Research Topics

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.

Biased Coefficients and Endogenous Regressors, or a Case of Collective Amnesia?

The possible endogeneity of labor and capital in production functions, and the consequent bias of the estimated elasticities, has been discussed and addressed in the literature in different ways since the 1940s. This paper revisits an argument first outlined in the 1950s, which questioned production function estimations. This argument is that output, capital, and employment are linked through a distribution accounting identity, a key point that the recent literature has overlooked. This identity can be rewritten as a form that resembles a production function (Cobb-Douglas, CES, translog). We show that this happens because the data used in empirical exercises are value (monetary) data, not physical quantities. The argument has clear predictions about the size of the factor elasticities and about what is commonly interpreted as the bias of the estimated elasticities. To test these predictions, we estimate a typical Cobb-Douglas function using five estimators and show that: (i) the identity is responsible for the fact that the elasticities must be the factor shares; (ii) the bias of the estimated elasticities (i.e., departure from the factor shares) is, in reality, caused by the omission of a term in the identity. However, unlike in the standard omitted-variable bias problem, here the omitted term is known; and (iii) the estimation method is a second-order issue. Estimation methods that theoretically deal with endogeneity, including the most recent ones, cannot solve this problem. We conclude that the use of monetary values rather than physical data poses an insoluble problem for the estimation of production functions. This is, consequently, far more serious than any supposed endogeneity problems.