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Frequently, however, real-world data do not exhibit enough dispersion to indicate the full range of increasing and then decreasing returns. In these cases, simpler functional specifications can be used to estimate production functions. The full generality of a cubic function may be unnecessary, and an alternative linear or log-linear model specification can be usefully applied in empirical estimation. The multiplicative production function described in the next section is one such approximation that has proven extremely useful in empirical studies of production relationships.

power production function

Multiplicative relation between input and output

Power Production Functions

One function commonly used in production studies is the power production function, a multiplicative relation between output and input that takes the form

Q = b0XbiYb2

Power functions have properties that are useful in empirical research. Power functions allow the marginal productivity of a given input to depend on the levels of all inputs used, a condition that often holds in actual production systems. Power functions are also easy to estimate

in log-linear form using least squares regression analysis because Equation 7.22 is mathematically equivalent to log Q = log b0 + b1log X + b2 log Y

Returns to scale are also easily calculated by summing the exponents of the power function or, alternatively, by summing the log-linear model coefficient estimates. As seen in Figure 7.10, if the sum of power function exponents is less than 1, diminishing returns are indicated. A sum greater than 1 indicates increasing returns. If the sum of exponents is exactly 1, returns to scale are constant, and the powerful tool of linear programming, described in Chapter 9, can be used to determine optimal input-output relations for the firm.

Power functions have been successfully used in a large number of empirical production studies since Charles W. Cobb and Paul H. Douglas's pioneering work in the late 1920s. The impact of their work is so great that power production functions are frequently referred to as Cobb-Douglas production functions.

The primary determinant of the functional form used to estimate any model of production depends on the relation hypothesized by the researcher. A simple linear approach will be adequate in many instances. In others, a power function or log-linear approach can be justified. When specification uncertainty is high, a number of plausible alternative model specifications can be fitted to the data to determine which form seems most representative of actual conditions.