Determine the marginal products of labor by differentiating the short-run pro- duction functions with respect to labor.

Determine the marginal products of labor by differentiating the short-run pro- duction functions with respect to labor.

124

Production

This chapter looks at an important set of decisions that managers, such as those of American Licorice, have to face. First, the firm must decide how to produce licorice. American Licorice now uses relatively more machines and fewer work- ers than in the past. Second, if a firm wants to expand its output, it must decide how to do that in both the short run and the long run. In the short run, American Licorice can expand output by hiring extra workers or extending the workweek (more shifts per day or more workdays per week) and using extra materials. To expand output even more, American Licorice would have to install more equipment and eventually build a new plant, all of which take time. Third, given its ability to change its output

5 John Nelson, American Licorice Company’s Union City plant manager, invested $10 million in new labor-saving equipment, such as an automated drying machine. This new equipment allowed the company to cut its labor force from 450 to 240 workers.

The factory produces 150,000 pounds of Red Vines licorice a day and about a tenth as much black licorice. The manufacturing process starts by combining flour and corn syrup (for red licorice) or molasses (for black licorice) to form a slurry in giant vats. The temperature is raised to 200° for several hours. Flavors are introduced and a dye is added for red licorice. Next the mixture is drained from the vats into barrels and cooled overnight, after which it is extruded through a machine to form long strands. Other machines punch an air hole through the center of the strands, after which the strands are twisted and cut. Then, the strands are dried in preparation for packaging.

Food manufacturers are usually less affected by recessions than are firms in other industries. Nonetheless during major economic downturns, the demand curve for licorice may shift to the left, and Mr. Nelson must con- sider whether to reduce production by laying off some of his workers. He needs to decide how many workers to lay off. To make this decision, he faces a managerial problem: How much will the output produced per worker rise or fall with each additional layoff?

Labor Productivity During Recessions

Managerial Problem

Hard work never killed anybody, but why take a chance? —Charlie McCarthy

1255.1 Production Functions

level, a firm must determine how large to grow. American Licorice determines how much to invest based on its expectations about future demand and costs.

Firms and the managers who run them perform the fundamental economic func- tion of producing output—the goods and services that consumers want. The main lesson of this chapter is that firms are not black boxes that mysteriously transform inputs (such as labor, capital, and materials) into outputs. Economic theory explains how firms make decisions about production processes, the types of inputs to use, and the volume of output to produce.

In this chapter, we examine five main topics

Main Topics 1. Production Functions: A production function summarizes how a firm converts inputs into outputs using one of possibly many available technologies.

2. Short-Run Production: In the short run, only some inputs can be varied, so the firm changes its output by adjusting its variable inputs.

3. Long-Run Production: In the long run, all factors of production can be varied and the firm has more flexibility than in the short run in how it produces and how it changes its output level.

4. Returns to Scale: How the ratio of output to input varies with the size of the firm is an important factor in determining the size of a firm.

5. Productivity and Technological Change: Technological progress increases pro- ductivity: the amount of output that can be produced with a given amount of inputs.

5.1 Production Functions A firm uses a technology or production process to transform inputs or factors of pro- duction into outputs. Firms use many types of inputs. Most of these inputs can be grouped into three broad categories:

◗ Capital (K). Services provided by long-lived inputs such as land, buildings (such as factories and stores), and equipment (such as machines and trucks)

◗ Labor (L). Human services such as those provided by managers, skilled workers (such as architects, economists, engineers, and plumbers), and less-skilled workers (such as custodians, construction laborers, and assembly-line workers)

◗ Materials (M). Natural resources and raw goods (e.g., oil, water, and wheat) and processed products (e.g., aluminum, plastic, paper, and steel)

The output can be a service, such as an automobile tune-up by a mechanic, or a physi- cal product, such as a computer chip or a potato chip.

Firms can transform inputs into outputs in many different ways. Companies that manufacture candy differ in the skills of their workforce and the amount of equip- ment they use. While all employ a chef, a manager, and some relatively unskilled workers, many candy firms also use skilled technicians and modern equipment. In small candy companies, the relatively unskilled workers shape the candy, decorate it, package it, and box it by hand. In slightly larger firms, relatively unskilled workers may use conveyor belts and other equipment that was invented decades ago. In mod- ern, large-scale plants, the relatively unskilled laborers work with robots and other state-of-the-art machines, which are maintained by skilled technicians. Before decid- ing which production process to use, a firm needs to consider its various options.

126 CHAPTER 5 Production

The various ways in which inputs can be transformed into output are summarized in the production function: the relationship between the quantities of inputs used and the maximum quantity of output that can be produced, given current knowledge about technology and organization. The production function for a firm that uses only labor and capital is

q = f(L, K), (5.1)

where q units of output (such as wrapped candy bars) are produced using L units of labor services (such as hours of work by assembly-line workers) and K units of capital (such as the number of conveyor belts).

The production function shows only the maximum amount of output that can be produced from given levels of labor and capital, because the production function includes only efficient production processes. A firm engages in efficient production (achieves technological efficiency) if it cannot produce its current level of output with fewer inputs, given existing knowledge about technology and the organization of production. A profit-maximizing firm is not interested in production processes that are inefficient and waste inputs: Firms do not want to use two workers to do a job that can be done as well by one worker.

A firm can more easily adjust its inputs in the long run than in the short run. Typically, a firm can vary the amount of materials and of relatively unskilled labor it uses comparatively quickly. However, it needs more time to find and hire skilled workers, order new equipment, or build a new manufacturing plant.

The more time a firm has to adjust its inputs, the more factors of production it can alter. The short run is a period of time so brief that at least one factor of produc- tion cannot be varied practically. A factor that cannot be varied practically in the short run is called a fixed input. In contrast, a variable input is a factor of produc- tion whose quantity can be changed readily by the firm during the relevant time period. The long run is a lengthy enough period of time that all relevant inputs can be varied. In the long run, there are no fixed inputs—all factors of production are variable inputs.

Suppose that a painting company’s customers all want the paint job on their homes to be finished by the end of the day. The firm could complete these projects on time if it had one fewer job. To complete all the jobs, it needs to use more inputs. Even if it wanted to do so, the firm does not have time to buy or rent an extra truck and buy another compressor to run a power sprayer; these inputs are fixed in the short run. To get the work done that afternoon, the firm uses the company’s one truck to pick up and drop off temporary workers, each equipped with only a brush and paint, at the last job. In the long run, however, the firm can adjust all its inputs. If the firm wants to paint more houses every day, it hires more full-time workers, gets a second truck, purchases more compressors to run the power sprayers, and uses a computer to keep track of all its projects.

How long it takes for all inputs to be variable depends on the factors a firm uses. For a janitorial service whose only major input is workers, the short run is a brief period of time. In contrast, an automobile manufacturer may need several years to build a new manufacturing plant or to design and construct a new type of machine. A pistachio farmer needs the better part of a decade before newly planted trees yield a substantial crop of nuts.

For many firms, materials and often labor are variable inputs over a month. How- ever, labor is not always a variable input. Finding additional highly skilled workers may take substantial time. Similarly, capital may be a variable or fixed input. A firm can rent small capital assets (such as trucks or office furniture) quickly, but it may

1275.2 Short-Run Production

take the firm years to obtain larger capital assets (buildings and large, specialized pieces of equipment).

To illustrate the greater flexibility that a firm has in the long run than in the short run, we examine the production function in Equation 5.1, in which output is a func- tion of only labor and capital. We look first at the short-run and then at the long-run production processes.

5.2 Short-Run Production The short run is a period in which there is at least one fixed input. Focusing on a production process in which capital and labor are the only inputs, we assume that capital is the fixed input and that labor is variable. The firm can therefore increase output only by increasing the amount of labor it uses. In the short run, the firm’s production function, Equation 5.1, becomes

q = f(L, K), (5.2)

where q is output, L is the amount of labor, and K is the firm’s fixed amount of capital.

To illustrate the short-run production process, we consider a firm that assembles computers for a manufacturing firm that supplies it with the necessary parts, such as computer chips and disk drives. If the assembly firm wants to increase its output in the short run, it cannot do so by increasing its capital (eight workbenches fully equipped with tools, electronic probes, and other equipment for testing computers). However, it can increase output in the short run by hiring extra workers or paying current workers extra to work overtime.

The Total Product Function The exact relationship between output or total product and labor can be illustrated by using a particular function, Equation 5.2, a table, or a figure. Table 5.1 shows the relationship between output and labor when a firm’s capital is fixed. The first col- umn lists the fixed amount of capital: eight fully equipped workbenches. The second column shows how much of the variable input, labor, the firm uses. In this example, the labor input is measured by the number of workers, as all work the same number of hours. Total output—the number of computers assembled in a day—is listed in the third column. As the number of workers increases, total output first increases and then decreases.

With zero workers, no computers are assembled. One worker with access to the firm’s equipment assembles five computers in a day. As the number of workers increases, so does output: 1 worker assembles 5 computers in a day, 2 workers assemble 18, 3 workers assemble 36, and so forth. The maximum number of com- puters that can be assembled with the capital on hand, however, is limited to 110 per day. That maximum can be produced with 10 or 11 workers. If the firm were to use 12 or more workers, the workers would get in each other’s way and production would be lower than with 11 workers. The dashed line in the table indicates that a firm would not use more than 11 workers, because it would be inefficient to do so. We can show how extra workers affect the total product by using two additional concepts: the marginal product of labor and the average product of labor.

128 CHAPTER 5 Production

The Marginal Product of Labor Before deciding whether to employ more labor, a manager wants to determine how much an extra unit of labor, ΔL = 1, will increase output, Δq. That is, the manager wants to know the marginal product of labor (MPL): the change in total output resulting from using an extra unit of labor, holding other factors (capital) constant. If output changes by Δq when the amount of labor increases by ΔL, the change in output per unit of labor is

MPL = Δq ΔL

.

As Table 5.1 shows, if the number of workers increases from 1 to 2, ΔL = 1, output rises by Δq = 13 = 18 – 5, so the marginal product of labor is 13.

Capital, K Labor, L

Output, Total Product of

Labor q

Marginal Product of Labor,

MPL = Δq/ΔL

Average Product of Labor,

APL = q/L

8 0 0

8 1 5 5 5 8 2 18 13 9 8 3 36 18 12 8 4 56 20 14 8 5 75 19 15 8 6 90 15 15 8 7 98 8 14 8 8 104 6 13 8 9 108 4 12 8 10 110 2 11 8 11 110 0 10

8 12 108 -2 9 8 13 104 -4 8

Labor is measured in workers per day. Capital is fixed at eight fully equipped workbenches.

TABLE 5.1 Total Product, Marginal Product, and Average Product of Labor with Fixed Capital

Using Calculus The short-run production function, q = f(L, K ) can be written as solely a function of L because capital is fixed: q = g(L). The calculus definition of the marginal product of labor is the derivative of this production function with respect to labor: MPL = dg(L)/dL.

In the long run, when both labor and capital are free to vary, the marginal product of labor is the partial derivative of the production function, Equation 5.1, q = f(L, K), with respect to labor:

MPL = 0q 0L

= 0 f(L,K)

0L .

Calculating the Marginal Product of Labor

1295.2 Short-Run Production

The Average Product of Labor Before hiring extra workers, a manager may also want to know whether output will rise in proportion to this extra labor. To answer this question, the firm determines how extra labor affects the average product of labor (APL): the ratio of output to the amount of labor used to produce that output,

APL = q

L .

Table 5.1 shows that 9 workers can assemble 108 computers a day, so the average product of labor for 9 workers is 12(= 108/9) computers a day. Ten workers can assemble 110 computers in a day, so the average product of labor for 10 workers is 11(= 110/10) computers. Thus, increasing the labor force from 9 to 10 workers low- ers the average product per worker.

Graphing the Product Curves Figure 5.1 and Table 5.1 show how output (total product), the average product of labor, and the marginal product of labor vary with the number of workers. (The figures are smooth curves because the firm can hire a “fraction of a worker” by

1Above, we defined the marginal product as the extra output due to a discrete change in labor, such as an additional worker or an extra hour of work. In contrast, the calculus definition of the marginal product—the partial derivative—is the rate of change of output with respect to the labor for a very small (infinitesimal) change in labor As a result, the numerical calculation of marginal products can differ slightly if derivatives rather than discrete changes are used.

Q&A 5.1 For a linear production function q = f(L, K) = 2L + K and a multiplicative production function q = LK, what are the short-run production functions given that capital is fixed at K = 100? What are the marginal products of labor for these short-run pro- duction functions?

Answer

1. Obtain the short-run production functions by setting K = 100. The short-run linear production function is q = 2L + 100 and the short-run multiplicative function is q = L * 100 = 100L.

2. Determine the marginal products of labor by differentiating the short-run pro- duction functions with respect to labor. The marginal product of labor is MPL = d(2L + 100)/dL = 2 for the short-run linear production function and MPL = d(100L)/dL = 100 for the short-run multiplicative production function.

We use the symbol 0q/0L instead of dq/dL to represent a partial derivative.1 We use partial derivatives when a function has more than one explanatory variable. Here, q is a function of both labor, L, and capital, K. To obtain a partial derivative with respect to one variable, say L, we differentiate as usual where we treat the other variables (here just K) as constants.

130 CHAPTER 5 Production

employing a worker for a fraction of a day.) The curve in panel a of Figure 5.1 shows how a change in labor affects the total product, which is the amount of output that can be produced by a given amount of labor. Output rises with labor until it reaches its maximum of 110 computers at 11 workers, point B; with extra workers, the number of computers assembled falls.

