If one worker can assemble 9 products per hour

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.