Estimating the Reliability of Quantities Derived from Empirical Production Functions

T HE estimation of production functions and the use of the empirically derived functions to estimate optimum levels and combinations of resource inputs has become quite common in agricultural economics research. Although measures of reliability (e.g. variances or confidence intervals) are generally given for the original coefficients, similar measures are seldom if ever reported for the derived quantities' (e.g. maximum profit points, isoclines, etc.). Since the derived quantities are often the end products of, as well as the motivation for, such research, it would seem highly desirable to have available measures of reliability for these estimates. It is the purpose of this paper to illustrate how techniques for variance estimation and setting of confidence regions may be applied to the estimates derived from estimated production functions. The derived quantities can in general be expressed as functions of the estimated coefficients of the production function. Thus the statistical properties of these quantities are determined by the statistical properties of the estimated production function coefficients. The following formulation for the variances of such quantities will be used extensively. Given that a,, a2, * , ar are r variables distributed with variance-covariance matrix estimated unbiasedly by 2; and zl, z2, ? * * , z,, are m variables defined by given functions of the a's, e.g. zl=fl(ai, a2, '* * , ar), z2=f2(ai, a2, ' ' , ar), etc.; then the variance-covariance matrix of the z's is estimated by D D' (1) where D is the m by r matrix: azi Zzl zzl'