is the per-unit subsidy and P is the competitive price. Explain your result intuitively. t P ¼ " 1 eQ, P, 14.9 Suppose a monopolist produces alkaline batteries that may have various useful lifetimes (X). Suppose also that consumers’ (inverse) demand depends on batteries’ lifetimes and quantity (Q) purchased according to the function where g 0 < 0. That is, consumers care only about the product of quantity times lifetime: They are willing to pay equally for many short-lived batteries or few long-lived ones. Assume also that battery costs are given by P Q where C 0(X) > 0. Show that, in this case, the monopoly will opt for the same level of X as does a competitive industry even though levels of output and prices may differ. Explain your result. Hint: Treat XQ as a composite commodity. C Q, X ð C X ð Q, Þ Analytical Problems 14.10 Taxation of a monopoly good The taxation of monopoly can sometimes produce results different from those that arise in the competitive case. This problem looks at some of those cases. Most of these can be analyzed by using the inverse elasticity rule (Equation 14.1). a. Consider ﬁrst an ad valorem tax on the price of a monopoly’s good. This tax reduces the net price received by the t)—where t is the proportional tax rate. Show that, with a linear demand curve and constant monopoly from P to P(1 marginal cost, the imposition of such a tax causes price to increase by less than the full extent of the tax. " b. Suppose that the demand curve in part (a) were a constant elasticity curve. Show that the price would now increase by pre- cisely the full extent of the tax. Explain the difference between these two cases. c. Describe a case where the imposition of an ad valorem tax on a monopoly would cause the price to increase by more than the tax. d
. A speciﬁc tax is a ﬁxed amount per unit of output. If the tax rate is t per unit, total tax collections are tQ. Show that the imposition of a speciﬁc tax on a monopoly will reduce output more (and increase price more) than will the imposition of an ad valorem tax that collects the same tax revenue. 14.11 More on the welfare analysis of quality choice An alternative way to study the welfare properties of a monopolist’s choices is to assume the existence of a utility function for the customers of the monopoly of the form utility U(Q, X), where Q is quantity consumed and X is the quality associated with that quantity. A social planner’s problem then would be to choose Q and X to maximize social welfare as represented by SW C(Q, X). U(Q, X) ¼ ¼ " Chapter 14: Monopoly 527 a. What are the ﬁrst-order conditions for a welfare maximum? b. The monopolist’s goal is to choose the Q and X that maximize p tions for this maximization? P(Q, X) Æ Q " ¼ C(Q, X). What are the ﬁrst-order condi- c. Use your results from parts (a) and (b) to show that, at the monopolist’s preferred choices, @SW/@Q > 0. That is, as we have already shown, prove that social welfare would be improved if more were produced. Hint: Assume that @U/@Q P. d. Show that, at the monopolist’s preferred choices, the sign of @SW/@X is ambiguous—that is, it cannot be determined (on the sole basis of the general theory of monopoly) whether the monopolist produces either too much or too little quality. ¼ 14.12 The welfare effects of third-degree price discrimination In an important 1985 article,18 Hal Varian shows how to assess third-degree price discrimination using only properties of the indirect utility function (see Chapter 3). This problem provides a simpliﬁed version of his approach. Suppose that a single good is sold in two separated markets. Quantities in the two markets are designated by q1, q2 with prices p1, p2. Consumers of the good are assumed to be characterized by an indirect utility function that takes a quasi
-linear form: V( p1, p2, I) I. þ Income is assumed to have an exogenous component (!I), and the monopoly earns proﬁts of p q2), where c is marginal and average cost (which is assumed to be constant). v( p1, p2) c(q1 þ ¼ p2q2 " p1q1 þ ¼ a. Given this setup, let’s ﬁrst show some facts about this kind of indirect utility function. (1) Use Roy’s identity (see the Extensions to Chapter 5) to show that the Marshallian demand functions for the two goods in this problem are given by qi ( p1, p2, I) @v/@pi. ¼ " (2) Show that the function v ( p1, p2) is convex in the prices. (3) Because social welfare (SW) can be measured by the indirect utility function of the consumers, show that the welfare Dp. How does this expression compare with the notion (intro- impact of any