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Solve for the hybrid equilibrium. c. Compute the players’ expected payoffs. Analytical Problems 8.9 Fairness in the Ultimatum Game Consider a simple version of the Ultimatum Game discussed in the text. The ﬁrst mover proposes a division of $1. Let r be the 1=2. Then the other player share received by the other player in this proposal (so the ﬁrst mover keeps 1 moves, responding by accepting or rejecting the proposal. If the responder accepts the proposal, the players are paid their r), where 0,!, r 294 Part 3: Uncertainty and Strategy shares; if the responder rejects it, both players receive nothing. Assume that if the responder is indifferent between accepting or rejecting a proposal, he or she accepts it. a. Suppose that players only care about monetary payoffs. Verify that the outcome mentioned in the text in fact occurs in the unique subgame-perfect equilibrium of the Ultimatum Game. b. Compare the outcome in the Ultimatum Game with the outcome in the Dictator Game (also discussed in the text), in which the proposer’s surplus division is implemented regardless of whether the second mover accepts or rejects (so it is not much of a strategic game!). c. Now suppose that players care about fairness as well as money. Following the article by Fehr and Schmidt cited in the text, suppose these preferences are represented by the utility function x1! where x1 is player 1’s payoff and x2 is player 2’s (a symmetric function holds for player 2). The ﬁrst term reﬂects the usual desire for more money. The second term reﬂects the desire for fairness, that the players’ payoffs not be too unequal. The parameter a measures how intense the preference for fairness is relative to the desire for more money. Assume a < 1=2. x1, x2Þ ¼ x1! U1ð x2j a j, 1. Solve for the responder’s equilibrium strategy in the Ultimatum Game. 2. Taking into account how the second mover will respond, solve for the proposer’s equilibrium strategy r’ in the Ultima- tum Game. (Hint: r’ will be a corner solution, which depends on the value of a.) 3. Continuing with
the fairness preferences, compare the outcome in the Ultimatum Game with that in the Dictator Game. Find cases that match the experimental results described in the text, in particular in which the split of the pot of money is more even in the Ultimatum Game than in the Dictator Game. Is there a limit to how even the split can be in the Ultimatum Game? 8.10 Rotten Kid Theorem In A Treatise on the Family (Cambridge, MA: Harvard University Press, 1981), Nobel laureate Gary Becker proposes his famous Rotten Kid Theorem as a sequential game between the potentially rotten child (player 1) and the child’s parent (player 2). The child moves ﬁrst, choosing an action r that affects his own income! 1ð and the income of the parent r! 2ð! 02ð < 0 r r. Later, the parent moves, leaving a monetary bequest L to the child. The child cares only for his own utility, Þ Þ½ aU1, where a > 0 reﬂects the parent’s altruism toward the child. Prove that,! 2!! 1 þ U1ð, but the parent maximizes U2ð L Þ in a subgame-perfect equilibrium, the child will opt for the value of r that maximizes! 1 þ! 2 even though he has no altruistic intentions. Hint: Apply backward induction to the parent’s problem ﬁrst, which will give a ﬁrst-order condition that implicitly determines L’; although an explicit solution for L’ cannot be found, the derivative of L’ with respect to r—required in the child’s ﬁrst-stage optimization problem—can be found using the implicit function rule.! 01ð r > 0 Þ þ Þ½ L Þ + + 8.11 Alternatives to Grim Strategy Suppose that the Prisoners’ Dilemma stage game (see Figure 8.1) is repeated for inﬁnitely many periods. a. Can players support the cooperative outcome by using tit-for-tat strategies, punishing deviation in a past period by reverting to the stage-game Nash equilibrium for just one period and then returning to cooperation? Are two periods of punishment enough? b. Suppose players use strategies that punish deviation from cooperation by reverting to the stage-game Nash
equilibrium for 10 periods before returning to cooperation. Compute the threshold discount factor above which cooperation is possible on the outcome that maximizes the joint payoffs. 8.12 Reﬁnements of perfect Bayesian equilibrium Recall the job-market signaling game in Example 8.9. a. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an educa- tion (NE) and where the ﬁrm offers an uneducated worker a job. Be sure to specify beliefs as well as strategies. b. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education (NE) and where the ﬁrm does not offer an uneducated worker a job. What is the lowest posterior belief that the worker is low-skilled conditional on obtaining an education consistent with this pooling equilibrium? Why is it more natural to think that a low-skilled worker would never deviate to E and thus an educated worker must be high-skilled? Cho and Kreps’s intuitive criterion is one of a series of complicated reﬁnements of perfect Bayesian equilibrium that rule out equilibria based on unreasonable posterior beliefs as identiﬁed in this part; see I. K. Cho and D. M. Kreps, ‘‘Signalling Games and Stable Equilibria,’’ Quarterly Journal of Economics 102 (1987): 179–221. Chapter 8: Game Theory 295 SUGGESTIONS FOR FURTHER READING Fudenberg, D., and J. Tirole. Game Theory. Cambridge, MA: MIT Press, 1991. Rasmusen, E. Games and Information, 4th ed. Malden, MA: Blackwell, 2007. A comprehensive survey of game theory at the graduate-student level, although selected sections are accessible to advanced undergraduates. Holt, C. A. Markets, Games, & Strategic Behavior. Boston: Pearson, 2007. An undergraduate games. text with emphasis on experimental An advanced undergraduate text with many real-world applications. Watson, Joel. Strategy: An Introduction to Game Theory. New York: Norton, 2002. An undergraduate text that balances rigor with simple exam2 games). Emphasis on bargaining and contractples (often 2 ) ing examples. EXTENSIONS EXISTENCE OF NASH EQUILIBRIUM This section will sketch John Nash’s original proof that all ﬁnite
games have at least one Nash equilibrium (in mixed if not in pure strategies). We will provide some of the details of the proof here; the original proof is in Nash (1950), and a clear textbook presentation of the full proof is provided in Fudenberg and Tirole (1991). The section concludes by mentioning a related existence theorem for games with continuous actions. Nash’s proof is similar to the proof of the existence of a general competitive equilibrium in Chapter 13. Both proofs rely on a ﬁxed point theorem. The proof of the existence of Nash equilibrium requires a slightly more powerful theorem. Instead of Brouwer’s ﬁxed point theorem, which applies to functions, Nash’s proof relies on Kakutani’s ﬁxed point theorem, which applies to correspondences—more general mappings than functions. E8.1 Correspondences versus functions A function maps each point in a ﬁrst set to a single point in a second set. A correspondence maps a single point in the ﬁrst set to possibly many points in the second set. Figure E8.1 illustrates the difference. An example of a correspondence that we have already seen is the best response, BRi(s–i). The best response need not map other players’ strategies si into a single strategy that is a best response for player i. There may be ties among several best responses. As shown in Figure 8.4, in the Battle of the Sexes, the husband’s best response to the wife’s playing the mixed strategy of going to ballet with probability 2/3 and boxing with probability 1/3 (or just w 2/3 for short) is not just a single point but the whole interval of possible mixed strategies. Both the husband’s and the wife’s best in this ﬁgure are correspondences, not functions. responses ¼ The reason Nash needed a ﬁxed point theorem involving correspondences is precisely than just because his proof works with players’ best responses to prove existence. functions rather FIGURE E8.1 Comparision of Functions and Correspondences The function graphed in (a) looks like a familiar curve. Each value of x is mapped into a single value of y. With the correspondence graphed in (b), each value of x may be mapped into many values of y. Thus, correspondences can have bulges as shown by the sh
aded regions in (b). y y (a) Function x (b) Correspondence x Chapter 8: Game Theory 297 FIGURE E8.2 Kakutani’s Conditions on Correspondences The correspondence in (a) is not convex because the dashed vertical segment between A and B is not inside the correspondence. The correspondence in (b) is not upper semicontinuous because there is a path (C) inside the correspondence leading to a point (D) that, as indicated by the open circle, is not inside the correspondence. Both (a) and (b) fail to have ﬁxed points. f(x) 1 f(x) 1 45° A B 45° C D (a) Correspondence that is not convex x 1 x (b) Correspondence that is not upper semicontinuous E8.2 Kakutani’s ﬁxed point theorem Here is the statement of Kakutani’s ﬁxed point theorem: Any convex, upper-semicontinuous correspondence [ f(x)] from a closed, bounded, convex set into itself has at least one ﬁxed point (x’) such that x’ f(x’). 2 Comparing the statement of Kakutani’s ﬁxed point theorem with Brouwer’s in Chapter 13, they are similar except for the substitution of ‘‘correspondence’’ for ‘‘function’’ and for the conditions on the correspondence. Brouwer’s theorem requires the function to be continuous; Kakutani’s theorem requires the correspondence to be convex and upper semicontinuous. These properties, which are related to continuity, are less familiar and worth spending a moment to understand. Figure E8.2 provides examples of correspondences violating (a) convexity and (b) upper semicontinuity. The ﬁgure shows why the two properties are needed to guarantee a ﬁxed point. Without both properties, the correspondence can ‘‘jump’’ across the 45! line and thus fail to have a ﬁxed point—that is, a point for which x f(x). ¼ E8.3 Nash’s proof We use R(s) to denote the correspondence that underlies Nash’s existence
proof. This correspondence takes any proﬁle (s1, s2, …, sn) (possibly mixed) and of players’ strategies s maps it into another mixed strategy proﬁle, the proﬁle of best responses: ¼ A ﬁxed point of the correspondence is a strategy for which s’ R(s’); this is a Nash equilibrium because each player’s strategy is a best response to others’ strategies. 2 The proof checks that all the conditions involved in Kakutani’s ﬁxed point theorem are satisﬁed by the best-response correspondence R(s). First, we need to show that the set of mixed-strategy proﬁles is closed, bounded, and convex. Because a strategy proﬁle is just a list of individual strategies, the set of strategy proﬁles will be closed, bounded, and convex if each player’s strategy set Si has these properties individually. As Figure E8.3 shows for the case of two and three actions, the set of mixed strategies over actions has a simple shape.1 The set is closed (contains its boundary), bounded (does not go off to inﬁnity in any direction), and convex (the segment between any two points in the set is also in the set). We then need to check that the best-response correspondence R(s) is convex. Individual best responses cannot look like Figure E8.2a because if any two mixed strategies such as A and B are best responses to others’ strategies, then mixed strategies between them must also be best responses. For example, in the Battle of the Sexes, if (1/3, 2/3) and (2/3, 1/3) are best responses for the husband against his wife’s playing (2/3, 1/3) (where, in each pair, the ﬁrst number is the probability of playing ballet and the second of playing boxing), then mixed strategies between the two such as (1/2, 1/2) must also be best responses for him. Figure 8.4 showed that in fact all s R ð BR1ð s Þ ¼ ð ; BR2ð s 1Þ!! s,..., BRnð : nÞÞ! 2
Þ (i) 1Mathematicians study them so frequently that they have a special name for such a set: a simplex. 298 Part 3: Uncertainty and Strategy FIGURE E8.3 Set of Mixed Strategies for an Individual Player 1’s set of possible mixed strategies over two actions is given by the diagonal line segment in (a). The set for three actions is given by the shaded triangle on the three-dimensional graph in (b). 2 p1 1 3 p1 1 0 1 2 p1 0 (a) Two actions 1 p1 1 1 1 p1 (b) Three actions possible mixed strategies for the husband are best responses to the wife’s playing (2/3, 1/3). Finally, we need to check that R(s) is upper semicontinuous. Individual best responses cannot look like in Figure E8.2b. They cannot have holes like point D punched out of them because payoff functions ui(si, s–i) are continuous. Recall that payoffs, when written as functions of mixed strategies, are actually expected values with probabilities given by the strategies si and s–i. As Equation 2.176 showed, expected values are linear functions of the underlying probabilities. Linear functions are, of course, continuous. E8.4 Games with continuous actions Nash’s existence theorem applies to ﬁnite games—that is, games with a ﬁnite number of players and actions per player. Nash’s theorem does not apply to games that feature continuous actions, such as the Tragedy of the Commons in Example 8.5. Is a Nash equilibrium guaranteed to exist for these games, too? Glicksberg (1952) proved that the answer is ‘‘yes’’ as long as payoff functions are continuous. References Fudenberg, D., and J. Tirole. Game Theory. Cambridge, MA: MIT Press, 1991, sec. 1.3. Glicksberg, I. L. ‘‘A Further Generalization of the Kakutani Fixed Point Theorem with Application to Nash Equilibrium Points.’’ Proceedings of the National Academy of Sciences 38 (1952): 170–74. Nash, John ‘‘Equilibrium Points in n-Person Games.’’ Proceedings of the National Academy of Sciences 36 (1950): 48–49. This page intentionally left blank Production and Supply P
A R T FOUR Chapter 9 Production Functions Chapter 10 Cost Functions Chapter 11 Proﬁt Maximization In this part we examine the production and supply of economic goods. Institutions that coordinate the transformation of inputs into outputs are called ﬁrms. They may be large institutions (such as Google, Sony, or the U.S. Department of Defense) or small ones (such as ‘‘Mom and Pop’’ stores or self-employed individuals). Although they may pursue different goals (Google may seek maximum proﬁts, whereas an Israeli kibbutz may try to make members of the kibbutz as well off as possible), all ﬁrms must make certain basic choices in the production process. The purpose of Part 4 is to develop some tools for analyzing those choices. In Chapter 9 we examine ways of modeling the physical relationship between inputs and outputs. We introduce the concept of a production function, a useful abstraction from the complexities of real-world production processes. Two measurable aspects of the production function are stressed: its returns to scale (i.e., how output expands when all inputs are increased) and its elasticity of substitution (i.e., how easily one input may be replaced by another while maintaining the same level of output). We also brieﬂy describe how technical improvements are reﬂected in production functions. The production function concept is then used in Chapter 10 to discuss costs of production. We assume that all ﬁrms seek to produce their output at the lowest possible cost, an assumption that permits the development of cost functions for the ﬁrm. Chapter 10 also focuses on how costs may differ between the short run and the long run. In Chapter 11 we investigate the ﬁrm’s supply decision. To do so, we assume that the ﬁrm’s manager will make input and output choices to maximize proﬁts. The chapter concludes with the fundamental model of supply behavior by proﬁt-maximizing ﬁrms that we will use in many subsequent chapters. 301 This page intentionally left blank C H A P T E R NINE Production Functions The principal activity of any ﬁrm is to turn inputs into outputs. Because economists are interested in the choices the ﬁrm makes in accomplishing this goal, but wish to avoid discussing many of the engineering intricacies involved, they have chosen to construct an abstract model of
production. In this model the relationship between inputs and outputs is formalized by a production function of the form q f k, l, m,... ð, Þ ¼ (9:1) where q represents the ﬁrm’s output of a particular good during a period,1 k represents the machine (i.e., capital) usage during the period, l represents hours of labor input, m represents raw materials used,2 and the notation indicates the possibility of other variables affecting the production process. Equation 9.1 is assumed to provide, for any conceivable set of inputs, the engineer’s solution to the problem of how best to combine those inputs to get output. Marginal Productivity In this section we look at the change in output brought about by a change in one of the productive inputs. For the purposes of this examination (and indeed for most of the purposes of this book), it will be more convenient to use a simpliﬁed production function deﬁned as follows Production function. The ﬁrm’s production function for a particular good, q, q f k, l ð, Þ ¼ (9:2) shows the maximum amount of the good that can be produced using alternative combinations of capital (k) and labor (l). Of course, most of our analysis will hold for any two inputs to the production process we might wish to examine. The terms capital and labor are used only for convenience. Similarly, it would be a simple matter to generalize our discussion to cases involving 1Here we use a lowercase q to represent one ﬁrm’s output. We reserve the uppercase Q to represent total output in a market. Generally, we assume that a ﬁrm produces only one output. Issues that arise in multiproduct ﬁrms are discussed in a few footnotes and problems. 2In empirical work, raw material inputs often are disregarded, and output, q, is measured in terms of ‘‘value added.’’ 303 304 Part 4: Production and Supply more than two inputs; occasionally, we will do so. For the most part, however, limiting the discussion to two inputs will be helpful because we can show these inputs on twodimensional graphs. Marginal physical product To study variation in a single input, we deﬁne marginal physical product as follows Marginal physical product. The marginal physical product of
an input is the additional output that can be produced by using one more unit of that input while holding all other inputs constant. Mathematically, marginal physical product of capital marginal physical product of labor MPk ¼ MPl ¼ ¼ ¼ @q @k ¼ @q @l ¼ f k, f l: (9:3) Notice that the mathematical deﬁnitions of marginal product use partial derivatives, thereby properly reﬂecting the fact that all other input usage is held constant while the input of interest is being varied. For example, consider a farmer hiring one more laborer to harvest the crop but holding all other inputs constant. The extra output this laborer produces is that farmhand’s marginal physical product, measured in physical quantities, such as bushels of wheat, crates of oranges, or heads of lettuce. We might observe, for example, that 50 workers on a farm are able to produce 100 bushels of wheat per year, whereas 51 workers, with the same land and equipment, can produce 102 bushels. The marginal physical product of the 51st worker is then 2 bushels per year. Diminishing marginal productivity We might expect that the marginal physical product of an input depends on how much of that input is used. Labor, for example, cannot be added indeﬁnitely to a given ﬁeld (while keeping the amount of equipment, fertilizer, and so forth ﬁxed) without eventually exhibiting some deterioration in its productivity. Mathematically, the assumption of diminishing marginal physical productivity is an assumption about the second-order partial derivatives of the production function: @MPk @k ¼ @MPl @l ¼ @ 2f @k2 ¼ @ 2 f @l2 ¼ fkk ¼ fll ¼ f11 < 0, f22 < 0: (9:4) The assumption of diminishing marginal productivity was originally proposed by the nineteenth-century economist Thomas Malthus, who worried that rapid increases in population would result in lower labor productivity. His gloomy predictions for the future of humanity led economics to be called the ‘‘dismal science.’’ But the mathematics of the production function suggests that such gloom may be misplaced. Changes in the marginal productivity of labor over time depend not only on how labor input is growing but also on changes in other inputs, such as capital. That is, we must also be concerned with
@MPl/@k flk. In most cases, flk > 0, thus, declining labor productivity as both l and k increase is not a foregone conclusion. Indeed, it appears that labor productivity has risen signiﬁcantly since Malthus’ time, primarily because increases in capital inputs (along with technical improvements) have offset the impact of decreasing marginal productivity alone. ¼ Chapter 9: Production Functions 305 Average physical productivity In common usage, the term labor productivity often means average productivity. When it is said that a certain industry has experienced productivity increases, this is taken to mean that output per unit of labor input has increased. Although the concept of average productivity is not nearly as important in theoretical economic discussions as marginal productivity is, it receives a great deal of attention in empirical discussions. Because average productivity is easily measured (say, as so many bushels of wheat per labor-hour input), it is often used as a measure of efﬁciency. We deﬁne the average product of labor (APl) to be APl ¼ output labor input ¼ q l ¼ f ð k, l l Þ : (9:5) Notice that APl also depends on the level of capital used. This observation will prove to be important when we examine the measurement of technical change at the end of this chapter. EXAMPLE 9.1 A Two-Input Production Function Suppose the production function for ﬂyswatters during a particular period can be represented by q f k, l ð ¼ Þ ¼ 600k2l2 k3l3: $ (9:6) To construct the marginal and average productivity functions of labor (l) for this function, we must assume a particular value for the other input, capital (k). Suppose k 10. Then the production function is given by ¼ 60,000l 2 q ¼ $ 1,000l 3: Marginal product. The marginal productivity function (when k 10) is given by ¼ MPl ¼ @q @l ¼ 120,000l 3,000l2, $ (9:7) (9:8) which diminishes as l increases, eventually becoming negative. This implies that q reaches a maximum value. Setting MPl equal to 0, yields or 120,000l 3,000l2 0 ¼ $ 40l l2 ¼ (9:9) (9:10)
¼ as the point at which q reaches its maximum value. Labor input beyond 40 units per period actually reduces total output. For example, when l 32 million ﬂyswatters, whereas when l 50, production of ﬂyswatters amounts to only 25 million. 40, Equation 9.7 shows that q ¼ ¼ l 40 (9:11) ¼ ¼ Average product. To ﬁnd the average productivity of labor in ﬂyswatter production, we divide q by l, still holding k 10: APl ¼ q l ¼ 60,000l $ 1,000l2: (9:12) 306 Part 4: Production and Supply Again, this is an inverted parabola that reaches its maximum value when @APl 60,000 2,000l 0, (9:13) ¼ which occurs when l 30. At this value for labor input, Equation 9.12 shows that APl ¼ ¼ 900,000, and Equation 9.8 shows that MPl is also 900,000. When APl is at a maximum, average and marginal productivities of labor are equal.3 @l ¼ $ Notice the relationship between total output and average productivity that is illustrated by this example. Even though total production of ﬂyswatters is greater with 40 workers (32 million) than with 30 workers (27 million), output per worker is higher in the second case. With 40 workers, each worker produces 800,000 ﬂyswatters per period, whereas with 30 workers each worker produces 900,000. Because capital input (ﬂyswatter presses) is held constant in this deﬁnition of productivity, the diminishing marginal productivity of labor eventually results in a declining level of output per worker. QUERY: How would an increase in k from 10 to 11 affect the MPl and APl functions here? Explain your results intuitively. Isoquant Maps And The Rate Of Technical Substitution To illustrate possible substitution of one input for another in a production function, we use its isoquant map. Again, we study a production function of the form q f (k, l), with the understanding that ‘‘capital’’ and ‘‘labor’’ are simply convenient examples of any two inputs that might happen to be of interest. An isoquant (from iso, meaning ‘�
�equal’’) records those combinations of k and l that are able to produce a given quantity of output. For example, all those combinations of k and l that fall on the curve labeled ‘‘q 10’’ in Figure 9.1 are capable of producing 10 units of output per period. This isoquant then records the fact that there are many alternative ways of producing 10 units of output. One way might be represented by point A: We would use lA and kA to produce 10 units of output. Alternatively, we might prefer to use relatively less capital and more labor and therefore would choose a point such as B. Hence we may deﬁne an isoquant as follows Isoquant. An isoquant shows those combinations of k and l that can produce a given level of output (say, q0). Mathematically, an isoquant records the set of k and l that satisﬁes f k, l ð Þ ¼ q0: (9:14) As was the case for indifference curves, there are inﬁnitely many isoquants in the k–l plane. Each isoquant represents a different level of output. Isoquants record successively higher levels of output as we move in a northeasterly direction. Presumably, using more 3This result is general. Because at a maximum l Æ MPl ¼ q or MPl ¼ APl. @APl @l ¼ l % MPl $ l2 q, Chapter 9: Production Functions 307 FIGURE 9.1 An Isoquant Map Isoquants record the alternative combinations of inputs that can be used to produce a given level of output. The slope of these curves shows the rate at which l can be substituted for k while keeping output constant. The negative of this slope is called the (marginal) rate of technical substitution (RTS). In the ﬁgure, the RTS is positive and diminishing for increasing inputs of labor. k per period kA A kB lA B lB q = 30 q = 20 q = 10 l per period ¼ 20 and of each of the inputs will permit output to increase. Two other isoquants (for q q 30) are shown in Figure 9.1. You will notice the similarity between an isoquant map and the individual’s indifference curve map discussed in Part 2. They are indeed similar
concepts because both represent ‘‘contour’’ maps of a particular function. For isoquants, however, the labeling of the curves is measurable—an output of 10 units per period has a quantiﬁable meaning. Therefore, economists are more interested in studying the shape of production functions than in examining the exact shape of utility functions. ¼ The marginal rate of technical substitution (RTS) The slope of an isoquant shows how one input can be traded for another while holding output constant. Examining the slope provides information about the technical possibility of substituting labor for capital. A formal deﬁnition follows Marginal rate of technical substitution. The marginal rate of technical substitution (RTS) shows the rate at which labor can be substituted for capital while holding output constant along an isoquant. In mathematical terms, RTS l for k ð Þ ¼ $ dk dl : q0 q! ¼!!! (9:15) In this deﬁnition, the notation is intended as a reminder that output is to be held constant as l is substituted for k. The particular value of this trade-off rate will depend not only on the level of output but also on the quantities of capital and labor being used. Its value depends on the point on the isoquant map at which the slope is to be measured. 308 Part 4: Production and Supply RTS and marginal productivities To examine the shape of production function isoquants, it is useful to prove the following result: The RTS (of l for k) is equal to the ratio of the marginal physical productivity of labor (MPl) to the marginal physical productivity of capital (MPk). Imagine using Equation 9.14 to graph the q0 isoquant. We would substitute a sequence of increasing values of l and see how k would have to adjust to keep output constant at q0. The graph of the isoquant is really the graph of the implicit function k(l) satisfying ð Just as we did in the section on implicit functions in Chapter 2 (see in particular Equation 2.22), we can use the chain rule to differentiate Equation 9.16, giving Þ q0 9:16) fk dk dl þ fl ¼ 0 ¼ MPk dk dl þ MPl, (9:17) where the initial 0 appears because q0 is being held constant; therefore, the
derivative of the left side of Equation 9.16 with respect to l equals 0. Rearranging Equation 9.17 gives RTS l for k ð Þ ¼ $ dk dl MPl MPk q0 ¼ : (9:18) q ¼ Hence the RTS is given by the ratio of the inputs’ marginal productivities.!!!! Equation 9.18 shows that those isoquants that we actually observe must be negatively sloped. Because both MPl and MPk will be non-negative (no ﬁrm would choose to use a costly input that reduced output), the RTS also will be positive (or perhaps zero). Because the slope of an isoquant is the negative of the RTS, any ﬁrm we observe will not be operating on the positively sloped portion of an isoquant. Although it is mathematically possible to devise production functions whose isoquants have positive slopes at some points, it would not make economic sense for a ﬁrm to opt for such input choices. Reasons for a diminishing RTS The isoquants in Figure 9.1 are drawn not only with a negative slope (as they should be) but also as convex curves. Along any one of the curves, the RTS is diminishing. For high ratios of k to l, the RTS is a large positive number, indicating that a great deal of capital can be given up if one more unit of labor becomes available. On the other hand, when a lot of labor is already being used, the RTS is low, signifying that only a small amount of capital can be traded for an additional unit of labor if output is to be held constant. This assumption would seem to have some relationship to the assumption of diminishing marginal productivity. A hasty use of Equation 9.18 might lead one to conclude that an increase in l accompanied by a decrease in k would result in a decrease in MPl, an increase in MPk, and, therefore, a decrease in the RTS. The problem with this quick ‘‘proof ’’ is that the marginal productivity of an input depends on the level of both inputs—changes in l affect MPk and vice versa. It is not possible to derive a diminishing RTS from the assumption of diminishing marginal productivity alone. To see why this is so mathematically, assume that q f(k, l) and that fk and fl are positive (i.
