# Marginal product

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In economics and in particular neoclassical economics, the marginal product or marginal physical product of an input (factor of production) is the extra output that can be produced by using one more unit of the input (for instance, the difference in output when a firm's labor usage is increased from five to six units), assuming that the quantities of no other inputs to production change. [1]

The marginal product of a given input can be expressed [2] as

$MP = \frac{\Delta Y}{\Delta X}$

where $\Delta X$ is the change in the firm's use of the input (conventionally a one-unit change) and $\Delta Y$ is the change in quantity of output produced. Note that the quantity $Y$ of the "product" is typically defined ignoring external costs and benefits.

If the output and the input are infinitely divisible, so the marginal "units" are infinitesimal, the marginal product is the mathematical derivative of the production function with respect to that input. Suppose a firm's output Y is given by the production function

$Y=F(K,L)$

where K and L are inputs to production (say, capital and labor). Then the marginal product of capital (MPK) and marginal product of labor (MPL) are given by:

$MPK=\frac{\partial F}{\partial K}$
$MPL=\frac{\partial F}{\partial L}$

In the "law" of diminishing marginal returns, the marginal product of one input is assumed to fall as one considers higher and higher starting points for the quantity of that input. The marginal product of labor is the slope of the total product curve, which is the production function plotted against labor usage for a fixed level of usage of the capital input.

In the neoclassical theory of competitive markets, the marginal product of labor equals the real wage. In aggregate models of perfect competition, in which a single good is produced and that good is used both in consumption and as a capital good, the marginal product of capital equals its rate of return. As was shown in the Cambridge capital controversy, this proposition about the marginal product of capital cannot generally be sustained in multicommodity models in which capital and consumption goods are distinguished.

## References

1. ^ Brewer, Anthony (2010). The Making of the Classical Theory of Economic Growth (in English). Routledge. ISBN 0415486203.
2. ^ Mukherjee, Sampat; Mukherjee, Mallinath; Ghose, Amitava (2003). Microeconomics (in English). New Delhi: Prentice-Hall of India. ISBN 81-203-2318-1.