Panel b of the figure shows how the average product of labor and marginal product of labor vary with the number of workers. We can line up the figures in panels a and b vertically because the units along the horizontal axes of both figures,

O ut

pu t,

q, U

ni ts

p er

d ay

A

B

1160 L, Workers per day

Marginal product, MPL

Average product, APL

A P

L, M

P L

110

90

(a)

a

b

1160

L, Workers per day

20

15

(b)

Total product

Slope of this line = 90/6 = 15

4

FIGURE 5.1 Production Relationships with Variable Labor

(a) The total product of labor curve shows how many computers, q, can be assembled with eight fully equipped workbenches and a varying number of workers, L, who work eight-hour days (see columns 2 and 3 in Table 5.1). Where extra workers reduce the number of computers assembled (beyond point B), the total product curve is a dashed line, which indicates that such production is inef- ficient and is thus not part of the production function. The

slope of the line from the origin to point A is the average product of labor for six workers. (b) Where the marginal product of labor (MPL = Δq/ΔL, column 4 of Table 5.1) curve is above the average product of labor (APL = q/L, column 5 of Table 5.1) curve, the APL must rise. Similarly, if the MPL curve is below the APL curve, the APL must fall. Thus, the MPL curve intersects the APL curve at the peak of the APL curve, point b, where the firm uses 6 workers.

1315.2 Short-Run Production

the number of workers per day, are the same. The vertical axes differ, however. The vertical axis is total product in panel a and the average or marginal product of labor—a measure of output per unit of labor—in panel b.

The Effect of Extra Labor. In most production processes, the average product of labor first rises and then falls as labor increases. One reason the APL curve initially rises in Figure 5.1 is that it helps to have more than two hands when assembling a computer. One worker holds a part in place while another one bolts it down. As a result, output increases more than in proportion to labor, so the average product of labor rises. Doubling the number of workers from one to two more than doubles the output from 5 to 18 and causes the average product of labor to rise from 5 to 9, as Table 5.1 shows.

Similarly, output may initially rise more than in proportion to labor because of greater specialization of activities. With greater specialization, workers are assigned to tasks at which they are particularly adept, and time is saved by not having work- ers move from task to task.

As the number of workers rises further, however, output may not increase by as much per worker because workers might have to wait to use a particular piece of equipment or get in each other’s way. In Figure 5.1, as the number of workers exceeds 6, total output increases less than in proportion to labor, so the average product falls.

If more than 11 workers are used, the total product curve falls with each extra worker as the crowding of workers gets worse. Because that much labor is not effi- cient, that section of the curve is drawn with a dashed line to indicate that it is not part of the production function, which includes only efficient combinations of labor and capital. Similarly, the dashed portions of the average and marginal product curves are irrelevant because no firm would hire additional workers if doing so meant that output would fall.

Relationships Among Product Curves. The three curves are geometrically related. First we use panel b to illustrate the relationship between the average and marginal product of labor curves. Then we use panels a and b to show the relation- ship between the total product of labor curve and the other two curves.

An extra hour of work increases the average product of labor if the marginal product of labor exceeds the average product. Similarly, if an extra hour of work generates less extra output than the average, the average product falls. Therefore, the average product rises with extra labor if the marginal product curve is above the average product curve, and the average product falls if the marginal product is below the average product curve. Consequently, the average product curve reaches its peak, point a in panel b of Figure 5.1, where the marginal product and average product are equal: where the curves cross.

The geometric relationship between the total product curve and the average and marginal product curves is illustrated in panels a and b of Figure 5.1. We can deter- mine the average product of labor using the total product of labor curve. The average product of labor for L workers equals the slope of a straight line from the origin to a point on the total product of labor curve for L workers in panel a. The slope of this line equals output divided by the number of workers, which is the definition of the average product of labor. For example, the slope of the straight line drawn from the origin to point A (L = 6, q = 90) is 15, which equals the “rise” of q = 90 divided by

132 CHAPTER 5 Production

the “run” of L = 6. As panel b shows, the average product of labor for 6 workers at point a is 15.

The marginal product of labor also has a geometric relationship to the total prod- uct curve. The slope of the total product curve at a given point equals the marginal product of labor. That is, the marginal product of labor equals the slope of a straight line that is tangent to the total output curve at a given point. For example, at point B in panel a where there are 11 workers, the line tangent to the total product curve is flat so the marginal product of labor is zero (point b in panel b): A little extra labor has no effect on output. The total product curve is upward sloping when there are fewer than 11 workers, so the marginal product of labor is positive. If the firm is fool- ish enough to hire more than 11 workers, the total product curve slopes downward (dashed line), so the MPL is negative: Extra workers lower output.

When there are 6 workers, the average product of labor equals the marginal product of labor. The reason is that the line from the origin to point A in panel a is tangent to the total product curve, so the slope of that line, 15, is the marginal product of labor and the average product of labor at point a in panel b, which is the peak of the APL curve.

The Law of Diminishing Marginal Returns Next to supply equals demand, the most commonly used economic phrase claims that there are diminishing marginal returns: If a firm keeps increasing an input, holding all other inputs and technology constant, the corresponding increases in output will eventually become smaller (diminish). As most observed production functions have this property, this pattern is often called the law of diminishing marginal returns. This law determines the shape of the marginal product of labor curves: if only one input is increased, the marginal product of that input will diminish eventually.

In Table 5.1, if the firm goes from 1 to 2 workers, the marginal product of labor of the second worker is 13. If 1 or 2 more workers are used, the marginal product rises: The marginal product for the third worker is 18, and the marginal product for the fourth worker is 20. However, if the firm increases the number of workers beyond 4, the marginal product falls: The marginal product of a fifth worker is 19, and that of the sixth worker is 15. Beyond 4 workers, each extra worker adds less and less extra output, so the total product of labor curve rises by smaller increments. At 11 workers, the marginal product is zero. This diminishing return to extra labor might be due to crowding, as workers get in each other’s way. As the amount of labor used grows large enough, the marginal product curve approaches zero and the total product curve becomes nearly flat.

Instead of referring to the law of diminishing marginal returns, some people talk about the law of diminishing returns—leaving out the word marginal. Making this change invites confusion as it is not clear if the phrase refers to marginal returns or total returns. If as labor increases the marginal returns fall but remain positive, the total return rises. In panel b of Figure 5.1, marginal returns start to diminish when the labor input exceeds 4 but total returns rise, as panel 1 shows, until the labor input exceeds 11, where the marginal returns become negative.

A second common misinterpretation of this law is to claim that marginal prod- ucts must fall as we increase an input without requiring that technology and other inputs stay constant. If we increase labor while simultaneously increasing other factors or adopting superior technologies, the marginal product of labor can continue to rise.

1335.2 Short-Run Production

Mini-Case In 1798, Thomas Malthus—a clergyman and professor of political economy— predicted that (unchecked) population would grow more rapidly than food pro- duction because the quantity of land was fixed. The problem, he believed, was that the fixed amount of land would lead to a diminishing marginal product of

labor, so output would rise less than in proportion to the increase in farm workers, possibly leading to widespread starvation and other “natural” checks on population such as disease and vio- lent conflict. Brander and Taylor (1998) argue that such a disaster might have occurred on Easter Island about 500 years ago.

Today the earth supports a population about seven times as large as when Malthus made his predictions. Why haven’t most of us starved to death? The answer is that a typical agricultural worker produces vastly more food today than was possible when Malthus was alive. The output of a U.S. farm worker today is more than double that of an average worker just 50 years ago. We do not see diminishing marginal returns to labor because the production function has changed due to substan- tial technological progress in agriculture and because farmers make greater use of other inputs such as fertilizers and capital.

Two hundred years ago, most of the world’s population had to work in agriculture to feed themselves. Today, less than 2% of the U.S. population works in agriculture. Over the last cen- tury, food production grew substantially faster than the popu- lation in most developed countries. For example, since World War II, the U.S. population doubled but U.S. food production tripled.

Of course, the risk of starvation is more severe in low-income countries than in the United States. Fortunately, agricultural pro- duction in these nations increased dramatically during the second half of the twentieth century, saving an estimated billion lives. This increased production was due to a set of innovations called the Green Revolution, which included development of drought- and insect-resistant crop varieties, improved irrigation, better use of fertilizer and pesticides, and improved equipment.

Perhaps the most important single contributor to the Green Revolution was U.S. agronomist Norman Borlaug, who won the Nobel Peace Prize in 1970. However, as he noted in his Nobel Prize speech, superior science is not the complete answer to pre-

venting starvation. A sound economic system and a stable political environment are also needed.

Economic and political failures such as the breakdown of economic pro- duction and distribution systems due to wars have caused per capita food production to fall, resulting in widespread starvation and malnutrition in sub-Saharan Africa. According to the United Nations Food and Agriculture Organization, about 27% of the population of sub-Saharan Africa suffer from significant undernourishment along with more than 17% of the population in South Asia (India, Pakistan, Bangladesh, and nearby countries)—harming over 500 million people in these two regions alone.

Malthus and the Green Revolution

134 CHAPTER 5 Production

5.3 Long-Run Production We started our analysis of production functions by looking at a short-run production function in which one input, capital, was fixed, and the other, labor, was variable. In the long run, however, both of these inputs are variable. With both factors variable, a firm can usually produce a given level of output by using a great deal of labor and very little capital, a great deal of capital and very little labor, or moderate amounts of both. That is, the firm can substitute one input for another while continuing to produce the same level of output, in much the same way that a consumer can main- tain a given level of utility by substituting one good for another.

Typically, a firm can produce in a number of different ways, some of which require more labor than others. For example, a lumberyard can produce 200 planks an hour with 10 workers using hand saws, with 4 workers using handheld power saws, or with 2 workers using bench power saws.

We illustrate a firm’s ability to substitute between inputs in Table 5.2, which shows the amount of output per day the firm produces with various combinations of labor per day and capital per day. The labor inputs are along the top of the table, and the capital inputs are in the first column. The table shows four combinations of labor and capital that the firm can use to produce 24 units of output (in bold numbers): The firm may employ (a) 1 worker and 6 units of capital, (b) 2 workers and 3 units of capital, (c) 3 workers and 2 units of capital, or (d) 6 workers and 1 unit of capital.

Isoquants These four combinations of labor and capital are labeled a, b, c, and d on the “q = 24” curve in Figure 5.2. We call such a curve an isoquant, which is a curve that shows the efficient combinations of labor and capital that can produce the same (iso) level of output (quantity). The isoquant shows the smallest amounts of inputs that will produce a given amount of output. That is, if a firm reduced either input, it could not produce as much output. If the production function is q = f(L, K), then the equation for an isoquant where output is held constant at q is

q = f(L, K).

An isoquant shows the flexibility that a firm has in producing a given level of out- put. Figure 5.2 shows three isoquants corresponding to three levels of output. These isoquants are smooth curves because the firm can use fractional units of each input.

Labor, L

Capital, K 1 2 3 4 5 6

1 10 14 17 20 22 24

2 14 20 24 28 32 35 3 17 24 30 35 39 42 4 20 28 35 40 45 49 5 22 32 39 45 50 55 6 24 35 42 49 55 60

TABLE 5.2 Output Produced with Two Variable Inputs

1355.3 Long-Run Production

We can use these isoquants to illustrate what happens in the short run when capi- tal is fixed and only labor varies. As Table 5.2 shows, if capital is constant at 2 units, 1 worker produces 14 units of output (point e in Figure 5.2), 3 workers produce 24 units (point c), and 6 workers produce 35 units (point f ). Thus, if the firm holds one factor constant and varies another factor, it moves from one isoquant to another. In contrast, if the firm increases one input while lowering the other appropriately, the firm stays on a single isoquant.

Properties of Isoquants. Isoquants have most of the same properties as indifference curves. The biggest difference between indifference curves and isoquants is that an isoquant holds quantity constant, whereas an indifference curve holds utility constant. We now discuss three major properties of isoquants. Most of these properties result from firms producing efficiently.

First, the farther an isoquant is from the origin, the greater the level of output. That is, the more inputs a firm uses, the more output it gets if it produces efficiently. At point e in Figure 5.2, the firm is producing 14 units of output with 1 worker and 2 units of capital. If the firm holds capital constant and adds 2 more workers, it pro- duces at point c. Point c must be on an isoquant with a higher level of output—here, 24 units—if the firm is producing efficiently and not wasting the extra labor.

Second, isoquants do not cross. Such intersections are inconsistent with the require- ment that the firm always produces efficiently. For example, if the q = 15 and q = 20 isoquants crossed, the firm could produce at either output level with the same com- bination of labor and capital. The firm must be producing inefficiently if it produces q = 15 when it could produce q = 20. So that labor-capital combination should not lie on the q = 15 isoquant, which should include only efficient combinations of inputs. Thus, efficiency requires that isoquants do not cross.

Third, isoquants slope downward. If an isoquant sloped upward, the firm could produce the same level of output with relatively few inputs or relatively many

K , U

ni ts

o f c

ap ita

l p er

d ay

e

b

a

d

fc

63210 L, Workers per day

6

3

2

1

q = 14

q = 24

q = 35

FIGURE 5.2 A Family of Isoquants

These isoquants show the combi- nations of labor and capital that produce 14, 24, or 35 units of output, q. Isoquants farther from the origin correspond to higher levels of output. Points a, b, c, and d are various combinations of labor and capital the firm can use to produce q = 24 units of output. If the firm holds capital constant at 2 and increases labor from 1 (point e on the q = 14 isoquant) to 3 (c), its output increases to q = 24 isoquant. If the firm then increases labor to 6 (f ), its output rises to q = 35.

