Another application of the partial derivative sheds light on one... gate differences in real wages (that is, the wage relative...

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Another application of the partial derivative sheds light on one source of aggregate differences in real wages (that is, the wage relative to the price level) across countries. In neoclassical theory, wages are equal to the marginal product of labor. We
consider the marginal product of labor for the Cobb-Douglas production function,
which takes the form
Q AK 1L,
where 0 1, K is the amount of capital, L is the amount of labor, and A is a measure of productivity.
The marginal product of labor is found by taking the partial derivative of output
with respect to labor. This partial derivative is calculated by using the rules for differentiation presented in the previous chapter and treating capital as a constant. Thus the
marginal product of labor is
K
Q
AK 1L 1 A
L
L
1
.
In neoclassical theory, the real wage is equated with the marginal product of
labor. The partial derivative shows that the real wage depends upon both the amount
of labor in the economy and the amount of capital. The term (KL)1 increases with
either an increase in K or a decrease in L. Thus the marginal product of labor and the
real wage increase with a larger capital stock. This model therefore predicts that countries with larger capital stocks have higher real wages, ceteris paribus.
In a manner completely analogous to the market for labor, neoclassical theory
predicts that the real return to capital, r, is equated to the marginal product of capital,
QK. In a Cobb-Douglas production function, the marginal product of capital is
L Q
(1 )AK L (1 )A
.
K
K
This suggests that the rate of return to capital is relatively high in countries that have a
relatively large labor-capital ratio. These high rates of return should prompt capital
inflows into capital-scarce countries. A puzzle, however, concerns why so little capital
seems to flow from rich to poor countries. This puzzle is addressed in the following
application.
Why Doesn’t Capital Flow to Poor Countries?
In an article published in 1990, Robert Lucas of the University of Chicago calculates
the size of the relative returns to capital in India and the United States with some simple statistics and the relative marginal products of capital.2 To follow Lucas’s argument, we multiply and divide the partial derivative of output with respect to capital
2
Robert E. Lucas, Jr., “Why doesn’t capital flow from rich to poor countries?” American Economic Review
80, no. 2 (May 1990): 92–96.
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given previously by Q(1) in order to have the marginal product expressed as a
function of output per worker, QL. This gives us
r
Q
Q
(1 ) . A 1(1 ) .
K
L
(1 )
.
Lucas considers a value of of 0.6, which is an average of estimated values of this
parameter for India and the United States. Noting that output per capita in the United
States is 15 times larger than output per capita in India and assuming that the parameter A is the same across countries, we find that the ratio of the real return on capital in
the United States and India is
1
rIndia
rU.S.
15
0.60.4
151.5 58.
That is, the return on capital should be about 58 times as large in India as in the
United States! Lucas states that “ . . . in the face of return differentials of this magnitude, investment goods would flow rapidly from the United States and other
wealthy countries to India and other poor countries. Indeed, one would expect no
investment to occur in the wealthy countries in the face of return differentials of this
magnitude.”
Of course, investment does take place in wealthy countries, and there is a relatively small amount of capital flow from rich to poor countries. Lucas addresses this
paradox by reconsidering his calculations. He first notes that workers in the United
States have greater skills (which economists call human capital) than workers in India.
The appropriate measure of QL is then output per effective worker rather than actual
output per worker. Lucas uses an estimate that the average American worker is five
times as productive as the average Indian worker. In this case the ratio of output per
effective worker in the two countries is 3, and the ratio of returns is (13) 0.60.4 5.
While this revised estimate partially addresses the puzzle, the ratio still seems high
given the lack of substantial capital flows.
Lucas next considers an alternative production function that incorporates external effects, whereby an increase in human capital per worker, h, has the additional
effect of increasing overall productivity. A production function that captures this
effect is
Q AK 1Lh.
The marginal product of capital in this case is
Q
Q
(1 ) . A 1(1) .
K
L
(1)
h (1),
where again we multiply and divide the partial derivative of output with respect to
capital by Q(1) in order to have the marginal product expressed as a function of
output per worker. With the estimate of human capital in the United States as five
times that in India, using output per effective worker, and employing an estimate of
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0.4, we find that the ratio of returns in India to those in the United States is3
rIndia
1
rU.S.
3
0.60.4
5
1
0.40.4
1.04.
According to this estimate, then, there is little difference in the rate of return in the two
countries. Thus the differences in skills across countries, along with the impact of these
skill differences on overall productivity, provide one potential reason why capital does
not flow from rich to poor countries.4
A Geometric Interpretation of Partial Derivatives
In the previous chapter, the derivative of a univariate function is illustrated with a twodimensional graph.There it is shown that the derivative of a function at a certain value of
its argument can be interpreted as the slope of a line tangent to the function at that value
of its argument. There is a similar geometric interpretation of the partial derivative.
As an example, consider the Cobb-Douglas production function discussed earlier. We can depict this as a three-dimensional graph linking the two inputs to the level
of output. Figure 8.2(a) depicts this relationship, with the amount of labor measured
along one axis, the amount of capital measured along another axis, and the resulting
level of output measured by the height of the surface above the labor-capital plane.
The depiction of the partial derivative of output with respect to labor is somewhat complicated by the difficulty in depicting three dimensions on a two-dimensional
piece of paper. We can get around this complication by considering the partial derivative as the tangent of a “slice” of the three-dimensional graph. These slices hold constant one factor while varying the level of the other factor. For example, Figure 8.2(b)
presents the slice of the production function for the level of capital K0, and Figure
8.2(c) presents the slice of the production function for the higher level of capital K1.
The marginal product of labor at any level of labor and capital is represented by the
slope of the line tangent to the production function at that level of labor for the slice
taken at that level of capital.
Figures 8.2(b) and 8.2(c) demonstrate the manner in which the marginal product
of labor changes with the amount of labor, with capital held constant. In both diagrams,
the tangent line at the level of labor L0 is steeper than the tangent line at the level of
labor L1. This shows that, with the level of capital held constant, the marginal product of
labor decreases with higher levels of labor. These diagrams also demonstrate that, with
the level of labor held constant, the marginal product of labor increases with the
amount of capital. This is shown by comparing the slopes of the tangents at a common
level of labor in each diagram. At both L0 and L1, the tangent line is steeper in Figure
8.2(c) than in Figure 8.2(b) because the level of capital is higher in Figure 8.2(c).
3
Lucas uses the estimate 0.4 based upon some calculations that draw from empirical research on growth
in the United States, though he acknowledges that he cannot verify the accuracy of the assumptions used to
derive this estimate.
4
Another possible explanation is that there is a paucity of capital flows, despite a large interest rate differential, because the interest differential compensates investors for the riskiness of factors such as the higher
likelihood of expropriation in developing countries.
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