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Cost Minimization: The Isoquant-Isocost Approach

Almost anything can be produced in many different ways. You have probably witnessed a simple example of this if you have rented videos from a variety of locations. In some stores the clerk uses an electric bar-code reader to transmit all of the relevant information about both the video and customer to the store's computer. In other shops, the clerk enters the information into the computer by hand--thereby eliminating the bar-code reader, but also taking more of the clerk's time. And in still other locations, usually small stores, even the computer is dispensed with, as the clerk records the information by hand on paper.

As these examples suggest, the feasible methods of producing any given output typically range from those that are very capital-intensive, (such as the system employing the bar-code reader) to those that are quite labor-intensive (as in the store that records all information with pen and paper). Clearly, each of these stores has made its choice of which means of production is best for it. Just as clearly, however, questions immediately arise. On what basis did each store make this decision? How is it possible for different systems such as these to coexist simultaneously? And what could cause a firm to change the means of production it uses? These and other questions will be answered in the course of this chapter.

THE PRODUCTION FUNCTION

If we were to completely describe all of the possible ways that a firm could produce output, we would have a description of the firm's production function. To see what this function might look like, consider Figure 22-1.

Figure 22-1: Isoquants

Along the horizontal axis we measure the amount of labor used by the firm, while along the vertical axis we measure the amount of capital used. Thus, if the firm is using LA units of labor and KA units of capital, we say that the firm is operating at point A. Given these amounts of capital and labor, the firm is producing an output that we shall call Q0. Thus, output is a function of the inputs, capital and labor, and given the amount of the inputs being used, Q0 is the specific amount of output that results.

Imagine now that the firm moves to point B, substituting labor for capital in a way that just happens to keep output constant at Q0. At B, we say that the firm is engaging in a more labor-intensive (less capital-intensive) means of production, because LB > LA and KB < KA. If the firm were to engage in even more substitution of labor for capital while holding output at Q0, we would find it at some point like C, which uses even more labor and less capital.

ISOQUANTS

Although points A, B, and C all involve different combinations of capital and labor, they all share one thing in common: Output is equal to Q0 with each of these combinations. Indeed, there are many other combinations of capital and labor that can be used to produce Q0, combinations that all lie along the line labeled Q0 in Figure 22-1. Because this line represents all of the possible combinations of capital and labor that yield the same level of output, Q0, we call this line an isoquant, a word that means "equal quantity"--iso signifying equal and quant signifying quantity.

Lying outward to the right of Q0 is another isoquant, labeled Q1, for which the level of output Q1 exceeds the level of output Q0. Thus, suppose we compare two points like B' and B. Because B' is to the northeast of B, more of both labor and capital are being used at B', yielding a higher level of output, Q1. A comparison of isoquants Q0 and Q1 reveals the first of three properties shared by isoquants: The isoquants we encounter as we move upward and outward to the northeast represent higher rates of output. Simply put, using more inputs yields more outputs, so that higher isoquants depict the input combinations required to produce greater amounts of output.

The second key feature of these isoquants is that they are downward sloping from left to right. The increase in labor from LA to LB, for example, makes it possible to reduce the amount of capital from KA to KB while still keeping output at Q0. In general, production functions exhibit the property that constant output can be maintained with less capital and more labor (moving down to the right), or with less labor and more capital (moving up to the left). As a result, each isoquant slopes downward from the northwest to the southeast.

The third key property of the isoquants we have drawn is that they are convex with respect to the origin--they bulge downward toward the left. To understand the nature of this convexity, consider what happens as the firm moves from A to B and then from B to C. Clearly, the amount of capital the firm uses is changing. If we let the Greek letter delta () indicate the (small) change in a variable, then the change in capital from A to B is given by K = KB - KA. Similarly, the change in capital from B to C is K = KC - KB.

As depicted, the change in capital from KA to KB is the same as the change in capital from KB to KC. Note, however, what happens to the amount of labor that must be substituted to keep output constant at Q0. To move from A to B, the change in labor (L) is the relatively modest amount LB - LA. But as the firm moves from B to C, holding output at Q0 requires a much larger increment in labor, LC - LB. Thus, as substitution of labor for capital takes place, it requires more and more extra labor (for given reductions in the amount of capital), to keep output constant; as a result, each isoquant gets progressively flatter as the firm moves downward from left to right.