136 CHAPTER 5 Production

inputs. Producing with relatively many inputs would be inefficient. Consequently, because isoquants show only efficient production, an upward-sloping isoquant is impossible. Virtually the same argument can be used to show that isoquants must be thin.

Shapes of Isoquants. The curvature of an isoquant shows how readily a firm can substitute one input for another. The two extreme cases are production processes in which inputs are perfect substitutes or in which they cannot be substituted for each other.

If the inputs are perfect substitutes, each isoquant is a straight line. Suppose either potatoes from Maine, x, or potatoes from Idaho, y, both of which are measured in pounds per day, can be used to produce potato salad, q, measured in pounds. The production function is

q = x + y.

One pound of potato salad can be produced by using 1 pound of Idaho potatoes and no Maine potatoes, 1 pound of Maine potatoes and no Idaho potatoes, or any combination that adds up to 1 pound in total. Panel a of Figure 5.3 shows the q = 1, 2, and 3 isoquants. These isoquants are straight lines with a slope of -1 because we need to use an extra pound of Maine potatoes for every pound fewer of Idaho potatoes used.2

Sometimes it is impossible to substitute one input for the other: Inputs must be used in fixed proportions. Such a production function is called a fixed-proportions production function. For example, the inputs needed to produce 12-ounce boxes of

2The isoquant for q = 1 pound of potato salad is 1 = x + y, or y = 1 – x. This equation shows that the isoquant is a straight line with a slope of -1.

y, Id

ah o

po ta

to es

p er

d ay

(a)

x, Maine potatoes per day

q = 3

q = 2

q = 1

B ox

es p

er d

ay

(b)

Cereal per day

q = 3

q = 2

q = 1

45° line q = 1

K , C

ap ita

l p er

u ni

t o f t

im e

(c)

L, Labor per unit of time

FIGURE 5.3 Substitutability of Inputs

(a) If inputs are perfect substitutes, each isoquant is a straight line. (b) If the inputs cannot be substituted at all, the isoquants are right angles (the dashed lines show that the isoquants would be right angles if we included

inefficient production). (c) Typical isoquants lie between the extreme cases of straight lines and right angles. Along a curved isoquant, the ability to substitute one input for another varies.

1375.3 Long-Run Production

cereal are cereal (in 12-ounce units per day) and cardboard boxes (boxes per day). If the firm has one unit of cereal and one box, it can produce one box of cereal. If it has one unit of cereal and two boxes, it can still make only one box of cereal. Thus, in panel b, the only efficient points of production are the large dots along the 45° line.3

Dashed lines show that the isoquants would be right angles if isoquants could include inefficient production processes.

Other production processes allow imperfect substitution between inputs. These processes have isoquants that are convex to the origin (so the middle of the isoquant is closer to the origin than it would be if the isoquant were a straight line). They do not have the same slope at every point, unlike the straight-line isoquants. Most iso- quants are smooth, slope downward, curve away from the origin, and lie between the extreme cases of straight lines (perfect substitutes) and right angles (fixed pro- portions), as panel c illustrates.

3This fixed-proportions production function is the minimum of g and b, q = min(g, b), where g is the number of 12-ounce measures of cereal, b is the number of boxes used in a day, and the min function means “the minimum number of g or b.” For example, if g is 4 and b is 3, q is 3.

Mini-Case We can show why isoquants curve away from the origin by deriving an isoquant for semiconductor integrated circuits (ICs, or “chips”)—the “brains” of com- puters and other electronic devices. Semiconductor manufacturers buy silicon wafers and then use labor and capital to produce the chips.

A chip consists of multiple layers of silicon wafers. A key step in the produc- tion process is to line up these layers. Three alternative alignment technologies are available, using different combinations of labor and capital. In the least capital- intensive technology, employees use machines called aligners, which require work- ers to look through microscopes and line up the layers by hand. A worker using an aligner can produce 25 ten-layer chips per day.

A second, more capital-intensive technology uses machines called steppers. The stepper aligns the layers automatically. This technology requires less labor: A single worker can produce 50 ten-layer chips per day.

A third, even more capital-intensive technology combines steppers with wafer-handling equipment, which further reduces the amount of labor needed. A single worker can produce 100 ten-layer chips per day. In the diagram the vertical axis measures the amount of capital used. An aligner represents less capital than a basic stepper, which in turn is less capital than a stepper with wafer-handling capabilities.

All three technologies use labor and capital in fixed proportions. To produce 200 chips takes 8 workers and 8 aligners, 3 workers and 6 basic steppers, or 1 worker and 4 steppers with wafer-handling capabilities. The accompanying graph shows the three right-angle isoquants corresponding to each of these three technologies.

Some plants employ a combination of these technologies, so that some workers use one type of machine while others use different types. By doing so, the plant can produce using intermediate combinations of labor and capital, as the solid- line, kinked isoquant illustrates. The firm does not use a combination of the aligner and the wafer-handling stepper technologies because those combinations

A Semiconductor Isoquant

138 CHAPTER 5 Production

are less efficient than using the basic stepper: The line connecting the aligner and wafer-handling stepper technologies is farther from the origin than the lines between those technolo- gies and the basic stepper technology.

New processes are con- stantly being invented. As they are introduced, the isoquant will have more and more kinks (one for each new process) and will begin to resemble the smooth, convex isoquants we’ve been drawing.

K , U

ni ts

o f c

ap ita

l p er

d ay

Aligner

Stepper

Wafer-handling stepper

200 ten-layer chips per day isoquant

81 3

L, Workers per day

0

Substituting Inputs The slope of an isoquant shows the ability of a firm to replace one input with another while holding output constant. Figure 5.4 illustrates this substitution using an estimated isoquant for a U.S. printing firm, which uses labor, L, and capital, K, to print its output, q.4 The isoquant shows various combinations of L and K that the firm can use to produce 10 units of output.

The firm can produce 10 units of output using the combination of inputs at a or b. At point a, the firm uses 2 workers and 16 units of capital. The firm could produce the same amount of output using ΔK = -6 fewer units of capital if it used one more worker, ΔL = 1, point b. If we drew a straight line from a to b, its slope would be ΔK/ΔL = -6. Thus, this slope tells us how many fewer units of capital (6) the firm can use if it hires one more worker.5

The slope of an isoquant is called the marginal rate of technical substitution (MRTS):

MRTS = change in capital

change in labor =

ΔK ΔL

.

The marginal rate of technical substitution tells us how many units of capital the firm can replace with an extra unit of labor while holding output constant. Because isoquants slope downward, the MRTS is negative. That is, the firm can produce a given level of output by substituting more capital for less labor (or vice versa).

4This isoquant for q = 10 is based on the estimated production function q = 2.35L0.5K0.4 (Hsieh, 1995), where the unit of labor, L, is a worker-day. Because capital, K, includes various types of machines, and output, q, reflects different types of printed matter, their units cannot be described by any common terms. This production function is an example of a Cobb-Douglas production function. 5The slope of the isoquant at a point equals the slope of a straight line that is tangent to the isoquant at that point. Thus, the straight line between two nearby points on an isoquant has nearly the same slope as that of the isoquant.

1395.3 Long-Run Production

Substitutability of Inputs Varies Along an Isoquant. The MRTS varies along a curved isoquant, as in Figure 5.4. If the firm is initially at point a and it hires one more worker, the firm can give up 6 units of capital and yet remain on the same isoquant (at point b), so the MRTS is -6. If the firm hires another worker, the firm can reduce its capital by 3 units and stay on the same isoquant, moving from point b to c, so the MRTS is -3. This decline in the MRTS (in absolute value) along an isoquant as the firm increases labor illustrates a diminishing MRTS. The more labor and less capital the firm has, the harder it is to replace remaining capital with labor and the flatter the isoquant becomes.

In the special case in which isoquants are straight lines, isoquants do not exhibit diminishing marginal rates of technical substitution because neither input becomes more valuable in the production process: The inputs remain perfect substitutes. Q&A 5.2 illustrates this result.

K , U

ni ts

o f c

ap ita

l p er

d ay

L, Workers per day

4 5

7

10

16 a

b

c d

e

q = 10

ΔK = –6

ΔL = 1

0 1

1

1

1

2 3

–3

–2

–1

4 5 6 7 8 9 10

FIGURE 5.4 How the Marginal Rate of Technical Substitution Varies Along an Isoquant

Moving from point a to b, a U.S. printing firm (Hsieh, 1995) can produce the same amount of output, q = 10, using six fewer units of capital, ΔK = -6, if it uses one more worker, ΔL = 1. Thus, its MRTS = ΔK/ΔL = -6. Moving from point b to c, its MRTS is -3. If it adds yet another worker, moving from c to d, its MRTS is -2. Finally, if it moves from d to e, its MRTS is -1. Thus, because the isoquant is convex to the origin, it exhibits a diminish- ing marginal rate of technical substitution. That is, each extra worker allows the firm to reduce capital by a smaller amount as the ratio of capital to labor falls.

Q&A 5.2 A manufacturer produces a container of potato salad using one pound of Idaho pota- toes, one pound of Maine potatoes, or one pound of a mixture of the two types of potatoes. Does the marginal rate of technical substitution vary along the isoquant? What is the MRTS at each point along the isoquant?

Answer

1. Determine the shape of the isoquant. As panel a of Figure 5.3 illustrates, the potato salad isoquants are straight lines because the two types of potatoes are perfect substitutes.

2. On the basis of the shape, conclude whether the MRTS is constant along the isoquant. Because the isoquant is a straight line, the slope is the same at every point, so the MRTS is constant.

3. Determine the MRTS at each point. Earlier, we showed that the slope of this isoquant was –1, so the MRTS is -1 at each point along the isoquant. That is, because the two inputs are perfect substitutes, 1 pound of Idaho potatoes can be replaced by 1 pound of Maine potatoes.

140 CHAPTER 5 Production

Substitutability of Inputs and Marginal Products. The marginal rate of technical substitution is equal to the ratio of marginal products. Because the marginal product of labor, MPL = Δq/ΔL, is the increase in output per extra unit of labor, if the firm hires ΔL more workers, its output increases by MPL * ΔL. For example, if the MPL is 2 and the firm hires one extra worker, its output rises by 2 units.

A decrease in capital alone causes output to fall by MPK * ΔK, where MPK = Δq/ΔK is the marginal product of capital—the output the firm loses from decreasing capital by one unit, holding all other factors fixed. To keep output constant, Δq = 0, this fall in output from reducing capital must exactly equal the increase in output from increasing labor:

(MPL * ΔL) + (MPK * ΔK) = 0.

Rearranging these terms, we find that

– MPL MPK

= ΔK ΔL

= MRTS. (5.3)

Thus the ratio of marginal products equals the MRTS (in absolute value). We can use Equation 5.3 to explain why marginal rates of technical substitution

diminish as we move to the right along the isoquant in Figure 5.4. As we replace capital with labor (move down and to the right along the isoquant), the marginal product of capital increases—when there are few pieces of equipment per worker, each remaining piece is more useful—and the marginal product of labor falls, so the MRTS = -MPL/MPK falls in absolute value.

Cobb-Douglas Production Functions. We can illustrate how to determine the MRTS for a particular production function, the Cobb-Douglas production function. It is named after its inventors, Charles W. Cobb, a mathematician, and Paul H. Douglas, an economist and U.S. Senator. Through empirical studies, economists have found that the production processes in a very large number of industries can be accurately summarized by the Cobb-Douglas production function, which is

q = ALαKβ, (5.4)

where A, α, and β are all positive constants. We used regression analysis to estimate the production function for the

BlackBerry smartphone, which was the first major smartphone.6 The estimated Cobb-Douglas production function is Q = 2.83L1.52K0.82. That is, A = 2.83, α = 1.52, and β = 0.82.

The constants α and β determine the relationships between the marginal and average products of labor and capital (as we show in the following section, Using Calculus). The marginal product of labor is α times the average product of labor, APL = q/L. That is, MPL = αq/L = αAPL. By dividing both sides of the expression by APL, we find that α equals the ratio of the marginal product of labor to the aver- age product of labor: α = MPL/APL. Similarly, the marginal product of capital is MPK = βq/K = βAPK, and β = MPK/APK.

6The data are from the annual and quarterly reports from 1999 through 2009 of Research In Motion (renamed BlackBerry in 2013), the company that manufactures BlackBerry phones.

1415.4 Returns to Scale

5.4 Returns to Scale So far, we have examined the effects of increasing one input while holding the other input constant (shifting from one isoquant to another) or decreasing the other input by an offsetting amount (moving along an isoquant). We now turn to the question of how much output changes if a firm increases all its inputs proportionately. The answer helps a firm determine its scale or size in the long run.

In the long run, a firm can increase its output by building a second plant and staffing it with the same number of workers as in the first one. Whether the firm chooses to do so depends in part on whether its output increases less than in proportion, in proportion, or more than in proportion to its inputs.