e., the marginal productivities are positive). Assume also that fkk < 0 and fll < 0 (that the marginal productivities are diminishing). To show that isoquants are convex, we would like to show that d(RTS)/dl < 0. Because RTS fl=fkÞ dl fl /fk, we have (9:19) dRTS dl ¼ ¼ ¼ d ð : Chapter 9: Production Functions 309 Because fl and fk are functions of both k and l, we must be careful in taking the derivative of this expression: dRTS dl ¼ fkð fll þ f lk % dk=dl flð 2 Þ $ fkÞ ð fkl þ fkk % dk=dl Þ : (9:20) Using the fact that dk/dl we have fl /fk along an isoquant and Young’s theorem ( fkl ¼ ¼ $ flk), f 2 k fll $ dRTS dl ¼ 2fk fl fkl þ fkÞ ð Because we have assumed fk > 0, the denominator of this function is positive. Hence the whole fraction will be negative if the numerator is negative. Because fll and fkk are both assumed to be negative, the numerator deﬁnitely will be negative if fkl is positive. If we can assume this, we have shown that dRTS/dl < 0 (that the isoquants are convex).4 (9:21) : 3 f 2 l fkk Importance of cross-productivity effects flk should be positive. Intuitively, it seems reasonable that the cross-partial derivative fkl ¼ If workers had more capital, they would have higher marginal productivities. Although this is probably the most prevalent case, it does not necessarily have to be so. Some production functions have fkl < 0, at least for a range of input values. When we assume a diminishing RTS (as we will throughout most of our discussion), we are therefore making a stronger assumption than simply diminishing marginal productivities for each input— speciﬁcally, we are assuming that marginal productivities diminish ‘‘rapidly enough’’ to compensate for any possible negative cross-productivity effects. Of course, as we shall see later, with three or more inputs,
things become even more complicated. EXAMPLE 9.2 A Diminishing RTS In Example 9.1, the production function for ﬂyswatters was given by ¼ General marginal productivity functions for this production function are Þ ¼ $ q f k, l ð 600k2l2 k3l3: MPl ¼ MPk ¼ fl ¼ fk ¼ @q @l ¼ @q @k ¼ 1,200k2l 3k3l2, $ 1,200kl2 3k2l3: $ (9:22) (9:23) Notice that each of these depends on the values of both inputs. Simple factoring shows that these marginal productivities will be positive for values of k and l for which kl < 400. Because and fll ¼ 1,200k2 6k3l $ fkk ¼ 1,200l2 6kl3, $ (9:24) it is clear that this function exhibits diminishing marginal productivities for sufﬁciently large values of k and l. Indeed, again by factoring each expression, it is easy to show that fll, fkk < 0 if 4As we pointed out in Chapter 2, functions for which the numerator in Equation 9.21 is negative are called (strictly) quasiconcave functions. 310 Part 4: Production and Supply kl > 200. However, even within the range 200 < kl < 400 where the marginal productivity relations for this function behave ‘‘normally,’’ this production function may not necessarily have a diminishing RTS. Cross-differentiation of either of the marginal productivity functions (Equation 9.23) yields fkl ¼ flk ¼ 2,400kl $ 9k2l2, (9:25) which is positive only for kl < 266. Therefore, the numerator of Equation 9.21 will deﬁnitely be negative for 200 < kl < 266, but for larger-scale ﬂyswatter factories the case is not so clear because fkl is negative. When fkl is negative, increases in labor input reduce the marginal productivity of capital. Hence the intuitive argument that the assumption of diminishing marginal productivities yields an unambiguous prediction about what will happen to the RTS ( fl/fk) as l increases and k decreases is incorrect. �
� It all depends on the relative effects on marginal productivities of diminishing marginal productivities (which tend to reduce fl and increase fk) and the contrary effects of cross-marginal productivities (which tend to increase fl and reduce fk). Still, for this ﬂyswatter case, it is true that the RTS is diminishing throughout the range of k and l where marginal productivities are positive. For cases where 266 < kl < 400, the diminishing marginal productivities exhibited by the function are sufﬁcient to overcome the inﬂuence of a negative value for fkl on the convexity of isoquants. QUERY: For cases where k l, what can be said about the marginal productivities of this production function? How would this simplify the numerator for Equation 9.21? How does this permit you to more easily evaluate this expression for some larger values of k and l? ¼ Returns to Scale We now proceed to characterize production functions. A ﬁrst question that might be asked about them is how output responds to increases in all inputs together. For example, suppose that all inputs were doubled: Would output double or would the relationship not be so simple? This is a question of the returns to scale exhibited by the production function that has been of interest to economists ever since Adam Smith intensively studied the production of pins. Smith identiﬁed two forces that came into operation when the conceptual experiment of doubling all inputs was performed. First, a doubling of scale permits a greater division of labor and specialization of function. Hence there is some presumption that efﬁciency might increase—production might more than double. Second, doubling of the inputs also entails some loss in efﬁciency because managerial overseeing may become more difﬁcult given the larger scale of the ﬁrm. Which of these two tendencies will have a greater effect is an important empirical question. These concepts can be deﬁned technically as followsk, l) and if all inputs are multiplied Returns to scale. If the production function is given by q by the same positive constant t (where t > 1), then we classify the returns to scale of the production function by ¼ Effect on Output Returns to Scale f (tk, tl ) tf (k, l ) ¼ f (tk, tl ) < tf (k, l ) f (tk, tl ) > tf (
k, l ) tq tq tq ¼ ¼ ¼ Constant Decreasing Increasing Chapter 9: Production Functions 311 In intuitive terms, if a proportionate increase in inputs increases output by the same proportion, the production function exhibits constant returns to scale. If output increases less than proportionately, the function exhibits diminishing returns to scale. And if output increases more than proportionately, there are increasing returns to scale. As we shall see, it is theoretically possible for a function to exhibit constant returns to scale for some levels of input usage and increasing or decreasing returns for other levels.5 Often, however, economists refer to returns to scale of a production function with the implicit understanding that only a fairly narrow range of variation in input usage and the related level of output is being considered. Constant returns to scale There are economic reasons why a ﬁrm’s production function might exhibit constant returns to scale. If the ﬁrm operates many identical plants, it may increase or decrease production simply by varying the number of them in current operation. That is, the ﬁrm can double output by doubling the number of plants it operates, and that will require it to employ precisely twice as many inputs. Empirical studies of production functions often ﬁnd that returns to scale are roughly constant for the ﬁrms studied (at least around for outputs close to the ﬁrms’ established operating levels—the ﬁrms may exhibit increasing returns to scale as they expand to their established size). For all these reasons, the constant returns-to-scale case seems worth examining in somewhat more detail. When a production function exhibits constant returns to scale, it meets the deﬁnition of ‘‘homogeneity’’ that we introduced in Chapter 2. That is, the production is homogeneous of degree 1 in its inputs because f (tk, tl) = t1f (k, l) = tq. (9:26) In Chapter 2 we showed that, if a function is homogeneous of degree k, its derivatives are homogeneous of degree k 1. In this context this implies that the marginal productivity functions derived from a constant returns-to-scale production function are homogeneous of degree 0. That is, $ MPk ¼ MPl ¼ @f @f k, l ð Þ @k ¼ k, l ð Þ @l ¼ @f @f tk, t
l ð @k tk, tl ð @l Þ, Þ for any t > 0. In particular, we can let t 1/ l in Equations 9.27 and get ¼ MPk ¼ MPl ¼ @f @f k=l, 1 ð @k k=l, 1 ð @l Þ, Þ : (9:27) (9:28) That is, the marginal productivity of any input depends only on the ratio of capital to labor input, not on the absolute levels of these inputs. This fact is especially important, for example, in explaining differences in productivity among industries or across countries. 5A local measure of returns to scale is provided by the scale elasticity, deﬁned as eq, t ¼ @f tk, tl ð @t Þ t tk, tl, Þ % f ð where this expression is to be evaluated at t level of input usage. For some examples using this concept, see Problem 9.9. ¼ 1. This parameter can, in principle, take on different values depending on the 312 Part 4: Production and Supply Homothetic production functions One consequence of Equations 9.28 is that the RTS ( MPl/MPk) for any constant returns-to-scale production function will depend only on the ratio of the inputs, not on their absolute levels. That is, such a function will be homothetic (see Chapter 2)—its isoquants will be radial expansions of one another. This situation is shown in Figure 9.2. Along any ray through the origin (where the ratio k/l does not change), the slopes of successively higher isoquants are identical. This property of the isoquant map will be useful to us on several occasions. ¼ A simple numerical example may provide some intuition about this result. Suppose a large bread order (consisting of, say, 200 loaves) can be ﬁlled in one day by three bakers working with three ovens or by two bakers working with four ovens. Therefore, the RTS of ovens for bakers is one for one—one extra oven can be substituted for one baker. If this production process exhibits constant returns to scale, two large bread orders (totaling 400 loaves) can be ﬁlled in one day, either by six bakers with six ovens or by four bakers
with eight ovens. In the latter case, two ovens are substituted for two bakers, so again the RTS is one for one. In constant returns-to-scale cases, expanding the level of production does not alter trade-offs among inputs; thus, production functions are homothetic. A production function can have a homothetic indifference curve map even if it does not exhibit constant returns to scale. As we showed in Chapter 2, this property of homotheticity is retained by any monotonic transformation of a homogeneous function. Hence increasing or decreasing returns to scale can be incorporated into a constant returns-toscale function through an appropriate transformation. Perhaps the most common such transformation is exponential. Thus, if f(k, l) is a constant returns-to-scale production FIGURE 9.2 Isoquant Map for a Constant Returns-toScale Production Function Because a constant returns-to-scale production function is homothetic, the RTS depends only on the ratio of k to l, not on the scale of production. Consequently, along any ray through the origin (a ray of constant k/l), the RTS will be the same on all isoquants. An additional feature is that the isoquant labels increase proportionately with the inputs. k per period per period Chapter 9: Production Functions 313 function, we can let F(k, l) = [ f (k, l)]g, (9:29) where g is any positive exponent. If g > 1, then F(tk, tl ) = [ f (tk, tl )]g = [tf (k, l )]g = tg[ f (k, l )]g = tgF(k, l ) > tF(k, l ) (9:30) ¼ for any t > 1. Hence this transformed production function exhibits increasing returns to scale. The exponent g captures the degree of the increasing returns to scale. A doubling of 2 but an eight-fold increase if inputs would lead to a four-fold increase in output if g 3. An identical proof shows that the function F exhibits decreasing returns to scale g for g < 1. Because this function remains homothetic through all such transformations, we have shown that there are important cases where the issue of returns to scale can be separated from issues involving the shape of an isoquant. In these cases, changes in the returns to scale will just change the labels on the is
oquants rather than their shapes. In the next section, we will look at how shapes of isoquants can be described. ¼ The n-input case The deﬁnition of returns to scale can be easily generalized to a production function with n inputs. If that production function is given by and if all inputs are multiplied by t > 1, we have q f x1, x2,..., xnÞ ð ¼ f (tx1, tx2,..., txn) = t kf (x1, x2,..., xn) = t kq (9:31) (9:32) 1, the production function exhibits constant returns to scale. for some constant k. If k Decreasing and increasing returns to scale correspond to the cases k < 1 and k > 1, respectively. ¼ The crucial part of this mathematical deﬁnition is the requirement that all inputs be increased by the same proportion, t. In many real-world production processes, this provision may make little economic sense. For example, a ﬁrm may have only one ‘‘boss,’’ and that number would not necessarily be doubled even if all other inputs were. Or the output of a farm may depend on the fertility of the soil. It may not be literally possible to double the acres planted while maintaining fertility because the new land may not be as good as that already under cultivation. Hence some inputs may have to be ﬁxed (or at least imperfectly variable) for most practical purposes. In such cases, some degree of diminishing productivity (a result of increasing employment of variable inputs) seems likely, although this cannot properly be called ‘‘diminishing returns to scale’’ because of the presence of inputs that are held ﬁxed. The Elasticity Of Substitution Another important characteristic of the production function is how ‘‘easy’’ it is to substitute one input for another. This is a question about the shape of a single isoquant rather than about the whole isoquant map. Along one isoquant, the rate of technical substitution will decrease as the capital–labor ratio decreases (i.e., as k/l decreases); now we wish to deﬁne some parameter that measures this degree of responsiveness. If the RTS does not change at all for changes in k/l
, we might say that substitution is easy because the ratio of the marginal productivities of the two inputs does not change as the input mix changes. Alternatively, if the RTS changes rapidly for small changes in k/l, we would say that 314 Part 4: Production and Supply substitution is difﬁcult because minor variations in the input mix will have a substantial effect on the inputs’ relative productivities. A scale-free measure of this responsiveness is provided by the elasticity of substitution, a concept we encountered informally in our discussion of CES utility functions. Here we will work on providing a more formal deﬁnition. For discrete changes, the elasticity of substitution is given by percent D k=l percent DRTS ¼ ð Þ r ¼ D k=l ð k=l Þ 4 DRTS RTS ¼ k=l D Þ ð DRTS % : RTS k=l ð Þ (9:33) More often, we will be interested in considering small changes; therefore, a modiﬁcation of Equation 9.33 will be of more interest: k=l d Þ ð d RTS % RTS k=l ¼ k=l d ln Þ ð d ln RTS : r ¼ (9:34) The logarithmic expression follows from mathematical derivations along the lines of Example 2.2 from Chapter 2. All these equations can be collected in the following formal deﬁnition Elasticity of substitution. For the production function q f (k, l), the elasticity of substitution (s) measures the proportionate change in k/l relative to the proportionate change in the RTS along an isoquant. That is, ¼ percent D percent DRTS ¼ k=l ð Þ r ¼ k=l d Þ ð d RTS % RTS k=l ¼ k=l d ln ð d ln RTS ¼ Þ k=l d ln ð Þ fl =fkÞ d ln ð : (9:35) Because along an isoquant k/l and RTS move in the same direction, the value of s is always positive. Graphically, this concept is illustrated in Figure 9.3 as a movement from point A to point B on an isoquant. In this movement,
both the RTS and the ratio k/l will change; we are interested in the relative magnitude of these changes. If s is high, then the RTS will not change much relative to k/l and the isoquant will be close to linear. On the other hand, a low value of s implies a rather sharply curved isoquant; the RTS will change by a substantial amount as k/l changes. In general, it is possible that the elasticity of substitution will vary as one moves along an isoquant and as the scale of production changes. Often, however, it is convenient to assume that s is constant along an isoquant. If the production function is also homothetic, then—because all the isoquants are merely radial blowups—s will be the same along all isoquants. We will encounter such functions later in this chapter and in many of the end of chapter problems.6 The n-input case Generalizing the elasticity of substitution to the many-input case raises several complications. One approach is to adopt a deﬁnition analogous to Equation 9.35; that is, to deﬁne 6The elasticity of substitution can be phrased directly in terms of the production function and its derivatives in the constant returns-to-scale case as But this form is cumbersome. Hence usually the logarithmic deﬁnition in Equation 9.35 is easiest to apply. For a compact summary, see P. Berck and K. Sydsaeter, Economist’s Mathematical Manual (Berlin, Germany: Springer-Verlag, 1999), chap. 5. r ¼ fl fk, l fk % f % Chapter 9: Production Functions 315 FIGURE 9.3 Graphic Description of the Elasticity of Substitution In moving from point A to point B on the q q0 isoquant, both the capital–labor ratio (k/l) and the RTS will change. The elasticity of substitution (s) is deﬁned to be the ratio of these proportional changes; it is a measure of how curved the isoquant is. ¼ k per period A RTSA RTSB (k /l ) A (k /l ) B B q = q0 l per period the elasticity of substitution between two inputs to be the proportionate change in the ratio of the two inputs to the proportionate change in the RTS between
them while holding output constant.7 To make this deﬁnition complete, it is necessary to require that all inputs other than the two being examined be held constant. However, this latter requirement (which is not relevant when there are only two inputs) restricts the value of this potential deﬁnition. In real-world production processes, it is likely that any change in the ratio of two inputs will also be accompanied by changes in the levels of other inputs. Some of these other inputs may be complementary with the ones being changed, whereas others may be substitutes, and to hold them constant creates a rather artiﬁcial restriction. For this reason, an alternative deﬁnition of the elasticity of substitution that permits such complementarity and substitutability in the ﬁrm’s cost function is generally used in the n-good case. Because this concept is usually measured using cost functions, we will describe it in the next chapter. 7That is, the elasticity of substitution between input i and input j might be deﬁned as rij ¼ xi=xjÞ @ ln ð @ ln fj=fiÞ ð for movements along f (x1, x2,..., xn) all inputs other than i and j be held constant when considering movements along the q0 isoquant. q0. Notice that the use of partial derivatives in this deﬁnition effectively requires that ¼ 316 Part 4: Production and Supply Four Simple Production Functions In this section we illustrate four simple production functions, each characterized by a different elasticity of substitution. These are shown only for the case of two inputs, but generalization to many inputs is easily accomplished (see the Extensions for this chapter). Case 1: Linear (r 5 ¥) Suppose that the production function is given by Þ ¼ It is easy to show that this production function exhibits constant returns to scale: For any t > 1, ¼ þ q f k, l ð ak bl: (9:36) f tk, tl ð Þ ¼ atk btl ak t ð þ bl ¼ þ Þ ¼ k, l tf ð : Þ (9:37) All isoquants for this production function are parallel straight lines with slope b/a. Such an isoquant map is pictured in Figure 9.4a. Because the R
TS is constant along any straight-line isoquant, the denominator in the deﬁnition of s (Equation 9.35) is equal to 0 and hence s is inﬁnite. Although this linear production function is a useful example, it is rarely encountered in practice because few production processes are characterized by such ease of substitution. Indeed, in this case, capital and labor can be thought of as perfect substitutes for each other. An industry characterized by such a production function could use only capital or only labor, depending on these inputs’ prices. It is hard to envision such a production process: Every machine needs someone to press its buttons, and every laborer requires some capital equipment, however modest. $ Case 2: Fixed proportions (r 5 0) Production functions characterized by s 0 have L-shaped isoquants as depicted in ¼ Figure 9.4b. At the corner of an L-shaped isoquant, a negligible increase in k/l causes an inﬁnite increase in RTS because the isoquant changes suddenly from horizontal to vertical there. Substituting 0 for the change in k/l in the numerator of the formula for s in Equation 9.33 and inﬁnity for the change in RTS in the denominator implies s 0. A ﬁrm would always operate at the corner of an isoquant. Operating anywhere else is inefﬁcient because the same output could be produced with fewer inputs by moving along the isoquant toward the corner. ¼ As drawn in Figure 9.4, the corners of the isoquants all lie along the same ray from the origin. This illustrates the important special case of a ﬁxed-proportions production function. Because the ﬁrm always operates at the corner of some isoquant, and all isoquants line up along the same ray, it must be the case that the ﬁrm uses inputs in the ﬁxed proportions given by the slope of this ray regardless of how much it produces.8 The inputs are perfect complements in that, starting from the ﬁxed proportion, an increase in one input is useless unless the other is increased as well. The mathematical form of the ﬁxed-proportions production function is given by q, a, b > 0, Þ where the operator ‘‘min’�
