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of production requires balanced increases in the necessary inputs. Like linear

production, fixed-proportions production should be thought of as an extreme

case. Rarely is there no opportunity for input substitution. (For example, it is

true that a crane needs an operator but, at a more general level, extra con-

struction workers can substitute for construction equipment.)

However, fixed-proportions production has an important implication. In

the face of an increase in an input’s price, the firm cannoteconomize on its

use, that is, substitute away from it. Thus, a petrochemical firm that uses fixed

proportions of different chemicals to produce its specialty products is at the

mercy of market forces that drive up the prices of some of these inputs.

Polynomial Functions

In the polynomial form, variables in the production function are raised to posi-

tive integer powers. As a simple example, consider the quadratic form

where a and b are positive coefficients. It is easy to check that each input shows

diminishing returns. (For example, MPL Q/ L aK 2bK^2 L, which

declines as L increases.) The quadratic form also displays decreasing returns to

scale. A more flexible representation is the cubic form,

where all coefficients are positive. We can show that this function displays

increasing returns for low-output levels and then decreasing returns for high-

output levels. The marginal product of an input (say, labor) takes the form

We see that marginal product is a quadratic function in the amount of labor;

that is, it is a parabola that rises, peaks, and then falls. Thus, this functional

form includes an initial region of increasing marginal productivity followed by

diminishing returns.

The Cobb-Douglas Function

Perhaps the most common production function specification is the Cobb-

Douglas function

QcL^ K^ , [5.6]

MPL    Q/ L(aKcK^2 eK^3 )2bKL3dKL^2.

QaLKbL^2 KcLK^2 dL^3 KeLK^3 ,

QaLKbL^2 K^2 ,

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