Constant, Increasing, and Decreasing Returns to Scale If, when all inputs are increased by a certain proportion, output increases by that same proportion, the production function is said to exhibit constant returns to scale (CRS). A firm’s production process, q = f(L, K), has constant returns to scale if, when

The marginal rate of technical substitution along an isoquant that holds output fixed at q is

MRTS = – MPL MPK

= – αq/L βq/K

= – α β

K L

. (5.5)

For example, given the BlackBerry’s production function, Q = 2.83L1.52K0.82, its MPL = αAPL = 1.52APL, and its MRTS = -(1.52/0.82)K/L ≈ -1.85K/L. The MRTS tells the firm’s managers the rate at which they can substitute capital for labor without reducing output.

Using Calculus To obtain the marginal product of labor for the Cobb-Douglas production func- tion, Equation 5.4, q = ALαKβ, we partially differentiate the production function with respect to labor, holding capital fixed:

MPL = 0q 0L

= αALα – 1Kβ = α ALαKβ

L = α

q

L .

We obtain the last equality by substituting q = ALαKβ. Similarly, we can derive the marginal product of capital by partially differentiating the production func- tion with respect to K:

MPK = 0q 0K

= βALαKβ – 1 = β ALαKβ

K = β

q

K .

Cobb-Douglas Marginal Products

142 CHAPTER 5 Production

the firm doubles its inputs—by, for example, building an identical second plant and using the same amount of labor and equipment as in the first plant—it doubles its output:

f(2L, 2K) = 2f(L, K) = 2q.

We can check whether the potato salad production function has constant returns to scale. If a firm uses x1 pounds of Idaho potatoes and y1 pounds of Maine potatoes, it produces q1 = x1 + y1 pounds of potato salad. If it doubles both inputs, using x2 = 2×1 Idaho and y2 = 2y1 Maine potatoes, it doubles its output:

q2 = x2 + y2 = 2×1 + 2y1 = 2(x1 + y1) = 2q1.

Thus, the potato salad production function exhibits constant returns to scale. If output rises more than in proportion to an equal

proportional increase in all inputs, the production function is said to exhibit increasing returns to scale (IRS). A technology exhibits increasing returns to scale if doubling inputs more than doubles the output:

f(2L, 2K) 7 2f(L, K) = 2q.

Why might a production function have increas- ing returns to scale? One reason is that, although it could build a copy of its original small factory and double its output, the firm might be able to more than double its output by building a single large plant, thereby allowing for greater specialization of labor or capital. In the two smaller plants, work- ers have to perform many unrelated tasks such as operating, maintaining, and fixing the machines they use. In the large plant, some workers may spe- cialize in maintaining and fixing machines, thereby increasing efficiency. Similarly, a firm may use specialized equipment in a large plant but not in a small one.

If output rises less than in proportion to an equal proportional increase in all inputs, the production function exhibits decreasing returns to scale (DRS). A tech- nology exhibits decreasing returns to scale if doubling inputs causes output to rise less than in proportion:

f(2L, 2K) 6 2f(L, K) = 2q.

One reason for decreasing returns to scale is that the difficulty of organizing, coordinating, and integrating activities increases with firm size. An owner may be able to manage one plant well but may have trouble running two plants. In some sense, the decreasing returns to scale stemming from the owner’s difficulties in running a larger firm may reflect our failure to take into account some fac- tor such as management skills in our production function. If a firm increases various inputs but does not increase the management input in proportion, the “decreasing returns to scale” may occur because one of the inputs to production, management skills, is fixed. Another reason is that large teams of workers may not function as well as small teams, in which each individual takes greater personal responsibility.

1435.4 Returns to Scale

Q&A 5.3 Under what conditions does a Cobb-Douglas production function, Equation 5.4, q = ALαKβ, exhibit decreasing, constant, or increasing returns to scale?

Answer

1. Show how output changes if both inputs are doubled. If the firm initially uses L and K amounts of inputs it produces q1 = ALαKβ. After the firm doubles the amount of both labor and capital it uses, it produces

q2 = A(2L)α(2K)β = 2α + βALαKβ = 2α + βq1. (5.6)

That is, q2 is 2α + β times q1. If we define γ = α + β, then Equation 5.6 tells us that

q2 = 2γq1. (5.7)

Thus, if the inputs double, output increases by 2γ. 2. Give a rule for determining the returns to scale. If γ = 1, we know from Equation

5.7 that q2 = 21q1 = 2q1. That is, output doubles when the inputs double, so the Cobb-Douglas production function has constant returns to scale. If γ 6 1, then q2 = 2γq1 6 2q1 because 2γ 6 2 if γ 6 1. That is, when input doubles, output increases less than in proportion, so this Cobb-Douglas production function exhibits decreasing returns to scale. Finally, the Cobb-Douglas pro- duction function has increasing returns to scale if γ 7 1 so that q2 7 2q1. Thus, the rule for determining returns to scale for a Cobb-Douglas production function is that the returns to scale are decreasing if γ 6 1, constant if γ = 1, and increasing if γ 7 1.

Comment: One interpretation of γ is that, as all inputs increase by 1%, output increases by γ%. Thus, for example, if γ = 1, a 1% increase in all inputs increases output by 1%.

Mini-Case Increasing, constant, and decreasing returns to scale are commonly observed. The table shows estimates of Cobb-Douglas production functions and returns to scale for various U.S. manufacturing industries (based on Hsieh, 1995).Returns to

Scale in U.S. Manufacturing

Labor, α Capital, β Returns to Scale,

γ=α + β

Decreasing Returns to Scale

Tobacco products 0.18 0.33 0.51

Food and kindred products 0.43 0.48 0.91

Transportation equipment 0.44 0.48 0.92

Constant Returns to Scale

Apparel and other textile products 0.70 0.31 1.01

Furniture and fixtures 0.62 0.40 1.02

Electronics and other electric equipment 0.49 0.53 1.02

Increasing Returns to Scale

Paper and allied products 0.44 0.65 1.09 Petroleum and coal products 0.30 0.88 1.18 Primary metal 0.51 0.73 1.24

144 CHAPTER 5 Production

The table shows that the estimated returns to scale measure γ for a tobacco firm is γ = 0.51: A 1% increase in the inputs causes output to rise by 0.51%. Because output rises less than in proportion to the inputs, the tobacco production function exhibits decreasing returns to scale. In contrast, firms that manufacture primary metals have increasing returns to scale production functions, in which a 1% increase in all inputs causes output to rise by 1.24%.

The accompanying graphs use isoquants to illustrate the returns to scale for the electronics, tobacco, and primary metal firms. We measure the units of labor, capital, and output so that, for all three firms, 100 units of labor and 100 units of capital produce 100 units of output on the q = 100 isoquant

in the three panels. For the constant returns to scale electronics firm, panel a, if both labor and capital are doubled from 100 to 200 units, output doubles to 200 (= 100 * 21, multiplying the original output by the rate of increase using Equation 5.7).

That same doubling of inputs causes output to rise to only 142 (≈ 100 * 20.51) for the tobacco firm, panel b. Because output rises less than in proportion to inputs, the production function has decreasing returns to scale. If the primary metal firm doubles its inputs, panel c, its out- put more than doubles, to 236 (≈ 100 * 21.24), so the production function has increasing returns to scale.

These graphs illustrate that the spacing of the isoquant determines the returns to scale. The closer together the q = 100 and q = 200 isoquants, the greater the returns to scale.

K , U

ni ts

o f c

ap ita

l p er

y ea

r

L, Units of labor per year

300

400

500

100

200

q = 100

q = 200

q = 236

0 100 200 300 400 500

(c) Primary Metal: Increasing Returns to Scale

K , U

ni ts

o f c

ap ita

l p er

y ea

r

L, Units of labor per year

300

400

500

100

200

q = 100

q = 200

0 100 200 300 400 500

(a) Electronics and Equipment: Constant Returns to Scale

K , U

ni ts

o f c

ap ita

l p er

y ea

r

L, Units of labor per year

300

400

500

100

200

q = 100

q = 142

q = 200

0 100 200 300 400 500

(b) Tobacco: Decreasing Returns to Scale

1455.4 Returns to Scale

Varying Returns to Scale When the production function is Cobb-Douglas, the returns to scale are the same at all levels of output. However, in other industries, a production function’s returns to scale may vary as the output level changes. A firm might, for example, have increasing returns to scale at low levels of output, constant returns to scale for some range of output, and decreasing returns to scale at higher levels of output.

Many production functions have increasing returns to scale for small amounts of output, constant returns for moderate amounts of output, and decreasing returns for large amounts of output. When a firm is small, increasing labor and capital allows for gains from cooperation between workers and greater specialization of workers and equipment—returns to specialization—so there are increasing returns to scale. As the firm grows, returns to scale are eventually exhausted. There are no more returns to specialization, so the production process has constant returns to scale. If the firm continues to grow, the owner starts having difficulty managing everyone, so the firm suffers from decreasing returns to scale.

We show such a pattern in Figure 5.5. Again, the spacing of the isoquants reflects the returns to scale. Initially, the firm has one worker and one piece of equipment, point a, and produces 1 unit of output on the q = 1 isoquant. If the firm doubles its inputs, it produces at b, where L = 2 and K = 2, which lies on the dashed line through the origin and point a. Output more than doubles to q = 3, so the produc- tion function exhibits increasing returns to scale in this range. Another doubling of inputs to c causes output to double to 6 units, so the production function has con- stant returns to scale in this range. Another doubling of inputs to d causes output to increase by only a third, to q = 8, so the production function has decreasing returns to scale in this range.

K , U

ni ts

o f c

ap ita

l p er

y ea

r

41 2

a

b

d

c

a → b: Increasing returns to scale

b → c: Constant returns to scale

c → d: Decreasing returns to scale

8 L, Work hours per year

4

2

1

0

8

q = 8

q = 6

q = 3 q = 1

FIGURE 5.5 Varying Scale Economies

The production function that corresponds to these isoquants exhibits varying returns to scale. Initially, the firm uses one worker and one unit of capital, point a. Point b has double the amount of labor and capital as does a. Similarly, c has double the inputs of b, and d has double the inputs of c. All these points lie along the dashed 45° line. The first time the inputs are dou- bled, a to b, output more than doubles from q = 1 to q = 3, so the production function has increasing returns to scale. The next doubling, b to c, causes a proportionate increase in out- put, constant returns to scale. At the last doubling, from c to d, the production function exhibits decreasing returns to scale.

146 CHAPTER 5 Production

5.5 Productivity and Technological Change Progress was all right. Only it went on too long. —James Thurber

Because firms may use different technologies and different methods of organizing production, the amount of output that one firm produces from a given amount of inputs may differ from that produced by another firm. Further, after a positive tech- nological or managerial innovation, a firm can produce more from a given amount of inputs than it could previously.

Relative Productivity Firms are not necessarily equally productive, in the sense that one firm might be able to produce more than another from a given amount of inputs. A firm may be more productive than others if its manager knows a better way to organize production or if it is the only firm with access to a new invention. Union-mandated work rules, government regulations, or other institutional restrictions that affect only some firms might also lower the relative productivity of those firms.

The industrial revolution of the late eighteenth century took advantage of econo- mies of scale to revolutionize production. Since then, the pursuit of scale econo- mies has driven firms to become larger and larger in most industries. However, a number of recent inventions may cause savvy managers to reverse this trend. Entrepreneurs and managers should consider whether new technologies make small-scale production economically attractive.

One of the most striking of these inventions is three-dimensional (3D) printing, which may greatly reduce the advantages of long production runs. An employee gives instructions (essentially a blueprint) to a 3D printer, presses Print, and the machine builds the object from the ground up, either by depositing material from a nozzle, or by selectively solidifying a thin layer of plastic or metal dust using drops of glue or a tightly focused beam. The final product can be a machine part, a bicycle frame, or a work of art.

This technology changes the relative positions of isoquants, potentially reduc- ing dramatically the extent of increasing returns to scale and allowing small entrepreneurs to compete effectively with larger firms. It may also allow greater customization at little additional cost.

Currently these machines work only with certain plastics, resins, and metals, and have a precision of around a tenth of a millimeter. Costs have fallen to the point where manufacturing using 3D printers is cost effective, and new uses seem virtually unlimited. For example, in 2012, scientists at the University of Glasgow demonstrated that 3D printing can be used to create existing and new chemi- cal compounds and, in 2013, a Dutch architect announced plans for the first 3D printed building. Moreover, 3D printing may lead to increased innovation and specialization. Any shape that you can design on a computer can be printed.

Managers should use this technology to experiment. Managers can produce small initial runs to determine the size of the market and consumers’ acceptance of the prod- uct. Based on information from early adopters, managers can determine if the market warrants further production and quickly modify designs to meet end-users’ desires.

Small Is Beautiful

Managerial Implication

1475.5 Productivity and Technological Change

Differences in productivity across markets may be due to differences in the degree of competition. In competitive markets, in which many firms can enter and exit the market easily, less productive firms lose money and are driven out of business, so the firms that are actually producing are equally productive. In a less competitive oligopoly market, with few firms and no possibility of entry by new firms, a less productive firm may be able to survive, so firms with varying levels of productivity are observed.

Mini-Case Prior to the mid-1990s, over 90% of the electricity was produced and sold to consumers by investor-owned utility monopolies that were subject to govern- ment regulation of the prices they charged. Beginning in the mid-1990s, some states mandated that electric production be restructured. In such a state, the util- ity monopoly was forced to sell its electric generation plants to several other firms. These new firms sell the electricity they generate to the utility monopoly, which delivers the electricity to final consumers. Because they expected these new electric generator firms to compete with each other, state legislators hoped that this increased competition would result in greater production efficiency.

