Chapter 9 Theory of the Firm 2: The Long-Run, Multiple-Input Model PDF

Chapter 9 Theory of the Firm 2: The Long-Run, Multiple-Input Model 9.1 Introduction In most of the last chapter we modeled a firm with one input and one output. However, assuming one input is unrealistic; most goods and services a...

Chapter 9 Theory of the Firm 2: The Long-Run, Multiple-Input Model 9.1 Introduction In most of the last chapter we modeled a firm with one input and one output. However, assuming one input is unrealistic; most goods and services are produced by firms with a variety of different inputs. The production of something as simple as corn really requires land, labor, trucks, tractors, combines, fertilizer, pesticides, possibly irrigation, and so on. Moreover, the single-input model fails to capture a basic economic problem. Normally there are many ways to combine inputs to produce a desired level of the output; some of the ways are expensive and some are cheap. How does the firm combine various inputs to produce a given level of output at the least cost? In this chapter, we assume there are two or more inputs that the firm combines in some way to produce its output. We analyze how the firm decides how much output to produce, and how much of each input to use, to minimize its costs and maximize its profits. That is, we will now develop the multiple-input/single-output model. As we indicated in the introduction to the last chapter, it is possible to learn about the most important results in the theory of the firm by studying either the single-input/single-output model or the multiple-input/single-output model. (The only important topic that the single-input/single-output model cannot handle is cost minimization.) This book differs somewhat from the typical textbook on microeconomics because it gives the reader the choice between these two models. We now turn to the second model. In the last chapter, x was the quantity of the (single) input and y was the quantity of the output. The production function was y = f (x). Now we assume there are two or more inputs. We let x1 represent the quantity of input 1 used by the firm, x2 the quantity of input 2, x3 the quantity of input 3, and so on. The production function now becomes y = f (x1 , x2 , x3 ,...). Most of the important implications of profit maximization with two or more inputs can be seen with just two inputs, 141 142 Chapter 9. Theory of the Firm 2: The Long-Run, Multiple-Input Model so we will focus on that case. In short, in this chapter the production function is assumed to be y = f (x1 , x2 ). The function f (x1 , x2 ) represents the technological constraints facing the firm, as did the function f (x) in the last chapter. The firm must work within the constraints imposed by nature, science, and technology. The firm also faces market constraints, involving the price of its output p, and the prices of its inputs w1 and w2. We consider another kind of constraint in this chapter and the next, having to do with the variability of the firm’s inputs. Some input quantities can be changed quickly and easily; others cannot. For example, if the firm’s inputs are electricity or phone service, the quantities used can easily be varied hour by hour, even minute by minute. However, if the inputs are, for example, acres of farmland planted in corn, or pharmaceutical research to develop new drugs, the input quantities can be varied only over periods of months, years, or even decades. Thus, time horizons and the degrees of variability of input levels within those time horizons create a new type of constraint on the firm. In this chapter we assume that the inputs are both (or all) freely variable. In the next chapter we will assume one or more of the inputs is fixed over the underlying time horizon, while one or more of the inputs is variable. Economists call a period of time that is so long that all of the firm’s inputs are freely variable the long run, and they call a period of time that is so short that one or more inputs is fixed the short run. Therefore, this chapter is about the theory of the firm in the long run. The next chapter is about the theory of the firm in the short run. We are doing the long-run theory first because it is simpler and more elegant than the short-run theory. We apologize for the inescapable vagueness about how long a time is long run, and how short a time is short run, but the study of economics is different from the study of, say, chemistry or physics. Moreover, as the great economist John Maynard Keynes (1883–1946) once quipped, “The long run is a misleading guide to current affairs. In the long run we are all dead.” (Keynes was writing about macroeconomic policy, rather than microeconomic theory, when he created that gem.) A complication of the long-run/short-run dichotomy is the possibility of bankruptcy. For example, in 2009, American automobile companies General Motors and Chrysler went through what is called Chapter 11 bankruptcy to escape the burdens of debt and union contracts, both of which create massive costs that are impossible to escape in the short run. Government aid facilitated the bankruptcy. Chapter 11 allows a firm to shield itself from its creditors in a court; the court has the power to rewrite or erase the firm’s contractual obligations and thus, modify or end such costs. In standard microeconomic theory, the long run is a period of time long enough to modify all the firm’s costs. In the real world, bankruptcy is another way to modify costs, and it may be faster than the “long run.” We end this very  9.2 The Production Function in the Long Run 143 brief discussion of bankruptcy by paraphrasing Keynes: In the long run we are all dead, and if not dead, perhaps bankrupt. 