change in prices is given by DSW Dv duced in Chapter 12) that any change in welfare is the sum of changes in consumer and producer surplus? b. Suppose now that we wish to compare the welfare associated with a single-price policy for these two markets, p1 ¼ with the welfare associated with different prices in the two markets, p1 ¼ change in social welfare from adopting a two-price policy is given by DSW order Taylor expansion for the function v around p!1, p!2 together with Roy’s identity and the fact that v is convex. p, p!2. Show that an upper bound to the p!1 and p2 ¼. Hint: Use a ﬁrstq2Þ q!1 þ c p * ð Þð p2 ¼ q!2 " q1 " þ " ¼ c. Show why the results of part (b) imply that, for social welfare to increase from the adoption of the two-price policy, total quantity demanded must increase. two-price policy is given by DSW d. Use an approach similar to that taken in part (b) to show that a lower bound to the change in social welfare from adopting a p!2 " e. Notice that the approach taken here never uses the fact that the price–quantity
combinations studied are proﬁt maximizing for the monopolist. Can you think of situations (other than third-degree price discrimination) where the analysis here might apply? Note: Varian shows that the bounds for welfare changes can be tightened a bit in the price discrimination case by using proﬁt maximization.. Can you interpret this lower bound condition? q1Þ þ ð q!1 " Þð p!1 " q!2 " q2Þ c Þð + ð c SUGGESTIONS FOR FURTHER READING Posner, R. A. Regulation.’’ 807–27. ‘‘The Social Costs of Monopoly and Journal of Political Economy 83 (1975): An analysis of the probability that monopolies will spend resources on the creation of barriers to entry and thus have higher costs than perfectly competitive ﬁrms. Schumpeter, J. A. Capitalism, Socialism and Democracy, 3rd ed. New York: Harper & Row, 1950. Classic defense of the role of the entrepreneur and economic proﬁts in the economic growth process. Spence, M. ‘‘Monopoly, Quality, and Regulation.’’ Bell Journal of Economics (April 1975): 417–29. Develops the approach to product quality used in this text and provides a detailed analysis of the effects of monopoly. Stigler, G. J. ‘‘The Theory of Economic Regulation.’’ Bell Journal of Economics and Management Science 2 (Spring 1971): 3. Early development of the ‘‘capture’’ hypothesis of regulatory behavior—that the industry captures the agency supposed to regulate it and uses that agency to enforce entry barriers and further enhance proﬁts. Tirole, J. The Theory of Industrial Organization. Cambridge, MA: MIT Press, 1989, chaps. 1–3. A complete analysis of the theory of monopoly pricing and product choice. Varian, H. R. Microeconomic Analysis, 3rd ed. New York: W. W. Norton, 1992, chap. 14. Provides a succinct analysis of the role of incentive compatibility constraints in second-degree price discrimination. 18H. R. Varian, ‘‘Price Discrimination and Social Welfare,’’ American Economic Review (September 1985): 870–75. EXTENSIONS OPTIMAL LINEAR TWO-PART TARIF
FS In Chapter 14 we examined a simple illustration of ways in which a monopoly may increase proﬁts by practicing seconddegree price discrimination—that is, by establishing price (or ‘‘outlay’’) schedules that prompt buyers to separate themselves into distinct market segments. Here we pursue the topic of linear tariff schedules a bit further. Nonlinear pricing schedules are discussed in Chapter 18. E14.1 Structure of the problem To examine issues related to price schedules in a simple context for each demander, we deﬁne the ‘‘valuation function’’ as (i) við q q pið q si, þ Þ ¼ Þ ’ where pi(q) is the inverse demand function for individual i and si is consumer surplus. Hence vi represents the total value to individual i of undertaking transactions of amount q, which includes total spending on the good plus the value of consumer surplus obtained. Here we will assume (a) there are only two demanders1 (or homogeneous groups of demanders) and (b) person 1 has stronger preferences for this good than person 2 in the sense that > v2ð q v1ð q (ii) Þ Þ for all values of q. The monopolist is assumed to have constant marginal costs (denoted by c) and chooses a tariff (revenue) schedule, T(q), that maximizes proﬁts given by c ð p T T ¼ q1 þ ð q1Þ þ ð q2Þ " where qi represents the quantity chosen by person i. In selecting a price schedule that successfully distinguishes among consumers, the monopolist faces two constraints. To ensure that the low-demand person (2) is served, it is necessary that, q2Þ (iii) 0: T (iv) v2ð q2Þ " q2Þ + ð That is, person 2 must derive a net beneﬁt from her optimal choice, q2. Person 1, the high-demand individual, must also obtain a net gain from his chosen consumption level (q1) and must prefer this choice to the output choice made by person 2: q1Þ + v1ð If the monopolist does not recognize this ‘‘incentive compatibility’’ constraint, it may ﬁnd that person 1 opts for the q2Þ " q1