Fabrizio, Rose, and Wolfram (2007) found that, in anticipation of greater competition, the generation plant operators in states that had restructured had reduced their labor and nonfuel expenses by 3% to 5% (holding output con- stant) relative to investor-owned utility monopoly plants in states that did not restructure. When compared to plants run by government-owned or coopera- tively owned utility monopolies that were not exposed to restructuring incen- tives, these gains were even greater: 6% in labor and 13% in nonfuel expenses.

U.S. Electric Generation Efficiency

Innovation In its production process, a firm tries to use the best available technological and managerial knowledge. An advance in knowledge that allows more output to be produced with the same level of inputs is called technological progress. The invention of new products is a form of technological innovation. The use of robotic arms increases the number of automobiles produced with a given amount of labor and raw materials. Better management or organization of the pro- duction process similarly allows the firm to produce more output from given levels of inputs.

Technological Progress. A technological innovation changes the production process. Last year a firm produced

q1 = f(L, K)

units of output using L units of labor services and K units of capital service. Due to a new invention that the firm uses, this year’s production function differs from last year’s, so the firm produces 10% more output with the same inputs:

q2 = 1.1 f(L, K).

Flath (2011) estimated the annual rate of technical innovation in Japanese manu- facturing firms to be 0.91% for electric copper, 0.87% for medicine, 0.33% for steel pipes and tubes, 0.19% for cement, and 0.08% for beer.

148 CHAPTER 5 Production

This type of technological progress reflects neutral technical change, in which more output is produced using the same ratio of inputs. However, technological progress may be nonneutral. Rather than increasing output for a given mix of inputs, techno- logical progress could be capital saving, where the firm can produce the same level of output as before using less capital and the same amount of other inputs. The Ameri- can Licorice Company’s automated drying machinery in the Managerial Problem is an example of labor-saving technological progress.

Alternatively, technological progress may be labor saving. Basker (2012) found that the introduction of barcode scanners in grocery stores increased the average product of labor by 4.5% on average across stores. Amazon bought Kiva Systems in 2012 with the intention of using its robots to move items in Amazon’s warehouses, partially replacing workers. Other robots help doctors perform surgery quicker and reduce patients’ recovery times.

Organizational Change. Organizational changes may also alter the produc- tion function and increase the amount of output produced by a given amount of inputs. In the early 1900s, Henry Ford revolutionized mass production of automo- biles through two organizational innovations. First, he introduced interchangeable parts, which cut the time required to install parts because workers no longer had to file or machine individually made parts to get them to fit.

Second, Ford introduced a conveyor belt and an assembly line to his produc- tion process. Before this change, workers walked around the car, and each worker performed many assembly activities. In Ford’s plant, each worker specialized in a single activity such as attaching the right rear fender to the chassis. A conveyor belt moved the car at a constant speed from worker to worker along the assembly line. Because his workers gained proficiency from specializing in only a few activi- ties and because the conveyor belts reduced the number of movements workers had to make, Ford could produce more automobiles with the same number of workers. These innovations reduced the ratio of labor to capital used. In 1908, the Ford Model T sold for $850, when rival vehicles sold for $2,000. By the early 1920s, Ford had increased production from fewer than a thousand cars per year to two million per year.

Mini-Case In 2009, the automotive world was stunned when India’s new Tata Motors started selling the Nano, its tiny, fuel-efficient four-passenger car. With a base price of less than $2,500, it is by far the world’s least expensive car. The next cheapest car in India, the Maruti 800, sold for about $4,800.

The Nano’s dramatically lower price is not the result of amazing new inven- tions; it is due to organizational innovations that led to simplifications and the use of less expensive materials and procedures. Although Tata Motors filed for 34 patents related to the design of the Nano (compared to the roughly 280 patents awarded to General Motors annually), most of these patents are for mun- dane items such as the two-cylinder engine’s balance shaft and the configuration of the transmission gears.

Instead of relying on innovations, Tata reorganized both production and dis- tribution to lower costs. It reduced manufacturing costs at every stage of the process with a no-frills design, decreased vehicle weight, and made other major production improvements.

Tata Nano’s Technical and Organizational Innovations

1495.5 Productivity and Technological Change

The Nano has a single windshield wiper, one side-view mirror, no power steering, a simplified door-opening lever, three nuts on the wheels instead of the customary four, and a trunk that does not open from the outside—it is accessed by folding down the rear seats. The Nano has smaller overall dimensions than the Maruti, but about 20% more seating capacity because of design decisions, such as putting the wheels at the extreme edges of the car. The Nano is much lighter than comparable models due to the reduced amount of steel, the use of lightweight steel, and the use of aluminum in the engine. The ribbed roof structure is not only a style element but also a strength structure,

which is necessary because the design uses thin-gauge sheet metal. Because the engine is in the rear, the driveshaft doesn’t need complex joints as in a front- engine car with front-wheel drive. To cut costs further, the company reduced the number of tools needed to make the components and thereby increased the life of the dies used by three times the norm. In consultation with their suppliers, Tata’s engineers determined how many useful parts the design required, which helped them identify functions that could be integrated in parts.

Tata’s plant can produce 250,000 Nanos per year and benefits from economies of scale. However, Tata’s major organizational innovation was its open distribu- tion and remote assembly. The Nano’s modular design enables an experienced mechanic to assemble the car in a workshop. Therefore, Tata Motors can dis- tribute a complete knock-down (CKD) kit to be assembled and serviced by local assembly hubs and entrepreneurs closer to consumers. The cost of transporting these kits, produced at a central manufacturing plant, is charged directly to the customer. This approach is expected to speed up the distribution process, par- ticularly in the more remote locations of India. The car has been a great success, selling more than 8,500 cars in May 2012.

Labor Productivity During Recessions

Managerial Solution

During a recession, a manager of the American Licorice Company has to reduce output and decides to lay off workers. Will the firm’s labor productivity— average product of labor—go up and improve the firm’s situation or go down and harm it?

Layoffs have the positive effect of freeing up machines to be used by remain- ing workers. However, if layoffs force the remaining workers to perform a wide variety of tasks, the firm will lose the benefits from specialization. When there are many workers, the advantage of freeing up machines is important and increased multitasking is unlikely to be a problem. When there are only a few workers, freeing up more machines does not help much (some machines might stand idle some of the time), while multitasking becomes a more serious problem.

Holding capital constant, a change in the number of workers affects a firm’s average product of labor. Labor productivity could rise or fall. For example, in panel b of Figure 5.1, the average product of labor rises up to 15 workers per day

150 CHAPTER 5 Production

SUMMARY 1. Production Functions. A production function

summarizes how a firm combines inputs such as labor, capital, and materials to produce output using the current state of knowledge about tech- nology and management. A production function shows how much output can be produced effi- ciently from various levels of inputs. A firm pro- duces efficiently if it cannot produce its current level of output with less of any one input, holding other inputs constant.

2. Short-Run Production. A firm can vary all its inputs in the long run but only some of them in the short run. In the short run, a firm cannot adjust the quantity of some inputs, such as capital. The firm varies its output in the short run by adjust- ing its variable inputs, such as labor. If all factors are fixed except labor, and a firm that was using very little labor increases its use of labor, its output may rise more than in proportion to the increase in labor because of greater specialization of workers.

and then falls as the number of workers increases. The average product of labor falls if the firm has 6 or fewer workers and lays 1 off, but rises if the firm initially has 7 to 11 workers and lays off a worker.

For some production functions, layoffs always raise labor productivity because the APL curve is downward sloping everywhere. For such a production function, the positive effect of freeing up capital always dominates any negative effect of layoffs on average product. For example, layoffs raise the APL for any Cobb-Douglas production function, q = ALαKβ, where α is less than one.7 All the estimated production functions listed in the “Returns to Scale in U.S. Manu- facturing” Mini-Case have this property.

Let’s return to our licorice manufacturer. According to Hsieh (1995), the Cobb-Douglas production function for food and kindred product plants is q = AL0.43K0.48, so α = 0.43 is less than one and the APL curve slopes down- ward at every quantity. We can illustrate how much the APL rises with a layoff for this particular production function. If A = 1 and L = K = 10 initially, then the firm’s output is q = 100.43 * 100.48 ≈ 8.13, and its average product of labor is APL = q/L ≈ 8.13/10 = 0.813. If the number of workers is reduced by one, then output falls to q = 90.43 * 100.48 ≈ 7.77, and the average product of labor rises to APL ≈ 7.77/9 ≈ 0.863. That is, a 10% reduction in labor causes output to fall by 4.4%, but causes the average product of labor to rise by 6.2%. The firm’s output falls less than 10% because each remaining worker is more productive.

Until recently, most large Japanese firms did not lay off workers during down- turns. Thus, in contrast to U.S. firms, their average product of labor fell during recessions because their output fell while labor remained constant. Similarly, European firms have 30% less employment volatility over time than do U.S. firms, at least in part because European firms that fire workers are subject to a tax (Veracierto, 2008). Consequently, with other factors held constant in the short run, recessions might be more damaging to the profit of a Japanese or European firm than to the profit of a comparable U.S. firm. However, retaining good work- ers over short-run downturns might be a good long-run policy.

7For this Cobb-Douglas production function, the average product of labor is APL = q/L = ALαKβ/L = ALα – 1Kβ. By partially differentiating this expression with respect to labor, we find that the change in the APL as the amount of labor rises is 0APL/0L = (α – 1)ALα – 2Kβ, which is negative if α 6 1. Thus, as labor falls, the average product of labor rises.

151Questions

Eventually, however, as more workers are hired, the workers get in each other’s way or must wait to share equipment, so output increases by smaller and smaller amounts. This latter phenomenon is described by the law of diminishing marginal returns: The marginal product of an input—the extra output from the last unit of input—eventually decreases as more of that input is used, holding other inputs fixed.

3. Long-Run Production. In the long run, when all inputs are variable, firms can substitute between inputs. An isoquant shows the combina- tions of inputs that can produce a given level of output. The marginal rate of technical substitution is the absolute value of the slope of the isoquant and indicates how easily the firm can substitute one factor of production for another. Usually, the more of one input the firm uses, the more difficult it is to substitute that input for another input. That is, there are diminishing marginal rates of technical substitution as the firm uses more of one input.

4. Returns to Scale. When a firm increases all inputs in proportion and its output increases by

QUESTIONS All exercises are available on MyEconLab; * = answer at the back of this book.

*2.3. Suppose that the production function is q = L0.75K0.25. (Hint: See Q&A 5.1.)

a. What is the average product of labor, holding capital fixed at K?

b. What is the marginal product of labor?

c. How is the marginal product of labor related to the average product of labor?

2.4. In the short run, a firm cannot vary its capital, K = 2, but can vary its labor, L. It produces output q. Explain why the firm will or will not experience diminishing marginal returns to labor in the short run if its production function is

a. q = 10L + K. b. q = L0.5K0.5.

2.5. Based on the information in the Mini-Case “Mal- thus and the Green Revolution,” how did the aver- age product of labor in food production change over time?

3. Long-Run Production 3.1. Why must isoquants be thin?

3.2. According to Card (2009), (a) workers with less than a high school education are perfect substitutes for those with a high school education, (b) “high school

1. Production Functions 1.1. What are the main types of capital, labor, and mate-

rials used to produce licorice?

*1.2. Suppose that for the production function q = f(L, K), if L = 3 and K = 5 then q = 10. Is it possible that L = 3 and K = 6 also yields q = 10 for this produc- tion function? Why or why not?

1.3. Consider Boeing (a producer of jet aircraft), General Mills (a producer of breakfast cereals), and Wacky Jack’s (which claims to be the largest U.S. provider of singing telegrams). For which of these firms is the short run the longest period of time? For which is the long run the shortest? Explain.

2. Short-Run Production *2.1. If each extra worker produces an extra unit of output,

how do the total product of labor, average product of labor, and marginal product of labor vary with labor? Plot these curves in a graph similar to Figure 5.1.

2.2. Each extra worker produces an extra unit of out- put up to six workers. As more workers are added, no additional output is produced. Draw the total product of labor, average product of labor, and marginal product of labor curves in a graph similar to Figure 5.1.

the same proportion, the production process is said to exhibit constant returns to scale. If output increases less than in proportion to the increase in inputs, the production process has decreas- ing returns to scale; if it increases more than in proportion, it has increasing returns to scale. All three types of returns to scale are commonly seen in actual industries. Many production processes exhibit first increasing, then constant, and finally decreasing returns to scale as the size of the firm increases.

5. Productivity and Technological Change. Even if all firms in an industry produce efficiently given what they know and the institutional and other constraints they face, some firms may be more productive than others, producing more output from a given bundle of inputs. More pro- ductive firms may have access to managerial or technical innovations not available to its rivals. Technological progress allows a firm to produce a given level of output using less inputs than it did previously. Technological progress changes the production function.

152 CHAPTER 5 Production

equivalent” and “college equivalent” workers are imperfect substitutes, and (c) within education groups, immigrants and natives are imperfect sub- stitutes. For each of these comparisons, draw the iso- quants for a production function that uses two types of workers. For example, in part (a), production is a function of workers with a high school diploma and workers with less education.

*3.3. To produce a recorded DVD, q = 1, a firm uses one blank disk, D = 1, and the services of a recording machine, M = 1, for one hour. (Hint: See Q&A 5.2.)

a. Draw the isoquants for this production function and explain its shape.

b. What is the MRTS at each point along the iso- quant corresponding to q = 100?

c. Draw the total product, average product, and marginal product of labor curves (you will probably want to use two diagrams) for this production function.