� means that q is given by the smaller of the two values in parentheses. For example, suppose that ak < bl; then q ak, and we would say that capital is the binding constraint in this production process. The employment of more labor would ak, bl ð (9:38) min ¼ ¼ 8Production functions with s along a nonlinear curve from the origin rather than lining up along a ray. ¼ 0 need not be ﬁxed proportions. The other possibility is that the corners of the isoquants lie Chapter 9: Production Functions 317 FIGURE 9.4 Isoquant Maps for Simple Production Functions with Various Values for s Three possible values for the elasticity of substitution are illustrated in these ﬁgures. In (a), capital and labor are perfect substitutes. In this case, the RTS will not change as the capital–labor ratio changes. In (b), the ﬁxed–proportions case, no substitution is possible. The capital–labor ratio is ﬁxed at b/a. A case of limited substitutability is illustrated in (c). k per period k per period σ = ∞ Slope = __−β α __q3 α q1 q2 q3 l per period __q3 β (b) (a) k per period σ = 0 q3 q2 q1 l per period σ = 1 q3 q2 q1 l per period (c) not increase output, and hence the marginal product of labor is zero; additional labor is superﬂuous in this case. Similarly, if ak > bl, then labor is the binding constraint on output, and additional capital is superﬂuous. When ak bl, both inputs are fully utilized. b/a, and production takes place at a vertex on the isoquant When this happens, k/l map. If both inputs are costly, this is the only cost-minimizing place to operate. The locus of all such vertices is a straight line through the origin with a slope given by b/a.9 ¼ ¼ 9With the form reﬂected by Equation 9.38, the ﬁxed-proportions production function exhibits constant returns to scale because Þ ¼ for any t > 1. As before, increasing or decreasing returns can be easily incorporated into the functions by using a nonlinear transformation of this functional
form—such as [ f (k, l)]g, where g may be greater than or less than 1. Þ ¼ Þ ¼ Þ % f tk, tl ð atk, btl min ð t min ð ak, bl tf k, l ð 318 Part 4: Production and Supply The ﬁxed-proportions production function has a wide range of applications. Many machines, for example, require a certain number of people to run them, but any excess labor is superﬂuous. Consider combining capital (a lawn mower) and labor to mow a lawn. It will always take one person to run the mower, and either input without the other is not able to produce any output at all. It may be that many machines are of this type and require a ﬁxed complement of workers per machine.10 Case 3: Cobb–Douglas (r 5 1) 1, called a Cobb–Douglas production function,11 The production function for which s provides a middle ground between the two polar cases previously discussed. Isoquants for the Cobb–Douglas case have the ‘‘normal’’ convex shape and are shown in Figure 9.4c. The mathematical form of the Cobb–Douglas production function is given by ¼ q = f (k, l ) = Akalb, (9:39) where A, a, and b are all positive constants. The Cobb–Douglas function can exhibit any degree of returns to scale, depending on the values of a and b. Suppose all inputs were increased by a factor of t. Then tk, tl f ð Þ ¼ ¼ A tk ð b ta þ a tl ð Þ k, l ð Ata bkalb þ ¼ b Þ : Þ (9:40) b ¼ þ 1, the Cobb–Douglas function exhibits constant returns to scale Hence if a b > 1, then the function exhibits because output also increases by a factor of t. If a b < 1 corresponds to the decreasing returnsincreasing returns to scale, whereas a to-scale case. It is a simple matter to show that the elasticity of substitution is 1 for the Cobb–Douglas function.12 This fact has led researchers to use the constant returnsto-scale version of the function for a general description of aggregate production relationships in
¼ Direct application of the deﬁnition of s to this function15 gives the important result that r ¼ 1 1 $ : q (9:43) Hence the linear, ﬁxed-proportions, and Cobb–Douglas cases correspond to r r Cobb–Douglas cases requires a limit argument. 1, 0, respectively. Proof of this result for the ﬁxed-proportions and, and r ¼ $1 ¼ $ ¼ Often the CES function is used with a distributional weight, a (0 the relative signiﬁcance of the inputs: a ) ) 1), to indicate q f k, l ð ¼ akq 1 lq =q: (9:44) With constant returns to scale and r form ¼ 0, this function converges to the Cobb–Douglas q f k, l ð ¼ Þ ¼ kal1 $ a: EXAMPLE 9.3 A Generalized Leontief Production Function Suppose that the production function for a good is given by q f k lp : k % ﬃﬃﬃﬃﬃﬃﬃ (9:45) (9:46) 14K. J. Arrow, H. B. Chenery, B. S. Minhas, and R. M. Solow, ‘‘Capital–Labor Substitution and Economic Efﬁciency,’’ Review of Economics and Statistics (August 1961): 225–50. 15For the CES function we have RTS fl fk ¼ ¼ g=q ð g=q ð Þ % Þ % q g qð g qð $ q $ =g Þ =g Þ % % 1 qlq $ qkq $ Applying the deﬁnition of the elasticity of substitution then yields @ ln k=l ð @ ln RTS ¼ Þ 1 r ¼ 1 $ : q Notice in this computation that the factor r cancels out of the marginal productivity functions, thereby ensuring that these marginal productivities are positive even when r is negative (as it is in many cases). This explains why r appears in two different places in the deﬁnition of the CES function. 320 Part 4: Production and Supply This function is a special case of a class
of functions named for the Russian-American economist Wassily Leontief.16 The function clearly exhibits constant returns to scale because (9:47) (9:48) Marginal productivities for the Leontief function are ﬃﬃﬃﬃ f tk, tl ð Þ ¼ tk tl þ þ 2t klp tf k, l ð : Þ ¼ 1 1 fk ¼ fl ¼ þ ð 0:5, k=l k=l $ Þ 0:5: Þ þ ð Hence marginal productivities are positive and diminishing. As would be expected (because this function exhibits constant returns to scale), the RTS here depends only on the ratio of the two inputs RTS fl fk ¼ ¼ 1 1 k=l k=l 0:5 0:9:49) This RTS diminishes as k/l falls, so the isoquants have the usual convex shape. There are two ways you might calculate the elasticity of substitution for this production function. First, you might notice that in this special case the function can be factored as q k l þ þ ¼ klp 2 kp ¼ ð lp 2 Þ þ k0:5 ¼ ð l 0:5 2, Þ þ (9:50) which makes clear that this function has a CES form with r elasticity of substitution here is s r) ﬃﬃﬃﬃ 1/(1 2. ﬃﬃﬃ ﬃﬃ 0.5 and g ¼ ¼ 1. Hence the Of course, in most cases it is not possible to do such a simple factorization. A more exhaustive approach is to apply the deﬁnition of the elasticity of substitution given in footnote 6 of this chapter: ¼ $ ¼ k=l 0:5 ( $ Þ r ¼ f fk fl fk l ¼ % 1 ½ þ ð k=l k=l 2 þ ð k=l 0:5 ð 0:5 Þ 0:5 Þ 0:5 Þ 1 (½ 0:5= q % ð þ ð klp 0:5 k=l 0:5 $ Þ k=l ð ﬃﬃﬃﬃ
ﬃ $ Þ Þ (9:51) 2: 1 þ ¼ þ ð þ Notice that in this calculation the input ratio (k/l) drops out, leaving a simple result. In other applications, one might doubt that such a fortuitous result would occur and hence doubt that the elasticity of substitution is constant along an isoquant (see Problem 9.7). But here the result 2 is intuitively reasonable because that value represents a compromise between the that s ) and its elasticity of substitution for this production function’s linear part (q Cobb–Douglas part (q 2k0.5l 0.5, s 0:5 ¼ ¼1 l, s 1). þ ¼ ¼ k ¼ ¼ QUERY: What can you learn about this production function by graphing the q Why does this function generalize the ﬁxed-proportions case? ¼ 4 isoquant? Technical Progress Methods of production improve over time, and it is important to be able to capture these improvements with the production function concept. A simpliﬁed view of such progress is provided by Figure 9.5. Initially, isoquant q0 records those combinations of capital and labor that can be used to produce an output level of q0. Following the development of superior production techniques, this isoquant shifts to q00. Now the same level of output 16Leontief was a pioneer in the development of input–output analysis. In input–output analysis, production is assumed to take place with a ﬁxed-proportions technology. The Leontief production function generalizes the ﬁxed-proportions case. For more details see the discussion of Leontief production functions in the Extensions to this chapter. Chapter 9: Production Functions 321 FIGURE 9.5 Technical Progress Technical progress shifts the q0 isoquant toward the origin. The new q0 isoquant, q00, shows that a given level of output can now be produced with less input. For example, with k1 units of capital it now only takes l1 units of labor to produce q0, whereas before the technical advance it took l2 units of labor. k per period k2 k1 q0 q′0 l1 l2 l per period can be produced with fewer inputs. One way to measure this improvement is by noting that with a level of capital input of, say,
k1, it previously took l2 units of labor to produce q0, whereas now it takes only l1. Output per worker has risen from q0/l2 to q0/l1. But one must be careful in this type of calculation. An increase in capital input to k2 would also have permitted a reduction in labor input to l1 along the original q0 isoquant. In this case, output per worker would also increase, although there would have been no true technical progress. Use of the production function concept can help to differentiate between these two concepts and therefore allow economists to obtain an accurate estimate of the rate of technical change. Measuring technical progress The ﬁrst observation to be made about technical progress is that historically the rate of growth of output over time has exceeded the growth rate that can be attributed to the growth in conventionally deﬁned inputs. Suppose that we let q A ¼ t ð k, l f Þ be the production function for some good (or perhaps for society’s output as a whole). The term A(t) in the function represents all the inﬂuences that go into determining q other than k (machine-hours) and l (labor-hours). Changes in A over time represent technical progress. For this reason, A is shown as a function of time. Presumably dA/dt > 0; particular levels of input of labor and capital become more productive over time. (9:52) Þ ð 322 Part 4: Production and Supply Growth accounting Differentiating Equation 9.52 with respect to time gives df k, l ð dt Þ % dq dt ¼ ¼ dA dt % dA dt % A f k, l f ð @f @k % dk dt þ @f @l % dl dt : & % Þ Dividing by q gives or dq=dt dA=dt q ¼ A þ @f =@k k, l f dk dt þ % @f =@l k, l f dl dt % ð Þ ð Þ dq=dt dA=dt q ¼ A þ @f @k % k k, l f ð % Þ dk=dt k þ @f @l % l k, l f ð % Þ dl=dt l : (9:53)
(9:54) (9:55) Now for any variable x, (dx/dt)/x is the proportional rate of growth of x per unit of time. We shall denote this by Gx.17 Hence Equation 9.55 can be written in terms of growth rates as Gq ¼ GA þ @f @k % k k, l f ð Þ Gk þ % @f @l % l k, l f ð % Þ Gt: (9:56) But and @f @k % k k, l f ð ¼ Þ @q @k % k q ¼ @f @l % l k, l f ð ¼ Þ @q @l % l q ¼ elasticity of output with respect to capital eq, k ¼ (9:57) elasticity of output with respect to labor eq, l: ¼ (9:58) Therefore, our growth equation ﬁnally becomes eq, kGk þ This shows that the rate of growth in output can be broken down into the sum of two components: growth attributed to changes in inputs (k and l) and other ‘‘residual’’ growth (i.e., changes in A) that represents technical progress. GA þ Gq ¼ eq, lGl: (9:59) Equation 9.59 provides a way of estimating the relative importance of technical progress (GA) in determining the growth of output. For example, in a pioneering study of the entire U.S. economy between the years 1909 and 1949, R. M. Solow recorded the following values for the terms in the equation18: Gq ¼ Gl ¼ Gk ¼ eq, l ¼ eq, k ¼ 2:75 percent per year, 1:00 percent per year, 1:75 percent per year, 0:65, 0:35: (9:60) 17Two useful features of this deﬁnition are: (1) Gx Æ y ¼ Gy. sum of each one’s growth rate; and (2) Gx/y ¼ 18R. M. Solow, ‘‘Technical Progress and the Aggregate Production Function,’’ Review of Economics and Statistics 39 (August 1957): 312–20. Gy—that is, the growth rate of a product of two variables is the Gx
þ Gx $ Chapter 9: Production Functions 323 0:35 1:75 ð Þ (9:61) Consequently, GA ¼ ¼ ¼ ¼ eq, lGl $ Gq $ 0:65 2:75 1:00 ð $ 0:65 2:75 1:50: $ $ eq, kGk Þ $ 0:60 The conclusion Solow reached then was that technology advanced at a rate of 1.5 percent per year from 1909 to 1949. More than half of the growth in real output could be attributed to technical change rather than to growth in the physical quantities of the factors of production. More recent evidence has tended to conﬁrm Solow’s conclusions about the relative importance of technical change. Considerable uncertainty remains, however, about the precise causes of such change. EXAMPLE 9.4 Technical Progress in the Cobb–Douglas Production Function The Cobb–Douglas production function provides an especially easy avenue for illustrating technical progress. Assuming constant returns to scale, such a production function with technical progress might be represented by q t A ð f Þ k, l ð ¼ Þ ¼ t A ð kal1 Þ a: $ (9:62) If we also assume that technical progress occurs at a constant exponential (y), then we can write A(t) Aeyt, and the production function becomes ¼ A particularly easy way to study the properties of this type of function over time is to use ‘‘logarithmic differentiation’’: a: Aeutkal1 $ q ¼ (9:63) @ ln q @t ¼ @ ln q @q u a % þ ¼ @q @t ¼ % @ ln k @q=@t q ¼ Gq ¼ ln A @ ½ þ ut þ a ln k @t 1 þ ð $ a Þ ln l ( 1 @t þ ð @ ln l Þ % @t ¼ a $ u aGk þ ð 1 þ $ a Gl: Þ (9:64) ¼ Thus, this derivation just repeats Equation 9.59 for the Cobb–Douglas case. Here the technical change factor is explicitly modeled, and the output elasticities are given by the values of the exponents in the Cobb–Douglas. The importance of technical progress can be illustrated numerically
allows for improving quality or (what amounts to the same thing) to use a multi-input production function. ¼ þ QUERY: Actual studies of production using the Cobb–Douglas tend to ﬁnd a 0.3. Use this ﬁnding together with Equation 9.67 to discuss the relative importance of improving capital and labor quality to the overall rate of technical progress. * SUMMARY In this chapter we illustrated the ways in which economists conceptualize the production process of turning inputs into outputs. The fundamental tool is the production function, form—assumes that output per which—in its simplest period (q) is a simple function of capital and labor inputs during that period, q f (k, l). Using this starting point, we developed several basic results for the theory of production. ¼ • If all but one of the inputs are held constant, a relationship between the single-variable input and output can be derived. From this relationship, one can derive the marginal physical productivity (MP) of the input as the change in output resulting from a one-unit increase in the use of the input. The marginal physical productivity of an input is assumed to decrease as use of the input increases. • The entire production function can be illustrated by its isoquant map. The (negative of the) slope of an isoquant is termed the marginal rate of technical substitution (RTS) because it shows how one input can be substituted for another while holding output constant. The RTS is the ratio of the marginal physical productivities of the two inputs. • Isoquants are usually assumed to be convex—they obey the assumption of a diminishing RTS. This assumption cannot be derived exclusively from the assumption of diminishing marginal physical productivities. One must also be concerned with the effect of changes in one input on the marginal productivity of other inputs. • The returns to scale exhibited by a production function record how output responds to proportionate increases in all inputs. If output increases proportionately with input use, there are constant returns to scale. If there are greater than proportionate increases in output, there are increasing returns to scale, whereas if there are less than proportionate increases in output, there are decreasing returns to scale. • The elasticity of substitution (s) provides a measure of how easy it is to substitute one input for another in production. A high s implies nearly linear isoquants, whereas a low s implies that isoquants are nearly L-shaped. • Technical progress shifts
the entire production function and its related isoquant map. Technical improveimproved, more ments may arise from the use of productive inputs or from better methods of economic organization. Chapter 9: Production Functions 325 PROBLEMS 9.1 Power Goat Lawn Company uses two sizes of mowers to cut lawns. The smaller mowers have a 22-inch deck. The larger ones combine two of the 22-inch decks in a single mower. For each size of mower, Power Goat has a different production function, given by the rows of the following table. Output per Hour (square feet) Capital Input (# of 2200 mowers) Labor Input Small mowers Large mowers 5000 8000 1 2 1 1 40,000 square feet isoquant for the ﬁrst production function. How much k and l would be used if these a. Graph the q ¼ factors were combined without waste? b. Answer part (a) for the second function. c. How much k and l would be used without waste if half of the 40,000-square-foot lawn were cut by the method of the ﬁrst production function and half by the method of the second? How much k and l would be used if one fourth of the lawn were cut by the ﬁrst method and three fourths by the second? What does it mean to speak of fractions of k and l? d. Based on your observations in part (c), draw a q ¼ 40,000 isoquant for the combined production functions. 9.2 Suppose the production function for widgets is given by kl q ¼ $ 0:8k2 $ 0:2l2, where q represents the annual quantity of widgets produced, k represents annual capital input, and l represents annual labor input. a. Suppose k 10; graph the total and average productivity of labor curves. At what level of labor input does this average productivity reach a maximum? How many widgets are produced at that point? ¼ b. Again assuming that k c. Suppose capital inputs were increased to k d. Does the widget production function exhibit constant, increasing, or decreasing returns to scale? 10, graph the MPl curve. At what level of labor input does MPl ¼ 20. How would your answers to parts (a) and (b) change? 0? ¼ ¼ 9.3 Sam Malone is considering renovating the bar stools at Cheers. The production function for
new bar stools is given by 0:1k0:2l 0:8, q ¼ where q is the number of bar stools produced during the renovation week, k represents the number of hours of bar stool lathes used during the week, and l represents the number of worker hours employed during the period. Sam would like to provide 10 new bar stools, and he has allocated a budget of $10,000 for the project. a. Sam reasons that because bar stool lathes and skilled bar stool workers both cost the same amount ($50 per hour), he might as well hire these two inputs in equal amounts. If Sam proceeds in this way, how much of each input will he hire and how much will the renovation project cost? b. Norm (who knows something about bar stools) argues that once again Sam has forgotten his microeconomics. He asserts that Sam should choose quantities of inputs so that their marginal (not average) productivities are equal. If Sam opts for this plan instead, how much of each input will he hire and how much will the renovation project cost? c. On hearing that Norm’s plan will save money, Cliff argues that Sam should put the savings into more bar stools to provide seating for more of his USPS colleagues. How many more bar stools can Sam get for his budget if he follows Cliff ’s plan? d. Carla worries that Cliff ’s suggestion will just mean more work for her in delivering food to bar patrons. How might she convince Sam to stick to his original 10-bar stool plan? 326 Part 4: Production and Supply 9.4 Suppose that the production of crayons (q) is conducted at two locations and uses only labor as an input. The production function in location 1 is given by q1 ¼ a. If a single ﬁrm produces crayons in both locations, then it will obviously want to get as large an output as possible given the labor input it uses. How should it allocate labor between the locations to do so? Explain precisely the relationship between l1 and l2. 1 and in location 2 by q2 ¼ 50l 0:5 2. 10l 0:5 b. Assuming that the ﬁrm operates in the efﬁcient manner described in part (a), how does total output (q) depend on the total amount of labor hired (l)? 9.5 As we have seen in many places, the general Cobb–
Douglas production function for two inputs is given by q k, l f ð ¼ Þ ¼ Akalb, where 0 < a < 1 and 0 < b < 1. For this production function: a. Show that fk > 0, f1 > 0, fkk < 0, fll < 0, and fkl ¼ flk > 0. b. Show that eq, k ¼ a and eq, l ¼ c. In footnote 5, we deﬁned the scale elasticity as b. eq, t ¼ @f tk, tl ð @t Þ % f t tk, tl ð Þ, where the expression is to be evaluated at t the scale elasticity and the returns to scale of the production function agree (for more on this concept see Problem 9.9). 1. Show that, for this Cobb–Douglas function, eq, t ¼ b. Hence in this case ¼ þ a d. Show that this function is quasi-concave. e. Show that the function is concave for a þ b ) 1 but not concave for a b > 1. þ 9.6 Suppose we are given the constant returns-to-scale CES production function q kq ¼ ½ 1=q: lq ( þ a. Show that MPk ¼ b. Show that RTS ¼ c. Determine the output elasticities for k and l; and show that their sum equals 1. d. Prove that (q/k)1–r and MPl ¼ (k/l )1–r; use this to show that s (q/l )1–r. 1/(1 r). $ ¼ and hence that q l ¼ r @q @l # $ ln q l ’ ( r ln ¼ @q @l # $ : Note: The latter equality is useful in empirical work because we may approximate @q/@l by the competitively determined wage rate. Hence s can be estimated from a regression of ln(q/l) on ln w. 9.7 Consider a generalization of the production function in Example 9.3: q b0 þ ¼ b1 klp b2k þ þ b3l, ﬃﬃﬃﬃ Chapter 9: Production Functions 327 where 0 bi ) ) 1, i ¼ 0
,..., 3: a. If this function is to exhibit constant returns to scale, what restrictions should be placed on the parameters b0,..., b3? b. Show that, in the constant returns-to-scale case, this function exhibits diminishing marginal productivities and that the marginal productivity functions are homogeneous of degree 0. c. Calculate s in this case. Although s is not in general constant, for what values of the b’s does s 0, 1, or? 1 ¼ 9.8 Show that Euler’s theorem implies that, for a constant returns-to-scale production function [q f (k, l )], ¼ q k fk % þ l: fl % ¼ Use this result to show that, for such a production function, if MPl > APl then MPk must be negative. What does this imply about where production must take place? Can a ﬁrm ever produce at a point where APl is increasing? Analytical Problems 9.9 Local returns to scale A local measure of the returns to scale incorporated in a production function is given by the scale elasticity eq, t ¼ evaluated at t a. Show that if the production function exhibits constant returns to scale, then eq, t ¼ b. We can deﬁne the output elasticities of the inputs k and l as ¼ 1. 1. @f (tk, tl )/@t Æ t/q @f @f eq, k ¼ eq, l ¼ eq, l. k, l ð @k k, l ð @l Þ % Þ % k q l q, : Show that eq, t ¼ eq, k þ c. A function that exhibits variable scale elasticity is Show that, for this function, eq,t > 1 for q < 0.5 and that eq, t < 1 for q > 0.5. d. Explain your results from part (c) intuitively. Hint: Does q have an upper bound for this production function? q 1 þ ¼ ð k$ 1 1l$ 1: $ Þ 9.10 Returns to scale and substitution Although much of our discussion of measuring the elasticity of substitution for various production functions has assumed constant returns to scale, often that assumption is not necessary. This problem illustrates some of these cases. a. In footnote 6 we pointed out that, in