MARGINAL RATE OF SUBSTITUTION

As the firm moves along an isoquant, it decreases the amount of one input that is used and increases the amount of the other input. The slope of the isoquant is given by the change in usage of one input divided by the change in the usage of the other input. In the case of the production process that utilizes capital and labor, we have

Isoquant slope = (K/L)|Q=Qo

This expression for the slope of an isoquant has two elements. First, the ratio of the change in capital to the change in labor (K/L) tells us the value of the slope. The second part of the expression (|Q=Qo) reminds us that we are measuring the slope along a given isoquant, and thus holding output (Q) constant at the specific level Qo. Thus, the slope of an isoquant at each point equals the change in capital divided by the change in labor, holding output constant at Qo. This slope is called the marginal rate of substitution.

DIMINISHING MARGINAL RATE OF SUBSTITUTION

Note how the marginal rate of substitution reflects the properties of the isoquants. Because we are moving along an isoquant, the terms K and L have different signs (one is positive while the other is negative), so that K/L < 0: Isoquants are negatively sloped. Also, notice that as the firm moves down an isoquant to the right, for any given K, the L required to keep output constant must get larger and larger. Thus, the magnitude of the slope (the marginal rate of substitution) gets smaller and smaller: Isoquants are convex toward the origin--they bulge downward to the left.

Because the isoquants are convex, more and more extra labor is required to keep output constant as the firm moves down along an isoquant. This is equivalent to saying that the firm's ability to substitute labor for capital diminishes as more and more substitution takes place. Thus, we say that the firm's production function obeys the law of diminishing marginal rate of substitution, which states that:

The marginal rate of substitution of labor for capital falls as the amount of capital decreases and the amount of labor increases.
MORE ON THE MARGINAL RATE OF SUBSTITUTION

The slope of the isoquant, K/L, is related in an important way to the marginal products of capital and labor. You will recall that the marginal product of labor, MPL, is defined as the change in output that results when the usage of labor is increased by one unit, holding the amount of capital fixed. In a similar vein, the marginal product of capital, MPK is defined as the change in output that results when the usage of capital is increased by one unit, holding the amount of labor fixed.

Consider now Figure 22-2, which illustrates output level Q0 being produced with an input combination shown by point A.

Figure 22-2: The Marginal Rate of Substitution

Consider a change in the input mix from A to C that leaves total output unchanged at Q0. We can think of the move from A to C as being composed of two parts--a move from A to D, followed by a move from D to C. The reduction in capital usage from KA to KD, holding labor constant, causes the move from A to D. The change in output that results is equal to the change in capital (K), multiplied by the marginal product of capital, MPK. As the firm moves from D to C, labor is increased from LA to LC, an amount that is just sufficient to bring output back up to it original level, Q0. This rise in output that occurs from D to C is equal to the change in labor (L), multiplied by the marginal product of labor, MPL.

Thus we have QAD = K • MPK as the firm moves from A to D, and QDC = L • MPL as it moves from D to C. Because the complete move from A to C leaves output unchanged, it must be true that

QAD = -QDC.

Thus,

K • MPK = - L • MPL

Rearranging this equation yields

MPL/MPK = - K/L

Thus we see that the marginal rate of substitution of labor for capital is equal to the ratio of the marginal products of the two factors. Algebraically, we have MRS = MPL/MPK.

ISOCOST LINES

Knowing the firm's production function is not enough information to know what mix of inputs the firm will choose; we must also have information about the prices the firm must pay to obtain those inputs. We utilize this information on input prices in the form of an isocost line, which shows all the combinations of capital and labor that can be purchased for a given total cost. Thus, if we let PK be the price per unit of capital, and PL be the price per unit of labor, then if the firm hires K units of capital and L units of labor, its total costs will be

C = PKK + PLL.

Rearranging yields the isocost equation

K = C/PK - (PL/PK) • L

which is drawn as the isocost line shown in Figure 22-3. Note that the vertical intercept of the isocost line is C/PK, while its slope is -(PL/PK).