3.4. The production function at Ginko’s Copy Shop is q = 1,000 * min(L, 3K), where q is the number of copies per hour, L is the number of workers, and K is the number of copy machines. As an example, if L = 4 and K = 1, then the minimum of L and 3K, min(L, 3K) = 3, and q = 3,000.

a. Draw the isoquants for this production function.

b. Draw the total product, average product, and marginal product of labor curves for this pro- duction function for some fixed level of capital.

3.5. Using the figure in the Mini-Case “A Semiconductor Isoquant,” show that as the firm employs additional fixed-proportion technologies, the firm’s overall iso- quant approaches a smooth curve similar to that in panel c of Figure 5.3.

*3.6. A laundry cleans white clothes using the produc- tion function q = B + 2G, where B is the number of cups of Clorox bleach, G is the number of cups of a generic bleach that is half as potent, and q is the bas- ketfuls of clothes that are cleaned. Draw an isoquant for one basketful of clothes. What is the marginal product of B? What is the marginal rate of technical substitution at each point on an isoquant?

*3.7. At L = 4, K = 4, the marginal product of labor is 2 and the marginal product of capital is 3. What is the marginal rate of technical substitution (MRTS)?

4. Returns to Scale 4.1. To speed relief to isolated South Asian communities

that were devastated by the December 2004 tsunami, the U.S. government doubled the number of heli- copters from 45 to 90 in early 2005. Navy admiral Thomas Fargo, head of the U.S. Pacific Command,

was asked if doubling the number of helicopters would “produce twice as much [relief].” He pre- dicted, “Maybe pretty close to twice as much.” (Vicky O’Hara, All Things Considered, National Public Radio, January 4, 2005, www.npr.org/dmg/ dmg.php?prgCode=ATC&showDate=04-Jan- 2005&segNum=10&NPRMediaPref=WM&ge tAd=1.) Identify the outputs and inputs and describe the production process. Is the admiral discussing a production process with nearly constant returns to scale, or is he referring to another property of the production process?

*4.2. The production function for the automotive and parts industry is q = L0.27K0.16M0.61, where M is energy and materials (based on Klein, 2003). What kind of returns to scale does this production function exhibit? What is the marginal product of materials?

4.3. Under what conditions do the following production functions exhibit decreasing, constant, or increasing returns to scale? (Hint: See Q&A 5.3.)

a. q = L + K. b. q = L + LαKβ + K.

4.4. A production function has the property that f(xL, xK) = x2f(L, K) for any positive value of x. What kind of returns to scale does this production function exhibit? If the firm doubles L and K, show that the marginal product of labor and the marginal product of capital also double.

*4.5. Show in a diagram that a production function can have diminishing marginal returns to a factor and constant returns to scale.

4.6. Is it possible that a firm’s production function exhibits increasing returns to scale while exhibit- ing diminishing marginal productivity of each of its inputs? To answer this question, calculate the mar- ginal productivities of capital and labor for the pro- duction of electronics and equipment, tobacco, and primary metal using the information listed in the “Returns to Scale in U.S. Manufacturing” Mini-Case.

*4.7. The BlackBerry production function indicated in the text is Q = 2.83L1.52K0.82. Epple et al. (2010) estimate that the production function for U.S. housing is q = 1.38L0.144M0.856, where L is land and M is an aggregate of all other mobile, nonland factors, which we call materials. Haskel and Sadun (2012) estimate the production function for U.K. supermarkets is Q = L0.23K0.10M0.66, where L is labor, K is capital, and M is materials. What kind of returns to scale do these production functions exhibit?

4.8. Michelle’s business produces ceramic cups using labor, clay, and a kiln. She produces cups using a fixed proportion of labor and clay, but regardless

153Questions

of how many cups she produces, she uses only one kiln. She can manufacture 25 cups a day with one worker and 35 with two workers. Does her produc- tion process illustrate decreasing returns to scale or a diminishing marginal product of labor? What is the likely explanation for why output doesn’t increase proportionately with the number of workers?

4.9. Does it follow that because we observe that the aver- age product of labor is higher for Firm 1 than for Firm 2, Firm 1 is more productive in the sense that it can produce more output from a given amount of inputs? Why?

5. Productivity and Technological Change *5.1. Firm 1 and Firm 2 use the same type of production

function, but Firm 1 is only 90% as productive as Firm 2. That is, the production function of Firm 2 is q2 = f(L, K), and the production function of Firm 1 is q1 = 0.9f(L, K). At a particular level of inputs, how does the marginal product of labor differ between the firms?

5.2. In a manufacturing plant, workers use a special- ized machine to produce belts. A new machine is invented that is laborsaving. With the new machine, the firm can use fewer workers and still produce the same number of belts as it did using the old machine. In the long run, both labor and capital (the machine) are variable. From what you know, what is the effect of this invention on the APL, MPL, and returns to scale? If you require more information to answer this question, specify what you need to know.

5.3. Until the mid-eighteenth century when spinning became mechanized, cotton was an expensive and relatively unimportant textile (Virginia Postrel, “What Separates Rich Nations from Poor Nations?” New York Times, January 1, 2004). Where it used to take a hand-spinner 50,000 hours to hand-spin 100 pounds of cotton, an operator of a 1760s-era hand- operated cotton mule-spinning machine could pro- duce 100 pounds of stronger thread in 300 hours. When the self-acting mule spinner automated the process after 1825, the time dropped to 135 hours, and cotton became an inexpensive, common cloth. In a figure, show how these technological changes affected isoquants. Explain briefly.

6. Managerial Problem 6.1. If a firm lays off workers during a recession, how

will the firm’s marginal product of labor change? (Hint: See Figure 5.1.)

*6.2. During recessions, U.S. firms lay off a larger propor- tion of their workers than Japanese firms do. (It has been claimed that Japanese firms continue to pro- duce at high levels and store the output or sell it at relatively low prices during the recession.) Assum- ing that the production function remains unchanged over a period that is long enough to include many recessions and expansions, would you expect the average product of labor to be higher in Japan or the United States? Why?

7. Spreadsheet Exercises 7.1. Labor, L, and capital, K, are the only inputs in each

of the following production functions:

a. q1 = (L + K)2. b. q2 = 12L + 2K22. c. q3 = 120 + 2L + 2K22.

For each production function, use a spreadsheet to find the output associated with the following out- put combinations: L = 2, K = 2; L = 4, K = 4; and L = 8, K = 8. Determine whether each production function exhibits increasing returns to scale, decreasing returns to scale, constant returns to scale, or variable returns to scale over this range.

7.2. The Green Revolution (see the Mini-Case “Malthus and the Green Revolution”) was based in part on extensive experimentation. The following data illustrates the relationship between nitrogen fertilizer (in pounds of nitrogen) and the output of a particular type of wheat (in bushels). Each observation is based on one acre of land and all other relevant inputs to production (such as water, labor, and capital) are held constant. The fertilizer levels are 20, 40, 60, 80, 100, 120, 140, and 160, and the associated output levels are 47, 86, 107, 131, 136, 148, 149, and 142.

a. Use Excel to estimate the short-run production function showing the relationship between fertilizer input and output. (Hint: As described in Chapter 3, use the Trendline option to regress output on fertilizer input. Try a linear function and try a quadratic func- tion and determine which function fits the data better.)

b. Does fertilizer exhibit the law of diminishing marginal returns? What is the largest amount of fertilizer that should ever be used, even if it is free?

154

Costs An economist is a person who, when invited to give a talk at a banquet, tells the audience there’s no such thing as a free lunch.

6 A manager of a semiconductor manufacturing firm, who can choose from many different production technologies, must determine whether the firm should use the same technology in its foreign plant that it uses in its domestic plant. U.S. semiconductor manufacturing firms have been moving much of their production abroad since 1961, when Fairchild Semiconductor built a plant in Hong Kong. According to the Semiconductor Industry Association, world-wide semiconductor billings from the Americas dropped from 66% in 1976, to 34% in 1998, and to 18% by April 2013. Firms are moving their production abroad

because of lower taxes, lower labor costs, and capital grant benefits. Capital grants are funds provided by a foreign government to firms to induce them to produce in that country. Such grants can reduce the cost of owning and operating an overseas semiconductor fabrication facility by as much as 25% compared to the costs of a U.S.-based plant.

The semiconductor manufacturer can produce a chip using sophisticated equipment and relatively few workers or many workers and less complex equipment. In the United States, firms use a relatively capital-intensive technology, because doing so minimizes their cost of producing a given level of output. Will that same technology be cost minimizing if they move their production abroad?

Technology Choice at Home Versus Abroad

Managerial Problem

A firm uses a two-step procedure to determine the most efficient way to produce a certain amount of output. First, the firm determines which production processes are technically efficient so that it can produce the desired level of output without any wasted or unnecessary inputs. As we saw in Chapter 5, the firm uses engineering and other information to determine its production function, which summarizes the many technically efficient production processes available. A firm’s production function shows the maximum output that can be produced with any specified combination of inputs or factors of production, such as labor, capital, energy, and materials.

The firm’s second step is to pick from these technically efficient production processes the one that is also economically efficient, minimizing the cost of producing a specified output level.1 To determine which process minimizes its cost

1Similarly, economically efficient production implies that the quantity of output is maximized for any given level of cost.

1556.1 The Nature of Costs

of production, the firm uses information about the production function and the cost of inputs.

Managers and economists need to understand the relationship between costs of inputs and production to determine the least costly way to produce. By minimizing the cost of producing a given level of output, a firm can increase its profit.

In this chapter, we examine five main topics

Main Topics 1. The Nature of Costs: When considering the cost of a proposed action, a good manager takes account of foregone alternative opportunities. 2. Short-Run Costs: To minimize costs in the short run, the firm adjusts its variable

factors of production (such as labor), but cannot adjust its fixed factors (such as capital).

3. Long-Run Costs: In the long run, all inputs are variable because the firm has the time to adjust all its factors of production.

4. The Learning Curve: A firm might be able to lower its costs of production over time as its workers and managers learn from experience about how best to produce a particular product.

5. The Costs of Producing Multiple Goods: If the firm produces several goods simultaneously, the cost of each may depend on the quantities of all the goods produced.

6.1 The Nature of Costs Too caustic? To hell with the costs, we’ll make the picture anyway. —Samuel Goldwyn

Making sound managerial decisions about investment and production requires information about the associated costs. Some cost information is provided in legally required financial accounting statements. However, such statements do not provide sufficient cost information for good decision making. Financial accounting state- ments correctly measure costs for tax purposes and to meet other legal requirements, but good managerial decisions require a different perspective on costs.

To produce a particular amount of output, a firm incurs costs for the required inputs such as labor, capital, energy, and materials. A firm’s manager (or accountant) determines the cost of labor, energy, and materials by multiplying the price of the factor times the number of units used. If workers earn $20 per hour and the firm hires 100 hours of labor per day, then the firm’s cost of labor is $20 * 100 = $2,000 per day. The manager can easily calculate these explicit costs, which are its direct, out-of-pocket payments for inputs to its production process during a given time period. While calculating explicit costs is straightforward, some costs are implicit in that they reflect only a foregone opportunity rather than explicit, current expenditure. Properly taking account of foregone opportunities requires particularly careful attention when deal- ing with durable capital goods, as past expenditures for an input may be irrelevant to current cost calculations if that input has no current, alternative use.

Opportunity Costs A fundamental principle of managerial decision making is that managers should focus on opportunity costs. The opportunity cost of a resource is the value of the best alternative use of that resource. Explicit costs are opportunity costs. If a firm

156 CHAPTER 6 Costs

purchases an input in a market and uses that input immediately, the input’s opportunity cost is the amount the firm pays for it, the market price. After all, if the firm did not use the input in its production process, its best alternative would be to sell it to someone else at the market price. The concept of an opportunity cost becomes particularly useful when the firm uses an input that it cannot purchase in a market or that was purchased in a market in the past.

A key example of such an opportunity cost is the value of a manager’s time. For example, Maoyong owns and manages a firm. He pays himself only a small monthly salary of $1,000 because he also receives the firm’s profit. However, Maoyong could work for another firm and earn $11,000 a month. Thus, the opportunity cost of his time is $11,000—from his best alternative use of his time—not the $1,000 he actually pays himself.

A financial statement may not include such an opportunity cost, but Maoyong needs to take account of this opportunity cost to make decisions that maximize his profit. Suppose that the explicit cost of operating his firm is $40,000, including the rent for work space, the cost of materials, the wage payments to an employee, and the $1,000 a month he pays himself. The full, opportunity cost of the firm is $50,000, which includes the extra $10,000 in opportunity cost for Maoyong’s time beyond the $1,000 that he already pays himself. If his firm’s revenue is $49,000 per month and he considers only his explicit costs of $40,000, it appears that his firm makes a profit of $9,000. In contrast, if he takes account of the full opportunity cost of $50,000, his firm incurs a loss of $1,000.

Another example of an opportunity cost is captured in the well-known phrase “There’s no such thing as a free lunch.” Suppose your parents come to town and offer to take you to lunch. Although they pay the explicit cost—the restaurant’s tab—for the lunch, you still incur the opportunity cost of your time. No doubt the best alternative use of your time is studying this textbook, but you could also consider working at a job for a wage or watching television as possible alternatives. In considering whether to accept the “free” lunch, you need to compare this true opportunity cost against the benefit of dining with your parents.