Þ " : q2Þ ð v1ð (v) T T ð 1Generalizations to many demanders are nontrivial. For a discussion, see Wilson (1993, chaps. 2–5). portion of the price schedule intended for person 2, thereby destroying the goal of obtaining self-selected market separation. Given this general structure, we can proceed to illustrate a number of interesting features of the monopolist’s problem. E14.2 Pareto superiority Permitting the monopolist to depart from a simple singleprice scheme offers the possibility of adopting ‘‘Pareto superior’’ tariff schedules under which all parties to the transaction are made better off. For example, suppose the monopolist’s proﬁt-maximizing price is pM. At this price, person 2 consumes qM 2 and receives a net value from this consumption of v2ð qM 2 Þ " pMqM 2 : A tariff schedule for which T q ð Þ ¼ pMq pq þ a % qM for q 2, * for q > qM 2, (vi) (vii) where a > 0 and c < p < pM, may yield increased proﬁts for the monopolist as well as increased welfare for person 1. Speciﬁcally, consider values of a and p such that a pqM 1 ¼ þ pMqM 1 or a pM " ¼ ð p qM 1, Þ (viii) where qM represents consumption of person 1 under a single1 price policy. In this case, a and p are set so that person 1 can still afford to buy qM 1 under the new price schedule. Because p < pM, however, he will opt for q!1 > qM 1. Because person 1 could have bought qM 1 but chose q!1 instead, he must be better off under the new schedule. The monopoly’s proﬁts are now given by p a pq1 þ þ ¼ pMqM 2 " q1 þ c ð qM 2 Þ and a p " pM ¼ pq1 þ q1 " c ð where pM is the monopoly’s single-price proﬁts qM 1 þ ð pM ". Substitution for a from Equation v
e.g., the Honda Accord comes in DX, LX, EX, and SX conﬁgurations) that act as tied goods in separating buyers into various market niches. A 1992 study by J. E. Kwoka examines one speciﬁc U.S. manufacturer (Chrysler) and shows how market segmentation is achieved through quality variation. The author calculates that signiﬁcant transfer from consumer surplus to ﬁrms occurs as a result of such segmentation. Generally, this sort of price discrimination in a tied good will be infeasible if that good is also produced under competitive conditions. In such a case the tied good will sell for marginal cost, and the only possibility for discriminatory behavior open to the monopolist is in the pricing of its basic good (i.e., by varying ‘‘entry fees’’ among demanders). In some special cases, however, choosing to pay the entry fee will confer monopoly power in the tied good on the monopolist even though it is otherwise reduced under competitive conditions. For example, Locay and Rodriguez (1992) examine the case of restaurants’ pricing of wine. Here group decirestaurant may confer sions monopoly power to the restaurant owner in the ability to practice wine price discrimination among buyers with strong grape preferences. Because the owner is constrained by the need to attract groups of customers to the restaurant, the power to price discriminate is less than under the pure monopoly scenario. to patronize a particular References Kwoka, J. E. ‘‘Market Segmentation by Price-Quality Schedules: Some Evidence from Automobiles.’’ Journal of Business (October 1992): 615–28. Locay, L., and A. Rodriguez. ‘‘Price Discrimination in Competitive Markets.’’ Journal of Political Economy (October 1992): 954–68. Oi, W. Y. ‘‘A Disneyland Dilemma: Two-Part Tariffs on a Mickey Mouse Monopoly.’’ Quarterly Journal of Economics (February 1971): 77–90. Smith, R. B. W. ‘‘The Conservation Reserve Program as a Least Cost Land Retirement Mechanism.’’ American Journal of Agricultural Economics (February 1995): 93–105. Willig, R. ‘‘Pareto Superior Non-Linear Outlay Schedules.’’ Bell Journal of