the constant returns-to-scale case, the elasticity of substitution for a two-input production function is given by r ¼ f : fk fl fkl % Suppose now that we deﬁne the homothetic production function F as F k, l ð Þ ¼ ½ g, f k, l ð Þ( where f (k, l) is a constant returns-to-scale production function and g is a positive exponent. Show that the elasticity of substitution for this production function is the same as the elasticity of substitution for the function f. b. Show how this result can be applied to both the Cobb–Douglas and CES production functions. i xifi ¼ kf, and P 328 Part 4: Production and Supply 9.11 More on Euler’s theorem Suppose that a production function f(x1, x2,..., xn) is homogeneous of degree k. Euler’s theorem shows that this fact can be used to show that the partial derivatives of f are homogeneous of degree k – 1. a. Prove that 1 ¼ b. In the case of n n i P ¼ P n 1 xixj fij ¼ j ¼ 2 and k ¼ 1 k ð k 1, what kind of restrictions does the result of part (a) impose on the second-order partial f. Þ $ derivative f12? How do your conclusions change when k > 1 or k < 1? c. How would the results of part (b) be generalized to a production function with any number of inputs? d. What are the implications of this problem for the parameters of the multivariable Cobb–Douglas production function f (x1, x2,..., xn) ¼ Q 1x ai i n i ¼ for ai + 0? SUGGESTIONS FOR FURTHER READING Clark, J. M. ‘‘Diminishing Returns.’’ In Encyclopaedia of the Social Sciences, vol. 5. New York: Crowell-Collier and Macmillan, 1931, pp. 144–46. Lucid discussion of the historical development of the diminishing returns concept. Douglas, P. H. ‘‘Are There Laws of Production?’’ American Economic Review 38 (March 1948): 1–41. A nice methodological analysis of the
uses and misuses of production functions. Ferguson, C. E. The Neoclassical Theory of Production and Distribution. New York: Cambridge University Press, 1969. A thorough discussion of production function theory (as of 1970). Good use of three-dimensional graphs. Fuss, M., and D. McFadden. Production Economics: A Dual Approach to Theory and Application. Amsterdam: NorthHolland, 1980. An approach with a heavy emphasis on the use of duality. Mas-Collell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. New York: Oxford University Press, 1995. Chapter 5 provides a sophisticated, if somewhat spare, review of production theory. The use of the proﬁt function (see Chapter 11) is sophisticated and illuminating. Shephard, R. W. Theory of Cost and Production Functions. Princeton, NJ: Princeton University Press, 1978. Extended analysis of the dual relationship between production and cost functions. Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGraw-Hill, 2001. Thorough analysis of the duality between production functions and cost curves. Provides a proof that the elasticity of substitution can be derived as shown in footnote 6 of this chapter. Stigler, G. J. ‘‘The Division of Labor Is Limited by the Extent of the Market.’’ Journal of Political Economy 59 (June 1951): 185–93. Careful tracing of the evolution of Smith’s ideas about economies of scale. MANY-INPUT PRODUCTION FUNCTIONS EXTENSIONS Most of the production functions illustrated in Chapter 9 can be easily generalized to many-input cases. Here we show this for the Cobb–Douglas and CES cases and then examine two ﬂexible forms that such production functions might take. In all these examples, the a’s are non-negative parameters and the n inputs are represented by x1,..., xn. 1. Because this parameter is so constrained in and sij, ¼ the Cobb–Douglas function, the function is generally not used in econometric analyses of microeconomic data on ﬁrms. However, the function has a variety of general uses in macroeconomics, as the next example illustrates. E9.1 Cobb–Douglas The many-input Cobb
–Douglas production function is given by q ¼ xai i : n 1 i Y ¼ a. This function exhibits constant returns to scale if n 1 i X ¼ ai ¼ 1: (i) (ii) b. In the constant-returns-to-scale Cobb–Douglas function, ai is the elasticity of q with respect to input xi. Because 0 ai < 1, each input exhibits diminishing marginal productivity. c. Any degree of increasing returns to scale can be incorpo- ) rated into this function, depending on e ¼ n ai: 1 i X ¼ (iii) d. The elasticity of substitution between any two inputs in this production function is 1. This can be shown by using the deﬁnition given in footnote 7 of this chapter: rij ¼ xi=xjÞ @ ln ð fj=fiÞ @ ln ð : Here Hence fj fi ¼ aj aix j aixai i 1 $ 1 $ j xai i aj j ¼ i x aj ai % xi xj : i 6¼ Q j 6¼ Q ln fj fi# $ ln ¼ aj ai# $ ln þ xi xj# $ The Solow growth model The many-input Cobb–Douglas production function is a primary feature of many models of economic growth. For example, Solow’s (1956) pioneering model of equilibrium growth can be most easily derived using a two-input constantreturns-to-scale Cobb–Douglas function of the form Akal1 $ a, q ¼ (iv) where A is a technical change factor that can be represented by exponential growth of the form Dividing both sides of Equation iv by l yields eat: A ¼ where eat ^ ka, ^q ¼ q^ ¼ ^ q=l and k ¼ k=l (v) (vi) Solow shows that economies will evolve toward an equilibrium value of ^k (the capital–labor ratio). Hence cross-country differences in growth rates can be accounted for only by differences in the technical change factor, a. Two features of Equation vi argue for including more inputs in the Solow model. First, the equation as it stands is incapable of explaining the large differences in per capita output (^q) that are observed around the world
analogs to some of these functions, which are more widely used than the production functions themselves. E9.3 Nested production functions In some applications, Cobb–Douglas and CES production functions are combined into a ‘‘nested’’ single function. To accomplish this, the original n primary inputs are categorized into, say, m general classes of inputs. The speciﬁc inputs in each of these categories are then aggregated into a single composite input, and the ﬁnal production function is a function of these m composites. For example, assume there are three primary inputs, x1, x2, x3. Suppose, however, that x1 and x2 are relatively closely related in their use by ﬁrms (e.g., capital and energy), whereas the third input (labor) is relatively distinct. Then one might want to use a CES aggregator function to construct a composite input for capital services of the form x4 ¼ ½ gxq 1 þ (1 1=q: g)xq 2 ( $ (ixx) This structure allows the elasticity of substitution between x1 and x2 to take on any value [s r)] but constrains the elasticity of substitution between x3 and x4 to be one. A variety of other options are available depending on how precisely the embedded functions are speciﬁed. 1/(1 $ ¼ The dynamics of capital/energy substitutability Nested production functions have been widely used in studies that seek to measure the precise nature of the substitutability between capital and energy inputs. For example, Atkeson and Kehoe (1999) use a model rather close to the one speciﬁed in Equations ix and x to try to reconcile two facts about the way in which energy prices affect the economy: (1) Over time, use of energy in production seems rather unresponsive to price (at least in the short run); and (2) across countries, energy prices seem to have a large inﬂuence over how much energy is used. By using a capital service equation of the form given in Equation ix with a low 2.3)—along with a Cobb– degree of substitutability (r Douglas production function that combines labor with capital services—they are able to replicate the facts about energy prices fairly well. They conclude, however, that this model implies a much more negative effect of
higher energy prices on economic growth than seems actually to have been the case. Hence they ultimately opt for a more complex way of modeling production that stresses differences in energy use among capital investments made at different dates. ¼ $ E9.4 Generalized Leontief aij xixjp, ﬃﬃﬃﬃﬃﬃﬃ aji. where aij ¼ a. The function considered in Problem 9.7 is a simple case of this function for the case n 3, the function ¼ would have linear terms in the three inputs along with three radical terms representing all possible cross-products of the inputs. 2. For n ¼ b. The function exhibits constant returns to scale, as can be shown by using txi. Increasing returns to scale can be incorporated into the function by using the transformation qe, q0 ¼ e > 1: c. Because each input appears both linearly and under the radical, the function exhibits diminishing marginal productivities to all inputs. d. The restriction aij ¼ aji is used to ensure symmetry of the second-order partial derivatives. E9.5 Translog n ln q a0 þ ¼ 1 i X ¼ ai ln xi þ 0: aij ln xi ln xj, aij ¼ aji: a. Note that the Cobb–Douglas function is a special case of aij ¼ b. As for the Cobb–Douglas, this function may assume any this function where a0 ¼ 0 for all i, j. degree of returns to scale. If n 1 i X ¼ ai ¼ 1 and n 1 j X ¼ aij ¼ 0 for all i, then this function exhibits constant returns to scale. The proof requires some care in dealing with the double summation. c. Again, the condition aij ¼ aji is required to ensure equality of the cross-partial derivatives. Immigration Because the translog production function incorporates a large number of substitution possibilities among various inputs, it has been widely used to study the ways in which newly arrived workers may substitute for existing workers. Of particular interest is the way in which the skill level of immigrants may lead to differing reactions in the demand for skilled and Chapter 9: Production Functions 331 unskilled workers in the domestic economy. Studies of the United States and many other countries (e.g., Canada, Germany, and France) have suggested that
the overall size of such effects is modest, especially given relatively small immigration ﬂows. But there is some evidence that unskilled immigrant workers may act as substitutes for unskilled domestic workers but as complements to skilled domestic workers. Hence increased immigration ﬂows may exacerbate trends toward increasing wage differentials. For a summary, see Borjas (1994). References Atkeson, Andrew, and Patrick J. Kehoe. ‘‘Models of Energy Use: Putty-Putty versus Putty-Clay.’’ American Economic Review (September 1999): 1028–43. Bairam, Erkin. ‘‘Elasticity of Substitution, Technical Progress and Returns to Scale in Branches of Soviet Industry: A New CES Production Function Approach.’’ Journal of Applied Economics (January–March 1991): 91–96. Borjas, G. J. ‘‘The Economics of Immigration.’’ Journal of Economic Literature (December 1994): 1667–717. Romer, David. Advanced Macroeconomics. New York: McGraw- Hill, 1996. Solow, R. M. ‘‘A Contribution to the Theory of Economic (February Growth.’’ Quarterly Journal of Economics 1956): 65–94. This page intentionally left blank C H A P T E R TEN Cost Functions In this chapter we illustrate the costs that a ﬁrm incurs when it produces output. In Chapter 11, we will pursue this topic further by showing how ﬁrms make proﬁt-maximizing input and output decisions. Deﬁnitions of Costs Before we can discuss the theory of costs, some difﬁculties about the proper deﬁnition of ‘‘costs’’ must be cleared up. Speciﬁcally, we must distinguish between (1) accounting cost and (2) economic cost. The accountant’s view of cost stresses out-of-pocket expenses, historical costs, depreciation, and other bookkeeping entries. The economist’s deﬁnition of cost (which in obvious ways draws on the fundamental opportunity-cost notion) is that the cost of any input is given by the size of the payment necessary to keep the resource in its present employment. Alternatively, the economic cost of using an input is what that input would be paid in
its next best use. One way to distinguish between these two views is to consider how the costs of various inputs (labor, capital, and entrepreneurial services) are deﬁned under each system. Labor costs Economists and accountants regard labor costs in much the same way. To accountants, expenditures on labor are current expenses and hence costs of production. For economists, labor is an explicit cost. Labor services (labor-hours) are contracted at some hourly wage rate (w), and it is usually assumed that this is also what the labor services would earn in their best alternative employment. The hourly wage, of course, includes costs of fringe beneﬁts provided to employees. Capital costs In the case of capital services (machine-hours), the two concepts of cost differ. In calculating capital costs, accountants use the historical price of the particular machine under investigation and apply some more-or-less arbitrary depreciation rule to determine how much of that machine’s original price to charge to current costs. Economists regard the historical price of a machine as a ‘‘sunk cost,’’ which is irrelevant to output decisions. They instead regard the implicit cost of the machine to be what someone else would be willing to pay for its use. Thus, the cost of one machine-hour is the rental rate for that machine in its best alternative use. By continuing to use the machine itself, the ﬁrm is 333 334 Part 4: Production and Supply implicitly forgoing what someone else would be willing to pay to use it. This rental rate for one machine-hour will be denoted by v.1 Suppose a company buys a computer for $2,000. An accountant applying a ‘‘straightline’’ depreciation method over ﬁve years would regard the computer as having a cost of $400 a year. An economist would look at the market value of the computer. The availability of much faster computers in subsequent years can cause the second-hand price of the original computer to decrease precipitously. If the second-hand price decreases all the way to, for example, $200 after the ﬁrst year, the economic cost will be related to this $200; the original $2,000 price will no longer be relevant. (All these yearly costs can easily be converted into computer-hour costs, of course.) The distinction between accounting and economic costs of capital largely disappears if the company rents it at a price of v
each period rather than purchasing. Then v reﬂects a current company expenditure that shows up directly as an accounting cost; it also reﬂects the market value of one period’s use of the capital and thus is an opportunity/economic cost. Costs of entrepreneurial services The owner of a ﬁrm is a residual claimant who is entitled to whatever extra revenues or losses are left after paying other input costs. To an accountant, these would be called proﬁts (which might be either positive or negative). Economists, however, ask whether owners (or entrepreneurs) also encounter opportunity costs by working at a particular ﬁrm or devoting some of their funds to its operation. If so, these services should be considered an input, and some cost should be imputed to them. For example, suppose a highly skilled computer programmer starts a software ﬁrm with the idea of keeping any (accounting) proﬁts that might be generated. The programmer’s time is clearly an input to the ﬁrm, and a cost should be attributed to it. Perhaps the wage that the programmer might command if he or she worked for someone else could be used for that purpose. Hence some part of the accounting proﬁts generated by the ﬁrm would be categorized as entrepreneurial costs by economists. Economic proﬁts would be smaller than accounting proﬁts and might be negative if the programmer’s opportunity costs exceeded the accounting proﬁts being earned by the business. Similar arguments apply to the capital that an entrepreneur provides to the ﬁrm. Economic costs In this book, not surprisingly, we use economists’ deﬁnition of cost Economic cost. The economic cost of any input is the payment required to keep that input in its present employment. Equivalently, the economic cost of an input is the remuneration the input would receive in its best alternative employment. Use of this deﬁnition is not meant to imply that accountants’ concepts are irrelevant to economic behavior. Indeed, accounting procedures are integrally important to any manager’s decision-making process because they can greatly affect the rate of taxation to be applied against proﬁts. Accounting data are also readily available, whereas data on economic costs must often be developed separately. Economists’ deﬁnitions, however, do have 1Sometimes the symbol r
is chosen to represent the rental rate on capital. Because this variable is often confused with the related but distinct concept of the market interest rate, an alternative symbol was chosen here. The exact relationship between v and the interest rate is examined in Chapter 17. Chapter 10: Cost Functions 335 the desirable features of being broadly applicable to all ﬁrms and of forming a conceptually consistent system. Therefore, they are best suited for a general theoretical analysis. Simplifying assumptions As a start, we will make two simpliﬁcations about the inputs a ﬁrm uses. First, we assume that there are only two inputs: homogeneous labor (l, measured in labor-hours) and homogeneous capital (k, measured in machine-hours). Entrepreneurial costs are included in capital costs. That is, we assume that the primary opportunity costs faced by a ﬁrm’s owner are those associated with the capital that the owner provides. Second, we assume that inputs are hired in perfectly competitive markets. Firms can buy (or sell) all the labor or capital services they want at the prevailing rental rates (w and v). In graphic terms, the supply curve for these resources is horizontal at the prevailing factor prices. Both w and v are treated as ‘‘parameters’’ in the ﬁrm’s decisions; there is nothing the ﬁrm can do to affect them. These conditions will be relaxed in later chapters (notably Chapter 16), but for the moment the price-taker assumption is a convenient and useful one to make. Therefore, with these simpliﬁcations, total cost C for the ﬁrm during the period is given by where, as before, l and k represent input usage during the period. total cost C ¼ ¼ wl þ vk (10:1) Relationship between proﬁt maximization and cost minimization Let’s look ahead to the next chapter on proﬁt maximization and compare the analysis here with the analysis in that chapter. We will deﬁne economic proﬁts (p) as the difference between the ﬁrm’s total revenues (R) and its total costs (C). Suppose the ﬁrm takes the market price ( p) for its total output (q) as given and that its production function is q f (k, l). Then its proﬁ
t can be written ¼ p R C pq wl vk pf k # wl # vk: (10:2) Equation 10.2 shows that the economic proﬁts obtained by this ﬁrm are a function of the amount of capital and labor employed. If, as we will assume in many places in this book, this ﬁrm seeks maximum proﬁts, then we might study its behavior by examining how k and l are chosen to maximize Equation 10.2. This would, in turn, lead to a theory of supply and to a theory of the ‘‘derived demand’’ for capital and labor inputs. In the next chapter we will take up those subjects in detail. Here, however, we wish to develop a theory of costs that is somewhat more general, applying not only to ﬁrms that are price-takers on their output markets (perfect competitors) but also to those whose output choice affects the market price (monopolies and oligopolies). The more general theory will even apply to nonproﬁts (as long as they are interested in operating efﬁciently). The other advantage of looking at cost minimization separately from proﬁt maximization is that it is simpler to analyze this small ‘‘piece’’ in isolation and only later add the insights obtained into the overall ‘‘puzzle’’ of the ﬁrm’s operations. The conditions derived for cost-minimizing input choices in this chapter will emerge again as a ‘‘by-product’’ of the analysis of the maximization of proﬁts as speciﬁed in Equation 10.2. Hence we begin the study of costs by ﬁnessing, for the moment, a discussion of output choice. That is, we assume that for some reason the ﬁrm has decided to produce a particular output level (say, q0). The ﬁrm will of course earn some revenue R from this output choice, but we will ignore revenue for now. We will focus solely on the question of how the ﬁrm can produce q0 at minimal cost. 336 Part 4: Production and Supply Cost-Minimizing Input Choices Mathematically, this is a constrained minimization problem. But before proceeding with a rigorous solution, it is
useful to state the result to be derived with an intuitive argument. To minimize the cost of producing a given level of output, a ﬁrm should choose that point on the q0 isoquant at which the rate of technical substitution of l for k is equal to the ratio w/v: It should equate the rate at which k can be traded for l in production to the rate at which they can be traded in the marketplace. Suppose that this were not true. 10, In particular, suppose that the ﬁrm were producing output level q0 using k ¼ and assume that the RTS were 2 at this point. Assume also that w $1, and hence that w/v 1 (which is unequal to 2). At this input combination, the cost of producing q0 is $20. It is easy to show this is not the minimal input cost. For example, q0 11; we can give up two units of k and keep can also be produced using k output constant at q0 by adding one unit of l. But at this input combination, the cost of producing q0 is $19, and hence the initial input combination was not optimal. A contradiction similar to this one can be demonstrated whenever the RTS and the ratio of the input costs differ. ¼ $1, v 8 and l 10, l ¼ ¼ ¼ ¼ ¼ Mathematical analysis Mathematically, we seek to minimize total costs given q Lagrangian f(k, l ) ¼ ¼ q0. Setting up the, Þ’ the ﬁrst-order conditions for a constrained minimum are q0 # k ½ k, l wl vk ¼ þ þ ð + f @f @l ¼ @f @k ¼ 0, 0, @+ @l ¼ @+ @k ¼ @+ @k ¼ w # k k v # q0 # k, l f ð Þ ¼ 0, or, dividing the ﬁrst two equations, w v ¼ @f =@l @f =@k ¼ RTS of l for k Þ : ð (10:3) (10:4) (10:5) This says that the cost-minimizing ﬁrm should equate the RTS for the two inputs to the ratio of their prices. Further interpretations These ﬁrst-order conditions for minimal costs can be
