We begin the analysis of production in the short run with the simplest kind of short-run situation: only one variable input and one fixed input
Q 5 f (L, __K )
The firm has chosen the level of capital (made its investment decision), so capital is fixed in amount. Once the level of capital is fixed, the only way the firm can change its output is by changing the amount of labor it employs.
Total Product
Suppose a firm with a production function of the form Q 5 f (L, K) can, in the long run, choose levels of both labor and capital between 0 and 10 units. A production function giving the maximum amount of output that can be produced from every possible combination of labor and capital is shown in Table 8.2. For example, from the table, 4 units of labor combined with 3 units of capital can produce a maximum of 325 units of output; 6 labor and 6 capital can produce a maximum of 655 units of output; and so on. Note that with 0 capital, no output can be produced regard- less of the level of labor usage. Likewise, with 0 labor, there can be no output.
T A B L E 8 .1
Inputs in Production Input type Payment Relation to output Avoidable or sunk? Employed in short run (SR) or long run (LR)?
Variable input Variable cost Direct Avoidable Both SR and LR
Fixed input Fixed costs Constant Sunk Only SR
Quasi-fixed input Quasi-fixed cost Constant Avoidable If required: SR and LR
Now try Technical Problem 5.
1Because quasi-fixed costs are avoidable costs, economists sometimes call the cost of quasi-fixed inputs either “avoidable fixed costs” or “nonsunk fixed costs.” We will always use the more traditional name, “quasi-fixed costs.”
Once the level of capital is fixed, the firm is in the short run, and output can be changed only by varying the amount of labor employed. Assume now that the capital stock is fixed at 2 units of capital. The firm is in the short run and can vary output only by varying the usage of labor (the variable input). The column in Table 8.2 under 2 units of capital gives the total output, or total product of labor, for 0 through 10 workers. This column, for which K 5 2, represents the short-run production function when capital is fixed at 2 units.
These total products are reproduced in column 2 of Table 8.3 for each level of labor usage in column 1. Thus, columns 1 and 2 in Table 8.3 define a production
T A B L E 8.2
A Production Function Units of capital (K)
Units of labor (L)
0 1 2 3 4 5 6 7 8 9 10
0 0 0 0 0 0 0 0 0 0 0 0
1 0 25 52 74 90 100 108 114 118 120 121
2 0 55 112 162 198 224 242 252 258 262 264
3 0 83 170 247 303 342 369 384 394 400 403
4 0 108 220 325 400 453 488 511 527 535 540
5 0 125 258 390 478 543 590 631 653 663 670
6 0 137 286 425 523 598 655 704 732 744 753
7 0 141 304 453 559 643 708 766 800 814 825
8 0 143 314 474 587 679 753 818 857 873 885
9 0 141 318 488 609 708 789 861 905 922 935
10 0 137 314 492 617 722 809 887 935 953 967
T A B L E 8.3 Total, Average, and Marginal Products of Labor (with capital fixed at 2 units)
(1) Number of workers (L)
(2)
Total product (Q)
(3)
Average product (AP 5 Q/L)
(4)
Marginal product (MP 5 DQ/DL)
0 0 — —
1 52 52 52
2 112 56 60
3 170 56.7 58
4 220 55 50
5 258 51.6 38
6 286 47.7 28
7 304 43.4 18
8 314 39.3 10
9 318 35.3 4
10 314 31.4 24
function of the form Q 5 f (L, __K ), where __K 5 2. In this example, total product (Q) rises with increases in labor up to a point (9 workers) and then declines. While total product does eventually fall as more workers are employed, managers would not (knowingly) hire additional workers if they knew output would fall. In Table 8.3, for example, a manager can hire either 8 workers or 10 workers to produce 314 units of output. Obviously, the economically efficient amount of labor to hire to produce 314 units is 8 workers.
Average and Marginal Products
Average and marginal products are obtained from the production function and may be viewed merely as different ways of looking at the same information. The average product of labor (AP) is the total product divided by the number of workers
AP 5 Q/L
In our example, average product, shown in column 3, first rises, reaches a maxi- mum at 56.7, then declines thereafter.
The marginal product of labor (MP) is the additional output attributable to using one additional worker with the use of all other inputs fixed (in this case, at 2 units of capital). That is,
MP 5 DQ/DL
where D means “the change in.” The marginal product schedule associated with the production function in Table 8.3 is shown in column 4 of the table. Because no output can be produced with 0 workers, the first worker adds 52 units of out- put; the second adds 60 units (i.e., increases output from 52 to 112); and so on.
