Further Details on Specific Factors

Further Details on Specific Factors
The specific factors model developed in this chapter is such a convenient tool of analysis
that we take the time here to spell out some of its details more fully. We give a fuller treatment
of two related issues: (1) the relationship between marginal and total product within
each sector; (2) the income distribution effects of relative price changes.
Marginal and Total Product
In the text we illustrated the production function of cloth in two different ways. In Figure 4-1
we showed total output as a function of labor input, holding capital constant. We then
observed that the slope of that curve is the marginal product of labor and illustrated that marginal
product in Figure 4-2. We now want to demonstrate that the total output is measured by
the area under the marginal product curve. (Students who are familiar with calculus will find
this obvious: Marginal product is the derivative of total, so total is the integral of marginal.
Even for these students, however, an intuitive approach can be helpful.)
In Figure 4A-1 we show once again the marginal product curve in cloth production.
Suppose that we employ person-hours. How can we show the total output of cloth?
Let’s approximate this using the marginal product curve. First, let’s ask what would happen
if we used slightly fewer person-hours, say fewer. Then output would be less. The
fall in output would be approximately
that is, the reduction in the work force times the marginal product of labor at the initial
level of employment. This reduction in output is represented by the area of the colored
dLC * MPLC,
dLC
LC
Marginal product
of labor, MPLC
Labor
input, LC
dLC
MPLC
Figure 4A-1
Showing that Output Is Equal to
the Area Under the Marginal
Product Curve
By approximating the marginal
product curve with a series of thin
rectangles, one can show that the
total output of cloth is equal to
the area under the curve.
rectangle in Figure 4A-1. Now subtract another few person-hours; the output loss will be
another rectangle. This time the rectangle will be taller, because the marginal product of
labor rises as the quantity of labor falls. If we continue this process until all the labor is
gone, our approximation of the total output loss will be the sum of all the rectangles shown
in the figure. When no labor is employed, however, output will fall to zero. So we can
approximate the total output of the cloth sector by the sum of the areas of all the rectangles
under the marginal product curve.
This is, however, only an approximation, because we used the marginal product of only
the first person-hour in each batch of labor removed. We can get a better approximation if
we take smaller groups—the smaller the better. As the groups of labor removed get infinitesimally
small, however, the rectangles get thinner and thinner, and we approximate ever
more closely the total area under the marginal product curve. In the end, then, we find that
the total output of cloth produced with labor , , is equal to the area under the marginal
product of labor curve up to .