manipulated in several different ways to yield interesting results. For example, cross-multiplying Equation 10.5 gives f k v ¼ f l w : (10:6) That is, for costs to be minimized, the marginal productivity per dollar spent should be the same for all inputs. If increasing one input promised to increase output by a greater amount per dollar spent than did another input, costs would not be minimal—the ﬁrm should hire more of the input that promises a bigger ‘‘bang per buck’’ and less of the more costly (in terms of productivity) input. Any input that cannot meet the common beneﬁt–cost ratio deﬁned in Equation 10.6 should not be hired at all. Chapter 10: Cost Functions 337 Equation 10.6 can, of course, also be derived from Equation 10.4, but it is more in- structive to derive its inverse: (10:7) This equation reports the extra cost of obtaining an extra unit of output by hiring either added labor or added capital input. Because of cost minimization, this marginal cost is the same no matter which input is hired. This common marginal cost is also measured by the Lagrange multiplier from the cost-minimization problem. As is the case for all constrained optimization problems, here the Lagrange multiplier shows how much in extra costs would be incurred by increasing the output constraint slightly. Because marginal cost plays an important role in a ﬁrm’s supply decisions, we will return to this feature of cost minimization frequently. Graphical analysis Cost minimization is shown graphically in Figure 10.1. Given the output isoquant q0, we wish to ﬁnd the least costly point on the isoquant. Lines showing equal cost are parallel straight lines with slopes w/v. Three lines of equal total cost are shown in Figure 10.1; C1 < C2 < C3. It is clear from the ﬁgure that the minimum total cost for producing q0 is given by C1, where the total cost curve is just tangent to the isoquant. The associated inputs are l c and k c, where the superscripts emphasize that these input levels are a solution to a cost-minimization problem. This combination will be a true minimum if the isoquant is convex (if the RTS diminishes for decreases in k/l ). The mathematical
and graphic analyses arrive at the same conclusion, as follows. # FIGURE 10.1 Minimization of Costs Given q q0 ¼ A ﬁrm is assumed to choose k and l to minimize total costs. The condition for this minimization is that the rate at which k and l can be traded technically (while keeping q q0) should be equal to the rate at which these inputs can be traded in the market. In other words, the RTS (of l for k) should be set equal to the price ratio w/v. This tangency is shown in the ﬁgure; costs are minimized at C1 by choosing inputs k c and l c. ¼ k per period C1 C2 kc lc C3 q0 l per period 338 Part 4: Production and Supply Cost minimization. To minimize the cost of any given level of output (q0), the ﬁrm should produce at that point on the q0 isoquant for which the RTS (of l for k) is equal to the ratio of the inputs’ rental prices (w/v). Contingent demand for inputs Figure 10.1 exhibits the formal similarity between the ﬁrm’s cost-minimization problem and the individual’s expenditure-minimization problem studied in Chapter 4 (see Figure 4.6). In both problems, the economic actor seeks to achieve his or her target (output or utility) at minimal cost. In Chapter 5 we showed how this process is used to construct a theory of compensated demand for a good. In the present case, cost minimization leads to a demand for capital and labor input that is contingent on the level of output being produced. Therefore, this is not the complete story of a ﬁrm’s demand for the inputs it uses because it does not address the issue of output choice. But studying the contingent demand for inputs provides an important building block for analyzing the ﬁrm’s overall demand for inputs, and we will take up this topic in more detail later in this chapter. The ﬁrm’s expansion path A ﬁrm can follow the cost-minimization process for each level of output: For each q, it ﬁnds the input choice that minimizes the cost of producing it. If input costs (w and v) remain constant for all amounts the ﬁrm may demand, we can easily trace this loc
us of cost-minimizing choices. This procedure is shown in Figure 10.2. The curve 0E records the cost-minimizing tangencies for successively higher levels of output. For example, the minimum cost for producing output level q1 is given by C1, and inputs k1 and l1 are used. Other tangencies in the ﬁgure can be interpreted in a similar way. The locus of these FIGURE 10.2 The Firm’s Expansion Path The ﬁrm’s expansion path is the locus of cost-minimizing tangencies. Assuming ﬁxed input prices, the curve shows how inputs increase as output increases. k per period E k1 q3 q2 q1 C1 C2 C3 0 l1 l per period Chapter 10: Cost Functions 339 tangencies is called the ﬁrm’s expansion path because it records how input expands as output expands while holding the prices of the inputs constant. As Figure 10.2 shows, the expansion path need not be a straight line. The use of some inputs may increase faster than others as output expands. Which inputs expand more rapidly will depend on the shape of the production isoquants. Because cost minimization requires that the RTS always be set equal to the ratio w/v, and because the w/v ratio is assumed to be constant, the shape of the expansion path will be determined by where a particular RTS occurs on successively higher isoquants. If the production function exhibits constant returns to scale (or, more generally, if it is homothetic), then the expansion path will be a straight line because in that case the RTS depends only on the ratio of k to l. That ratio would be constant along such a linear expansion path. It would seem reasonable to assume that the expansion path will be positively sloped; that is, successively higher output levels will require more of both inputs. This need not be the case, however, as Figure 10.3 illustrates. Increases of output beyond q2 cause the quantity of labor used to decrease. In this range, labor would be said to be an inferior input. The occurrence of inferior inputs is then a theoretical possibility that may happen, even when isoquants have their usual convex shape. Much theoretical discussion has centered on the analysis of factor inferiority. Whether inferiority is likely to occur in real-world production functions is a difﬁcult empirical question
to produce q0 ¼ 40. The ﬁrst-order condition for a minimum requires that k 4l. Inserting that into the production function (the ﬁnal requirement in Equation 10.9), we have q0 ¼ k0.5l0.5 40 80, and ¼ total costs are given by vk 480. That this is a true cost minimum 12(20) is suggested by looking at a few other input combinations that also are capable of producing 40 units of output: 2l. Thus, the cost-minimizing input combination is l 20 and k 3(80) 12, v wl 40, l 10, l 160, l ¼ ¼ ¼ 40, C 160, C 10, C ¼ ¼ ¼ 600, 2,220, 600: (10:11) Any other input combination able to produce 40 units of output will also cost more than 480. Cost minimization is also suggested by considering marginal productivities. At the optimal point MPk ¼ MPl ¼ f k ¼ f l ¼ 0:5k# 0:5k0:5l # 0:5l 0:5 0:5 ¼ 0:5 0:5 20=80 ð 80=20 ð 0:5 Þ 0:5 Þ 0:25, ¼ 1:0; (10:12) ¼ hence at the margin, labor is four times as productive as capital, and this extra productivity precisely compensates for the higher unit price of labor input. ¼ 2. CES: q f (k, l ) (kr ¼ þ ¼ l r)g/r. Again we set up the Lagrangian expression + vk wl k ½ kq q0 # ð þ l q g=q Þ, ’ þ þ ¼ and the ﬁrst-order conditions for a minimum are @+ @k ¼ @+ @l ¼ @+ @g ¼ v g=q k ð # kq Þð l q ð Þ þ g q # ==q k # ð kq q0 # ð þ kq Þð l q ð Þ þ g # =: 0, 0, ¼ ¼ (10:13) (10:14) Chapter 10: Cost Functions 341 Dividing the ﬁrst
two of these equations causes a lot of this mass of symbols to drop out, leaving =r, or 10:15) ¼ 1). With the Cobb where s 1/(1 – r) is the elasticity of substitution. Because the CES function is also homothetic, the cost-minimizing input ratio is independent of the absolute level of production. Douglas result (when The result in Equation 10.15 is a simple generalization of the Cobb Douglas, the cost-minimizing capital–labor ratio changes directly s in proportion to changes in the ratio of wages to capital rental rates. In cases with greater substitutability (s > 1), changes in the ratio of wages to rental rates cause a greater than proportional increase in the cost-minimizing capital–labor ratio. With less substitutability (s < 1), the response is proportionally smaller. # ¼ # QUERY: In the Cobb–Douglas numerical example with w/v minimizing input ratio for producing 40 units of output was k/l value change for s would total costs be? 4, we found that the cost¼ 4. How would this 80/20 ¼ 0.5? What actual input combinations would be used? What 2 or s ¼ ¼ ¼ Cost Functions We are now in a position to examine the ﬁrm’s overall cost structure. To do so, it will be convenient to use the expansion path solutions to derive the total cost function Total cost function. The total cost function shows that, for any set of input costs and for any output level, the minimum total cost incurred by the ﬁrm is C C v, w, q ð : Þ ¼ (10:16) Figure 10.2 makes clear that total costs increase as output, q, increases. We will begin by analyzing this relationship between total cost and output while holding input prices ﬁxed. Then we will consider how a change in an input price shifts the expansion path and its related cost functions. Average and marginal cost functions Although the total cost function provides complete information about the output–cost relationship, it is often convenient to analyze costs on a per-unit of output basis because that approach corresponds more closely to the analysis of demand, which focused on the price per unit of a commodity. Two different unit cost measures are widely used in economics: (1) average cost, which is the cost per unit of output; and (2) marginal cost, which
is the cost of one more unit of output Average and marginal cost functions. The average cost function (AC) is found by computing total costs per unit of output: average cost AC v, w, q ð Þ ¼ ¼ C v, w, q ð q Þ : (10:17) 342 Part 4: Production and Supply The marginal cost function (MC ) is found by computing the change in total costs for a change in output produced: marginal cost MC v, w, q ð Þ ¼ ¼ @C v, w, q ð @q Þ : (10:18) Notice that in these deﬁnitions, average and marginal costs depend both on the level of output being produced and on the prices of inputs. In many places throughout this book, we will graph simple two-dimensional relationships between costs and output. As the definitions make clear, all such graphs are drawn on the assumption that the prices of inputs remain constant and that technology does not change. If input prices change or if technology advances, cost curves generally will shift to new positions. Later in this chapter, we will explore the likely direction and size of such shifts when we study the entire cost function in detail. Graphical analysis of total costs Figures 10.4a and 10.5a illustrate two possible shapes for the relationship between total cost and the level of the ﬁrm’s output. In Figure 10.4a, total cost is simply proportional to output. Such a situation would arise if the underlying production function exhibits constant returns to scale. In that case, suppose k1 units of capital input and l1 units of labor input are required to produce one unit of output. Then C q ð ¼ 1 Þ ¼ v k1 þ w l1: (10:19) To produce m units of output requires mk1 units of capital and ml1 units of labor because of the constant returns-to-scale assumption.2 Hence wml1 ¼, 1 Þ ¼ vmk1 þ q C m ð ( vk1 þ ð Þ ¼ ¼ wl1Þ (10:20) q ð m m ¼ C and the proportionality between output and cost is established. The situation in Figure 10.5a is more complicated. There it is assumed that initially the total cost curve is concave; although initially costs increase rapidly for increases in output, that rate of increase slows
as output expands into the midrange of output. Beyond this middle range, however, the total cost curve becomes convex, and costs begin to increase progressively more rapidly. One possible reason for such a shape for the total cost curve is that there is some third factor of production (say, the services of an entrepreneur) that is ﬁxed as capital and labor usage expands. In this case, the initial concave section of the curve might be explained by the increasingly optimal usage of the entrepreneur’s services—he or she needs a moderate level of production to use his or her skills fully. Beyond the point of inﬂection, however, the entrepreneur becomes overworked in attempting to coordinate production, and diminishing returns set in. Hence total costs increase rapidly. A variety of other explanations have been offered for the cubic-type total cost curve in Figure 10.5a, but we will not examine them here. Ultimately, the shape of the total cost curve is an empirical question that can be determined only by examining realworld data. In the Extensions to this chapter, we illustrate some of the literature on cost functions. 2The input combination (ml1, mk1) minimizes the cost of producing m units of output because the ratio of the inputs is still k1/l1 and the RTS for a constant returns-to-scale production function depends only on that ratio. FIGURE 10.4 Total, Average, and Marginal Cost Curves for the Constant Returns-to-Scale Case Chapter 10: Cost Functions 343 In (a) total costs are proportional to output level. Average and marginal costs, as shown in (b), are equal and constant for all output levels. Total costs (a) Average and marginal costs (b) C Output per period AC = MC Output per period Graphical analysis of average and marginal costs Information from the total cost curves can be used to construct the average and marginal cost curves shown in Figures 10.4b and 10.5b. For the constant returns-to-scale case (Figure 10.4), this is simple. Because total costs are proportional to output, average and marginal costs are constant and equal for all levels of output.3 These costs are shown by the horizontal line AC MC in Figure 10.4b. For the cubic total cost curve case (Figure 10.5), computation of the average and marginal cost curves requires some geometric intuition. As the deﬁnition in Equation 10.18 makes clear, marginal cost is simply
the slope of the total cost curve. Hence because of ¼ 3Mathematically, because C ¼ aq (where a is the cost of one unit of output), AC C q ¼ a ¼ ¼ @C @q ¼ MC: 344 Part 4: Production and Supply FIGURE 10.5 Total, Average, and Marginal Cost Curves for the Cubic Total Cost Curve Case If the total cost curve has the cubic shape shown in (a), average and marginal cost curves will be U-shaped. In (b) the marginal cost curve passes through the low point of the average cost curve at output level q). Total costs (a) Average and marginal costs C Output per period MC AC (b) q* Output per period the assumed shape of the curve, the MC curve is U-shaped, with MC falling over the concave portion of the total cost curve and rising beyond the point of inﬂection. Because the slope is always positive, however, MC is always greater than 0. Average costs (AC) start out being equal to marginal cost for the ‘‘ﬁrst’’ unit of output.4 As output expands, however, AC exceeds MC because AC reﬂects both the marginal cost of the last unit produced 4Technically, AC MC at q ¼ ¼ 0. This can be shown by L’Hoˆpital’s rule, which states that if f (a) g (a) ¼ ¼ 0, then In this case, C 0 at q ¼ ¼ 0, and thus or which was to be shown. lim lim 0ð Þ AC lim 0 q! lim 0 q! ¼ C q ¼ lim 0 q! @C=@q 1 ¼ MC lim 0 q! AC ¼ MC at q 0, ¼ Chapter 10: Cost Functions 345 and the higher marginal costs of the previously produced units. As long as AC > MC, average costs must be decreasing. Because the lower costs of the newly produced units are below average cost, they continue to pull average costs downward. Marginal costs increase, however, and eventually (at q)) equal average cost. Beyond this point, MC > AC, and average costs will increase because they are pulled upward by increasingly higher marginal costs. Consequently, we have shown that the AC curve also has a Ushape and that it reaches a low point at q), where AC and
MC intersect.5 In empirical studies of cost functions, there is considerable interest in this point of minimum average cost. It reﬂects the minimum efﬁcient scale (MES) for the particular production process being examined. The point is also theoretically important because of the role it plays in perfectly competitive price determination in the long run (see Chapter 12). Cost Functions and Shifts in Cost Curves The cost curves illustrated in Figures 10.4 and 10.5 show the relationship between costs and quantity produced on the assumption that all other factors are held constant. Speciﬁcally, construction of the curves assumes that input prices and the level of technology do not change.6 If these factors do change, the cost curves will shift. In this section, we delve further into the mathematics of cost functions as a way of studying these shifts. We begin with some examples. EXAMPLE 10.2 Some Illustrative Cost Functions In this example we calculate the cost functions associated with three different production functions. Later we will use these examples to illustrate some of the general properties of cost functions. ¼ 1. Fixed Proportions: q f (k, l ) min(ak, bl ). The calculation of cost functions from their underlying production functions is one of the more frustrating tasks for economics students. Thus, let’s start with a simple example. What we wish to do is show how total costs depend on input costs and on quantity produced. In the ﬁxed-proportions case, we know bl. Hence that production will occur at a vertex of the L-shaped isoquants where q total costs are ak ¼ ¼ ¼ C q a # $ This is indeed the sort of function we want because it states total costs as a function of v, w, and q only together with some parameters of the underlying production function. q b! " v, w, q ð v a þ q! (10:21) w b Þ ¼ wl vk þ ¼ ¼ þ " w v : 5Mathematically, we can ﬁnd the minimum AC by setting its derivative equal to 0: or @AC @q ¼ @ C=q Þ ð @q ¼ q ( ð @C=@q q2 C 1 ( Þ # q ( MC q2 C # ¼ 0, ¼ MC q ( C # ¼ 0 or MC C=q AC:
¼ ¼ ¼ AC when AC is minimized. Thus, MC 6For multiproduct ﬁrms, an additional complication must be considered. For such ﬁrms it is possible that the costs associated with producing one output (say, q1) are also affected by the amount of some other output being produced (q2). In this case the ﬁrm is said to exhibit ‘‘economies of scope,’’ and the total cost function will be of the form C(v, w, q1, q2). Hence q2 must also be held constant in constructing the q1 cost curves. Presumably increases in q2 shift the q1 cost curves downward. 346 Part 4: Production and Supply Because of the constant returns-to-scale nature of this production function, it takes the special form C v, w, q ð Þ ¼ q C : v, w, 1 Þ ð (10:22) That is, total costs are given by output times the cost of producing one unit. Increases in input prices clearly increase total costs with this function, and technical improvements that take the form of increasing the parameters a and b reduce costs. 2. Cobb–Douglas: q f (k, l ) kalb. This is our ﬁrst example of burdensome computation, but we can clarify the process by recognizing that the ﬁnal goal is to use the results of cost minimization to replace the inputs in the production function with costs. From Example 10.1 we know that cost minimization requires that ¼ ¼ and so : k ¼ (10:23) (10:24) Substitution into the production function permits a solution for labor input in terms of q, v, and w as kal 10:25) or lc v, w, q ð Þ ¼ q1=a þ a= ð a þ b Þ w# a= ð b b a! " a b Þva= ð a b Þ: þ (10:26) þ A similar set of manipulations gives kc v, w, q ð Þ ¼ b= b a q1= wb= ð Now we are ready to derive total costs as a b Þv# b= ð a þ b Þ: (10:27) þ C v, w, q ð a
/(a b)a# vk c wl c þ ¼ Þ ¼ q1= ð a þ ÞBva= b ð a b Þw b= ð a þ b Þ, (10:28) þ þ þ ¼ þ b/(a (a b)b# b)—a constant that involves only the parameters a and b. where B Although this derivation was a bit messy, several interesting aspects of this Cobb–Douglas cost function are readily apparent. First, whether the function is a convex, linear, or concave b < 1), function of output depends on whether there are decreasing returns to scale (a þ b > 1). Second, an constant returns to scale (a increase in any input price increases costs, with the extent of the increase being determined by the relative importance of the input as reﬂected by the size of its exponent in the production function. Finally, the cost function is homogeneous of degree 1 in the input prices—a general feature of all cost functions, as we shall show shortly. 1), or increasing returns to scale (a ¼ þ þ b 3. CES: q f (k, l ) l r)g/r. For this case, your authors will mercifully spare you the algebra. To derive the total cost function, we use the cost-minimization condition speciﬁed in Equation 10.15, solve for each input individually, and eventually get (kr ¼ ¼ þ C v, w, q ð Þ ¼ ¼ vk þ q1=g wl v1 # ð ¼ r q1=g vq= ð w1 # = Þ þ þ r Þ, wq= ð q # 1 Þ q # =q 1 Þ ð Þ (10:29) where the elasticity of substitution is given by s 1/(1 – r). Once again the shape of the total cost is determined by the scale parameter (g) for this production function, and the cost ¼ Chapter 10: Cost Functions 347 function increases in both of the input prices. The function is also homogeneous of degree 1 in those prices. One limiting feature of this form of the CES function is that the inputs are given equal weights—hence their prices are equally important in the cost function. This feature of the CES is easily generalized, however (see Problem 10.