9.2 The Production Function in the Long Run We are now assuming that y = f (x1 , x2 ), and that both inputs are free to vary. Suppose a firm can produce 1 unit of its output by using 2 units of input 1, which we will call “workers,” and 3 units of input 2, “raw materials.” If 2 workers and 3 units of raw materials can produce 1 unit of the output, it is easy to imagine that 3 workers and 3 units of raw materials can also produce 1 unit of output. Just ask the third worker to sit at a table in the shop or office and send text messages to her children, while the original 2 workers make the 1 unit of output! Naturally, we want to refine our notion of the production function to rule out this possibility, so we will impose the assumption of technological efficiency. When various combinations of inputs produce the same level of output, we call such combinations production techniques. For instance, in the preceding example, (x1 , x2 ) = (2, 3) and (x1 , x2 ) = (3, 3) are alternative production techniques, both resulting in output y = 1. We will say that a production technique is technologically inefficient if there is another production technique (or combination of techniques) that results in the same level of output, but uses less of one of the inputs and no more of the other input or inputs. Otherwise, it is technologically efficient. In our example, (3, 3) is technologically inefficient. Because firms generally have to pay for the inputs they use, they do not want to use technologically inefficient production techniques. Therefore, we will confine the application of the production function f to efficient production techniques. That is, when we write y = f (x1 , x2 ) in what follows, it is understood that (x1 , x2 ) is an efficient production technique. We now define an isoquant. The prefix “iso” is Greek for “the same” or “equal,” and “quant” is short for “quantity.” An isoquant is a set of efficient production techniques that result in the same quantity of output. To graph an isoquant, we start with a picture that has quantities of the inputs x1 and x2 on the horizontal and vertical axes, respectively, and we identify a locus of points (x1 , x2 ) that all produce a fixed quantity of output y. When we do this for different y’s, we get different isoquants. Isoquants for y = 1, 2, and 3 are shown in Figure 9.1. The isoquants in Figure 9.1 also happen to be “evenly spaced”; we will explain the meaning of this later. The reader may think that isoquants look vaguely familiar – and so they should. Isoquants in the theory of the firm play a role very similar to that of indifference curves in the theory of the consumer, introduced back in Chapter 2. Recall that a consumer wants to get to higher and higher indifference curves, all else being equal. Similarly, a firm wants to get to higher and higher isoquants, all else being equal. (Of course, the exact meanings of these statements depends on what we mean by “all else being equal.”) There is, however, one important 144 Chapter 9. Theory of the Firm 2: The Long-Run, Multiple-Input Model Input 2 y=3 y=2 y=1 Input 1 Fig. 9.1. A map of isoquants. difference between isoquants and indifference curves, which we should point out immediately. In consumer theory, utility is an ordinal measure; only relative utilities matter, and in the statement “I am on the indifference curve for u = 3,” the number 3 has no intrinsic meaning. On the other hand, in the theory of the firm, output is a cardinal measure. In the statement “the firm is at y = 3,” the number 3 does have significance. It means that the firm is producing 3 cars, or 3 bushels of wheat, or 3 units of whatever the firm produces. Moreover, the u = 6 indifference curve is better for the consumer than the u = 3 indifference curve, but not twice as good. In contrast, the y = 6 isoquant really does produce twice the output of the y = 3 isoquant. Marginal Products of the Inputs and Assumptions about Production Functions and Isoquants In Chapter 8, on the single-input/single-output model, we defined the marginal product of the input. Intuitively, the marginal product is the extra output resulting from another unit of the input. Formally, it is the derivative of the production function f (x) with respect to x. When there are two (or more) inputs, the definition is quite similar. Intuitively, the marginal product of input 1, for instance, is the extra output resulting from an additional unit of input 1 and zero additional units of input 2. Formally, the marginal product of input 1 is the derivative of the production function with respect to x1 , holding x2 constant, or the partial derivative of f (x1 , x2 ) with respect to x1. That is, ∂f (x1 , x2 ) MP1 =. ∂x1  9.2 The Production Function in the Long Run 145 This may also be written as ∂f/∂x1 , or ∂f (x1 , x2 )/∂x1 , or f1 (x1 , x2 ). If we need to emphasize where the partial derivative is being computed, we will write MP1 (x1 , x2 ) if it is being evaluated at (x1 , x2 ). The marginal product for input 2 is defined similarly. Note the strong resemblance between the marginal product of input 1 in the theory of the firm, and the marginal utility of good 1 in the theory of the consumer, as defined in Chapter 2. The reader may remember that, in the theory of the consumer, the marginal rate of substitution was equal to the marginal utility of good 1 divided by the marginal utility of good 2. As we shall see, there is a close parallel in the theory of the firm. We will now turn to the assumptions we make about the production function and its associated isoquants. Assumption 1: Monotonicity. If a firm increases one input without decreasing the other, output