Figure 22-3: The Isocost Line

For any given set of input prices, an isocost line can be drawn for each and every possible level of total costs. We have drawn three of these isocost lines in Figure 22-4a, for cost levels C0, C1 and C2.

Figure 22-4: The Isocost Map

Note that C0 < C1 < C2; that is, higher levels of cost are represented by isocost lines that are farther out to the right. This set of isocost lines is called an isocost map; because the lines are drawn holding input prices constant, higher cost isocost lines are those for which larger amounts of the inputs are being used.

Another feature of isocost lines is that their slopes change when relative input prices change. In Figure 22-4b we have drawn two isocost lines that represent the same level costs, with different relative prices. In particular, (PL/PK)* > (PL/PK); note that when labor becomes more expensive relative to capital, the isocost line becomes steeper.

THE LEAST-COST TECHNIQUE

It is now possible to combine isocost lines and isoquant curves to determine the least-cost technique--that combination of inputs that minimizes the total cost of producing a given level of output. This is illustrated in Figure 22-5, where we consider three alternative methods of producing output level Q0, involving input choices at A, B, and C.

Figure 22-5: The Least-Cost Technique

Associated with each of these input bundles is a total cost, shown by the isocost lines labeled C0, C1, and C2. Given the input prices implied by the (common) slope of the isocost lines, input bundle C clearly involves the highest total cost, C2, while bundle A involves the lowest cost C0. Note that there is no isocost line lower than C0 that would enable the firm to reach Q0. Thus, we say input combination A is the least-cost technique for producing Q0.

Note also that at point A, the isocost line and isoquant are just touching--they are tangent at this point, meaning that their slopes are equal. Recall now the values of the slopes of these two functions. At any point, the magnitude of the slope of an isoquant is MRS = MPL/MPK, while the magnitude of the slope of an isocost line is PL/PK. Thus, at a least-cost technique such as A, where the isoquant and isocost line are tangent, we have

MRS = MPL/MPK = PL/PK

Thus, the least-cost combination of inputs occurs at the point where the marginal rate of substitution between the inputs equals the ratio of their prices.

This solution can also be rearranged to yield this expression:

PL/MPL = PK/MPK = Marginal Cost

This simply tells the firm that the least-cost combination of labor and capital is such that equal increments of output should be obtained for the last dollar spent on each of the inputs. (Note that PK is not the price of machine, but the implicit rental cost per unit time period (i.e., the user cost of capital).)

CHANGES IN INPUT PRICES

The firm's choice of how to produce Q0 clearly depends on the prices of capital and labor. To explore the exact nature of this dependence, consider Figure 22-6, which demonstrates the least-cost methods of producing Q0, for two sets of relative input prices.

Figure 22-6: When Input Prices Change

When relative input prices are given by (PL/PK)*, the relevant isocost line is shown by C, which has a slope of -(PL/PK)*, and the firm's input choice is shown by the bundle A. Imagine now that the price of labor falls relative to the price of capital. Thus, if the firm wished to produce Q0, it would face the new isocost line C', with slope -(PL/PK). Note that because the relative price of labor is now lower, the isocost line for producing Q0 is flatter. As a result, the firm chooses input combination B, which involves more labor and less capital. Thus, we may conclude that when the relative price of labor falls, the firm will substitute toward a production technique that utilizes more labor and less capital. As with other goods, the demand curves for factors of production are downward sloping.

WHY DO FIRMS CHOOSE DIFFERENT INPUT COMBINATIONS?

We began this chapter by noting that different video stores often choose different input combinations. We can now inquire into why we observe these differences. Although we obviously don't have enough information to determine how a particular video store chooses exactly the combination of inputs that it does, we do know enough to understand the basic principles involved. First, we know that the video stores will choose a mix of inputs that minimizes the total cost of producing the level of output at which they are operating. This will require them to set the marginal rate of substitution between the inputs equal to the ratio of the input prices. Second, if stores all have access to the same technology, and yet we observe them operating with different mixes of inputs, we know that they must face different input prices. Finally, we know that if the prices of the store's inputs change relative to one another, the store owner will rearrange the use of inputs so that the one that is now cheaper will be used more intensively, while the one that is more expensive will be used less intensively.




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