9). QUERY: How are the various substitution possibilities inherent in the CES function reﬂected in the CES cost function in Equation 10.29? Properties of cost functions These examples illustrate some properties of total cost functions that are general. 1. Homogeneity. The total cost functions in Example 10.2 are all homogeneous of degree 1 in the input prices. That is, a doubling of input prices will precisely double the cost of producing any given output level (you might check this out for yourself). This is a property of all proper cost functions. When all input prices double (or are increased by any uniform proportion), the ratio of any two input prices will not change. Because cost minimization requires that the ratio of input prices be set equal to the RTS along a given isoquant, the cost-minimizing input combination also will not change. Hence the ﬁrm will buy exactly the same set of inputs and pay precisely twice as much for them. One implication of this result is that a pure, uniform inﬂation in all input costs will not change a ﬁrm’s input decisions. Its cost curves will shift upward in precise correspondence to the rate of inﬂation. 2. Total cost functions are nondecreasing in q, v, and w. This property seems obvious, but it is worth dwelling on it a bit. Because cost functions are derived from a costminimization process, any decrease in costs from an increase in one of the function’s arguments would lead to a contradiction. For example, if an increase in output from q1 to q2 caused total costs to decrease, it must be the case that the ﬁrm was not minimizing costs in the ﬁrst place. It should have produced q2 and thrown away an output of q2 # q1, thereby producing q1 at a lower cost. Similarly, if an increase in the price of an input ever reduced total cost, the ﬁrm could not have been minimizing its costs in the ﬁrst place. To see this, suppose the ﬁrm was using the input combination (l1, k1) and that w increases. Clearly that will increase the cost of the initial input combination. But if changes in input choices caused total costs to decrease, that must imply that there was a lower-cost input mix than (l1, k1) initially. Hence we have a contradiction, and this property
of cost functions is established.7 3. Total cost functions are concave in input prices. It is probably easiest to illustrate this property with a graph. Figure 10.6 shows total costs for various values of an input price, say, w, holding q and v constant. Suppose that initially input prices w0 and v0 prevail 7A formal proof could also be based on the envelope theorem as applied to constrained minimization problems. Consider the Lagrangian in Equation 10.3. As was pointed out in Chapter 2, we can calculate the change in the objective in such an expression (here, total cost) with respect to a change in a variable by differentiating the Lagrangian. Performing this differentiation yields @C @q ¼ @C @v ¼ @C @w ¼ @+ @q ¼ @+ @v ¼ @+ @w ¼ MC k ð¼ Þ * 0, k c l c 0, * 0: * Not only do these envelope results prove this property of cost functions, but they also are useful in their own right, as we will show later in this chapter. 348 Part 4: Production and Supply FIGURE 10.6 Cost Functions Are Concave in Input Prices With input prices w 0 and v 0, total costs of producing q0 are C (v 0, w 0, q0). If the ﬁrm does not change its input mix, costs of producing q0 would follow the straight line CPSEUDO. With input substitution, actual costs C (v 0, w, q0) will fall below this line, and hence the cost function is concave in w. Costs C(v′, w′, q0) CPSEUDO C(v′, w, q0) w′ w and that total output q0 is produced at total cost C(v0, w0, q0) using cost-minimizing inputs l 0 and k 0. If the ﬁrm did not change its input mix in response to changes in wages, then its total cost curve would be linear as reﬂected by the line CPSEUDO(v 0, w, q0) v 0k 0 wl 0 in the ﬁgure. But a cost-minimizing ﬁrm probably would change the input mix it þ uses to produce q0 when wages change, and these actual costs C(v 0, w, q
0) would fall below the ‘‘pseudo’’ costs. Hence the total cost function must have the concave shape shown in Figure 10.6. One implication of this ﬁnding is that costs will be lower when a ﬁrm faces input prices that ﬂuctuate around a given level than when they remain constant at that level. With ﬂuctuating input prices, the ﬁrm can adapt its input mix to take advantage of such ﬂuctuations by using a lot of, say, labor when its price is low and economizing on that input when its price is high. ¼ 4. Average and marginal costs. Some, but not all, of these properties of total cost functions carry over to their related average and marginal cost functions. Homogeneity is one property that carries over directly. Because C(tv, tw, q) tC(v, w, q), we have ¼ AC ð tv, tw, q Þ ¼ C tv, tw, q ð q Þ tC v, w, q ð q Þ ¼ tAC ð ¼ v, w, q Þ (10:30) and8 MC tv, tw, q ð Þ ¼ @C ð tv, tw, q Þ @q ¼ t@C v, w, q Þ ð @q tMC ð ¼ v, w, q : Þ (10:31) 8This result does not violate the theorem that the derivative of a function that is homogeneous of degree k is homogeneous of 1 because we are differentiating with respect to q and total costs are homogeneous with respect to input prices only. degree k # Chapter 10: Cost Functions 349 The effects of changes in q, v, and w on average and marginal costs are sometimes ambiguous, however. We have already shown that average and marginal cost curves may have negatively sloped segments, so neither AC nor MC is nondecreasing in q. Because total costs must not decrease when an input price increases, it is clear that average cost is increasing in w and v. But the case of marginal cost is more complex. The main complication arises because of the possibility of input inferiority. In that (admittedly rare) case, an increase in an inferior input’s price will actually cause marginal cost to decrease. Although the proof of this is relatively straightforward,9
an intuitive explanation for it is elusive. Still, in most cases, it seems clear that the increase in the price of an input will increase marginal cost as well. Input substitution A change in the price of an input will cause the ﬁrm to alter its input mix. Hence a full study of how cost curves shift when input prices change must also include an examination of substitution among inputs. The previous chapter provided a concept measuring how substitutable inputs are—the elasticity of substitution. Here we will modify the deﬁnition, using some results from cost minimization, so that it is expressed only in terms of readily observable variables. The modiﬁed deﬁnition will turn out to be more useful for empirical work. Recall the formula for the elasticity of substitution from Chapter 9, repeated here: d k=l ð Þ d RTS ( RTS k=l ¼ d ln k=l Þ ð d ln RTS : r ¼ (10:32) But the cost-minimization principle says that RTS stituting gives a new version of the elasticity of substitution:10 of l for k ð Þ ¼ w=v at an optimum. Sub- s ¼ d k=l Þ ð d w=v Þ ð w=v k=l ¼ ( d ln k=l Þ ð d ln w=v Þ ð, (10:33) distinguished by changing the label from s to s. The elasticities differ in two respects. Whereas s applies to any point on any isoquant, s applies only to a single point on a single isoquant (the equilibrium point where there is a tangency between the isoquant and an equal total cost line). Although this would seem to be a drawback of s, the big advantage of focusing on the equilibrium point is that s then involves only easily observable variables: input amounts and prices. By contrast, s involves the RTS, the slope of an isoquant. Knowledge of the RTS would require detailed knowledge of the production process that even the ﬁrm’s engineers may not have, let alone an outside observer. In the two-input case, s must be non-negative; an increase in w/v will be met by an increase in k/l (or, in the limiting ﬁxed-proportions case
, k/l will stay constant). Large values of s indicate that ﬁrms change their input proportions signiﬁcantly in response to changes in relative input prices, whereas low values indicate that changes in input prices have relatively little effect. 9The proof follows the envelope theorem results presented in footnote 7. Because the MC function can be derived by differentiation from the Lagrangian for cost minimization, we can use Young’s theorem to show @MC @v ¼ @ ð @+=@q Þ @v ¼ @ 2L @v@q ¼ @ 2+ @q@v ¼ @k @q : Hence, if capital is a normal input, an increase in v will raise MC whereas, if capital is inferior, an increase in v will actually reduce MC. 10This deﬁnition is usually attributed to R. G. D. Allen, who developed it in an alternative form in his Mathematical Analysis for Economists (New York: St. Martin’s Press, 1938), pp. 504–9. 350 Part 4: Production and Supply Substitution with many inputs Instead of just the two inputs k and l, now suppose there are many inputs to the production process (x1, x2, …, xn) that can be hired at competitive rental rates (w1, w2, …, wn). Then the elasticity of substitution between any two inputs (sij) is deﬁned as follows Elasticity of substitution. The elasticity of substitution between inputs xi and xj is given by sij ¼ @ @ xi=xjÞ ð wj=wiÞ ð wj=wi xi=xj ¼ @ ln @ ln ( xi=xjÞ ð wj=wiÞ ð, (10:34) where output and all other input prices are held constant. A subtle point that did not arise in the two-input case regards what is assumed about the ﬁrm’s usage of the other inputs besides i and j. Should we perform the thought experiment of holding them ﬁxed as are other input prices and output? Or should we take into account the adjustment of these other inputs to achieve cost minimization? The latter assumption has proved to be more useful in economic analysis; therefore, that is the one we will take to be embodied in Equation 10.
34.11 For example, a major topic in the theory of ﬁrms’ input choices is to describe the relationship between capital and energy inputs. The deﬁnition in Equation 10.34 would permit a researcher to study how the ratio of energy to capital input changes when relative energy prices increase while permitting the ﬁrm to make any adjustments to labor input (whose price has not changed) that would be required for cost minimization. Hence this would give a realistic picture of how ﬁrms behave with regard to whether energy and capital are more like substitutes or complements. Later in this chapter we will look at this deﬁnition in a bit more detail because it is widely used in empirical studies of production. Quantitative size of shifts in cost curves We have already shown that increases in an input price will raise total, average, and (except in the inferior input case) marginal costs. We are now in a position to judge the extent of such increases. First, and most obviously, the increase in costs will be inﬂuenced importantly by the relative signiﬁcance of the input in the production process. If an input constitutes a large fraction of total costs, an increase in its price will raise costs signiﬁcantly. An increase in the wage rate would sharply increase home-builders’ costs because labor is a major input in construction. On the other hand, a price increase for a relatively minor input will have a small cost impact. An increase in nail prices will not raise home costs much. A less obvious determinant of the extent of cost increases is input substitutability. If ﬁrms can easily substitute another input for the one that has increased in price, there may be little increase in costs. Increases in copper prices in the late 1960s, for example, had little impact on electric utilities’ costs of distributing electricity because they found they could easily substitute aluminum for copper cables. Alternatively, if the ﬁrm ﬁnds it difﬁcult or impossible to substitute for the input that has become more costly, then costs may increase rapidly. The cost of gold jewelry, along with the price of gold, rose rapidly during the early 1970s because there was simply no substitute for the raw input. 11This deﬁnition is attributed to the Japanese economist M. Morishima, and these elasticities are sometimes referred to as Morishima
elasticities. In this version, the elasticity of substitution for substitute inputs is positive. Some authors reverse the order of subscripts in the denominator of Equation 10.31, and in this usage the elasticity of substitution for substitute inputs is negative. Chapter 10: Cost Functions 351 It is possible to give a precise mathematical statement of the quantitative sizes of all these effects by using the elasticity of substitution. To do so, however, would risk further cluttering the book with symbols.12 For our purposes, it is sufﬁcient to rely on the previous intuitive discussion. This should serve as a reminder that changes in the price of an input will have the effect of shifting ﬁrms’ cost curves, with the size of the shift depending on the relative importance of the input and on the substitution possibilities that are available. Technical change Technical improvements allow the ﬁrm to produce a given output with fewer inputs. Such improvements obviously shift total costs downward (if input prices stay constant). Although the actual way in which technical change affects the mathematical form of the total cost curve can be complex, there are cases where one may draw simple conclusions. Suppose, for example, that the production function exhibits constant returns to scale and that technical change enters that function as described in Chapter 9 [i.e., q A(t)f(k, l ) where A(0) 1]. In this case, total costs in the initial period are given by ¼ ¼ qC0ð Because the same inputs that produced one unit of output in period 0 are also the costminimizing way of producing A(t) units of output in period t, we know that v, w, 1 Þ v, w, q C0ð (10:35) Þ ¼ : v, w, 1 C0ð v, w, A t Ctð ð ÞÞ ¼ Þ ¼ Therefore, we can compute the total cost function in period t as v, w, q C0ð t A ð Þ Hence total costs decrease over time at the rate of technical change.13 v, w, 1 qC0ð t A ð qCtð v, w, q v, w, 1 Ctð Ctð v, w, 1 Þ : (10:36) Þ : (10:37) Note that in this case technical change is ‘‘neutral’’ in that it does not
affect the ﬁrm’s input choices (as long as input prices stay constant). This neutrality result might not hold in cases where technical progress takes a more complex form or where there are variable returns to scale. Even in these more complex cases, however, technical improvements will cause total costs to decrease. EXAMPLE 10.3 Shifting the Cobb–Douglas Cost Function In Example 10.2 we computed the Cobb–Douglas cost function as C v, w, q ð Þ ¼ q1= a ð ÞBva= b ð þ a þ Þw b= b a ð b Þ, þ (10:38) 12For a complete statement, see C. Ferguson, Neoclassical Theory of Production and Distribution (Cambridge, UK: Cambridge University Press, 1969), pp. 154–60. 13To see that the indicated rates of change are the same, note ﬁrst that the rate of change of technical progress is while the rate of change in total cost is using Equation 10.34. r t ð Þ ¼ A0 t ð t A ð Þ Þ ; @Ct @t ( 1 Ct ¼ t C0A0ð 2 t A Þ ð 1 Ct ¼ Þ ( t A0ð 352 Part 4: Production and Supply where B (a assume that a ¼ b)a# þ b ¼ ¼ a/(a b)b# þ b/(a b). As in the numerical illustration in Example 10.1, let’s þ 0.5, in which case the total cost function is greatly simpliﬁed: Þ ¼ This function will yield a total cost curve relating total costs and output if we specify particular values for the input prices. If, as before, we assume v 12, then the relationship is 3 and w C v, w, q ð 2qv0:5w0:5: (10:39) 3, 12, q C ð Þ ¼ 2q 12q, ¼ (10:40) ¼ 36p ¼ and, as in Example 10.1, it costs 480 to produce 40 units of output. Here average and marginal costs are easily computed as ﬃﬃﬃﬃﬃ 12, AC MC ¼ ¼ C q ¼ @C @q �
� 12: (10:41) As expected, average and marginal costs are constant and equal to each other for this constant returns-to-scale production function. Changes in input prices. If either input price were to change, all these costs would change also. For example, if wages were to increase to 27 (an easy number with which to work), costs would become C ð 3, 27, q AC MC Þ ¼ ¼ ¼ 2q 18, 18: 81p ﬃﬃﬃﬃﬃ 18q, ¼ (10:42) Notice that an increase in wages of 125 percent increased costs by only 50 percent here, both because labor represents only 50 percent of all costs and because the change in input prices encouraged the ﬁrm to substitute capital for labor. The total cost function, because it is derived from the cost-minimization assumption, accomplishes this substitution ‘‘behind the scenes’’— reporting only the ﬁnal impact on total costs. Technical progress. Let’s look now at the impact that technical progress can have on costs. Speciﬁcally, assume that the Cobb–Douglas production function is q t A ð k0:5l 0:5 Þ ¼ ¼ e:03tk0:5l 0:5: (10:43) That is, we assume that technical change takes an exponential form and that the rate of technical change is 3 percent per year. Using the results of the previous section (Equation 10.37) yields v, w, q Ctð Þ ¼ v, w, q C0ð t A ð Þ Þ ¼ 2qv0:5w 0:5e# :03t: (10:44) if input prices remain the same, then total costs decrease at the rate of technical Thus, improvement—that is, at 3 percent per year. After, say, 20 years, costs will be (with v 36p :60 e# ( 12q 0:55 ( ð Þ ¼ ¼ 6:6q, C20ð 3, 12, q Þ ¼ AC20 ¼ MC20 ¼ 2q 6:6, 6:6: ﬃﬃﬃﬃﬃ 3, w 12) ¼ ¼ (10:45) Consequently,
costs will have decreased by nearly 50 percent as a result of the technical change. This would, for example, more than have offset the wage increase illustrated previously. QUERY: In this example, what are the elasticities of total costs with respect to changes in input costs? Is the size of these elasticities affected by technical change? Chapter 10: Cost Functions 353 Contingent demand for inputs and Shephard’s lemma As we described earlier, the process of cost minimization creates an implicit demand for inputs. Because that process holds quantity produced constant, this demand for inputs will also be ‘‘contingent’’ on the quantity being produced. This relationship is fully reﬂected in the ﬁrm’s total cost function and, perhaps surprisingly, contingent demand functions for all the ﬁrm’s inputs can be easily derived from that function. The process involves what has come to be called Shephard’s lemma,14 which states that the contingent demand function for any input is given by the partial derivative of the total cost function with respect to that input’s price. Because Shephard’s lemma is widely used in many areas of economic research, we will provide a relatively detailed examination of it. The intuition behind Shephard’s lemma is straightforward. Suppose that the price of labor (w) were to increase slightly. How would this affect total costs? If nothing it seems that costs would increase by approximately the amount of else changed, labor (l) that the ﬁrm was currently hiring. Roughly speaking then, @C/@w l, and that is what Shephard’s lemma claims. Figure 10.6 makes roughly the same point graphically. Along the ‘‘pseudo’’ cost function all inputs are held constant; therefore, an increase in the wage increases costs in direct proportion to the amount of labor used. Because the true cost function is tangent to the pseudo-function at the current wage, its slope (i.e., its partial derivative) also will show the current amount of labor input demanded. ¼ Technically, Shephard’s lemma is one result of the envelope theorem that was ﬁrst discussed in Chapter 2. There we showed that the change in the optimal value in a constrained optimization problem with respect to one of the parameters of the problem can be found by differentiating the
Lagrangian for that optimization problem with respect to this changing parameter. In the cost-minimization case, the Lagrangian is + vk wl k q ½ # k, l f ð Þ’ þ þ ¼ and the envelope theorem applied to either input is @C @C ð ð v, w, q @v v, w, q @w Þ Þ ¼ ¼ @+ @+ ð ð v, w, q, k @v v, w, q, k @w Þ Þ ¼ ¼ kc v, w, q ð Þ, lc v, w, q ð Þ, (10:46) (10:47) where the notation is intended to make clear that the resulting demand functions for capital and labor input depend on v, w, and q. Because quantity produced enters these functions, input demand is indeed contingent on that variable. This feature of the demand functions is also reﬂected by the ‘‘c’’ in the notation.15 Hence the demand relations in Equation 10.47 do not represent a complete picture of input demand because they still depend on a variable that is under the ﬁrm’s control. In the next chapter, we will complete the study of input demand by showing how the assumption of proﬁt maximization allows us to effectively replace q in the input demand relationships with the market price of the ﬁrm’s output, p. 14Named for R. W. Shephard, who highlighted the important relationship between cost functions and input demand functions in his Cost and Production Functions (Princeton, NJ: Princeton University Press, 1970). 15The notation mirrors that used for compensated demand curves in Chapter 5 (which were derived from the expenditure function). In that case, such demand functions were contingent on the utility target assumed. 354 Part 4: Production and Supply EXAMPLE 10.4 Contingent Input Demand Functions In this example, we will show how the total cost functions derived in Example 10.2 can be used to derive contingent demand functions for the inputs capital and labor. 1. Fixed Proportions: C(v, w, q) functions are simple: q(v/a þ ¼ w/b). For this cost function, contingent demand k c l c v, w, q ð Þ ¼ v, w,
q ð Þ ¼ @C @C v, w, q ð @v v, w, q ð @10:48) To produce any particular output with a ﬁxed proportions production function at minimal cost, the ﬁrm must produce at the vertex of its isoquants no matter what the inputs’ prices are. Hence the demand for inputs depends only on the level of output, and v and w do not enter the contingent input demand functions. Input prices may, however, affect total input demands in the ﬁxed proportions case because they may affect how much the ﬁrm decides to sell. 2. Cobb–Douglas: C(v, w, q) ¼ ier but also more instructive: q1/(a þ b) Bva/(a þ b)wb/(a þ b). In this case, the derivation is mess- q1= a ð b ÞBv# þ b= a ð b Þw b, b= a ð b ÞB w v # $ ÞBv a= b a q1= þ ð a ð b Þw# a= ð þ (10:49) a þ b Þ k c v, w, q ð Þ ¼ ¼ l c v, w, q ð Þ ¼ b ( @C @v ¼ a a þ @C @w ¼ b a a b ( þ q1= ð a þ a b ( b þ q1= ð ¼ a b ( þ a= ð # a þ b Þ: a b ÞB þ w v # $ Consequently, the contingent demands for inputs depend on both inputs’ prices. If we assume a 2), these reduce to 0.5 (so :5 0:5 0:5 ¼ ¼ ¼ l c k c 0:5 0:5 Þ ¼ Þ ¼ 3, w ¼ 0:5 (10:50) 12, and q v, w, q ð v, w # $ 40, Equations 10.50 yield the result we obtained previously: With v that the ﬁrm should choose the input combination k 20 to minimize the cost of producing 40 units of output. If the wage were to increase to, say, 27, the ﬁrm would choose the input
combination k 40/3 to produce 40 units of output. Total costs would increase from 480 to 520, but the ability of the ﬁrm to substitute capital for the now more expensive labor does save considerably. For example, the initial input combination would now cost 780. 3. CES: C(v, w, q) w 1–s)1/(1–s). The importance of input substitution is shown even more clearly with the contingent demand functions derived from the CES function. For that function, q1/g(v 1–s 120, l 80, w, q ð Þ ¼ ¼ l c v, w, q ð Þ ¼ ¼ @C @v ¼ 1 q1=g v1 ð @C @w ¼ 1 q1=g v1 ð 1 r ( # r # þ 1 w1 r q1=g v1 # ð r= Þ q1= ( w1 r # þ v1 ð r= Þ þ r Þv# þ r Þw# r: 1 ð # w1 # r 1 # r Þ ð r= Þ 1 ð # r r v# Þ r, (10:51) w1 1 ð r # r # r= Þ Þ 1 ð # r r w# Þ Chapter 10: Cost Functions 355 These functions collapse when s with either more (s 2) or less (s middle ground. If we assume constant returns to scale (g 40, then contingent demands for the inputs when s q 1 (the Cobb–Douglas case), but we can study examples 0.5) substitutability and use Cobb–Douglas as the 12, and 1) and v ¼ 2 are 3, 12, 40 ð 3, 12, 40 ð Þ ¼ 40 40 3# ð 3# ð 1 1 þ 1 12# 1 12# 2 2 # Þ # Þ ¼ 3# 2 ( 12# ¼ 2 25:6, 1:6: (10:52) (10:53) þ That is, the level of capital input is 16 times the amount of labor input. With less substitutability (s 0.5), contingent input demands are, 12, 40 ð 3, 12, 40 ð Þ ¼ 40 40 30:5 ð 30:5 ð þ