Note that increasing the amount of labor from 9 to 10 actually decreases output from 318 to 314. Thus the marginal product of the 10th worker is negative. In this example, marginal product first increases as the amount of labor increases, then decreases, and finally becomes negative. This is a pattern frequently assumed in economic analysis.
In this example, the production function assumes that labor, the variable input, is increased one worker at a time. But we can think of the marginal product of an input when more than 1 unit is added. At a fixed level of capital, suppose that 20 units of labor can produce 100 units of output and that 30 units of labor can produce 200 units of output. In this case, output increases by 100 units as labor increases by 10. Thus
MP 5 DQ____
DL 5 100 ____ 10 5 10
Output increases by 10 units for each additional worker hired.
We might emphasize that we speak of the marginal product of labor, not the marginal product of a particular laborer. We assume that all workers are the same, in the sense that if we reduce the number of workers from 8 to 7 in Table 8.3, total
average product of labor (AP) Total product (output) divided by the number of workers (AP 5 Q/L).
marginal product of labor (MP) The additional output attributable to using one additional worker with the use of all other inputs fixed (MP 5 DQ/DL).
product falls from 314 to 304 regardless of which of the 8 workers is released. Thus the order of hiring makes no difference; a third worker adds 58 units of output no matter who is hired.
Figure 8.1 shows graphically the relations among the total, average, and marginal products set forth in Table 8.3. In Panel A, total product increases up to 9 workers, then decreases. Panel B incorporates a common assumption made in production theory: Average product first rises then falls. When average product is increasing, marginal product is greater than average product (after the first worker, at which they are equal). When average product is decreasing, marginal product is less than
50
Total product
10 9 8 7 6 5 4 3 2 100
150 200 250 300 350
Units of labor Panel A
Total product (K = 2)–
0 1
10
Average and marginal products 0
10 9 8 7 6 5 4 3 2 1 20 30 40 50 60
Panel B –10
Marginal product (K = 2)–
Average product (K = 2)–
Units of labor F I G U R E 8.1
Total, Average, and Marginal Products ( K 5 2)__
average product. This result is not peculiar to this particular production function;
it occurs for any production function for which average product first increases then decreases.
An example might help demonstrate that for any average and marginal schedule, the average must increase when the marginal is above the aver- age and decrease when the marginal is below the average. If you have taken two tests and made grades of 70 and 80, your average grade is 75. If your third test grade is higher than 75, the marginal grade is above the average, so your average grade increases. Conversely, if your third grade is less than 75—the marginal grade is below the average—your average falls. In produc- tion theory, if each additional worker adds more than the average, average product rises; if each additional worker adds less than the average, average product falls.
As shown in Figure 8.1, marginal product first increases then decreases, becoming negative after 9 workers. The maximum marginal product occurs before the maximum average product is attained. When marginal product is increasing, total product increases at an increasing rate. When marginal prod- uct begins to decrease (after 2 workers), total product begins to increase at a decreasing rate. When marginal product becomes negative (10 workers), total product declines.
We should note another important relation between average and marginal product that is not obvious from the table or the graph but does follow directly from the discussion. If labor is allowed to vary continuously rather than in discrete units of one, as in the example, marginal product equals average product when average is at its maximum. This follows because average product must increase when marginal is above average and decrease when marginal is below average. The two, therefore, must be equal when average is at its maximum.
Law of Diminishing Marginal Product
The slope of the marginal product curve in Panel B of Figure 8.1 illustrates an important principle, the law of diminishing marginal product. As the number of units of the variable input increases, other inputs held constant, there exists a point beyond which the marginal product of the variable input declines. When the amount of the variable input is small relative to the fixed inputs, more intensive utilization of fixed inputs by variable inputs may initially increase the marginal product of the variable input as this input is increased. Nonetheless, a point is reached beyond which an increase in the use of the variable input yields progres- sively less additional output. Each additional unit has, on average, fewer units of the fixed inputs with which to work.
To illustrate the concept of diminishing marginal returns, consider the kitchen at Mel’s Hot Dogs, a restaurant that sells hot dogs, french fries, and soft drinks.
