Marginal and Average Physical Products of Labour

Suppose that your work force is becoming so specialized that you decide to apply some concepts you learned in college. The data in columns 1 and 2 of Table 1 relate production and various levels of labor inputs, holding other resources constant. This data represents the total product curve. Note that total product curves and production functions are not the same things. A production function allows all inputs to vary, while the total product curve assumes that only one input changes. We are using labor as the variable input, but had we held labor constant and varied capital (or land), the analysis would be quite similar, although the specific curves would differ.

Table 1 Total Output and the Average and Marginal Physical Products of Labor (Sand and Gravel Operation)

 (1) LABOR Workers per 8-hr. Shift      (L) (2) OUTPUT Tons of Sand and Gravel Removed Daily           (q) (3) APPL Average Physical Product of Labor            (q/L) (4) MPPL Marginal Physical Product of Labor (Dq/DL) 0 0 0 0 1 10 10.00 10 2 22 11.00 12 3 36 12.00 14 4 52 13.00 16 5 70 14.00 18 6 86 14.33 16 7 100 14.28 14 8 112 14.00 12 9 122 13.55 10 10 130 13.00 8 11 137 12.45 7 12 143 11.92 6 13 148 11.38 5 14 152 10.85 4 15 155 10.33 3 16 157 9.81 2 17 158 9.29 1 18 158 8.78 0 19 157 8.26 -1

Table 2           Total Output, Total Costs, and Fixed and Variable Costs (Sand and Gravel Example)

 (1)     Workers per 8-hr Shift (L) (2)   Tons of Sand & Gravel Removed Daily (q) (3)   Wages per worker (\$) (8 hrs. Daily) (w) (4) Total Variable Cost (\$) (w x L) (TVC) (5)     Total Fixed Cost (\$) (TFC) (6)       Total Costs (\$) (TVC+TFC=TC) 0 0 50 0 100 100 1 10 50 50 100 150 2 22 50 100 100 200 3 36 50 150 100 250 4 52 50 200 100 300 5 70 50 250 100 350 6 86 50 300 100 400 7 100 50 350 100 450 8 112 50 400 100 500 9 122 50 450 100 550 10 130 50 500 100 600 11 137 50 550 100 650 12 143 50 600 100 700 13 148 50 650 100 750 14 152 50 700 100 800 15 155 50 750 100 850 16 157 50 800 100 900 17 158 50 850 100 950 18 158 50 900 100 1000 19 157 50 950 100 1050

If you know total output for each level of labor hired (columns 1 and 2), output per worker is calculated by dividing total output (q) by labor (L).

The average physical product of labor (APPL) equals total output divided by labor (q/L).

These figures are entered in column 3 of the table. You will also want to know how much each extra worker adds to total output.

The marginal physical product of labor (MPPL) is the additional output produced by an additional unit of labor   -   computed by dividing the change in total output (Dq) by the change in labor (DL): Dq/DL.

Hiring decisions intended to maximize profit hinge on labor's marginal physical product. Extra workers will not be hired unless the extra revenue from their marginal physical products would exceed the extra costs of hiring them. Only workers generating at least as much revenue as it costs to hire them will be employed. In most cases, each worker's productivity (the MPPL) will be higher as the amounts of other resources used rise---a worker operating a bulldozer on a dry riverbed will produce more sand and gravel than a shovel wielder digging on a city lot.

The MPPL is calculated by looking at small changes in labor hired and the resulting changes in output. With large numbers of workers (as at a steel mill), a given change in the amount of labor (DL) is divided into the resulting change in output (Dq) to approximate the MPPL. One worker = DL for a small firm like your operation.  Labor's marginal physical products (Dq/DL) for your firm are listed in column 4 of Table 1.

The total product curve graphed in Panel A of Figure 1 (from columns 1 and 2 of Table 1) for your sand-and-gravel operation relates production and various levels of labor inputs, holding other resources constant. Panel B shows the corresponding marginal and average physical products of labor.

As more and more labor is employed, total output (Panel A) initially rises at an increasing rate because of gains from specialization. In this range, the marginal and average physical products of labor both grow (Panel B). As congestion begins to emerge, total output continues to grow, but at a falling rate. In this range, average physical product continues to climb, but marginal productivity diminishes. Once marginal physical product falls below average physical product, average physical product begins to fall. However, total output continues to rise until the marginal physical product is zero. This occurs when congestion or other problems are so severe that further labor inputs cause output to fall.

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