increases. That is, the marginal products MP1 = ∂f (x1 , x2 )/∂x1 and MP2 = ∂f (x1 , x2 )/∂x2 , or the partial derivatives of f with respect to x1 and x2 , are both positive. This rules out instances of technological inefficiency such as that described in the earlier example. It also rules out the possibility of “fat” isoquants; that is, isoquants that are other than thin lines. And it rules out isoquants that are horizontal, vertical, or upward sloping. That is, it forces isoquants to be downward sloping. Assumption 2: Convexity. The isoquants are convex. There are several reasons why this is a plausible assumption. First, in many cases, it is reasonable to assume that production techniques can operate at different levels, with output levels scaled proportionately, and that a firm can use two or more production techniques simultaneously, without the techniques interfering with one another. For instance, suppose (x1 , x2 ) = (2, 3) is an efficient production technique that gives y = 1, and suppose (x1 , x2 ) = (3, 2) is another efficient technique that gives y = 1. Consider running the first technique at half level, and the second also at half level. It is reasonable to assume that this would produce at least 1/2 + 1/2 = 1 unit of output. Running the two techniques at these levels would require (1/2)(2, 3) + (1/2)(3, 2) = (2.5, 2.5) units of inputs 1 and 2. Now think of the y = 1 isoquant. It has to pass through (2, 3) and (3, 2). It would fail to be convex if it passed above the point midway between these two points – that is, (2.5, 2.5). However, it cannot pass above (2.5, 2.5), because at that point the firm can produce at least y = 1 just by running those two production techniques at half level. There are likely to be other techniques available that would transform inputs of (2.5, 2.5) into output of y > 1. This implies that the isoquant running through (2, 3) and (3, 2) should pass below the point (2.5, 2.5). This would mean that the isoquant is (strictly) convex. Second, if isoquants are not convex, a firm will use “extreme” input bundles (extreme in the sense that one of the input levels is zero). That is, the inputs would 146 Chapter 9. Theory of the Firm 2: The Long-Run, Multiple-Input Model not be used in combination. Because we observe firms using combinations of inputs, it is plausible to assume convexity. Technical Rate of Substitution The reader will recall that in the theory of the consumer, a crucial concept is the marginal rate of substitution of good 2 for good 1, or MRSx1 ,x2 , or MRS for short. The intuition is this: If the consumer gives up a unit of good 1, how much good 2 does he need to replace it, and remain on the same indifference curve? The marginal rate of substitution of good 2 for good 1 is minus 1 times the slope of an indifference curve. Also recall from Chapter 2 the relationship between the marginal rate of substitution and the ratio of the marginal utilities: MRS = MU1 /MU2. In the theory of the firm, we have a concept exactly analogous to marginal rate of substitution in the theory of the consumer, and we have a relationship just like MRS = MU1 /MU2 in the theory of the consumer. The technical rate of substitution of input 2 for input 1, formally written T RSx1 ,x2 , or T RS for short, is defined as x2 T RSx1 ,x2 = − , x1 where x1 and x2 are small (more precisely, infinitesimal) increments in inputs x1 and x2 , one negative and the other positive, that leave the firm on the same isoquant. In other words, T RS is minus 1 times the slope of the isoquant. The intuition is this: If the firm uses a unit less of input 1, how much more of input 2 does it need to use in order to keep output constant; that is, to remain on the same isoquant? To put it another way, T RS is the value of a unit of input 1 in the production process, measured in terms of units of input 2 needed to replace it. We can establish the relationship among T RS, MP1 , and MP2 with an argument very similar to the one we made in Chapter 2 on the theory of the consumer. Imagine that we start at an input bundle (x1 , x2 ) and we simultaneously reduce x1 and increase x2 in a way that leaves y unchanged. The increments are x1 < 0 and x2 > 0. Then output changes by MP1 x1 < 0 because of the reduction in input 1, and it simultaneously changes by MP2 x2 > 0 because of the increase in input 2. But we end up on the same isoquant, and the net effect is zero. This implies that MP1 x1 + MP2 x2 = 0. Therefore, x2 MP1 ∂f (x1 , x2 ) ∂f (x1 , x2 ) T RS = − = = /. x1 MP2 ∂x1 ∂x2 Now, let us reconsider the idea of convexity for an isoquant. Consider moving toward the right and down along a single isoquant. Convexity means that minus  9.2 The Production Function in the Long Run 147 1 times the slope of the isoquant, or the absolute value of the slope, is declining as we move to the right and down, or T RS declines as we move to the right and down. That is, as x1 , the quantity of input 1, gets greater and greater, the value of an extra unit of input 1 in the production process gets smaller and smaller. This seems a very plausible assumption for T RS, and another reason to view convexity for isoquants as a reasonable basic assumption. Returns to Scale The reader will recall that in Chapter 8, on the single-input/single-output model of production, we assumed that the production function f (x) was either (1) concave, or more realistically, (2) at first convex and then concave. The intuition was that a concave production function, with a negative second derivative f  (x), represents diminishing returns, and that a production function that starts convex and then becomes concave represents the more realistic real-world case of increasing