Mel’s kitchen has one gas range for cooking the hot dogs, one deep-fryer for cooking french fries, and one soft-drink dispenser. One cook in the kitchen can
Now try Technical Problems 6, 7.
law of diminishing marginal product The principle that as the number of units of the variable input increases, other inputs held constant, a point will be reached beyond which the marginal product decreases.
prepare 15 meals (consisting of a hot dog, fries, and soft drink) per hour. Two cooks can prepare 35 meals per hour. The marginal product of the second cook is 20 meals per hour, five more than the marginal product of the first cook. One cook possibly concentrates on making fries and soft drinks while the other cook prepares hot dogs. Adding a third cook results in 50 meals per hour being pro- duced, so the marginal product of the third worker is 15 (5 50 2 35) additional meals per hour.
Therefore, after the second cook, the marginal product of additional cooks begins to decline. The fourth cook, for example, can increase the total num- ber of meals prepared to 60 meals per hour—a marginal product of just 10 additional meals. A fifth cook adds only five extra meals per hour, an in- crease to 65 meals. Even though the third, fourth, and fifth cooks increase the total number of meals prepared each hour, their marginal contribution is di- minishing because the amount of space and equipment in the kitchen is fixed (i.e., capital is fixed). Mel could increase the size of the kitchen or add more cooking equipment to increase the productivity of all workers. The point at which diminishing returns set in would then possibly occur at a higher level of employment.
The marginal product of additional cooks can even become negative. For ex- ample, adding a sixth cook reduces the number of meals from 65 to 60. The mar- ginal product of the sixth cook is 25. Do not confuse negative marginal product with diminishing marginal product. Diminishing marginal product sets in with the third cook, but marginal product does not become negative until the sixth cook is hired. Obviously, the manager would not want to hire a sixth cook, because out- put would fall. The manager would hire the third, or fourth, or fifth cook, even though marginal product is decreasing, if more than 35, 50, or 60 meals must be prepared. As we will demonstrate, managers do in fact employ variable inputs beyond the point of diminishing returns but not to the point of negative marginal product.
The law of diminishing marginal product is a simple statement concerning the relation between marginal product and the rate of production that comes from observing real-world production processes. While the eventual diminishing of marginal product cannot be proved or refuted mathematically, it is worth noting that a contrary observation has never been recorded. That is why the relation is called a law.
Changes in Fixed Inputs
The production function shown in Figure 8.1 and also in Table 8.3 was derived from the production function shown in Table 8.2 by holding the capital stock fixed at 2 units ( __K 5 2). As can be seen in Table 8.2, when different amounts of capital are used, total product changes for each level of labor usage. Indeed, each column in Table 8.2 represents a different short-run production function, each corresponding to the particular level at which capital stock is fixed. Because the output associated with every level of labor usage changes when capital stock
changes, a change in the level of capital causes a shift in the total product curve for labor. Because total product changes for every level of labor usage, average product and marginal product of labor also must change at every level of labor usage.
Referring once more to Table 8.2, notice what happens when the capital stock is increased from 2 to 3 units. The total product of 3 workers increases from 170 to 247, as shown in column 3. The average product of three workers increases from 56.7 to 82.3 (5 247/3). The marginal product of the third worker increases from 58 to 85 [DQ/DL 5 (247 2 162)/1 5 85]. Table 8.4 shows the total, average, and mar- ginal product schedules for two levels of capital stock, __K 5 2 and __K 5 3. As you can see, TP, AP, and MP all increase at each level of labor usage as K increases from 2 to 3 units. Figure 8.2 shows how a change in the fixed amount of capital shifts the product curves. In Panel A, increasing __K causes the total product curve to shift upward, and in Panel B, the increase in __K causes both AP and MP to shift upward.
Note that the two capital levels in Figure 8.2 represent two of the 10 possible short- run situations (see Table 8.2) comprising the firm’s long-run planning horizon.
We are now ready to derive the cost structure of the firm in the short run. For any level of output the manager wishes to produce, the economically efficient amount of labor to combine with the fixed amount of capital is found from the total product curve. In Figure 8.1, if the manager wishes to produce 220 units of output, the amount of labor that will produce 220 units at the lowest total cost is 4 units. The total cost of producing 220 units of output is found by multiplying the price of labor per unit by 4 to get the total expenditure on labor; this amount is then added to the cost of the fixed input. This computation can be done for every level of output to get the short-run total cost of production. We turn now to the costs of production in the short run.
Now try Technical Problem 8.
