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Measuring Production Functions 207

This is exactly the condition established in Equation 5.4. The marginal prod-

uct per dollar of input should be the same across all inputs.

MEASURING PRODUCTION FUNCTIONS

In this section, we briefly discuss ways in which managers can estimate and

measure production functions based on engineering or economic data. Let us

begin by considering four common specifications.

Linear Production

As the term suggests, a linear production functiontakes the form

[5.5]

where a, b, and c are coefficients that must be estimated from the data. An

immediate implication of linearity is that each input’s marginal product is con-

stant: MPLa and MPKb. Constant marginal productivity may approximate

production over a limited range of input usage, but at sufficiently high levels

of inputs, it is at odds with the law of diminishing marginal productivity. In this

sense, the linear form is too simple and should be viewed as a somewhat

extreme case.

Because of the constant marginal products, the inputs are perfect substitutes

for one another. Suppose, for example, that the production function is Q 

20L 40K. In this case, one can always substitute two units of labor for one of

capital to maintain the same level of production, and vice versa. Given fixed

input prices, production will be “all or nothing” in the long run. If the unit

cost of capital is less than twice the wage per unit of labor, the firm’s least-cost

means of production is to use only capital. In contrast, if labor is the less expen-

sive option, production should use labor exclusively. In general, as long as

MPK/PKMPL/PL, the firm should use capital exclusively (and vice versa if the

inequality is reversed).

Production with Fixed Proportions

Production with fixed proportionsis the opposite extreme from linear pro-

duction; fixed-proportions production allows no input substitution. Output

can only be produced with a fixed proportion of inputs. Simple examples

include a taxi and its driver or a construction crane and its operator. In both

cases, the required mix of labor to capital is one to one. An excess of either

input—a machine without an operator or vice versa—does no good. Expansion

QaLbKc,

c05Production.qxd 9/5/11 5:49 PM Page 207