returns when the firm is small, eventually becoming diminishing returns when the firm is large. We will now consider similar notions applied to the firm with two (or more) inputs. For this purpose, it is conventional practice to consider what happens when all inputs are scaled up proportionately, rather than to consider what happens as each input is modified incrementally. Therefore, we proceed as follows. Assume that both inputs are scaled up by a constant t > 1. (Remember, we are talking about production in the long run, so there are no input quantities that are fixed and cannot be scaled up.) Then, if output changes by the same scale factor t, the production function is said to have the property of constant returns to scale. For instance, if the firm doubles all its inputs, or increases them by 100 percent, output should double – that is, rise by 100 percent. More formally, f (x1 , x2 ) is a constant returns to scale production function, if, for any t > 1 and any (x1 , x2 ), f (tx1 , tx2 ) = tf (x1 , x2 ). If scaling up the inputs results in output increasing, but by less than the scale factor, the production function is said to have the property of decreasing returns to scale. For instance, the firm might double all its inputs and see its output rise by 50 percent as a result. More formally, f (x1 , x2 ) is a decreasing returns to scale production function, if, for any t > 1 and any (x1 , x2 ), f (tx1 , tx2 ) < tf (x1 , x2 ). If scaling up the inputs results in output increasing, and by more than the scale factor, the production function is said to have the property of increasing returns to scale. For instance, the firm might double all its inputs and see its output rise by 150 percent as a result. More formally, f (x1 , x2 ) is an increasing returns to scale production function, if, for any t > 1 and any (x1 , x2 ), f (tx1 , tx2 ) > tf (x1 , x2 ). When the isoquants for a constant returns to scale production function are graphed, they are “evenly spaced,” in the sense that the y = 2 isoquant is twice as far from the origin as the y = 1 isoquant, the y = 3 isoquant is three times as 148 Chapter 9. Theory of the Firm 2: The Long-Run, Multiple-Input Model Input 2 (3, 3) y=3 (2, 2) y=2 (1, 1) y=1 Input 1 Fig. 9.2. Decreasing returns to scale. far from the origin as the y = 1 isoquant, and so on. Figure 9.1 was drawn under the constant returns to scale assumption. Under increasing returns to scale, as one moves away from the origin, isoquants get closer and closer to each other, and under decreasing returns to scale, as one moves away from the origin, isoquants get farther and farther apart. Figure 9.2 illustrates decreasing returns to scale. Decreasing returns to scale is the scale assumption that corresponds to our Chapter 8 assumption of concavity for the production function f (x). In the Chapter 8 real-world case, the production function f (x) starts convex and then becomes concave. The isoquant-spacing assumption that corresponds to this case is the following: when output is low, successive isoquants get closer and closer to each other, but when output is high, successive isoquants get farther and farther from each other. Loosely speaking, this is increasing returns to scale at the start, but becoming decreasing returns to scale at the end. Figure 9.3 shows the real-world case. Marginal Products and TRS in the Constant Returns to Scale Case A constant returns to scale production function f (x1 , x2 ) scales output propor- tionately when both the inputs are scaled up by a factor t > 1, so f (tx1 , tx2 ) = tf (x1 , x2 ). Taking the partial derivative of the left side of this equation (that is, f (tx1 , tx2 )) with respect to x1 (that is, differentiating with respect to x1 while holding x2 constant) gives ∂f (tx1 , tx2 ) d(tx1 ) ∂f (tx1 , tx2 ) =t = tMP1 (tx1 , tx2 ). ∂tx1 dx1 ∂tx1  9.2 The Production Function in the Long Run 149 Input 2 y=6 y=7 y=5 y=4 y=3 y=2 y=1 Input 1 Fig. 9.3. Returns to scale in the real-world case. Taking the partial derivative of the right side of the equation (that is, tf (x1 , x2 )) with respect to x1 gives ∂f (x1 , x2 ) t = tMP1 (x1 , x2 ). ∂x1 Setting the partial derivative of the left-hand side equal to the partial derivative of the right-hand side, and canceling out the t’s on the left and on the right, we get MP1 (tx1 , tx2 ) = MP1 (x1 , x2 ). We conclude that for a constant returns to scale production function, if both inputs are scaled up proportionately, the marginal products of the inputs do not change. Scaling (x1 , x2 ) up or down is graphically equivalent to moving up or down a ray from the origin, in a graph with inputs 1 and 2 on the horizontal and vertical axes. Therefore, for a constant returns to scale production function, MP1 and MP2 are constant as the firm moves along rays from the origin, or, to put it another way, MP1 and MP2 remain constant as long as the input ratio x2 /x1 remains constant. Now, consider what happens to the technical rate of substitution as we vary x1 and x2 , but keep the ratio x2 /x1 fixed. Because MP1 T RS = , MP2 and because the marginal products don’t change, T RS remains constant. In other words, for a constant returns to scale production function, if one scales up both inputs proportionately, the technical rate of substitution stays constant. In a graph 150 Chapter 9. Theory of the Firm 2: The Long-Run, Multiple-Input Model with inputs 1 and 2 on the horizontal and vertical axes, if you move out a ray from the origin, the slopes of the isoquants crossing that ray are all the same. 9.3 Cost Minimization in the Long Run We now turn to the topic of cost minimization. We continue to assume that the time horizon is long run, and that