. [David Besanko, Ronald Braeutigam, Ronald R. Braeu PDF

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accessible, affordable, active learning

«PRWLYDWHVVWXGHQWVZLWK FRQÀGHQFHERRVWLQJ IHHGEDFNDQGSURRIRI SURJUHVV

«VXSSRUWVLQVWUXFWRUVZLWK UHOLDEOHUHVRXUFHVWKDW UHLQIRUFHFRXUVHJRDOV LQVLGHDQGRXWVLGHRI WKHFODVVURRP

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$//7+(+(/35(6285&(6$1'3(5621$/6833257 APL 0

6

12

APL is decreasing so MPL < APL 18

24

30

APL 36

L, thousands of man-hours per day −5

−10

MPL

FIGURE 6.3

Average and Marginal Product Functions APL is the average product function. MPL is the marginal product function. The marginal product function rises in the region of increasing marginal returns (L  12) and falls in the region of diminishing marginal returns (12  L  24). It becomes negative in the region of diminishing total returns (L  24). At point A, where APL is at a maximum, APL  MPL.

The other notion of productivity is the marginal product of labor, which we write as MPL. The marginal product of labor is the rate at which total output changes as the firm changes its quantity of labor: MPL 

change in total product change in quantity of labor



¢Q ¢L

The marginal product of labor is analogous to the concept of marginal utility from consumer theory, and just as we could represent that curve graphically, we can also represent the marginal product curve graphically, as shown in Figure 6.3. Marginal product, like average product, is not a single number but varies with the quantity of labor. In the region of increasing marginal returns, where 0  L  12, the marginal product function is increasing. When diminishing marginal returns set in, at L  12, the marginal product function starts decreasing. When diminishing total returns set in, at L  24, the marginal product function cuts through the horizontal axis and becomes negative. As shown in the upper panel in Figure 6.4, the marginal product corresponding to any particular amount of labor L1 is the slope of the line that is tangent to the total product function at L1 (line BC in the figure). Since the slopes of these tangent lines vary as we move along the production function, the marginal product of labor must also vary. In most production processes, as the quantity of one input (e.g., labor) increases, with the quantities of other inputs (e.g., capital and land) held constant, a point will be reached beyond which the marginal product of that input decreases. This phenomenon, which reflects the experience of real-world firms, seems so pervasive that economists call it the law of diminishing marginal returns.

marginal product of labor The rate at which total output changes as the quantity of labor the firm uses is changed.

law of diminishing marginal returns Principle that as the usage of one input increases, the quantities of other inputs being held fixed, a point will be reached beyond which the marginal product of the variable input will decrease.

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Q, thousands of chips per day

C

Marginal product at L1 equals slope of line BC Average product at L0 equals slope of ray 0A

Q0

FIGURE 6.4

Relationship among Total, Average, and Marginal Product Functions The marginal product of labor at any point equals the slope of the total product curve at that point. The average product at any point is equal to the slope of the ray from the origin to the total product curve at that point.

A P P L I C A T I O N

APL, MPL , chips per man-hour

0

B

Total product function

A

L0

18 L1 24 L, thousands of man-hours per day

APL 0

L0

18 L1

24

L, thousands of man-hours per day

MPL

6.2

The Resurgence of Labor Productivity in the United States When the average product of labor is computed for an entire economy—say, that of the United States— what we get is a measure of overall labor productivity in the economy. Labor productivity is an important

indicator of the overall well-being of an economy. Rising labor productivity implies that more output can be produced from a given amount of labor, and when that is the case, the standard of living in the economy rises over time. By contrast, when the growth of labor productivity stalls, improvements in the standard of living will slow down as well.

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6.2 PRODUCTION FUNCTIONS WITH A SINGLE INPUT The accompanying table shows the average annual growth in labor productivity in the United States between 1947 and 2009.5 The table reveals a striking pattern: from 1947 though the mid-1970s, labor productivity grew at a rate of about 2.5 to 3 percent per year. However, from the mid-1970s through the mid-1990s, the growth of labor productivity slowed significantly, falling to a rate of about 1.4 to 1.5 percent annually. Beginning in the late 1990s, there was a resurgence of labor productivity, with annual growth rates between 1995 and 2005 averaging over 2.9 percent. Keeping in mind that the 1995–2005 period encompassed 9/11, the “Dot Bomb” technology crash, the recession of 2001, and numerous corporate governance scandals, the growth of labor productivity over this period is impressive indeed. Growth in Labor Productivity in the United States, 1947–2009 Years

Annual Growth Rate in Labor Productivity

1947–1955 1955–1965 1965–1975 1975–1985 1985–1995 1995–2005 2005–2009

3.21% 2.61% 2.18% 1.38% 1.51% 2.94% 1.90%

What explains the slowdown in labor productivity beginning in the mid-1970s? Based on the study of detailed industry-level data on labor productivity, William Nordhaus finds that the largest slowdowns in productivity growth were in energy-reliant industries such as pipelines, oil and gas extraction, and automobile repair services.6 This suggests, then, that the primary culprits in the slowdown of productivity growth in the United States were the oil shocks of 1973 and 1979. As Nordhaus puts it, “In a sense, the energy