Economics (January 1978): 56–69. Wilson, W. Nonlinear Pricing. Oxford: Oxford University Press, 1993. This page intentionally left blank C H A P T E R FIFTEEN Imperfect Competition This chapter discusses oligopoly markets, falling between the extremes of perfect competition and monopoly Oligopoly. A market with relatively few ﬁrms but more than one. Oligopolies raise the possibility of strategic interaction among ﬁrms. To analyze this strategic interaction rigorously, we will apply the concepts from game theory that were introduced in Chapter 8. Our game-theoretic analysis will show that small changes in details concerning the variables ﬁrms choose, the timing of their moves, or their information about market conditions or rival actions can have a dramatic effect on market outcomes. The ﬁrst half of the chapter deals with short-term decisions such as pricing and output, and the second half covers longer-term decisions such as investment, advertising, and entry. Short-Run Decisions: Pricing And Output It is difﬁcult to predict exactly the possible outcomes for price and output when there are few ﬁrms; prices depend on how aggressively ﬁrms compete, which in turn depends on which strategic variables ﬁrms choose, how much information ﬁrms have about rivals, and how often ﬁrms interact with each other in the market. For example, consider the Bertrand game studied in the next section. The game involves two identical ﬁrms choosing prices simultaneously for their identical products in their one meeting in the market. The Bertrand game has a Nash equilibrium at point C in Figure 15.1. Even though there may be only two ﬁrms in the market, in this equilibrium they behave as though they were perfectly competitive, setting price equal to marginal cost and earning zero proﬁt. We will discuss whether the Bertrand game is a realistic depiction of actual ﬁrm behavior, but an analysis of the model shows that it is possible to think up rigorous game-theoretic models in which one extreme—the competitive outcome—can emerge in concentrated markets with few ﬁrms. At the other extreme, as indicated by point M in Figure 15.1, ﬁrms as a group may act as a cartel, recognizing that they can affect price and coordinate their decisions. Indeed, they may be able to act as
a perfect cartel and achieve the highest possible proﬁts— namely, the proﬁt a monopoly would earn in the market. One way to maintain a cartel is to bind ﬁrms with explicit pricing rules. Such explicit pricing rules are often prohibited by antitrust law. But ﬁrms need not resort to explicit pricing rules if they interact on the market repeatedly; they can collude tacitly. High collusive prices can be maintained with 531 532 Part 6: Market Power FIGURE 15.1 Pricing and Output under Imperfect Competition Market equilibrium under imperfect competition can occur at many points on the demand curve. In the ﬁgure, which assumes that marginal costs are constant over all output ranges, the equilibrium of the Bertrand game occurs at point C, also corresponding to the perfectly competitive outcome. The perfect cartel outcome occurs at point M, also corresponding to the monopoly outcome. Many solutions may occur between points M and C, depending on the speciﬁc assumptions made about how ﬁrms compete. For example, the equilibrium of the Cournot game might occur at a point such as A. The deadweight loss given by the shaded triangle increases as one moves from point C to M. Price PM PA PC M 1 2 A 3 C MC D MR QM QA QC Quantity the tacit threat of a price war if any ﬁrm undercuts. We will analyze this game formally and discuss the difﬁculty of maintaining collusion. The Bertrand and cartel models determine the outer limits between which actual prices in an imperfectly competitive market are set (one such intermediate price is represented by point A in Figure 15.1). This band of outcomes may be wide, and given the plethora of available models there may be a model for nearly every point within the band. For example, in a later section we will show how the Cournot model, in which ﬁrms set quantities rather than prices as in the Bertrand model, leads to an outcome (such as point A) somewhere between C and M in Figure 15.1. It is important to know where the industry is on the line between points C and M because total welfare (as measured by the sum of consumer surplus and ﬁrms’ proﬁts; see Chapter 12) depends on the location of this point. At point C, total welfare is as high as possible; at point A, total welfare is lower by the