both inputs are freely variable. Cost minimization was not an issue in Chapter 8, in which we discussed profit maximization by a firm using only one input. This is because if a firm produces y with one input x, according to the production function y = f (x), there is only one way to produce a given level of output y 0 , and only one possible cost: use x 0 = f −1 (y 0 ) and pay wx 0 for it. In the single-input case, it is a waste of time to search for a cheaper way to produce y 0. Now, though, we are assuming that the firm is producing its output y, and it is using two inputs in quantities x1 and x2 to do so. The prices for the inputs are w1 and w2 , respectively. The firm is, of course, constrained by its production function y = f (x1 , x2 ). But for any given level of output, say y 0 , there may be infinitely many ways to produce that output; all the input combinations on the y = y 0 isoquant will do it. The firm wants to maximize its profits, but to maximize profits, it must minimize costs. At the risk of belaboring the obvious, let us emphasize this point. The firm’s profit equals its revenue less its cost, or, in our notation, π = py − C(y) = py − (w1 x1 + w2 x2 ). If, for a given y, the cost C(y) is not at the minimum, then profit can obviously be increased by switching to a lower-cost method of producing the given y. In other words, cost minimization is a necessary condition for profit maximization. Isocost Lines and the Condition for Cost Minimization To facilitate the analysis of cost minimization, we use isocost lines. (Remember, “iso” means “the same” or “equal.”) An isocost line is a set of input combinations (x1 , x2 ), all with the same cost. For a given level of cost, say C 0 , the isocost line is the graph of w1 x1 + w2 x2 = C 0. This equation may seem vaguely familiar to the reader, because it looks like the equation for the budget line in the theory of the consumer: p1 x1 + p2 x2 = M.  9.3 Cost Minimization in the Long Run 151 Input 2 Tangency of isoquant and isocost line y = y0 C0 C1 C2 Input 1 Fig. 9.4. Several isocost lines and one isoquant. Cost is minimized at the tangency point. Of course, the consumer tries to get to the highest indifference curve, for a given budget line. The firm tries to get to the lowest isocost line, for a given isoquant. Figure 9.4 shows several isoquants corresponding to cost levels C 0 , C 1 , and 2 C. A lower isocost line in the figure corresponds to a lower level of cost, as w1 and w2 are both assumed to be positive. The slope of an isocost line is very much analogous to the slope of the consumer’s budget line; it equals w1 /w2 in absolute value. Note that Figure 9.4 includes one isoquant, which happens to be tangent to the C 1 isocost line. Figure 9.4 makes it clear that, in order to minimize the cost of producing y 0 , the firm will try to find a point where the y 0 isoquant is tangent to an isocost line. We noted earlier that cost minimization is a necessary condition for profit max- imization. We now describe the tangency condition that must hold for cost min- imization. Suppose the firm is producing y 0. Suppose the production function is differentiable, the isoquants are smooth and convex, and there is an isoquant/isocost line tangency. To produce y 0 at least cost, the firm must choose the input combina- tion (x1 , x2 ) at which the isoquant is tangent to an isocost line. Because the slope of the isoquant is −T RS and the slope of the isocost line is −w1 /w2 , we have the following basic cost-minimization condition: w1 T RS =. w2 This condition should look familiar; it is exactly like the corresponding tangency condition in the theory of the consumer. The consumer tangency condition says 152 Chapter 9. Theory of the Firm 2: The Long-Run, Multiple-Input Model that to get to the highest indifference curve subject to the budget constraint, the consumer must satisfy p1 MRS =. p2 Now let us consider a variable output level y instead of a particular level y 0. The firm is facing input prices (w1 , w2 ). To produce an arbitrary y at the least cost, the firm will solve for a pair of input levels, which we call x1∗ and x2∗ , that satisfy (1) the cost minimization tangency condition w1 T RS = w2 and (2) the production function equation y = f (x1 , x2 ). The desired input levels (x1∗ , x2∗ ) now depend on the input prices (w1 , w2 ) and on the output level y, and so we write them as x1∗ (w1 , w2 , y) and x2∗ (w1 , w2 , y). These two functions show how much of the two inputs the firm wants to hire, given the input prices, and given the level of output. These are called the conditional factor demands or conditional input demands for the firm. In a typical application of this kind of analysis, w1 and w2 are fixed, and the word “conditional” is used here because the amounts of the inputs that the firm demands depend on the level of output the firm will choose. With the conditional input demands, we can define the firm’s long-run cost function or long-run cost curve. This shows, for any level of output y, the least cost of producing y (assuming fixed input prices, and assuming efficient production techniques). The long-run cost function is C(y) = w1 x1∗ (w1 , w2 , y) + w2 x2∗ (w1 , w2 , y). The Relation between Long-Run Cost Curves and Returns to Scale Loosely speaking, the returns of a technology and the costs of production are mirror images. That is, when returns are high, costs are low, and vice versa. Somewhat more precisely, when returns are increasing, costs are falling, and vice versa. As we will see next, this is what we find for constant, increasing, and decreasing returns to scale. Suppose a firm’s production function satisfies constant returns to scale. Let (x1 , x2∗ ) be the cost-minimizing input combination that results in 1 unit of output. ∗ Then C(1) = w1 x1∗ + w2 x2∗. Now, if the firm wants to produce an arbitrary y units of output, by constant returns to scale, it can do so by multiplying by y both the input quantities that gave 1 unit of output. Moreover, as we saw earlier, scaling up both the inputs in this fashion will leave MP1 , MP2 , and T RS unchanged.  