120:5 120:5 1 1 Þ ( 0:5 3# 0:5 12# ¼ 120, 60: ( Þ ¼ Þ ¼ þ Thus, in this case, capital input is only twice as large as labor input. Although these various cases cannot be compared directly because different values for s scale output differently, we can, as an example, look at the consequence of a increase in w to 27 in the lowsubstitutability case. With w 53.3. In this case, the 27, the ﬁrm will choose k cost savings from substitution can be calculated by comparing total costs when using the initial input combination ( 1,980) to total costs with the optimal combination ( 1,919). Hence moving to the optimal input combination reduces total costs by only about 3 percent. In the Cobb–Douglas case, cost savings are over 20 percent. ¼ 27 Æ 53.3 3 Æ 160 3 Æ 120 27 Æ 60 160 QUERY: How would total costs change if w increased from 12 to 27 and the production function took the simple linear form q 4l? What light does this result shed on the other cases in this example? þ ¼ k Shephard’s Lemma and the Elasticity of Substitution One especially nice feature of Shephard’s lemma is that it can be used to show how to derive information about input substitution directly from the total cost function through differentiation. Using the deﬁnition in Equation 10.34 yields Ci=CjÞ ð wj=wiÞ ð xi=xjÞ ð wj=wiÞ ð @ ln @ ln @ ln @ ln sij ¼ (10:54) ¼, where Ci and Cj are the partial derivatives of the total cost function with respect to the input prices. Once the total cost function is known (perhaps through econometric estimation), information about substitutability among inputs can thus be readily obtained from it. In the Extensions to this chapter, we describe some of the results that have been obtained in this way. Problems 10.11 and 10.12 provide some additional details about ways in which substitutability among inputs can be measured. Short-Run, Long-Run Distinction It is traditional in economics to make a distinction between the ‘‘short run’’ and the ‘‘
long run.’’ Although no precise temporal deﬁnition can be provided for these terms, the general purpose of the distinction is to differentiate between a short period during which 356 Part 4: Production and Supply economic actors have only limited ﬂexibility in their actions and a longer period that provides greater freedom. One area of study in which this distinction is important is in the theory of the ﬁrm and its costs because economists are interested in examining supply reactions over differing time intervals. In the remainder of this chapter, we will examine the implications of such differential response. To illustrate why short-run and long-run reactions might differ, assume that capital input is held ﬁxed at a level of k1 and that (in the short run) the ﬁrm is free to vary only its labor input.16 Implicitly, we are assuming that alterations in the level of capital input are inﬁnitely costly in the short run. As a result of this assumption, the short-run production function is where this notation explicitly shows that capital inputs may not vary. Of course, the level of output still may be changed if the ﬁrm alters its use of labor. q f k1, l ð, Þ ¼ (10:55) Short-run total costs Total cost for the ﬁrm continues to be deﬁned as C vk wl (10:56) ¼ SC þ for our short-run analysis, but now capital input is ﬁxed at k1. To denote this fact, we will write vk1 þ where the S indicates that we are analyzing short-run costs with the level of capital input ﬁxed. Throughout our analysis, we will use this method to indicate short-run costs. Usually we will not denote the level of capital input explicitly, but it is understood that this input is ﬁxed. The cost concepts introduced earlier—C, AC, MC—are in fact long-run concepts because, in their deﬁnitions, all inputs were allowed to vary freely. Their longrun nature is indicated by the absence of a leading S. (10:57) wl, ¼ Fixed and variable costs The two types of input costs in Equation 10.57 are given special names. The term vk1 is referred to as (short-run) ﬁxed costs; because k
1 is constant, these costs will not change in the short run. The term wl is referred to as (short-run) variable costs—labor input can indeed be varied in the short run. Hence we have the following deﬁnitions Short-run ﬁxed and variable costs. Short-run ﬁxed costs are costs associated with inputs that cannot be varied in the short run. Short-run variable costs are costs of those inputs that can be varied to change the ﬁrm’s output level. The importance of this distinction is to differentiate between variable costs that the ﬁrm can avoid by producing nothing in the short run and costs that are ﬁxed and must be paid regardless of the output level chosen (even zero). Nonoptimality of short-run costs It is important to understand that total short-run costs are not the minimal costs for producing the various output levels. Because we are holding capital ﬁxed in the short run, 16Of course, this approach is for illustrative purposes only. In many actual situations, labor input may be less ﬂexible in the short run than is capital input. Chapter 10: Cost Functions 357 the ﬁrm does not have the ﬂexibility of input choice that we assumed when we discussed cost minimization earlier in this chapter. Rather, to vary its output level in the short run, the ﬁrm will be forced to use ‘‘nonoptimal’’ input combinations. The RTS will not necessarily be equal to the ratio of the input prices. This is shown in Figure 10.7. In the short run, the ﬁrm is constrained to use k1 units of capital. To produce output level q0, it will use l0 units of labor. Similarly, it will use l1 units of labor to produce q1 and l2 units to produce q2. The total costs of these input combinations are given by SC0, SC1, and SC2, respectively. Only for the input combination k1, l1 is output being produced at minimal cost. Only at that point is the RTS equal to the ratio of the input prices. From Figure 10.7, it is clear that q0 is being produced with ‘‘too much’’ capital in this short-run situation. Cost minimization should suggest a southeasterly movement along the q0
isoquant, indicating a substitution of labor for capital in production. Similarly, q2 is being produced with ‘‘too little’’ capital, and costs could be reduced by substituting capital for labor. Neither of these substitutions is possible in the short run. Over a longer period, however, the ﬁrm will be able to change its level of capital input and will adjust its input usage to the cost-minimizing combinations. We have already discussed this ﬂexible case earlier in this chapter and shall return to it to illustrate the connection between long-run and short-run cost curves. FIGURE 10.7 ‘‘Nonoptimal’’ Input Choices Must Be Made in the Short Run Because capital input is ﬁxed at k, in the short run the ﬁrm cannot bring its RTS into equality with the ratio of input prices. Given the input prices, q0 should be produced with more labor and less capital than it will be in the short run, whereas q2 should be produced with more capital and less labor than it will be. k per period k1 SC2 SC0 SC1 = C q2 q1 q0 l 0 l1 l2 l per period 358 Part 4: Production and Supply Short-run marginal and average costs Frequently, it is more useful to analyze short-run costs on a per-unit of output basis rather than on a total basis. The two most important per-unit concepts that can be derived from the short-run total cost function are the short-run average total cost function (SAC) and the short-run marginal cost function (SMC). These concepts are deﬁned as total costs total output ¼ change in total costs SC q, SAC ¼ SMC ¼ change in output ¼ @SC @q, (10:58) where again these are deﬁned for a speciﬁed level of capital input. These deﬁnitions for average and marginal costs are identical to those developed previously for the long-run, fully ﬂexible case, and the derivation of cost curves from the total cost function proceeds in exactly the same way. Because the short-run total cost curve has the same general type of cubic shape as did the total cost curve in Figure 10.5, these short-run average and marginal cost curves will also be U-shaped. Relationship
between short-run and long-run cost curves It is easy to demonstrate the relationship between the short-run costs and the fully ﬂexible long-run costs that were derived previously in this chapter. Figure 10.8 shows this relationship for both the constant returns-to-scale and cubic total cost curve cases. Short-run total costs for three levels of capital input are shown, although of course it would be possible to show many more such short-run curves. The ﬁgures show that long-run total costs (C) are always less than short-run total costs, except at that output level for which the assumed ﬁxed capital input is appropriate to long-run cost minimization. For example, as in Figure 10.7, with capital input of k1 the ﬁrm can obtain full cost minimization when q1 is produced. Hence short-run and long-run total costs are equal at this point. For output levels other than q1, however, SC > C, as was the case in Figure 10.7. Technically, the long-run total cost curves in Figure 10.8 are said to be an ‘‘envelope’’ of their respective short-run curves. These short-run total cost curves can be represented parametrically by short-run total cost SC v, w, q, k ð, Þ ¼ (10:59) and the family of short-run total cost curves is generated by allowing k to vary while holding v and w constant. The long-run total cost curve C must obey the short-run relationship in Equation 10.59 and the further condition that k be cost minimizing for any level of output. A ﬁrst-order condition for this minimization is that @SC(v, w, q, k) @k 0: ¼ (10:60) Solving Equations 10.59 and 10.60 simultaneously then generates the long-run total cost function. Although this is a different approach to deriving the total cost function, it should give precisely the same results derived earlier in this chapter—as the next example illustrates. Chapter 10: Cost Functions 359 FIGURE 10.8 Two Possible Shapes for Long-Run Total Cost Curves By considering all possible levels of capital input, the long-run total cost curve (C ) can be traced. In (a), the underlying production function exhibits constant returns to
scale: In the long run, although not in the short run, total costs are proportional to output. In (b), the long-run total cost curve has a cubic shape, as do the short-run curves. Diminishing returns set in more sharply for the short-run curves, however, because of the assumed ﬁxed level of capital input. Total costs SC(k1) SC(k2) C SC(k0) q0 q1 q2 Output per period (a) Constant returns to scale Total costs C SC(k2) SC(k1) SC(k0) q0 q1 q2 Output per period (b) Cubic total cost curve case 360 Part 4: Production and Supply EXAMPLE 10.5 Envelope Relations and Cobb–Douglas Cost Functions Again we start with the Cobb–Douglas production function q input constant at k1. Thus, in the short run, ¼ kalb, but now we hold capital and total costs are given by 1 l b ka q ¼ or l ¼ q1=bk# 1 a=b, SC v, w, q, k1Þ ¼ ð vk1 þ wl vk1 þ ¼ wq1=bk# 1 a=b : (10:61) (10:62) Notice that the ﬁxed level of capital enters into this short-run total cost function in two ways: (1) k1 determines ﬁxed costs; and (2) k1 also in part determines variable costs because it determines how much of the variable input (labor) is required to produce various levels of output. To derive long-run costs, we require that k be chosen to minimize total costs: @SC v, w, q, k Þ ð @k v þ ¼ a # b ( wq1=bk#ð a þ b)=b 0: ¼ (10:63) Although the algebra is messy, this equation can be solved for k and substituted into Equation 10.62 to return us to the Cobb–Douglas cost function: C v, w, q ð Þ ¼ Bq1= ð a þ Þva= b ð a þ Þw b= b a ð b Þ: þ (10:64) Numerical example. If we again let a function is b
¼ ¼ 0.5, v ¼ 3, and w ¼ 12, then the short-run cost SC 3, 12, q; k1Þ ¼ ð 3k1 þ 12q2k# 1 1 : (10:65) In Example 10.1 we found that the cost-minimizing level of capital input for q 40 was k Equation 10.65 shows that short-run total costs for producing 40 units of output with k1 ¼ ¼ 80. ¼ 80 is SC 3, 12, q, 80 ð Þ ¼ 3:80 12 q2 ( þ 240 þ ¼ 240 ¼ 240 3q2 20 þ 1 80 ¼ ( 480, (10:66) which is just what we found before. We can also use Equation 10.65 to show how costs differ in 40, short-run the short and long run. Table 10.1 shows that, for output levels other than q costs are larger than long-run costs and that this difference is proportionally larger the farther one gets from the output level for which k 80 is optimal. ¼ ¼ ¼ 240 3q2/20 TABLE 10.1 DIFFERENCE BETWEEN SHORT-RUN AND LONG-RUN TOTAL COST, k 80 q 10 20 30 40 50 60 70 80 SC ¼ C 12q ¼ 120 240 360 480 600 720 840 960 þ 255 300 375 480 615 780 975 1,200 Chapter 10: Cost Functions 361 TABLE 10.2 UNIT COSTS IN THE LONG RUN AND THE SHORT RUN, k 80 ¼ AC 12 12 12 12 12 12 12 12 MC 12 12 12 12 12 12 12 12 SAC 25.5 15.0 12.5 12.0 12.3 13.0 13.9 15.0 SMC 3 6 9 12 15 18 21 24 q 10 20 30 40 50 60 70 80 It is also instructive to study differences between the long-run and short-run per-unit costs 12. We can compute the short-run equivalents (when MC in this situation. Here AC k 80) as ¼ ¼ ¼ SAC SMC ¼ ¼ SC q ¼ @SC @q ¼ 240 q þ 6q 20 : 3q 20, (10:67) Both of these short-run unit costs are equal to 12 when q 40. However, as Table 10.
2 shows, short-run unit costs can differ signiﬁcantly from this ﬁgure, depending on the output level that the ﬁrm produces. Notice in particular that short-run marginal cost increases rapidly as output 40 because of diminishing returns to the variable input (labor). This expands beyond q conclusion plays an important role in the theory of short-run price determination. ¼ ¼ QUERY: Explain why an increase in w will increase both short-run average cost and short-run marginal cost in this illustration, but an increase in v affects only short-run average cost. Graphs of per-unit cost curves The envelope total cost curve relationships exhibited in Figure 10.8 can be used to show geometric connections between short-run and long-run average and marginal cost curves. These are presented in Figure 10.9 for the cubic total cost curve case. In the ﬁgure, shortrun and long-run average costs are equal at that output for which the (ﬁxed) capital input is appropriate. At q1, for example, SAC(k1) AC because k1 is used in producing q1 at ¼ minimal costs. For movements away from q1, short-run average costs exceed long-run average costs, thus reﬂecting the cost-minimizing nature of the long-run total cost curve. Because the minimum point of the long-run average cost curve (AC) plays a major role in the theory of long-run price determination, it is important to note the various curves that pass through this point in Figure 10.9. First, as is always true for average and marginal cost curves, the MC curve passes through the low point of the AC curve. At q1, long-run average and marginal costs are equal. Associated with q1 is a certain level of capital input (say, k1); the short-run average cost curve for this level of capital input is tangent to the AC curve at its minimum point. The SAC curve also reaches its minimum at output level q1. For movements away from q1, the AC curve is much ﬂatter than the SAC curve, and this reﬂects the greater ﬂexibility open to ﬁrms in the long run. Short-run costs increase rapidly because capital inputs are ﬁxed. In the long run, such inputs are 362 Part
4: Production and Supply FIGURE 10.9 Average and Marginal Cost Curves for the Cubic Cost Curve Case This set of curves is derived from the total cost curves shown in Figure 10.8. The AC and MC curves have the usual U-shapes, as do the short-run curves. At q1, long-run average costs are minimized. The conﬁguration of curves at this minimum point is important. Costs MC SMC(k2) SAC(k2) AC SAC(k1) SMC(k1) SAC(k0) SMC(k0) q0 q1 q2 Output per period not ﬁxed, and diminishing marginal productivities do not occur so abruptly. Finally, because the SAC curve reaches its minimum at q1, the short-run marginal cost curve (SMC) also passes through this point. Therefore, the minimum point of the AC curve brings together the four most important per-unit costs: At this point, AC ¼ MC ¼ SAC ¼ SMC: (10:68) For this reason, as we shall show in Chapter 12, the output level q1 is an important equilibrium point for a competitive ﬁrm in the long run. SUMMARY In this chapter we examined the relationship between the level of output a ﬁrm produces and the input costs associated with that level of production. The resulting cost curves should generally be familiar to you because they are widely used in most courses in introductory economics. Here we have shown how such curves reﬂect the ﬁrm’s underlying production function and the ﬁrm’s desire to minimize costs. By developing cost curves from these basic foundations, we were able to illustrate a number of important ﬁndings. • A ﬁrm that wishes to minimize the economic costs of producing a particular level of output should choose that input combination for which the rate of technical substitution (RTS) is equal to the ratio of the inputs’ rental prices. • Repeated application of this minimization procedure yields the ﬁrm’s expansion path. Because the expansion path shows how input usage expands with the level of output, it also shows the relationship between output level and total cost. That relationship is summarized by the total cost function, C(v, w, q), which shows production costs as a function of output levels and input prices. Chapter 10:
Cost Functions 363 • • Input demand functions can be derived from the ﬁrm’s total cost function through partial differentiation. These input demand functions will depend on the quantity of output that the ﬁrm chooses to produce and are therefore called ‘‘contingent’’ demand functions. In the short run, the ﬁrm may not be able to vary some inputs. It can then alter its level of production only by changing its employment of variable inputs. In so doing, it may have to use nonoptimal, higher-cost input combinations than it would choose if it were possible to vary all inputs. • The ﬁrm’s average cost (AC C/q) and marginal cost @C/@q) functions can be derived directly from (MC the total cost function. If the total cost curve has a general cubic shape, then the AC and MC curves will be U-shaped. ¼ ¼ • All cost curves are drawn on the assumption that the input prices are held constant. When input prices change, cost curves will shift to new positions. The extent of the shifts will be determined by the overall importance of the input whose price has changed and by the ease with which the ﬁrm may substitute one input for another. Technical progress will also shift cost curves. PROBLEMS 10.1 Suppose that a ﬁrm produces two different outputs, the quantities of which are represented by q1 and q2. In general, the ﬁrm’s C(0, q2) > C(q1, q2) for all total costs can be represented by C(q1, q2). This function exhibits economies of scope if C(q1, 0) output levels of either good. þ a. Explain in words why this mathematical formulation implies that costs will be lower in this multiproduct ﬁrm than in two single-product ﬁrms producing each good separately. b. If the two outputs are actually the same good, we can deﬁne total output as q C/q) decreases as q increases. Show that this ﬁrm also enjoys economies of scope under the deﬁnition provided here. q1 þ ¼ q2. Suppose that in this case average cost ( ¼ 10.2 Professor Smith and Professor Jones are going to produce a new introductory textbook. As true scientists, they have laid out the production function for
the book as where q hours spent working by Jones. ¼ the number of pages in the ﬁnished book, S S1=2J 1=2, q ¼ the number of working hours spent by Smith, and J the number of ¼ ¼ After having spent 900 hours preparing the ﬁrst draft, time which he valued at $3 per working hour, Smith has to move on to other things and cannot contribute any more to the book. Jones, whose labor is valued at $12 per working hour, will revise Smith’s draft to complete the book. a. How many hours will Jones have to spend to produce a ﬁnished book of 150 pages? Of 300 pages? Of 450 pages? b. What is the marginal cost of the 150th page of the ﬁnished book? Of the 300th page? Of the 450th page? 10.3 Suppose that a ﬁrm’s ﬁxed proportion production function is given by q min 5k, 10l ð : Þ ¼ a. Calculate the ﬁrm’s long-run total, average, and marginal cost functions. b. Suppose that k is ﬁxed at 10 in the short run. Calculate the ﬁrm’s short-run total, average, and marginal cost functions. c. Suppose v 3. Calculate this ﬁrm’s long-run and short-run average and marginal cost curves. 1 and w ¼ ¼ 10.4 A ﬁrm producing hockey sticks has a production function given by klp : 2 q ¼ ﬃﬃﬃﬃ 364 Part 4: Production and Supply ¼ In the short run, the ﬁrm’s amount of capital equipment is ﬁxed at k for l is w $4. 100. The rental rate for k is v $1, and the wage rate ¼ ¼ a. Calculate the ﬁrm’s short-run total cost curve. Calculate the short-run average cost curve. b. What is the ﬁrm’s short-run marginal cost function? What are the SC, SAC, and SMC for the ﬁrm if it produces 25 hockey sticks? Fifty hockey sticks? One hundred hockey sticks? Two hundred hockey sticks
? c. Graph the SAC and the SMC curves for the ﬁrm. Indicate the points found in part (b). d. Where does the SMC curve intersect the SAC curve? Explain why the SMC curve will always intersect the SAC curve at its lowest point. Suppose now that capital used for producing hockey sticks is ﬁxed at k in the short run. e. Calculate the ﬁrm’s total costs as a function of q, w, v, and k. f. Given q, w, and v, how should the capital stock be chosen to minimize total cost? g. Use your results from part (f) to calculate the long-run total cost of hockey stick production. h. For w $4, v short-run curves computed in part (e) by examining values of k of 100, 200, and 400. ¼ ¼ $1, graph the long-run total cost curve for hockey stick production. Show that this is an envelope for the 10.5 An enterprising entrepreneur purchases two factories to produce widgets. Each factory produces identical products, and each has a production function given by q ¼ kili, 1, 2: i ¼ ﬃﬃﬃﬃﬃﬃ p The factories differ, however, in the amount of capital equipment each has. In particular, factory 1 has k1 ¼ factory 2 has k2 ¼ a. If the entrepreneur wishes to minimize short-run total costs of widget production, how should output be allocated between 100. Rental rates for k and l are given by w 25, whereas $1. ¼ ¼ v the two factories? b. Given that output is optimally allocated between the two factories, calculate the short-run total, average, and marginal cost curves. What is the marginal cost of the 100th widget? The 125th widget? The 200th widget? c. How should the entrepreneur allocate widget production between the two factories in the long run? Calculate the long-run total, average, and marginal cost curves for widget production. d. How would your answer to part (c) change if both factories exhibited diminishing returns to scale? 10.6 Suppose the total-cost function for a ﬁrm is given by a. Use Shephard’s lemma to compute the (constant output) demand functions for inputs l and k. b. Use
your results from part (a) to calculate the underlying production function for q. qw2=3v1=3: C ¼ 10.7 Suppose the total-cost function for a ﬁrm is given by C q v ð þ ¼ 2 vwp w : Þ þ a. Use Shephard’s lemma to compute the (constant output) demand function for each input, k and l. b. Use the results from part (a) to compute the underlying production function for q. c. You can check the result by using results from Example 10.2 to show that the CES cost function with s ﬃﬃﬃﬃﬃﬃ generates this total-cost function. 0.5, r 1 ¼ # ¼ 10.8 In a famous article [J. Viner, ‘‘Cost Curves and Supply Curves,’’ Zeitschrift fur Nationalokonomie 3 (September 1931): 23–46], Viner criticized his draftsman who could not draw a family of SAC curves whose points of tangency with the U-shaped AC curve were also the minimum points on each SAC curve. The draftsman protested that such a drawing was impossible to construct. Whom would you support in this debate? Chapter 10: Cost Functions 365 Analytical Problems 10.9 Generalizing the CES cost function The CES production function can be generalized to permit weighting of the inputs. In the two-input case, this function is q f k, l ð ¼ Þ ¼ ½ð ak Þ q g=q: bl q Þ ’ þ ð a. What is the total-cost function for a ﬁrm with this production function? Hint: You can, of course, work this out from scratch; easier perhaps is to use the results from Example 10.2 and reason that the price for a unit of capital input in this production function is v/a and for a unit of labor input is w/b. 1, it can be shown that this production function converges to the Cobb–Douglas form q b. If g 1 and a ¼ b þ ¼ kal b as ¼ r ﬁ 0. What is the total cost function for this particular version of the CES function? c. The relative labor cost share for a two-input production function is given by wl/