area of the shaded triangle 3. In Chapter 12, this shortfall in total welfare relative to the highest possible level was called deadweight loss. At point M, deadweight loss is even greater and is given by the area of shaded regions 1, 2, and 3. The closer the imperfectly competitive outcome to C and the farther from M, the higher is total welfare and the better off society will be.1 1Because this section deals with short-run decision variables (price and quantity), the discussion of total welfare in this paragraph focuses on short-run considerations. As discussed in a later section, an imperfectly competitive market may produce considerably more deadweight loss than a perfectly competitive one in the short run yet provide more innovation incentives, leading to lower production costs and new products and perhaps higher total welfare in the long run. The patent system intentionally impairs competition by granting a monopoly right to improve innovation incentives. Chapter 15: Imperfect Competition 533 Bertrand Model The Bertrand model is named after the economist who ﬁrst proposed it.2 The model is a game involving two identical ﬁrms, labeled 1 and 2, producing identical products at a constant marginal cost (and constant average cost) c. The ﬁrms choose prices p1 and p2 simultaneously in a single period of competition. Because ﬁrms’ products are perfect subp2. Let stitutes, all sales go to the ﬁrm with the lowest price. Sales are split evenly if p1 ¼ D( p) be market demand. We will look for the Nash equilibrium. The game has a continuum of actions, as does Example 8.5 (the Tragedy of the Commons) in Chapter 8. Unlike Example 8.5, we cannot use calculus to derive best-response functions because the proﬁt functions are not differentiable here. Starting from equal prices, if one ﬁrm lowers its price by the smallest amount, then its sales and proﬁt would essentially double. We will proceed by ﬁrst guessing what the Nash equilibrium is and then spending some time to verify that our guess was in fact correct. Nash equilibrium of the Bertrand game c. That is, The only pure-strategy Nash equilibrium of the Bertrand game is p"1 ¼ the Nash equilibrium involves both ﬁrms charging marginal cost. In saying that this is the only Nash equilibrium, we are making two statements that need to
be veriﬁed: This outcome is a Nash equilibrium, and there is no other Nash equilibrium. p"2 ¼ To verify that this outcome is a Nash equilibrium, we need to show that both ﬁrms are playing a best response to each other—or, in other words, that neither ﬁrm has an incentive to deviate to some other strategy. In equilibrium, ﬁrms charge a price equal to marginal cost, which in turn is equal to average cost. But a price equal to average cost means ﬁrms earn zero proﬁt in equilibrium. Can a ﬁrm earn more than the zero it earns in equilibrium by deviating to some other price? No. If it deviates to a higher price, then it will make no sales and therefore no proﬁt, not strictly more than in equilibrium. If it deviates to a lower price, then it will make sales but will be earning a negative margin on each unit sold because price would be below marginal cost. Thus, the ﬁrm would earn negative proﬁt, less than in equilibrium. Because there is no possible proﬁtable deviation for the ﬁrm, we have succeeded in verifying that both ﬁrms’ charging marginal cost is a Nash equilibrium. p2. The same conclusions would be reached taking 2 to be the low-price ﬁrm. It is clear that marginal cost pricing is the only pure-strategy Nash equilibrium. If prices exceeded marginal cost, the high-price ﬁrm would gain by undercutting the other c p"2 ¼ slightly and capturing all the market demand. More formally, to verify that p"1 ¼ is the only Nash equilibrium, we will go one by one through an exhaustive list of cases for various values of p1, p2, and c, verifying that none besides p1 ¼ c is a Nash equilibrium. To reduce the number of cases, assume ﬁrm 1 is the low-price ﬁrm—that is, p l # p1. Case (i) cannot c on every unit it sells, and be a Nash equilibrium. Firm 1 earns a negative margin pl $ because it makes positive sales, it must earn negative proﬁt. It could earn higher proﬁt by deviating to a higher price. For example, ﬁrm 1 could guarantee itself
zero proﬁt by deviating to p1 ¼ Case (ii) cannot be a Nash equilibrium either. At best, ﬁrm 2 gets only half of market p2) and at worst gets no demand (if p1 < p2). Firm 2 could capture all demand (if p1 ¼ the market demand by undercutting ﬁrm 1’s price by a tiny amount e. This e could be There are three exhaustive cases: (i) c > p1, (ii) c < p1, and (iii) c p2 ¼ ¼ c. 2J. Bertrand, ‘‘The´orie Mathematique de la Richess Sociale,’’ Journal de Savants (1883): 499–508. 