9.3 Cost Minimization in the Long Run 153 $ C(y) MC = AC = C(1) y (output) 1 Fig. 9.5. Total cost C(y) with constant returns to scale. Therefore, because (x1∗ , x2∗ ) was a point of tangency between an isoquant and an isocost line, (yx1∗ , yx2∗ ) will also be a tangency point. Therefore, C(y) = yC(1) is the least-cost way to produce y units of output. That is, for constant returns to scale, the long-run cost function is a very simple linear function that makes C(y) directly proportional to output y. We show this kind of linear cost function in Figure 9.5. Remember that average cost is total cost divided by quantity, or AC(y) = C(y)/y. For our constant returns to scale case, average cost is constant. This is because AC(y) = C(y)/y = yC(1)/y = C(1). Also recall that marginal cost is the derivative of the cost function, or, intuitively, the extra cost per additional unit of output. That is, MC(y) = dC(y)/dy. In the constant returns to scale case, MC(y) = d(C(1)y)/dy = C(1) = AC(y). In short, with constant returns to scale, total cost is a linear function of y, and both average and marginal cost are constant, equal to each other, and equal to the cost of producing just 1 unit of output. Next, we assume that the firm’s production function satisfies increasing returns to scale. The cost of producing 1 unit of output is C(1). If the firm wants to produce y units of output, it can scale up the input quantities by less than y, because if it scaled up by y, by the increasing returns to scale assumption, output would rise to more than y. Therefore, the cost of producing y will be less than yC(1). In short, C(y) < yC(1). This implies that the total cost curve is concave, as shown in Figure 9.6. In Figure 9.6, we have included a line l2 that is tangent to the total cost curve at a point P , as well as a line l1 going from the origin through the point P. The slope of the tangent line is dC(y)/dy, or marginal cost, and the slope of the line from the origin is C(y)/y, or average cost. This is similar to what we did in Figure 8.3. 154 Chapter 9. Theory of the Firm 2: The Long-Run, Multiple-Input Model $ l2 l1 C(y) P y (output) Fig. 9.6. Total cost C(y) with increasing returns to scale. Average cost at the point P is the slope of l1 , and marginal cost is the slope of l2. It is clear from Figure 9.6 that at any point on the total cost curve, the slope of the tangent line, or marginal cost, is less than the slope of the line from the origin, or average cost. In short, for the increasing returns to scale case, MC(y) < AC(y). Also, as y increases, both marginal cost and average cost are decreasing. Now we assume that the firm’s production function satisfies decreasing returns to scale. We continue to assume that the cost of producing 1 unit of output is C(1). Now, if the firm wants to produce y units of output, it must scale up the input quantities by more than y. Therefore, the cost of producing y will be more than yC(1). In short, C(y) > yC(1). This implies that the total cost curve is convex, as shown in Figure 9.7. In Figure 9.7, we have again included a line l2 that is tangent to the total cost curve at a point P , as well as a line l1 going from the origin through the point P. The slope of the tangent line is dC(y)/dy, or marginal cost, and the slope of the line from the origin is C(y)/y, or average cost. It is clear from Figure 9.7 that at any point on the total cost curve, the slope of the tangent line, or marginal cost, is greater than the slope of the line from the origin, or average cost. In short, for the decreasing returns to scale case, MC(y) > AC(y). Also, as y increases, both marginal cost and average cost are increasing. Finally, let us suppose, loosely speaking, that the production function satisfies increasing returns to scale at the beginning, changes to constant returns to scale, and then changes to decreasing returns to scale at the end. We say “loosely speaking” here because, as we have defined constant, increasing, and decreasing returns, the properties are universal, and not attached to particular scales of operation for the  9.4 Profit Maximization in the Long Run 155 $ l2 C(y) l1 P y (output) Fig. 9.7. Total cost C(y) with decreasing returns to scale. Average cost at the point P is the slope of l1 , and marginal cost is the slope of l2. (C(y) position approximate.) firm. In terms of isoquants, we are now assuming that the isoquants first get closer and closer together, and ultimately get farther and farther apart. Now the total cost curve is first concave, and then turns to convex, as in Figure 9.8. Note that this figure is just like Figure 8.3, on the single-input model. The reader should refer to Figure 8.4 to recall the appearance of the U-shaped average and marginal cost curves in the real-world case for the single-input model. A similar figure applies here in the multi-input model. Both average and marginal costs first decline and then rise, and the marginal cost curve passes through the minimum of the average cost curve. When average cost is declining, marginal cost is below average cost, and when average cost is rising, marginal cost is above average cost. 9.4 Profit Maximization in the Long Run Throughout this chapter, we have been assuming that the firm is competitive in the market for its inputs. That is, it takes the input prices w1 and w2 as given and fixed. It is too small to affect the input prices. We now also assume that our firm is competitive in the market for its output. That is, its choice of y does not affect the output price p. (We will return to a careful analysis of the competitive market assumption in a subsequent chapter.) We now derive the conditions for profit maximization in the long run, as well as the firm’s long-run supply curve. This section should look very familiar, because what we do here is very similar to what we already did in Chapter 8, the single-input model. 