5

209

shocks were the earthquake, and the industries with the largest slowdown were nearest the epicenter of the tectonic shifts in the economy.” To explain the resurgence of labor productivity since 1995, it is useful to identify factors that would tend to make workers more productive. One important factor that can affect labor productivity is the amount of sophistication of the capital equipment available to workers. The period between 1995 and 2005 was one of rapid growth in the sophistication and ubiquity of information and communications technologies. Thus the hypothesis that the post–1995 resurgence of labor productivity is attributable to increases in the quantity and quality of capital (what economists call “capital deepening”) is quite plausible. A second factor affecting the productivity of labor is the increase in the quality of labor itself. Improvements in aggregate labor quality occur primarily when the ratio of high-skill to lower-skill workers increases, which in turn occurs as firms demand higher levels of experience and education from their workers (which, of course, is related to the increased sophistication of the capital that workers use in their jobs). So what does explain the resurgence of U.S. productivity growth since 1995? According to an analysis by Dale Jorgenson, Mun Ho, and Kevin Stiroh (JHS), the most important factor was capital deepening.7 Indeed, JHS find that capital deepening explains more than half of the jump in the labor productivity growth rate in the period after 1975. As one might expect, much of the capital deepening was due to improvements in information and communications technology. On the other hand, JHS find that changes in labor quality played a relatively small role in driving productivity growth upward, suggesting that changes in the mix between high- and low-skill workers have not been responsible for the increases in the growth of labor productivity since 1995.

The growth rates were calculated from changes in output per hour in all nonfarm business in the United States, using data from the Bureau of Labor Statistics website www.bls.gov/data/. Data for 2005–2009 are calculated through the second quarter of 2009 only. 6 William Nordhaus, “Retrospective on the 1970s Productivity Slowdown,” NBER Working Paper No. W10950 (December 2004), available at SSRN, http://ssrn.com/abstract=629592. 7 Dale Jorgenson, Mun Ho, and Kevin Stiroh, “Will the U.S. Productivity Resurgence Continue?” Current Issues in Economics & Finance 10, no. 13, Federal Reserve Bank of New York (December 2004): 1–7.

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R E L AT I O N S H I P B E T W E E N M A R G I N A L A N D AV E R AG E P R O D U C T As with other average and marginal concepts you will study in this book (e.g., average cost versus marginal cost), there is a systematic relationship between average product and marginal product. Figure 6.3 illustrates this relationship: • When average product is increasing in labor, marginal product is greater than average product. That is, if APL increases in L, then MPL  APL. • When average product is decreasing in labor, marginal product is less than average product. That is, if APL decreases in L, then MPL  APL. • When average product neither increases nor decreases in labor because we are at a point at which APL is at a maximum (point A in Figure 6.3), then marginal product is equal to average product. The relationship between marginal product and average product is the same as the relationship between the marginal of anything and the average of anything. To illustrate this point, suppose that the average height of students in your class is 160 cm. Now Mike Margin joins the class, and the average height rises to 161 cm. What do we know about Mike’s height? Since the average height is increasing, the “marginal height” (Mike Margin’s height) must be above the average. If the average height had fallen to 159 cm, it would have been because his height was below the average. Finally, if the average height had remained the same when Mike joined the class, his height would have had to exactly equal the average height in the class. The relationship between average and marginal height in your class is the same as the relationship between average and marginal product shown in Figure 6.3. It is also the relationship between average and marginal cost that we will study in Chapter 8 and the relationship between average and marginal revenue that we will see in Chapter 11.

6.3

The single-input production function is useful for developing key concepts, such as

PRODUCTION FUNCTIONS WITH MORE THAN ONE INPUT

marginal and average product, and building intuition about the relationships between these concepts. However, to study the trade-offs facing real firms, such as semiconductor companies thinking about substituting robots for humans, we need to study multiple-input production functions. In this section, we will see how to describe a multiple-input production function graphically, and we will study a way to characterize how easily a firm can substitute among the inputs within its production function.

TOTA L P R O D U C T A N D M A R G I N A L P R O D U C T WITH TWO INPUTS To illustrate a production function with more than one input, let’s consider a situation in which the production of output requires two inputs: labor and capital. This might broadly illustrate the technological possibilities facing a semiconductor manufacturer contemplating the use of robots (capital) or humans (labor).

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TABLE 6.3

Production Function for Semiconductors* K **

L**

0 6 12 18 24 30

0

6

12

18

24

30

0 0 0 0 0 0

0 5 15 25 30 23

0 15 48 81 96 75

0 25 81 137 162 127

0 30 96 162 192 150

0 23 75 127 150 117

*Numbers in table equal the output that can be produced with various combinations of labor and capital. **L is expressed in thousands of man-hours per day; K is expressed in thousands of machinehours per day; and Q is expressed in thousands of semiconductor chips per day.