Mini-Case During major economic downturns, do applications to MBA programs fall, hold steady, or take off like tech stocks during the first Internet bubble? Knowledge of opportunity costs helps us answer this question.

The biggest cost of attending an MBA program is often the opportunity cost of giving up a well-paying job. Someone who leaves a job paying $6,000 per month to attend an MBA program is, in effect, incurring a $6,000 per month opportunity cost, in addition to the tuition and cost of textbooks (though this one is well worth the money).

Thus, it is not surprising that MBA applications rise in bad economic times when outside opportunities decline. People thinking of going back to school face a reduced opportunity cost of entering an MBA program if they think they might be laid off or might not be promoted during an economic downturn. As Stacey Kole, deputy dean for the MBA program at the University of Chicago Graduate School of Business, observed, “When there’s a go-go economy, fewer people decide to go back to school. When things go south the opportunity cost of leaving work is lower.”

The Opportunity Cost of an MBA

1576.1 The Nature of Costs

Q&A 6.1 Meredith’s firm has sent her to a conference for managers and paid her registration fee. Included in the registration fee is free admission to a class on how to price deriva- tive securities, such as options. She is considering attending, but her most attractive alternative opportunity is to attend a talk at the same time by Warren Buffett on his investment strategies. She would be willing to pay $100 to hear his talk, and the cost of a ticket is $40. Given that attending either talk involves no other costs, what is Meredith’s opportunity cost of attending the derivatives talk?

Answer To determine her opportunity cost, determine the benefit that Meredith would forego by attending the derivatives class. Because she incurs no additional fee to attend the derivatives talk, Meredith’s opportunity cost is the foregone benefit of hearing the Buffett speech. Because she values hearing the Buffett speech at $100, but only has to pay $40, her net benefit from hearing that talk is $60 (= $100 – $40). Thus, her opportunity cost of attending the derivatives talk is $60.

Costs of Durable Inputs Determining the opportunity cost of capital such as land, buildings, or equipment is more complex than calculating the cost of inputs that are bought and used in the same period such as labor services, energy, or materials. Capital is a durable good: a product that is usable for a long period, perhaps for many years. Two problems may arise in measuring the cost of a firm’s capital. The first is how to allocate the initial purchase cost over time. The second is what to do if the value of the capital changes over time.

We can avoid these two measurement problems if capital is rented instead of purchased. For example, suppose a firm can rent a small pick-up truck for $400 a month or buy it outright for $20,000. If the firm rents the truck, the rental payment is the relevant opportunity cost per month. The truck is rented month by month, so the firm does not have to worry about how to allocate the purchase cost of a truck over time. Moreover, the rental rate would adjust if the cost of trucks changes over time. Thus, if the firm can rent capital for short periods of time, it calculates the cost of this capital in the same way that it calculates the cost of nondurable inputs such as labor services or materials.

The firm faces a more complicated problem in determining the opportunity cost of the truck if it purchases the truck. The firm’s accountant may expense the truck’s purchase price by treating the full $20,000 as a cost when the truck is purchased, or the accountant may amortize the cost by spreading the $20,000 over the life of the

In 2008, when U.S. unemployment rose sharply and the economy was in poor shape, the number of people seeking admission to MBA programs rose sharply. The number of applicants to MBA programs for the class of 2008–2009 increased over the previous year by 79% in the United States, 77% in the United Kingdom, and 69% in other European programs. Applicants increased substantially for 2009–2010 as well in Canada and Europe. However, as economic conditions improved, global applications fell in 2011 and were relatively unchanged in 2012.

158 CHAPTER 6 Costs

truck, following rules set by an accounting organization or by a relevant government authority such as the Internal Revenue Service (IRS).

A manager who wants to make sound decisions about operating the truck does not expense or amortize the truck using such rules. The firm’s opportunity cost of using the truck is the amount that the firm would earn if it rented the truck to others. Thus, even though the firm owns the truck, the manager should view the opportunity cost of this capital good as a rent per time period. If the value of an older truck is less than that of a newer one, the rental rate for the truck falls over time.

If no rental market for trucks exists, we must determine the opportunity cost in another way. Suppose that the firm has two choices: It can choose not to buy the truck and keep the truck’s purchase price of $20,000, or it can use the truck for a year and sell it for $17,000 at the end of the year. If the firm did not purchase the truck it would deposit the $20,000 in a bank account that pays, for example, 2% per year, earning $400 in interest and therefore having $20,400 at the end of the year. Thus, the oppor- tunity cost of capital of using the truck for a year is $20,400 – $17,000 = $3,400.2 This $3,400 opportunity cost equals the depreciation of the truck of $3,000 (= $20,000 – $17,000) plus the $400 in foregone interest that the firm could have earned over the year if the firm had invested the $20,000.

The value of trucks, machines, and other equipment declines over time, leading to declining rental values and therefore to declining opportunity costs. In contrast, the value of some land, buildings, and other forms of capital may rise over time. To maximize its economic profit, a firm must properly measure the opportunity cost of a piece of capital even if its value rises over time. If a beauty parlor buys a building when similar buildings in that area rent for $1,000 per month, then the opportunity cost of using the building is $1,000 a month. If land values rise causing rents in the area to rise to $2,000 per month, the beauty parlor’s opportunity cost of its building rises to $2,000 per month.

Sunk Costs An opportunity cost is not always easy to observe but should always be taken into account in deciding how much to produce. In contrast, a sunk cost—a past expendi- ture that cannot be recovered—though easily observed is not relevant to a manager when deciding how much to produce now. If an expenditure is sunk, it is not an opportunity cost. Nonetheless, a sunk cost paid for a specialized input should still be deducted from income before paying taxes even if that cost is sunk, and must therefore appear in financial accounts.

If a firm buys a forklift for $25,000 and can resell it for the same price, then the expenditure is not sunk, and the opportunity cost of using the forklift is $25,000. If instead the firm buys a specialized piece of equipment for $25,000 and cannot resell it, then the original expenditure is a sunk cost—it cannot be recovered. Because this equipment has no alternative use—it cannot be resold—its opportunity cost is zero, and hence should not be included in the firm’s current cost calculations. If the specialized equipment that originally cost $25,000 can be resold for $10,000, then only $15,000 of the original expenditure is sunk, and the opportunity cost is $10,000.

2The firm would also pay for gasoline, insurance, and other operating costs, but these items would all be expensed as operating costs and would not appear in the firm’s accounts as capital costs.

1596.2 Short-Run Costs

6.2 Short-Run Costs When making short-run and long-run production and investment decisions, managers must take the relevant costs into account. As noted in Chapter 5, the short run is the period over which some inputs, such as labor, can be varied while other inputs, such as capital, are fixed. In contrast, in the long run, the firm can vary all its inputs. For simplicity in our graphs, we concentrate on firms that use only two inputs, labor and capital. We focus on the case in which labor is the only variable input in the short run, and both labor and capital are variable in the long run. However, we can generalize our analysis to examine a firm that uses any number of inputs.

We start by examining various measures of cost and cost curves that can be used to analyze costs in both the short run and the long run. Then we show how the shapes of the short-run cost curves are related to the firm’s production function.

Common Measures of Cost All firms use the same basic cost measures for making both short-run and long-run decisions. The measures should be based on inputs’ opportunity costs.

Fixed Cost, Variable Cost, and Total Cost. A fixed cost (F) is a cost that does not vary with the level of output. Fixed costs, which include expenditures on land, office space, production facilities, and other overhead expenses, cannot be avoided by reducing output and must be incurred as long as the firm stays in business.

Fixed costs are often sunk costs, but not always. For example, a restaurant rents space for $2,000 per month on a month-to-month lease. This rent does not vary with the number of meals served (its output level), so it is a fixed cost. Because the restau- rant has already paid this month’s rent, this fixed cost is a sunk cost: the restaurant cannot get the $2,000 back even if it goes out of business. Next month, if the res- taurant stays open, it will have to pay the fixed, $2,000 rent. If the restaurant has a month-to-month rental agreement, this fixed cost of $2,000 is an avoidable cost, not a

A manager should ignore sunk costs when making current decisions. To see why, consider a firm that paid $300,000 for a parcel of land for which the market value has fallen to $200,000, which is the land’s current opportunity cost. The $100,000 difference between the $300,000 purchase price and the current market value of $200,000 is a sunk cost that has already been incurred and cannot be recovered. The land is worth $240,000 to the firm if it builds a plant on this parcel. Is it worth carrying out production on this land or should the land be sold for its market value of $200,000? A manager who uses the original purchase price in the decision-making process would falsely conclude that using the land for production will result in a $60,000 loss: the value to using the land of $240,000 minus the purchase price of $300,000. Instead, the firm should use the land because it is worth $40,000 more as a production facility than the firm’s next best alternative of selling the land for $200,000. Thus, the firm should use the land’s opportunity cost in making its decisions and ignore the land’s sunk cost. In short, “there’s no use crying over spilt milk.”

Ignoring Sunk Costs

Managerial Implication

160 CHAPTER 6 Costs

sunk cost. The restaurant can shut down, cancel its rental agreement, and avoid paying this fixed cost. Therefore, in planning for next month, the restau- rant should treat the $2,000 rent as a fixed cost but not as a sunk cost. Thus, the fixed cost of $2,000 per month is a fixed cost in both the short run (this month) and in the long run. However, it is a sunk cost only in the short run.

A variable cost (VC) is a cost that changes as the quantity of output changes. Variable costs are the costs of variable inputs, which are inputs that the firm can adjust to alter its output level, such as labor and materials.

A firm’s cost (or total cost), C, is the sum of a firm’s variable cost and fixed cost:

C = VC + F.

Because variable costs change as the output level changes, so does total cost. For example, in Table 6.1, if the fixed cost is F = $48 and the firm produces 5 units of output, its variable cost is VC = $100, so its total cost is C = $148.

Average Cost. Firms use three average cost measures corresponding to fixed, variable, and total costs. The average fixed cost (AFC) is the fixed cost divided by the units of output produced: AFC = F/q. The average fixed cost falls as output rises because the fixed cost is spread over more units. The average fixed cost falls from $48 for 1 unit of output to $4 for 12 units of output in Table 6.1.

The average variable cost (AVC), or variable cost per unit of output, is the variable cost divided by the units of output produced: AVC = VC/q. Because the variable

Output, q Fixed Cost, F Variable Cost, VC Total Cost, C

Marginal Cost, MC

Average Fixed Cost, AFC = F/q

Average Variable Cost, AVC = VC/q

Average Cost, AC = C/q

0 48 0 48

1 48 25 73 25 48 25 73

2 48 46 94 21 24 23 47

3 48 66 114 20 16 22 38

4 48 82 130 16 12 20.5 32.5

5 48 100 148 18 9.6 20 29.6

6 48 120 168 20 8 20 28

7 48 141 189 21 6.9 20.1 27

8 48 168 216 27 6 21 27

9 48 198 246 30 5.3 22 27.3

10 48 230 278 32 4.8 23 27.8

11 48 272 320 42 4.4 24.7 29.1

12 48 321 369 49 4.0 26.8 30.8

TABLE 6.1 How Cost Varies with Output

1616.2 Short-Run Costs

cost increases with output, the average variable cost may either increase or decrease as output rises. In Table 6.1, the average variable cost is $25 at 1 unit, falls until it reaches a minimum of $20 at 6 units, and then rises.

The average cost (AC)—or average total cost—is the total cost divided by the units of output produced: AC = C/q. Because total cost is C = VC + F, if we divide both sides of the equation by q, we find that average cost is the sum of the average fixed cost and the average variable cost:

AC = C q

= F q

+ VC q

= AFC + AVC.

In Table 6.1, because AFC falls with output and AVC eventually rises with output, average cost falls until output is 8 units and then rises.

Marginal Cost. A firm’s marginal cost (MC) is the amount by which a firm’s cost changes if the firm produces one more unit of output. The marginal cost is

MC = ΔC Δq

where ΔC is the change in cost when the change in output, Δq, is 1 unit in Table 6.1. If the firm increases its output from 2 to 3 units (Δq = 1), its total cost rises from $94 to $114 so ΔC = $20. Thus its marginal cost is ΔC/Δq = $20.

Because only variable cost changes with output, marginal cost also equals the change in variable cost from a one-unit increase in output:

MC = ΔVC Δq

.

As the firm increases output from 2 to 3 units, its variable cost increases by ΔVC = $20 = $66 – $46, so its marginal cost is MC = ΔVC/Δq = $20. A firm takes account of its marginal cost curve to decide whether it pays to change its output level.

Using Calculus Using calculus, we may alternatively define the marginal cost as MC = dC/dq,which is the rate of change of cost as we make an infinitesimally small change in out- put. Because C = VC + F, it follows that MC = dVC/dq + dF/dq = dVC/dq, because fixed costs do not change as output changes: dF/dq = 0.

For example, suppose that the variable cost is VC = 4q + 6q2 and the fixed cost is F = 10, so the total cost is C = VC + F = 4q + 6q2 + 10. Using the variable cost, the marginal cost is dVC/dq = d(4q + 6q2)/dq = 4 + 12q. We get the same expression for marginal cost if we use the total cost: dC/dq = d(4q + 6q2 + 10)/dq = 4 + 12q.