156 Chapter 9. Theory of the Firm 2: The Long-Run, Multiple-Input Model $ C(y) l1 Q l2 P y (output) y0 y* Fig. 9.8. Total cost C(y) under increasing returns to scale followed by decreasing returns to scale. Average cost at P is the slope of l1 , and marginal cost is the slope of l2. In the long run, both (or all) inputs are free to vary. If the firm produces nothing, y, x1 , and x2 are all zero, revenue is zero, total cost is zero, and profit is zero. Therefore, whenever the firm chooses a y to maximize its profit, it will consider only output quantities for which profit is nonnegative. Therefore, π (y) = py − C(y) ≥ 0. Dividing both sides of the inequality by y leads to p ≥ min AC(y). That is, the firm will operate only if the market price equals or exceeds minimum average cost. In the long run, it will not be in business if being in business means losing money. If the price p is high enough for the firm to operate without a loss, then it must decide how much to produce. Its profit is π (y) = py − C(y). The first-order condition for maximizing this function is dπ (y) dC(y) =p− = p − MC(y) = 0. dy dy  9.4 Profit Maximization in the Long Run 157 This gives p = MC(y), or price equals marginal cost. The second-order condition for maximizing profit is d 2 π (y) d(p − MC(y)) 2 = ≤ 0, dy dy which gives dMC(y) ≥ 0. dy That is, at the profit-maximizing point, price equals marginal cost, and marginal cost is rising (or at least not falling). We refer the reader to Figure 8.6 to view a graph illustrating the profit- maximizing choice in the real-world case of a firm with U-shaped average and marginal cost curves. That graph includes a horizontal line at a price p that is greater than the minimum of the average cost curve. The horizontal line intersects the marginal cost curve at two points; both points satisfy the first-order condition, but only one satisfies the second-order condition. We also refer the reader to Figure 8.7 to view the graph of the profit-maximizing firm’s supply curve in the real-world case. In summary, in the typical competitive firm case, with U-shaped average and marginal cost curves, in the long run, the firm supplies nothing if p < min AC(y). But if p ≥ min AC(y), the amount the firm supplies is given by the upward-sloping part of the marginal cost curve. Returns to Scale and Long-Run Supply We now analyze the firm’s profit-maximization decision and its supply curve in the cases of constant, decreasing, and increasing returns to scale. If the firm’s production function is constant returns to scale, its total cost, average cost, and marginal cost curves are as illustrated in Figure 9.5. In particular, marginal cost and average cost are constant (that is, horizontal lines) at C(1). For p > C(1), the firm would want to supply an unlimited amount, and would have unlimited profit. For p = C(1), the firm would supply any quantity; all would result in zero profit. For p < C(1), the firm would supply nothing, and would have a profit of zero. Next we assume an increasing returns to scale production function and the corresponding total cost curve, as in Figure 9.6. We can see from that figure that average cost (the slope of the ray l1 , from the origin through a point on C(y)) is always greater than marginal cost (the slope of the tangent l2 to C(y)), and both 158 Chapter 9. Theory of the Firm 2: The Long-Run, Multiple-Input Model $ Increase π without limit p AC MC y (output) y* y′ Fig. 9.9. Unbounded supply under increasing returns to scale. average cost and marginal cost are declining as y increases. Figure 9.9 shows the average and marginal cost curves corresponding to this case. At the price/quantity combination (p, y ∗ ) illustrated, the first-order condition p = MC(y) is satisfied, but the second-order condition is not. In fact, that point is a profit minimum, rather than a profit maximum. Given the price p, if the firm increases its output to y  , it will break even, and moving to the right of y  allows increasing profit without limit. In other words, given that average cost and marginal cost continue to decline, the firm wants to produce an infinite amount of its output and earn infinite profits. This, of course, would eventually result in the firm becoming so large that its choice of y would affect the price p. In other words, the assumption of increasing returns to scale, over all levels of output, is ultimately inconsistent with the assumption of competitive behavior. Finally, we assume a decreasing returns to scale production function, and the corresponding total cost curve, as in Figure 9.7. In that figure, we can see that average cost is less than marginal cost, and both are increasing as y increases. Figure 9.10 shows the average and marginal cost curves corresponding to this case. Both curves are upward sloping, and the marginal cost curve lies above the average cost curve. Given a market price p, as shown, the firm will maximize profit by producing the y ∗ shown. The firm’s profit will then be π (y ∗ ) = py ∗ − C(y ∗ ) = py ∗ − AC(y ∗ )y ∗ , or the area of the crosshatched rectangle. The marginal cost curve is also the firm’s supply curve, at least for prices greater than or equal to the intercept of MC(y) with the vertical axis.  9.5 A Solved Problem 159 $ MC = Supply curve AC Profit p y (output) y* Fig. 9.10. The long-run supply with decreasing returns to scale. 