vk. Show that this share is constant for the Cobb–Douglas function in part (b). How is the relative labor share affected by the parameters a and b? d. Calculate the relative labor cost share for the general CES function introduced above. How is that share affected by changes in w/v? How is the direction of this effect determined by the elasticity of substitution, s? How is it affected by the sizes of the parameters a and b? 10.10 Input demand elasticities The own-price elasticities of contingent input demand for labor and capital are deﬁned as elc, w ¼ @l c @w ( w lc, ekc, v ¼ @kc @v ( v kc : a. Calculate elc, w and ekc, v for each of the cost functions shown in Example 10.2. b. Show that, in general, elc, w þ 0. c. Show that the cross-price derivatives of contingent demand functions are equal—that is, show that @lc/@v this fact to show that slelc, v ¼ total cost (vk/C). ekc, v ¼ @kc/@w. Use skekc, w where sl, sk are, respectively, the share of labor in total cost (wl/C) and of capital in ¼ d. Use the results from parts (b) and (c) to show that slel c,w þ e. Interpret these various elasticity relationships in words and discuss their overall relevance to a general theory of input skekc, w ¼ 0. demand. 10.11 The elasticity of substitution and input demand elasticities The deﬁnition of the (Morishima) elasticity of substitution sij in Equation 10.54 can be recast in terms of input demand elasticities. This illustrates the basic asymmetry in the deﬁnition. a. Show that if only wj changes, sij ¼ b. Show that if only wi changes, sji ¼ c. Show that if the production function takes the general CES form q elasticities are the same: sij ¼ 1/(1 n xq i s. This is the only case in which the Morishima deﬁnition is symmetric. ex c i,wj # ex c j,wi # r) 0, then all of
the Morishima 1=q for r j,wj : i,wi : ex c ex c # ¼ ¼ 6¼ ’ & P 10.12 The Allen elasticity of substitution Many empirical studies of costs report an alternative deﬁnition of the elasticity of substitution between inputs. This alternative deﬁnition was ﬁrst proposed by R. G. D. Allen in the 1930s and further clariﬁed by H. Uzawa in the 1960s. This deﬁnition builds directly on the production function-based elasticity of substitution deﬁned in footnote 6 of Chapter 9: Aij ¼ CijC/CiCj, where the subscripts indicate partial differentiation with respect to various input prices. Clearly, the Allen deﬁnition is symmetric. a. Show that Aij ¼ b. Show that the elasticity of si with respect to the price of input j is related to the Allen elasticity by esi, pj ¼ c. Show that, with only two inputs, Akl ¼ 1 for the Cobb–Douglas case and Akl ¼ d. Read Blackorby and Russell (1989: ‘‘Will the Real Elasticity of Substitution Please Stand Up?’’) to see why the Morishima i,wj =sj, where sj is the share of input j in total cost. s for the CES case. Aij # : 1 Þ sjð ex c deﬁnition is preferred for most purposes. 366 Part 4: Production and Supply SUGGESTIONS FOR FURTHER READING Allen, R. G. D. Mathematical Analysis for Economists. New York: St. Martin’s Press, 1938, various pages—see index. Complete (though dated) mathematical analysis of substitution possibilities and cost functions. Notation somewhat difﬁcult. Blackorby, C., and R. R. Russell. ‘‘Will the Real Elasticity of Substitution Please Stand Up? (A Comparison of the Allen/ Uzawa and Morishima Elasticities).’’ American Economic Review (September 1989): 882–88. A nice clariﬁcation of the proper way to measure substitutability among many inputs in production. Argues that the Allen/ Uzawa de�
�nition is largely useless and that the Morishima deﬁnition is by far the best. Ferguson, C. E. The Neoclassical Theory of Production and Distribution. Cambridge: Cambridge University Press, 1969, Chap. 6. Nice development of cost curves; especially strong on graphic analysis. Fuss, M., and D. McFadden. Production Economics: A Dual Approach to Theory and Applications. Amsterdam: NorthHolland, 1978. Difﬁcult and quite complete treatment of the dual relationship between production and cost functions. Some discussion of empirical issues. Knight, H. H. ‘‘Cost of Production and Price over Long and Short Periods.’’ Journal of Political Economics 29 (April 1921): 304–35. Classic treatment of the short-run, long-run distinction. Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGraw-Hill, 2001. Chapters 7–9 have a great deal of material on cost functions. Especially recommended are the authors’ discussions of ‘‘reciprocity effects’’ and their treatment of the short-run-long, run distinction as an application of the Le Chatelier principle from physics. Sydsaeter, K., A. Strom, and P. Berck. Economists’ Mathematical Manual, 3rd ed. Berlin: Springer-Verlag, 2000. Chapter 25 provides a succinct summary of the mathematical concepts in this chapter. A nice summary of many input cost functions, but beware of typos. THE TRANSLOG COST FUNCTION EXTENSIONS ¼ The two cost functions studied in Chapter 10 (the Cobb– Douglas and the CES) are very restrictive in the substitution possibilities they permit. The Cobb–Douglas implicitly assumes that s 1 between any two inputs. The CES permits s to take any value, but it requires that the elasticity of substitution be the same between any two inputs. Because empirical economists would prefer to let the data show what the actual substitution possibilities among inputs are, they have tried to ﬁnd more ﬂexible functional forms. One especially popular such form is the translog cost function, ﬁrst made popular by Fuss and McFadden (1978). In this extension we will look at this function. E10.1 The translog with two inputs In Example
10.2, we calculated the Cobb–Douglas cost b) multi function in the two-input case as C(v, w, q) b). If we take the natural logarithm of this we b)wb/(a þ Bq1/(a þ va/(a þ ¼ + have b ln q Þ ¼ ln B ln C v, w, q ð a 1 That is, the log of total costs is linear in the logs of output and the input prices. The translog function generalizes this by permitting second-order terms in input prices: Þ’ ln v ln w: þ Þ’ Þ’ þ ½ þ ½ a= b= (i) ln C v, w, q ð Þ ¼ ln q a0 þ þ a3(ln v)2 + a4ð a5 ln v ln w, þ a1 ln v + a2 ln w ln w)2 þ (ii) where this function implicitly assumes constant returns to scale (because the coefﬁcient of ln q is 1.0)—although that need not be the case. Some of the properties of this function are: • • • 0. a2 ¼ a5 ¼ a4 ¼ For the function to be homogeneous of degree 1 in 1 and input prices, it must be the case that a1 þ a3 þ a4 þ This function includes the Cobb–Douglas as the special case a3 ¼ 0. Hence the function can be used to test statistically whether the Cobb–Douglas is appropriate. Input shares for the translog function are especially easy (@ ln C)/(@ ln wi). to compute using the result that si ¼ In the two-input case, this yields a5 ¼ • @ ln C @ ln v ¼ @ ln C @ ln w ¼ a1 þ a2 þ sk ¼ sl ¼ 2a3 ln v a5 ln w, þ 2a4 ln w a5 ln v: þ (iii) 0) these In the Cobb–Douglas case (a3 ¼ shares are constant, but with the general translog function they are not. a5 ¼ a4 ¼ • Calculating the elasticity of substitution in the translog case proceeds by using the result given in Problem 10
.11 that skl ¼ calculation is el c,w. Making this straightforward (provided one keeps track of how to use logarithms): ek c,w # (iv) ek c,w ¼ @ ln Cv @ ln w ¼ @ ln C ¼ h # @ ln C @ ln v @ ln C v ( # @ ln w $ ln @ ln C @ ln v # @2 ln C @v@w ¼ $ i ln v þ @ ln w @ ln sk @sk a5 sk : sl þ 0 sl # ¼ þ ( Observe that, in the Cobb–Douglas case (a5 ¼ 0), the contingent price elasticity of demand for k with respect to the wage has a simple form: ek c,w ¼ sl. A 2a4=sl similar set of manipulations yields el c,w ¼ # sk þ sk. Bringing in the Cobb–Douglas case, el c,w ¼ # and, these two elasticities together yields skl ¼ ¼ ¼ el c,w a5 sk # ek c,w # sk þ sl þ sla5 # 1 sksl þ 2a4 sl 2ska4 : (v) 1, as Again, in the Cobb–Douglas case we have skl ¼ should have been expected. The Allen elasticity of substitution (see Problem 10.12) for the translog function is Akl ¼ a5 /sk sl. This function can also be used to calculate that the (contingent) crossa5=sk, price elasticity of demand is ek c,w ¼ slAk l ¼ sl þ as was shown previously. Here again, Ak l ¼ 1 in the Cobb–Douglas case. In general, however, the Allen and Morishima deﬁnitions will differ even with just two inputs. þ 1 368 Part 4: Production and Supply E10.2 The many-input translog cost function Most empirical studies include more than two inputs. The translog cost function is especially easy to generalize to these situations. If we assume there are n inputs, each with a price of wi (i 1, …, n) then this function is ¼ C w1,..., wn, q ð Þ ¼ ln q a
deregulation had signiﬁcant welfare beneﬁts. Doucouliagos and Hone (2000) provide a similar analysis of deregulation of dairy prices in Australia. They show that changes in the price of raw milk caused dairy processing ﬁrms to undertake signiﬁcant changes in input usage. They also show that the industry adopted signiﬁcant new technologies in response to the price change. An interesting study that uses the translog primarily to judge returns to scale is Latzko’s (1999) analysis of the U.S. mutual fund industry. He ﬁnds that the elasticity of total costs with respect to the total assets managed by the fund is less than 1 for all but the largest funds (those with more than $4 billion in assets). Hence the author concludes that money management exhibits substantial returns to scale. A number of other studies that use the translog to estimate economies of scale focus on municipal services. For example, Garcia and Thomas (2001) look at water supply systems in local French communities. They conclude that there are signiﬁcant operating economies of scale in such systems and that some merging of systems would make sense. Yatchew (2000) reaches a similar conclusion about electricity distribution in small communities in Ontario, Canada. He ﬁnds that there are economies of scale for electricity distribution systems serving up to about 20,000 customers. Again, some efﬁciencies might be obtained from merging systems that are much smaller than this size. References Doucouliagos, H., and P. Hone. ‘‘Deregulation and Subequilibrium in the Australian Dairy Processing Industry.’’ Economic Record (June 2000): 152–62. Fuss, M., and D. McFadden, Eds. Production Economics: A Dual Approach to Theory and Applications. Amsterdam: North Holland, 1978. Garcia, S., and A. Thomas. ‘‘The Structure of Municipal Water Supply Costs: Application to a Panel of French Local Communities.’’ Journal of Productivity Analysis (July 2001): 5–29. Latzko, D. ‘‘Economies of Scale in Mutual Fund Administra- tion.’’ Journal of Financial Research (Fall 1999): 331–39. Sydsæter, K., A. Strøm, and P. Berck. Economists
’ Mathemati- cal Manual, 3rd ed. Berlin: Springer-Verlag, 2000. Westbrook, M. D., and P. A. Buckley. ‘‘Flexible Functional Forms and Regularity: Assessing the Competitive Relationship between Truck and Rail Transportation.’’ Review of Economics and Statistics (November 1990): 623–30. Yatchew, A. ‘‘Scale Economies in Electricity Distribution: A Semiparametric Analysis.’’ Journal of Applied Econometrics (March/April 2000): 187–210. This page intentionally left blank C H A P T E R ELEVEN Profit Maximization In Chapter 10 we examined the way in which ﬁrms minimize costs for any level of output they choose. In this chapter we focus on how the level of output is chosen by proﬁtmaximizing ﬁrms. Before investigating that decision, however, it is appropriate to discuss brieﬂy the nature of ﬁrms and the ways in which their choices should be analyzed. The Nature and Behavior of Firms In this chapter, we delve deeper into the analysis of decisions made by suppliers in the market. The analysis of the supply/ﬁrm side of the market raises questions that did not come up in our previous analysis of the demand/consumer side. Whereas consumers are easy to identify as single individuals, ﬁrms come in all shapes and sizes, ranging from a corner ‘‘mom and pop’’ grocery store to a vast modern corporation, supplying hundreds of different products produced in factories operating across the globe. Economists have long puzzled over what determines the size of ﬁrms, how their management is structured, what sort of ﬁnancial instruments should be used to fund needed investment, and so forth. The issues involved turn out to be rather deep and philosophical. To make progress in this chapter, we will continue to analyze the standard ‘‘neoclassical’’ model of the ﬁrm, which brushes most of these deeper issues aside. We will provide only a hint of the deeper issues involved, returning to a fuller discussion in the Extensions to this chapter. Simple model of a ﬁrm Throughout Part 4, we have been examining a simple model of the ﬁrm without being explicit about the assumptions involved. It is worth being a bit more explicit here. The
ﬁrm has a technology given by the production function, say f (k, l). The ﬁrm is run by an entrepreneur who makes all the decisions and receives all the proﬁts and losses from the ﬁrm’s operations. The combination of these elements—production technology, entrepreneur, and inputs used (labor l, capital k, and others)—together constitutes what we will call the ‘‘ﬁrm.’’ The entrepreneur acts in his or her own self-interest, typically leading to decisions that maximize the ﬁrm’s proﬁts, as we will see. Complicating factors Before pushing ahead further with the analysis of the simple model of the ﬁrm, which will occupy most of this chapter, we will hint at some complicating factors. In the simple model just described, a single party—the entrepreneur—makes all the decisions and receives all the returns from the ﬁrm’s operations. With most large corporations, decisions and returns are separated among many parties. Shareholders are really the owners 371 372 Part 4: Production and Supply of the corporation, receiving returns in the form of dividends and stock returns. But shareholders do not run the ﬁrm; the average shareholder may own hundreds of different ﬁrms’ stock through mutual funds and other holdings and could not possibly have the time or expertise to run all these ﬁrms. The ﬁrm is run on shareholders’ behalf usually by the chief executive ofﬁcer (CEO) and his or her management team. The CEO does not make all the decisions but delegates most to managers at one of any number of levels in a complicated hierarchy. The fact that ﬁrms are often not run by the owner leads to another complication. Whereas the shareholders may like proﬁts to be maximized, the manager may act in his or her own interest rather than the interests of the shareholders. The manager may prefer the prestige from expanding the business empire beyond what makes economic sense, may seek to acquire expensive perks, and may shy away from proﬁtable but uncomfortable actions such as ﬁring redundant workers. Different mechanisms may help align the manager’s interests with those of the shareholder. Managerial compensation in the form of stock and stock options may provide incentives for proﬁt maximization as might the threat of ﬁring if
a poorly performing ﬁrm goes bankrupt or is taken over by a corporate raider. But there is no telling that such mechanisms will work perfectly. Even a concept as simple as the size of the ﬁrm is open to question. The simple deﬁnition of the ﬁrm includes all the inputs it uses to produce its output, for example, all the machines and factories involved. If part of this production process is outsourced to another ﬁrm using its machines and factories, then several ﬁrms rather than one are responsible for supply. A classic example is provided by the automaker, General Motors (GM).1 Initially GM purchased the car bodies from another ﬁrm, Fisher Body, who designed and made these to order; GM was only responsible for ﬁnal assembly of the body with the other auto parts. After experiencing a sequence of supply disruptions over several decades, GM decided to acquire Fisher Body in 1926. Overnight, much more of the production—the construction of the body and ﬁnal assembly—was concentrated in a single ﬁrm. What then should we say about the size of a ﬁrm in the auto-making business? Is the combination of GM and Fisher Body after the acquisition or the smaller GM beforehand a better deﬁnition of the ‘‘ﬁrm’’ in this case? Should we expect the acquisition of Fisher Body to make any real economic difference to the auto market, say, reducing input supply disruptions, or is it a mere name change? These are deep questions we will touch on in the Extensions to this chapter. For now, we will take the size and nature of the ﬁrm as given, speciﬁed by the production function f(k, l). Relationship to consumer theory Part 2 of this book was devoted to understanding the decisions of consumers on the demand side of the market; this Part 4 is devoted to understanding ﬁrms on the supply side. As we have already seen, there are many common elements between the two analyses, and much of the same mathematical methods can be used in both. There are two essential differences that merit all the additional space devoted to the study of ﬁrms. First, as just discussed, ﬁrms are not individuals but can be much more complicated organizations. We will mostly ‘‘ﬁnesse’’
this difference by assuming that the ﬁrm is represented by the entrepreneur as an individual decision-maker, dealing with the complications in more detail in the Extensions. 1GM’s acquisition of Fisher Body has been extensively analyzed by economists. See, for example, B. Klein, ‘‘Vertical Integration as Organization Ownership: the Fisher-Body–General Motors Relationship Revisited,’’ Journal of Law, Economics and Organization (Spring 1988): 199–213. Chapter 11: Prof it Maximization 373 Another difference between ﬁrms and consumers is that we can be more concrete about the ﬁrm’s objectives than a consumer’s. With consumers, there is ‘‘no accounting for taste.’’ There is no telling why one consumer likes hot dogs more than hamburgers and another consumer the opposite. By contrast, it is usually assumed that ﬁrms do not have an inherent preference regarding the production of hot dogs or hamburgers; the natural assumption is that it produces the product (or makes any number of other decisions) earning the most proﬁt. There are certainly a number of caveats with the proﬁt-maximization assumption, but if we are willing to make it, we can push the analysis farther than we did with consumer theory. Proﬁt Maximization Most models of supply assume that the ﬁrm and its manager pursue the goal of achieving the largest economic proﬁts possible. The following deﬁnition embodies this assumption and also reminds the reader of the deﬁnition of economic proﬁts Proﬁt-maximizing ﬁrm. The ﬁrm chooses both its inputs and its outputs with the sole goal of maximizing economic proﬁts, the difference between its total revenues and its total economic costs. This assumption—that ﬁrms seek maximum economic proﬁts—has a long history in economic literature. It has much to recommend it. It is plausible because ﬁrm owners may indeed seek to make their asset as valuable as possible and because competitive markets may punish ﬁrms that do not maximize proﬁts. This assumption comes with caveats. We already noted in the previous section that if the manager is not the owner of the ﬁrm, he or she may act in a self-interested way
and not try to maximize owner wealth. Even if the manager is also the owner, he or she may have other concerns besides wealth, say, reducing pollution at a power plant or curing illness in developing countries in a pharmaceutical lab. We will put such other objectives aside for now, not because they are unrealistic but rather because it is hard to say exactly which of the broad set of additional goals are most important to people and how much they matter relative to wealth. The social goals may be addressed more efﬁciently by maximizing the ﬁrm’s proﬁt and then letting the owners use their greater wealth to fund other goals directly through taxes or charitable contributions. In any event, a rich set of theoretical results explaining actual ﬁrms’ decisions can be derived using the proﬁt-maximization assumption; thus, we will push ahead with it for most of the rest of the chapter. Proﬁt maximization and marginalism If ﬁrms are strict proﬁt maximizers, they will make decisions in a ‘‘marginal’’ way. The entrepreneur will perform the conceptual experiment of adjusting those variables that can be controlled until it is impossible to increase proﬁts further. This involves, say, looking at the incremental, or ‘‘marginal,’’ proﬁt obtainable from producing one more unit of output, or at the additional proﬁt available from hiring one more laborer. As long as this incremental proﬁt is positive, the extra output will be produced or the extra laborer will be hired. When the incremental proﬁt of an activity becomes zero, the entrepreneur has pushed that activity far enough, and it would not be proﬁtable to go further. In this chapter, we will explore the consequences of this assumption by using increasingly sophisticated mathematics. 374 Part 4: Production and Supply Output choice First we examine a topic that should be familiar: what output level a ﬁrm will produce to obtain maximum proﬁts. A ﬁrm sells some level of output, q, at a market price of p per unit. Total revenues (R) are given by q p ð where we have allowed for the possibility that the selling price the ﬁrm receives might be affected by how much it sells. In the production of q, certain economic costs are
incurred and, as in Chapter 10, we will denote these by C (q). q Þ ¼ ð (11:1) Þ $ q, R The difference between revenues and costs is called economic proﬁts (p). We will recap this deﬁnition here for reference Economic proﬁt. A ﬁrm’s economic proﬁts are the difference between its revenues and costs: economic profits 11:2) Because both revenues and costs depend on the quantity produced, economic proﬁts will also. The necessary condition for choosing the value of q that maximizes proﬁts is found by setting the derivative of Equation 11.2 with respect to q equal to 0:2 dp dq ¼ p 0 q ð Þ ¼ dR dq % dC dq ¼ 0, so the ﬁrst-order condition for a maximum is that dR dq ¼ dC dq : (11:3) (11:4) In the previous chapter, the derivative dC/dq was deﬁned to be marginal cost, MC. The other derivative, dR/dq, can be deﬁned analogously as follows Marginal revenue. Marginal revenue is the change in total revenue R resulting from a change in output q: marginal revenue MR ¼ dR dq : ¼ (11:5) With the deﬁnitions of MR and MC in hand, we can see that Equation 11.4 is a mathematical statement of the ‘‘marginal revenue equals marginal cost’’ rule usually studied in introductory economics courses. The rule is important enough to be highlighted as an optimization principle Proﬁt maximization. To maximize economic proﬁts, the ﬁrm should choose output q& at which marginal revenue is equal to marginal cost. That is, MR q& ð Þ ¼ MC q& ð : Þ (11:6) 2Notice that this is an unconstrained maximization problem; the constraints in the problem are implicit in the revenue and cost functions. Speciﬁcally, the demand curve facing the ﬁrm determines the revenue function, and the ﬁrm’s production function (together with input prices) determines its costs. Chapter 11: Prof it Maximization 375
Second-order conditions Equation 11.4 or 11.5 is only a necessary condition for a proﬁt maximum. For sufﬁciency, it is also required that d2p dq2 q dp 0 ð dq Þ ¼ q q, q& q ¼ (11:7) or that ‘‘marginal’’ proﬁt must decrease at the optimal level of output, q&. For q less than q&, proﬁt must increase [p 0(q) > 0]; for q greater than q&, proﬁt must decrease [p 0(q) < 0]. Only if this condition holds has a true maximum been achieved. Clearly the condition holds if marginal revenue decreases (or remains constant) in q and marginal cost increases in q. Graphical analysis These relationships are illustrated in Figure 11.1, where the top panel depicts typical cost and revenue functions. For low levels of output, costs exceed revenues; thus, economic proﬁts are negative. In the middle ranges of output, revenues exceed costs; this means that proﬁts are positive. Finally, at high levels of output, costs rise sharply and again exceed revenues. The vertical distance between the revenue and cost curves (i.e., proﬁts) is shown in Figure 11.1b. Here proﬁts reach a maximum at q&. At this level of output it is also true that the slope of the revenue curve (marginal revenue) is equal to the slope of the cost curve (marginal cost). It is clear from the ﬁgure that the sufﬁcient conditions for a maximum are also satisﬁed at this point because proﬁts are increasing to the left of q& and decreasing to the right of q&. Therefore, output level q& is a true proﬁt maximum. This is not so for output level q&&. Although marginal revenue is equal to marginal cost at this output, proﬁts are in fact at a local minimum there. Marginal Revenue Marginal revenue is simple to compute when a ﬁrm can sell all it wishes without having any effect on market price. The extra revenue obtained from selling one more unit is just this market price. A ﬁrm may not always be able to sell all it wants at the prevailing

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