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Estimating Production Functions
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
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As noted, capital is twice as expensive as labor. As a result, for the CobbDouglas function, the firm employs half the number of units of capital as it does of labor. [.5L .5K.5]/12 [.5L.5K .5]/24.
K .5L.
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 production 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 engineering 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 production 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
9Another 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.