9.5 A Solved Problem The Problem Consider the production function β y = f (x1 , x2 ) = x1α x2 , where α and β are positive constants. Assume the input prices are w1 and w2. This production function is called a Cobb-Douglas production function, after the people who first studied it, Charles Cobb (1875–1949) and Paul Douglas (1892– 1976). Cobb was a mathematician and economist; Douglas was an economist at the University of Chicago who became an important and influential Democratic senator from Illinois. (a) Show that if α + β < 1, the production function has the property of decreasing returns to scale. (We will say “the production function is decreasing returns to scale” for short.) Show that if α + β = 1, the production function is constant returns to scale; and show that if α + β > 1, the production function is increasing returns to scale. (b) Find the marginal products MP1 and MP2 , and the technical rate of substitution T RS. (c) Now assume w1 = 1 and w2 = 1, and also assume α = 1 and β = 1. Find the long-run cost function C(y), the average cost function AC(y), and the marginal cost function MC(y). 160 Chapter 9. Theory of the Firm 2: The Long-Run, Multiple-Input Model The Solution (a) Note that β f (tx1 , tx2 ) = (tx1 )α (tx2 )β = (t α+β )x1α x2 , and β tf (x1 , x2 ) = tx1α x2. A production function is decreasing returns to scale if for any t > 1, f (tx1 , tx2 ) < tf (x1 , x2 ); it is constant returns to scale if for any t > 1, f (tx1 , tx2 ) = tf (x1 , x2 ); and it is increasing returns to scale if for any t > 1, f (tx1 , tx2 ) > tf (x1 , x2 ). Therefore, the Cobb-Douglas production function is decreasing returns to scale if β β (t α+β )x1α x2 < tx1α x2 ⇔ t α+β < t ⇔ α + β < 1. Similarly, the Cobb-Douglas production function is constant returns to scale if β β (t α+β )x1α x2 = tx1α x2 ⇔ t α+β = t ⇔ α + β = 1. Finally, the Cobb-Douglas production function is increasing returns to scale if β β (t α+β )x1α x2 > tx1α x2 ⇔ t α+β > t ⇔ α + β > 1. (b) To find MP1 , we take the partial derivative of f (x1 , x2 ) with respect to x1. This gives β ∂x1α x2 β MP1 = = αx1α−1 x2. ∂x1 MP2 is found similarly: β ∂x1α x2 β−1 MP2 = = βx1α x2. ∂x2 The technical rate of substitution is MP1 divided by MP2 , or β MP1 αx1α−1 x2 αx2 T RS = = β−1 =. MP2 α βx1 x2 βx1 (c) Now we are assuming w1 = w2 = 1, and we are also assuming α = β = 1. These two assumptions will make things a lot easier. For cost minimization, in the general case, the firm finds input combinations where T RS = w1 /w2. This gives αx2 x2 w1 T RS = = = = 1. βx1 x1 w2 Therefore, x2 = x1. Substituting back in the production function, we now get y = x1α x2 = x1 x2 = x12. Therefore, the cost-minimizing inputs are x1∗ = y 1/2 and x2∗ = y 1/2. β The cost function is thus, C(y) = w1 x1∗ + w2 x2∗ = y 1/2 + y 1/2 = 2y 1/2.  Exercises 161 The average cost and marginal cost functions are 2y 1/2 AC(y) = C(y)/y = = 2y −1/2 , y dC(y) d(2y 1/2 ) MC(y) = = = y −1/2. dy dy Exercises 1. Explain why the concepts of constant, increasing, and decreasing returns to scale make sense when applied to isoquants, but would not make sense in the theory of the consumer, if applied to indifference curves. That is, why does the spacing between successive isoquants make sense, whereas the spacing of successive indifference curves does not? 2. If the price of the output of a profit-maximizing firm rises, how will the firm’s output change? 1/4 1/4 3. Suppose a firm’s production function is y = x1 x2. The prices of the inputs are w1 = 1 and w2 = 2. √ 2 ∗ (a) Show √ that the long-run conditional factor demands are x1 (y) = 2y and x2∗ (y) = 2 y / 2. √ (b) Show that the long-run cost function is C(y) = 2 2y 2. √ (c) Show that the long-run supply curve for the firm is given by y ∗ (p) = p/(4 2). 4. A firm produces computers with two factors of production: labor L and capital K. Its production function is y = LK/10. Suppose the factor prices are wL = 10 and wK = 100. (a) Graph the isoquants for y equal to 1, 2, and 3. Does this technology show increasing, constant, or decreasing returns to scale? Why? (b) Derive the conditional factor demands. (c) Derive the long-run cost function C(y). (d) If the firm wants to produce one computer, how many units of labor and how many units of capital should it use? How much will it cost? What if the firm wants to produce two computers? (e) Derive the firm’s long-run average cost function AC(y) and long-run marginal cost function MC(y). Graph AC(y) and MC(y). What is the firm’s long-run supply curve? 5. Let the firm’s production function be given by y = x1 + x2. Suppose w1 = 2 and w2 = 1. (a) Derive the conditional factor demands and use them to find the long-run cost function for this firm. (b) For these factor prices, derive and graph the firm’s long-run supply curve. (c) Suppose the price of the second input, w2 , rises to $2 per unit. What is the long-run cost curve? Derive and graph the new supply curve. Hint: Because these isoquants are straight lines, cost minimization cannot require tangencies of isoquants and isocost lines. 162 Chapter 9. Theory of the Firm 2: The Long-Run, Multiple-Input Model 1/5 1/5 1/5 6. Consider a production function that uses three inputs: y = x1 x2 x3. Suppose the factor prices are w1 = w2 = w3 = 1. (a) What are the conditional factor demands x1∗ (y), x2∗ (y), and x3∗ (y)? (b) Find the long-run cost function C(y). (c) Find the long-run supply curve y ∗ (p).