Calculating Marginal Cost

Cost Curves We illustrate the relationship between output and the various cost measures in Figure 6.1. Panel a shows the variable cost, fixed cost, and total cost curves that cor- respond to Table 6.1. The fixed cost, which does not vary with output, is a horizontal line at $48. The variable cost curve is zero at zero units of output and rises with

162 CHAPTER 6 Costs

output. The total cost curve, which is the vertical sum of the variable cost curve and the fixed cost line, is $48 higher than the variable cost curve at every output level, so the variable cost and total cost curves are parallel.

Panel b shows the average fixed cost, average variable cost, average cost, and mar- ginal cost curves. The average fixed cost curve falls as output increases. It approaches zero as output gets large because the fixed cost is spread over many units of output. The average cost curve is the vertical sum of the average fixed cost and average variable cost curves. For example, at 6 units of output, the average variable cost is 20 and the average fixed cost is 8, so the average (total) cost is 28.

The relationships between the average and marginal cost curves and the total cost curve are similar to those between the average and marginal product curves and the total product curve (as discussed in Chapter 5). The average cost at a particular

120

216

400

48

0 6 10

10

42 8

Quantity, q, Units per day 6

b

a

B

A

42 8

C

F

1

1

27

20

VC

MC

AC

AVC

AFC

C os

t, $

C os

t p er

u ni

t, $

(a)

(b)

60

28 27

20

8

0

Quantity, q, Units per day

FIGURE 6.1 Cost Curves

(a) Because the total cost differs from the variable cost by the fixed cost, F, of $48, the total cost curve, C, is parallel to the variable cost curve, VC. (b) The marginal cost curve, MC, cuts the average variable cost, AVC, and average cost, AC, curves at their minimums. The height of the AC curve at point a equals the slope of the line from the origin to the cost curve at A. The height of the AVC at b equals the slope of the line from the origin to the variable cost curve at B. The height of the marginal cost is the slope of either the C or VC curve at that quantity.

1636.2 Short-Run Costs

output level is the slope of a line from the origin to the corresponding point on the total cost curve. The slope of that line is the rise (the cost at that output level) divided by the run (the output level), which is the definition of the average cost. In panel a, the slope of the line from the origin to point A is the average cost for 8 units of out- put. The height of the cost curve at A is 216, so the slope is 216/8 = 27, which is the height of the average cost curve at the corresponding point a in panel b.

Similarly, the average variable cost is the slope of a line from the origin to a point on the variable cost curve. The slope of the dashed line from the origin to B in panel a is 20 (the height of the variable cost curve, 120, divided by the number of units of output, 6), which is also the height of the average variable cost curve at 6 units of output, point b in panel b.

The marginal cost is the slope of either the cost curve or the variable cost curve at a given output level. Because the total cost and variable cost curves are parallel, they have the same slope at any given output. The difference between total cost and variable cost is fixed cost, which does not affect marginal cost.

The thin black line from the origin is tangent to the cost curve at A in panel a. Thus, the slope of the thin black line equals both the average cost and the marginal cost at point a (8 units of output). This equality occurs at the corresponding point a in panel b, where the marginal cost curve intersects the average cost.

Where the marginal cost curve is below the average cost, the average cost curve declines with output. Because the average cost of 47 for 2 units is greater than the marginal cost of the third unit, 20, the average cost for 3 units falls to 38.3 Where the marginal cost is above the average cost, the average cost curve rises with output. At 8 units, the marginal cost equals the average cost (at point a in panel b, the minimum point of the average cost curve), so the average is unchanging.

Because the dashed line from the origin is tangent to the variable cost curve at B in panel a, the marginal cost equals the average variable cost at the corresponding point b in panel b. Again, where marginal cost is above average variable cost, the average variable cost curve rises with output; where marginal cost is below average variable cost, the average variable cost curve falls with output. Because the average cost curve is above the average variable cost curve everywhere and the marginal cost curve is rising where it crosses both average curves, the minimum of the average variable cost curve, b, is at a lower output level than the minimum of the average cost curve, a.

Production Functions and the Shapes of Cost Curves The production function determines the shape of a firm’s cost curves. The produc- tion function shows the amount of inputs needed to produce a given level of output (Chapter 5). The firm calculates its variable cost by multiplying the quantity of each input by its price and summing the costs of the variable inputs.

In this section, we focus on cost curves in the short run. If a firm produces output using capital and labor, and its capital is fixed in the short run, the firm’s variable cost is its cost of labor. Its labor cost is the wage per hour, w, times the number of hours of labor, L, employed by the firm: VC = wL.

In the short run, when the firm’s capital is fixed, the only way the firm can increase its output is to use more labor. If the firm increases its labor enough, it reaches the

3From Table 6.1, we know that the average cost of the first two units is 47. If we add a third unit with a marginal cost of 20, the new average can be calculated by adding the average values of the first two units plus the marginal cost of the third unit and dividing by 3: (47 + 47 + 20)/3 = 38. Thus, if we add a marginal cost that is less than the old average cost, the new average cost must fall.

164 CHAPTER 6 Costs

point of diminishing marginal returns to labor, at which each extra worker increases output by a smaller amount. We can use this information about the relationship between labor and output—the production function—to determine the shape of the variable cost curve and its related curves.

The Variable Cost Curve. If input prices are constant, the firm’s production function determines the shape of the variable cost curve. We illustrate this rela- tionship in Figure 6.2. The firm faces a constant input price for labor, the wage, of $10 per hour.

The total product of labor curve in Figure 6.2 shows the firm’s short-run produc- tion function relationship between output and labor when capital is held fixed. At point a, the firm uses 5 hours of labor to produce 1 unit of output. At point b, it takes 20 hours of labor to produce 5 units of output. Here, output increases more than in proportion to labor: Output rises 5 times when labor increases 4 times. In contrast, as the firm moves from b to c, output increases less than in proportion. Output doubles to 10 as a result of increasing labor from 20 to 46—an increase of 2.3 times. The move- ment from c to d results in even a smaller increase in output relative to labor. This flattening of the total product curve at higher levels of labor reflects diminishing marginal returns to labor.

This curve shows both the production relation of output to labor and the variable cost relation of output to cost. Because each hour of work costs the firm $10, we can relabel the horizontal axis in Figure 6.2 to show the firm’s variable cost, its cost of labor. To produce 5 units of output takes 20 hours of labor, so the firm’s variable cost

Q ua

nt ity

,q , U

ni ts

p er

d ay

46 460

20 200

5 50

77 770

Total product, Variable cost

0

5

1

10

13

L, Hours of labor per day VC = wL, Variable cost, $

c

d

b

a

FIGURE 6.2 Variable Cost and Total Product

The firm’s short-run variable cost curve and its total prod- uct curve have the same shape. The total product curve uses the horizontal axis measuring hours of work. The

variable cost curve uses the horizontal axis measuring labor cost, which is the only variable cost.

1656.2 Short-Run Costs

is $200. By using the variable cost labels on the horizontal axis, the total product of labor curve becomes the variable cost curve.

As output increases, the variable cost increases more than proportionally due to the diminishing marginal returns. Because the production function determines the shape of the variable cost curve, it also determines the shape of the marginal, aver- age variable, and average cost curves. We now examine the shape of each of these cost curves in detail, because when making decisions, managers rely more on these per-unit cost measures than on total variable cost.

The Marginal Cost Curve. The marginal cost is the change in variable cost as output increases by one unit: MC = ΔVC/Δq. In the short run, capital is fixed, so the only way the firm can produce more output is to use extra labor. The extra labor required to produce one more unit of output is ΔL/Δq. The extra labor costs the firm w per unit, so the firm’s cost rises by w(ΔL/Δq). As a result, the firm’s marginal cost is

MC = ΔVC Δq

= w ΔL Δq

.

The marginal cost equals the wage times the extra labor necessary to produce one more unit of output. To increase output by one unit from 5 to 6 units takes 4 extra hours of work in Figure 6.2. If the wage is $10 per hour, the marginal cost is $40.

How do we know how much extra labor we need to produce one more unit of output? That information comes from the production function. The marginal prod- uct of labor—the amount of extra output produced by another unit of labor, holding other inputs fixed—is MPL = Δq/ΔL. Thus, the extra labor we need to produce one more unit of output, ΔL/Δq, is 1/MPL, so the firm’s marginal cost is

MC = w

MPL . (6.1)

Equation 6.1 says that the marginal cost equals the wage divided by the marginal product of labor. If the firm is producing 5 units of output, it takes 4 extra hours of labor to produce 1 more unit of output in Figure 6.2, so the marginal product of an hour of labor is 14 unit of output. Given a wage of $10 an hour, the marginal cost of the sixth unit is $10 divided by 14, or $40.

Equation 6.1 shows that the marginal product of labor and marginal cost move in opposite directions as output changes. At low levels of labor, the marginal product of labor commonly rises with additional labor because extra workers help the original workers and they can collectively make better use of the firm’s equipment. As the marginal product of labor rises, the marginal cost falls.

Eventually, however, as the number of workers increases, workers must share the fixed amount of equipment and may get in each other’s way. As more workers are added, the marginal product of each additional worker begins to fall and the marginal cost of each additional unit of product rises. As a result, the marginal cost curve slopes upward because of diminishing marginal returns to labor. Thus, the marginal cost first falls and then rises.

The Average Cost Curves. Because they determine the shape of the variable cost curve, diminishing marginal returns to labor also determine the shape of the average variable cost curve. The average variable cost is the variable cost divided

166 CHAPTER 6 Costs

by output: AVC = VC/q. For the firm we’ve been examining, whose only variable input is labor, variable cost is wL, so average variable cost is

AVC = VC q

= wL q

.

Because the average product of labor, APL, is q/L, average variable cost is the wage divided by the average product of labor:

AVC = w

APL . (6.2)

In Figure 6.2, at 6 units of output, the average product of labor is 14 (= q/L = 6/24), so the average variable cost is $40, which is the wage, $10, divided by the average product of labor, 14.

With a constant wage, the average variable cost moves in the opposite direction of the average product of labor in Equation 6.2. As we discussed in Chapter 5, the average product of labor tends to rise and then fall, so the average cost tends to fall and then rise, as in panel b of Figure 6.1.

The average cost curve is the vertical sum of the average variable cost curve and the average fixed cost curve, as in panel b of Figure 6.1. If the average variable cost curve is U-shaped, adding the strictly falling average fixed cost makes the average cost fall more steeply than the average variable cost curve at low output levels. At high output levels, the average cost and average variable cost curves differ by ever smaller amounts, as the average fixed cost, F/q, approaches zero. Thus, the average cost curve is also U-shaped.

Using Calculus If we know the production function for a product (Chapter 5) and the factor prices, we can use math to derive the various cost functions. Based on the esti- mates of Flath (2011), the Cobb-Douglas production function of a typical Japanese beer manufacturer (Chapter 5) is

q = 1.52L0.6K0.4,

where labor, L, is measured in hours, K is the num- ber of units of capital, and q is the amount of output.

We assume that the firm’s capital is fixed at K = 100 units in the short run. If the rental rate of a unit of capital is $8, the fixed cost, F, is $800, the average fixed cost is

AFC = F/q = 800/q,

which falls as output increases. We can use the production function to derive the

variable cost. Given that capital is fixed in the short run, the short-run production function is solely a function of labor:

q = 1.52L0.61000.4 ≈ 9.59L0.6.

Calculating Cost Curves

1676.2 Short-Run Costs

Short-Run Cost Summary We use cost curves to illustrate three cost level concepts—total cost, fixed cost, and variable cost—and four cost-per-unit cost concepts—average cost, average fixed cost, average variable cost, and marginal cost. Understanding the shapes of these curves and the relationships among them is crucial to the analysis of firm behavior in the rest of this book. Fortunately, we can derive most of what we need to know about the shapes and the relationships between the short-run curves using four basic concepts:

1. In the short run, the cost associated with inputs that cannot be adjusted is fixed, while the cost from inputs that can be adjusted is variable.

2. Given that input prices are constant, the shapes of the variable cost and the cost- per-unit curves are determined by the production function.

3. Where a variable input exhibits diminishing marginal returns, the variable cost and cost curves become relatively steep as output increases, so the average cost, average variable cost, and marginal cost curves rise with output.

Rearranging this expression, we can write the number of workers, L, needed to pro- duce q units of output as a function solely of output:

L(q) = ¢ q 9.59

≤ 1

0.6

= ¢ 1 9.59

≤1.67q1.67 ≈ 0.023q1.67. (6.3)

Now that we know how labor and output are related, we can calculate variable cost directly. The only variable input is labor, so if the wage is $24, the firm’s variable cost is

VC(q) = wL(q) = 24L(q). (6.4)

Substituting for L(q) using Equation 6.3 into the variable cost Equation 6.4, we see how variable cost varies with output:

VC(q) = 24L(q)

= 24(0.023q1.67) ≈ 0.55q1.67. (6.5)

Using this expression for variable cost, we can construct the other cost measures. To obtain the equation for marginal cost as a function of output, we differenti-

ate the variable cost, VC(q), with respect to output:

MC(q) = dVC(q)

dq ≈

d(0.55q1.67)

dq = 1.67 * 0.55q0.67 ≈ 0.92q0.67.

We can also calculate total cost, C = F + VC, average cost, AC = C/q, and aver- age variable cost, AVC = VC/q using algebra. The figure plots the beer firm’s AFC, AVC, AC, and MC curves.

C os

ts p

er u

ni t,

$

100 200 300

q, Units per year

AFC

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