Higher isocost lines correspond to higher

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.