The Estimation of Production Functions with Monetary Values

For decades, the literature on the estimation of production functions has focused on the elimination of endogeneity biases through different estimation procedures to obtain the correct factor elasticities and other relevant parameters. Theoretical discussions of the problem correctly assume that production functions are relationships among physical inputs and output. However, in practice, they are most often estimated using deflated monetary values for output (value added or gross output) and capital. This introduces two additional problems—an errors-in-variables problem, and a tendency to recover the factor shares in value added instead of their elasticities.  The latter problem derives from the fact that the series used are linked through the accounting identity that links value added to the sum of the wage bill and profits. Using simulated data from a cross-sectional Cobb-Douglas production function in physical terms from which we generate the corresponding series in monetary values, we show that the coefficients of labor and capital derived from the monetary series will be (a) biased relative to the elasticities by simultaneity and by the error that results from proxying physical output and capital with their monetary values; and (b) biased relative to the factor shares in value added as a result of a peculiar form of omitted variables bias. We show what these biases are and conclude that estimates of production functions obtained using monetary values are likely to be closer to the factor shares than to the factor elasticities. An alternative simulation that does not assume the existence of a physical production function confirms that estimates from the value data series will converge to the factor shares when cross-sectional variation in the factor prices is small. This is, again, the result of the fact that the estimated relationship is an approximation to the distributional accounting identity.