Constrained Joint Estimation of Factor Demand and Production Functions

Abstract

FIRMS determine their demands for factors of production by finding the combination of factor inputs that will maximize profits or minimize costs, subject to a technological constraint the production function. Because the production function is common to decisions for all factors, the factor demands must be interrelated; each factor demand function contains parameters from the underlying production function which also appear in other demand functions. Yet factor demand equations are commonly estimated independently, with the consequence that production function parameters implied by the different equation estimates may well be inconsistent. If the demand functions are to be included in a complete model of an economy, it is certainly desirable, if not essential, that they be consistent, i.e., that they imply a single set of parameters for the underlying production function. The authors are currently engaged in constructing a medium-term macro-model of the United States economy, an important subsector of which consists of aggregative labor and investment demand functions. This paper summarizes our efforts at estimating these two demand functions with data extending back to the early 1920's, treating relative prices and financial variables as exogenous. To illuminate the methodological issue just raised, we present results for three estimation procedures: (1) independent estimation of both relations, (2) a two-step approach in which production function parameters implied by independent estimates of one function are imposed in estimating the other relation, (3) joint estimation of both functions. By the nature of our approach, we obtain estimates of the production function from estimates of the factor demand relations. Indirect estimation of the production function in this way has also been suggested, though not carried out, by Dhrymes [7] and Nerlove [15]. It seems to us to be preferable to several alternatives. The production function is a constraint relating desired or expected (long-run equilibrium) output to desired (long-run equilibrium) factor inputs, and these are generally not observable variables. Expected output may be related to current and past levels of output, and observed factor inputs are likely to be disequilibrium values associated with lagged adjustment to desired levels. Use of actual current output and actual current inputs to estimate the production function directly is objectionable for these reasons. The frequent practice of correcting measured capital stock for its utilization (usually by assuming that capital is unemployed to the same extent as labor) recognizes that the observed capital stock is not the equilibrium quantity to which the production function refers, but it seems an inadequate way of accounting for expectations and adjustment lags in capital stock decisions. Also, if lagged adjustment characterizes the labor input so that labor, like capital, is a partially fixed factor, then labor input (employment) should also be corrected for utilization. The procyclical behavior of labor productivity suggests that this is the case. A second shortcoming of many direct estimates of the production function is that they ignore information contained in marginal productivity conditions. This information can usually be summarized in a so-called expansion path equation, i.e., an equation relating the ratio of desired factor inputs to the ratio of their prices. If the production function is estimated independently of the expansion path equation, the estimates will be statistically inconsistent. For this reason, among others, the expansion path is sometimes estimated first, and the information so obtained is then used in *This research was supported by National Science Foundation Grant No. GS1686. The authors wish to acknowledge the assistance of Richard Freeman, Margaret Simms, and Robert Willig on much of the computational work. A preliminary version of the paper was read at the Winter Meetings of the Econometric Society, December 28, 1968. We are indebted to Sherwin Rosen for his constructive suggestions, especially as concerns the clarification of our hypotheses on the adjustment process.