T A B L E 8.4
The Effect of Changes in Capital Stock
__K 5 2 __K 5 3
L Q AP MP Q AP MP
0 0 — — 0 — —
1 52 52 52 74 74 74
2 112 56 60 162 81 88
3 170 56.7 58 247 82.3 85
4 220 55 50 325 81.3 78
5 258 51.6 38 390 78 65
6 286 47.7 28 425 70.8 35
7 304 43.4 18 453 64.7 28
8 314 39.3 10 474 59.3 21
9 318 35.3 4 488 54.2 14
10 314 31.4 24 492 49.2 4
Output
10 9 8 7 6 5 4 3 2 100
200 400
300 500
Labor
Panel A — Shift in total product when K increases Total product
(K = 2)–
L
Q Total product
(K = 3)–
0 1
10
Marginal and average products
0 1 2 3 4 5 6 7 8 9 10
20 30 40 50 60
Panel B — Shifts in MP and AP when K increases 90
Labor AP, MP
L 70
80
Marginal product (K = 2)–
Average product (K = 2)– Average product
(K = 3)–
Marginal product (K = 3)– F I G U R E 8.2
Shifts in Total, Average, and Marginal Product Curves
I L L U S T R AT I O N 8 . 1 Employing More and Better Capital Boosts
Productivity in U.S. Petroleum and Chemical Industries
One of the important relations of production theory established in this chapter holds that increasing the amount of capital employed by a firm increases the productivity of the other inputs employed by the firm.
Recall in Panel A of Figure 8.2, an increase in capital from 2 to 3 units resulted in higher total output at every level of labor usage, because using more capital causes the total product curve to shift upward. With 3 units of capital, each level of output can be pro- duced with less labor than if the firm employs only 2 units of capital. While increasing the quantity of capital boosts labor productivity and output at every level of labor usage, the productivity gained will be even greater if the additional capital input embodies advanced, state-of-the-art technology. Technological progress makes newly acquired capital equipment more productive than the firm’s existing stock of capital because the older, currently employed capital is designed around less advanced technology. Thus purchasing better capital magnifies the increase in productivity that results from having more capital.
This strategy of combining more and better capital to sharply increase productivity is exactly what has happened in two very important industries: crude oil production and petrochemical refining.
Petroleum-producing firms in the United States are experiencing a period of high productivity in explora- tion operations (the process of finding underground and undersea oil deposits) and development and production operations (the process of getting the oil to the earth’s surface). The amount of new crude oil discovered and produced is increasing rapidly, even as the amount of time, labor, energy, and number of wells drilled have decreased. To achieve the impres- sive gains in productivity, oil-producing firms in the United States invested heavily during the 1990s in new technologies that promised to lower both the cost of finding oil deposits and the cost of getting oil out of the ground. These new technologies involve add- ing new and better types of capital to the exploration and production process. Three of the most important
new and better technologies are three-dimensional seismology, horizontal drilling, and new deepwater drilling technologies.
Three-dimensional (3D) views of underground rock formations provide a tremendous advantage over two-dimensional (2D) seismology techniques.
Even though 3D seismological analysis costs twice as much as 2D analysis, the success rate in exploration is more than doubled and average costs of exploration decrease more than 20 percent. The 3D seismology method, coupled with horizontal drilling techniques and so-called geosteering drill bits, makes it pos- sible to recover more of the oil in the newly discov- ered deposits. Deepwater drilling, which on average yields five times as much crude oil as onshore drill- ing does, is booming now. Advances in deepwater drilling platform technology—such as computer- controlled thrusters using coordinate readings from satellites to keep floating platforms in place—have made deep deposits in the Gulf of Mexico accessible.
Some deep deposits in the Gulf of Mexico are as large as some of the oil fields in the Middle East. Accord- ing to The Wall Street Journal, “Reserves at depths ap- proaching a mile or more now represent the biggest single new old resource since the Middle East came on line in the 1930s.”a
In the petrochemical industry, equally impressive advancements in technology are changing the meth- ods and processes for obtaining fuels and valuable chemicals from crude oil. An article in Fortune maga- zine explains how this is being accomplished: “Instead of highly trained technicians manually monitoring hundreds of complex processes, the work is now done faster, smarter, and more precisely by computer. . . . The result: greater efficiency . . . and significant savings.”b In one such project, BP (formerly British Petroleum), purchased $75 million of new capital to renovate an old petrochemical plant in Texas City. By adding computer-controlled digital automation to the old plant and equipment, the old plant was trans- formed into a leading producer of specialty chemicals.
Adding more (and better) capital increased the pro- ductivity of the other refinery resources, making it possible to decrease the usage of some of these inputs.