Table 6.3 shows a production function (or, equivalently, the total product function) for semiconductors, where the quantity of output Q depends on the quantity of labor L and the quantity of capital K employed by the semiconductor firm. Figure 6.5 shows this production function as a three-dimensional graph. The graph in Figure 6.5 is called a total product hill––a three-dimensional graph that shows the relationship between the quantity of output and the quantity of the two inputs employed by the firm.8

total product hill A three-dimensional graph of a production function.

C Q (thousands of semiconductor chips per day) = height of hill at any point

thousands K of machine30 hours per day North

B

A 24 18 12 6 0

6

12

18

L, thousands 30 of man-hours per day

24 East

FIGURE 6.5

Total Product Hill The height of the hill at any point is equal to the quantity of output Q attainable from the quantities of labor L and capital K corresponding to that point.

8

In Figure 6.5, we show the “skeleton,” or frame, of the total product hill, so that we can draw various lines underneath it. Figure 6.6 shows the same total product hill as a solid surface.

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The height of the hill at any point is equal to the quantity of output Q the firm produces from the quantities of inputs it employs. We could move along the hill in any direction, but it is easiest to imagine moving in either of two directions. Starting from any combination of labor and capital, we could move eastward by increasing the quantity of labor, or we could move northward by increasing the quantity of capital. As we move either eastward or northward, we move to different elevations along the total product hill, where each elevation corresponds to the particular quantity of output. Let’s now see what happens when we fix the quantity of capital at a particular level, say K  24, and increase the quantity of labor. The outlined column in Table 6.3 shows that when we do this, the quantity of output initially increases but then begins to decrease (when L  24). In fact, notice that the values of Q in Table 6.3 are identical to the values of Q for the total product function in Table 6.1. This shows that the total product function for labor can be derived from a two-input production function by holding the quantity of capital fixed at a particular level (in this case, at K  24) and varying the quantity of labor. We can make the same point with Figure 6.5. Let’s fix the quantity of capital at K  24 and move eastward up the total product hill by changing the quantity of labor. As we do so, we trace out the path ABC, with point C being at the peak of the hill. This path has the same shape as the total product function in Figure 6.2, just as the K  24 column in Table 6.3 corresponds exactly to Table 6.1. Just as the concept of total product extends directly to the multiple input case, so too does the concept of marginal product. The marginal product of an input is the rate at which output changes as the firm changes the quantity of one of its inputs, holding the quantities of all other inputs constant. The marginal product of labor is given by: change in quantity of output Q change in quantity of labor L Q  L K is held constant

MPL 

K is held constant

(6.2)

Similarly, the marginal product of capital is given by: change in quantity of output Q change in quantity of capital K Q  K L is held constant

MPK 

L is held constant

(6.3)

The marginal product tells us how the steepness of the total product hill varies as we change the quantity of an input, holding the quantities of all other inputs fixed. The marginal product at any particular point on the total product hill is the steepness of the hill at that point in the direction of the changing input. For example, in Figure 6.5, the marginal product of labor at point B—that is, when the quantity of labor is 18 and the quantity of capital is 24—describes the steepness of the total product hill at point B in an eastward direction.

I S O Q UA N T S To illustrate economic trade-offs, it helps to reduce the three-dimensional graph of the production function (the total product hill) to two dimensions. Just as we used a contour plot of indifference curves to represent utility functions in consumer theory,

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6.3 PRODUCTION FUNCTIONS WITH MORE THAN ONE INPUT

TABLE 6.4

Production Function for Semiconductors* K**

0 6 12 18 24 30

L**

0

6

12

18

24

30

0 0 0 0 0 0

0 5 15 25 30 23

0 15 48 81 96 75

0 25 81 137 162 127

0 30 96 162 192 150

0 23 75 127 150 117

*Numbers in table equal the output that can be produced with various combinations of labor and capital. **L is expressed in thousands of man-hours per day; K is expressed in thousands of machine-hours per day; and Q is expressed in thousands of semiconductor chips per day.

we can also use a contour plot to represent the production function. However, instead of calling the contour lines indifference curves, we call them isoquants. Isoquant means “same quantity”: any combination of labor and capital along a given isoquant allows the firm to produce the same quantity of output. To illustrate, let’s consider the production function described in Table 6.4 (the same function as in Table 6.3). From this table we see that two different combinations of labor and capital—(L  6, K  18) and (L  18, K  6)—result in an output of Q  25 units (where each “unit” of output represents a thousand semiconductors). Thus, each of these input combinations is on the Q  25 isoquant. The same isoquant is shown in Figure 6.6 (equivalent to Figure 6.5), illustrating the total product hill for the production function in Table 6.4. Suppose that you started

isoquant A curve that shows all of the combinations of labor and capital that can produce a given level of output.

All combinations of L and K along path ABCDE produce...