534 Part 6: Market Power p2 chosen small enough that market price and total market proﬁt are hardly affected. If p1 ¼ before the deviation, the deviation would essentially double ﬁrm 2’s proﬁt. If pl < p2 before the deviation, the deviation would result in ﬁrm 2 moving from zero to positive proﬁt. In either case, ﬁrm 2’s deviation would be proﬁtable. c, which we saw is a Nash equilibrium. p2 ¼ Case (iii) includes the subcase of p1 ¼ p1 < p2. This subcase cannot be a p2 is c The only remaining subcase in which p1 # Nash equilibrium: Firm 1 earns zero proﬁt here but could earn positive proﬁt by deviating to a price slightly above c but still below p2. ¼ c. % p"n ¼ p"2 ¼ & & & ¼ Although the analysis focused on the game with two ﬁrms, it is clear that the same 2. The Nash equilibrium of the n-ﬁrm outcome would arise for any number of ﬁrms n Bertrand game is p"1 ¼ Bertrand paradox The Nash equilibrium of the Bertrand model is the same as the perfectly competitive outcome. Price is set to marginal cost, and ﬁrms earn zero proﬁt. This result—that the Nash equilibrium in the Bertrand model is the same as in perfect competition even though there may be only two ﬁrms in
the market—is called the Bertrand paradox. It is paradoxical that competition between as few as two ﬁrms would be so tough. The Bertrand paradox is a general result in the sense that we did not specify the marginal cost c or the demand curve; therefore, the result holds for any c and any downward-sloping demand curve. In another sense, the Bertrand paradox is not general; it can be undone by changing various of the model’s other assumptions. Each of the next several sections will present a different model generated by changing a different one of the Bertrand assumptions. In the next section, for example, we will assume that ﬁrms choose quantity rather than price, leading to what is called the Cournot game. We will see that ﬁrms do not end up charging marginal cost and earning zero proﬁt in the Cournot game. In subsequent sections, we will show that the Bertrand paradox can also be avoided if still other assumptions are changed: if ﬁrms face capacity constraints rather than being able to produce an unlimited amount at cost c, if products are slightly differentiated rather than being perfect substitutes, or if ﬁrms engage in repeated interaction rather than one round of competition. Cournot Model The Cournot model, named after the economist who proposed it,3 is similar to the Bertrand model except that ﬁrms are assumed to simultaneously choose quantities rather than prices. As we will see, this simple change in strategic variable will lead to a big change in implications. Price will be above marginal cost, and ﬁrms will earn positive proﬁt in the Nash equilibrium of the Cournot game. It is somewhat surprising (but nonetheless an important point to keep in mind) that this simple change in choice variable matters in the strategic setting of an oligopoly when it did not matter with a monopoly: The monopolist obtained the same proﬁt-maximizing outcome whether it chose prices or quantities. We will start with a general version of the Cournot game with n ﬁrms indexed by 1,..., n. Each ﬁrm chooses its output qi of an identical product simultaneously. qn, i The outputs are combined into a total industry output Q ¼ q1 þ q2 þ & & & þ ¼ 3A. Cournot, Researches into the Mathematical Principles of the Theory of Wealth, trans.
N. T. Bacon (New York: Macmillan, 1897). Although the Cournot model appears after Bertrand’s in this chapter, Cournot’s work, originally published in 1838, predates Bertrand’s. Cournot’s work is one of the ﬁrst formal analyses of strategic behavior in oligopolies, and his solution concept anticipated Nash equilibrium. Chapter 15: Imperfect Competition 535 resulting in market price P(Q). Observe that P(Q) is the inverse demand curve corresponding to the market demand curve Q D(P). Assume market demand is down¼ ward sloping and so inverse demand is, too; that is, P 0(Q) < 0. Firm i’s proﬁt equals its total revenue, P(Q)qi, minus its total cost, Ci(qi): : qiÞ qi $ Þ pi ¼ (15:1) Cið Q ð P Nash equilibrium of the Cournot game Unlike the Bertrand game, the proﬁt function (15.1) in the Cournot game is differentiable; hence we can proceed to solve for the Nash equilibrium of this game just as we did in Example 8.5, the Tragedy of the Commons. That is, we ﬁnd each ﬁrm i’s best response by taking the ﬁrst-order condition of the objective function (15.1) with respect to qi: @pi @qi ¼ P Q ð Þ þ P 0 qi Q ð Þ C 0i ð qiÞ $ ¼ 0: MR MC (15:2) ¼ Equation 15.2 must hold for all i \|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} 1,..., n in the Nash equilibrium. \|ﬄﬄ{zﬄﬄ} According to Equation 15.2, the familiar condition for proﬁt maximization from Chapter 11—marginal revenue (MR) equals marginal cost (MC

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