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EXAMPLE 4 Suppose the firm faces the production function Q L.5K.5

and input prices are PL $12 and PK $24. (The inputs are equally

productive, but capital is twice as expensive as labor.) The optimal input mix

satisfies Equation 5.4 so that

After collecting terms, we get K.5/K.5(12/24)L.5/L.5, or

As noted, capital is twice as expensive as labor. As a result, for the Cobb-

Douglas function, the firm employs half the number of units of capital as it

does of labor.

Estimating Production Functions

Data for estimating production functions come in a number of forms.

Engineering data can provide direct answers to a number of production

questions: On average, how much output can be produced by a certain type

of machine under different operating conditions? How many bushels of a

particular crop can be grown and harvested on land (of known quality)

using specified amounts of labor, capital, and materials (such as fertilizer)?

Such information usually is based on experience with respect to similar (or

not so similar) production processes. Consequently, the estimated produc-

tion function is only as accurate as the past production experience on which

it is based. The development of new weapons systems is a case in point.

Although production and cost estimates are based on the best available engi-

neering estimates (and possibly on tests of prototypes), they nonetheless are

highly uncertain.^9

A second source of production information is production data. For

example, in a production time-series analysis, the firm’s managers compile

a production history, month by month or year by year, recording the

amounts of inputs (capital, labor, land, materials, and so on) used in pro-

duction and the resulting level of output. Alternatively, the economic data

may come in the form of a cross section. In this case, information is gathered

for different plants and firms in a given industry during a single period of

time. For instance, by observing production in the auto industry, one can

address a number of important questions: For plants of fixed size (possibly

K.5L.

[.5L.5K.5]/12[.5L.5K.5]/24.

210 Chapter 5 Production

(^9) Another limitation of engineering data is that they apply only to parts of the firm’s activities,
typically physical production operations. Thus, such data shed little light on the firm’s marketing,
advertising, or financial activities.
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