A Methodology for Estimating Production Parameters

Abstract

T HE algebraic production function is the classical method of specifying physical relationships between inputs and output. The traditional approach in determining coefficients for an aggregate production function has been a direct least-squares estimation, usually based on time-series data. An important problem that develops is highly correlated "independent" variables. It is not possible to obtain reliable estimates of the parameters in a production function such as (1) (if n is more than two or three) because of the high correlations between the inputs X i .l Therefore, output is usually regressed on a limited number of highly aggregated variables; i.e., output Y is "described," but the structural validity and usefulness of the parameter estimates is questionable.A second approach-the use of factor shares as production elasticitiestrades the intercorrelation problem for the dubious assumption that economic equilibrium prevails.t Use of production coefficients derived in such a manner to determine optimum resource allocation and output is anachronistic.The approach used in this paper is an attempt to combine favorable features of the two approaches described above. RationaleThe factor share is defined as the ratio of expenditure on a resource to the value of output. If equilibrium quantities of all resources are used, then the factor share and elasticity of production are equal, as shown below. The factor share is (2) by definition: PiXI(2) F, = --. P 7YIn equilibrium the marginal product of XI is equal to the factor-product price ratio, as in (3): 1 Simple correlation coefficients between the nine input categories used in this study (there are 9!/2!7!=86 correlations) were .78•.88•••.•.99. The matrix of sums of squares is then so nearly singular as to give highly dubious results.I For published work on factor shares see

Elasticities of Production and Resource Earnings

“…Then substituting the estimated adjustment coefficients in equation (2) and solving for Fit* yields production elasticity estimates. One problem encountered in fitting equation (4) in logarithm form is that large percentage changes in the factor shares from year t -1 to year t sometimes result in unreasonably large annual fluctations in estimated production elasticities. This problem could be alleviated by using moving averages of factor shares in the computations.…”

“…The resulting elasticities, with pricequantity components of factor shares, estimates of product price expectations, and estimates of actual and expected overhead costs, are used to estimate (1) ex post and ex ante MVP's, both gross and net of overhead costs, (2) ex post and ex ante present values of discounted future net MVP's of durables, (8) capital gains from durables, and (4) economic rents on expendables.The equilibrium factor-share adjustment coefficient technique of estimating Cobb-Douglas production parameters is advocated by Tyner and Tweeten [4]. Its advantages over traditional direct least-squares estimation of production parameters are that (1) the multicollinearity problem encountered from time series clustering around expansion paths is avoided, and (2) while the direct least-squares production parameter estimates are constant over the course of time and different input combinations, the technique of adjusting factor shares yields different elasticity estimates over the course of time and varying input combinations.…”

“…The equilibrium factor-share adjustment coefficient technique of estimating Cobb-Douglas production parameters is advocated by Tyner and Tweeten [4]. Its advantages over traditional direct least-squares estimation of production parameters are that (1) the multicollinearity problem encountered from time series clustering around expansion paths is avoided, and (2) while the direct least-squares production parameter estimates are constant over the course of time and different input combinations, the technique of adjusting factor shares yields different elasticity estimates over the course of time and varying input combinations.…”

The Elasticity of Aggregate Australian Agricultural Supply: Estimates and Policy Implications

“…Three methods for estimating input shares have been used in past studies and all three are used here: the non-normalised gross output method, the normalised gross output method (normalised over capital inputs and labour only) and the total expenditure method (normalised over all inputs). Discussions of these methods can be found in Tyner and Tweeten (1965), Rayner (19706), Young (1971), Rosine and Helmberger (1974), Shumway et al (1979) and Pandey (1981).…”

The Factor Share Approach to Production Function “Estimation”: Actual or Estimated Equilibrium Shares?

“…In an effort to circumvent the multicollinearity barrier posed by direct estimation, Tyner and Tweeten (1965) indirectly estimated aggregate production elasticities for U.S. agriculture using a factor share approach. Noting that in equilibrium, factor shares equal factor elasticities for an industry comprised of profitmaximizing firms, Tyner and Tweeten synthesized a Cobb-Douglas production function by estimating equilibrium factor shares based on historical data and the Nerlove partial adjustment model.…”