Profit Function Approximations and Duality Applications to Agriculture

Abstract

Monte Carlo simulations of profit functions indicate that flexible functional forms may be ranked nearly the same with respect to Allen-Uzawa partial substitution elasticities or price and fixed factor elasticities. Flexible functional form approximations of output price elasticities are generally worse than approximations of input price or fixed factor elasticities. Minflex Laurent functional forms did not appear to approximate underlying elasticities any better than did their translog and generalized Leontief counterparts.Monte Carlo studies have been employed to assess the accuracy of flexible functional forms (FFF) in approximating known underlying technologies and preferences. Monte Carlo production studies have compared Allen-V zawa partial elasticities of substitution generated from cost functions (Guilkey and Lovell; Guilkey, Lovell, and Sickles; Chalfant and Gallant) or profit functions (Dixon, Garcia, and Anderson) with Allen-Uzawa partial substitution elasticities (AVES) derived from a known underlying production function.While the Monte Carlo production studies have measured dual approximations solely in technical terms, many recent agricultural economics studies have utilized FFF specifications of the profit function to derive price elasticities (Sidhu and Baanante, Weaver, Shumway, Lopez, Antle). These applications of FFF raise the methodological issue of the extent to which conclusions about the performance of FFF in relation to AVES can be extrapolated to price elasticities. The purpose of this study is to perform quantitative comparisons of different functional forms with respect to the two types of measures-AVES and price elasticities-that are of interest in empirical studies.A Monte Carlo simulation is used to measure the performance of five FFF in recovering AVES and price elasticities from samples Gary D. Thompson is an assistant professor, and Mark Langworthy is an assistant research scientist, Department of Agricultural Economics, University of Arizona.where these parameters are known. Three commonly used FFF-translog, generalized Leontief, and quadratic-and two newer FFF-minflex Laurent translog and minflex Laurent generalized Leontief-are compared for the single output normalized profit function. V se of Allen-V zawa partial substitution elasticities and price elasticities for evaluating dual profit function approximations provides information about the FFF's ability to track both technical and economic relationships. Price and Allen-Uzawa ElasticitiesThe theoretical link between price elasticities and Allen-V zawa partial elasticities of substitution can be derived from Lau's conjugate duality development of the single output normalized profit function. With certain regularity conditions satisfied (see Lau,, the primal production function and the dual profit function are related directly through Hessian matrices. In particular, F u ] and [G u ] are the (n x n) Hessian matrices of the production function y = f(x; z) and the normalized profit function G = G(w; z), respectively, y is outp...

Empirical properties of duality theory

“…The effect of choice of functional form on the ability to recover dual results has also been studied (e.g. Thompson and Langworthy 1989). The normalised quadratic form was chosen as both: (i) the true underlying technological data-generating process; and (ii) the flexible functional form used for estimating the profit, restricted profit and production functions.…”

“…However, Chambers and Quiggin (1998) illustrated that, in some cases, cost functions can be theoretically derived from stochastic production functions, and Coyle (1999) and Pope and Just (1998) suggested alternative methods of estimating cost and profit functions under risk aversion. Thompson and Langworthy (1989) illustrated that elasticities calculated from primal and dual approaches depend on the choice of functional form. They showed that identical results will never be obtained from primal and dual approaches unless the flexible functional form is selfdual (such as the quadratic) and the underlying data-generating process identically matches the chosen functional form.…”

How Reliable is Duality Theory in Empirical Work?

“…There are two main reasons for adopting the normalized quadratic functional form (4) in the present study. First, this function has been broadly employed in applications of the duality approach (Huffman and Evenson 1989; Thompson and Langworthy 1989;Shumway and Lim 1993;Lusk et al 2002;Arnade and Kelch 2007;Schuring, Huffman, and Fan 2011). Second and most important, the normalized quadratic function is self-dual, with the Hessian matrix of the profit and production functions depending on parameters only.…”

“…Considerable effort has been put into testing the most appropriate flexible functional form (FFF) for a given dataset (Guilkey, Lovell, and Sickles 1983;Dixon, Garcia, and Anderson 1987;Thompson and Langworthy 1989), because the utilized FFF drives the results. Analyses of this type usually consist of the following steps.…”

Are Dual and Primal Estimations Equivalent in the Presence of Stochastic Errors in Input Demand?

“…Pope (1980) and Pope (1982a) suggested that duality results do not hold for stochastic models. Thompson and langworthy (1989) and Pope and Just (2002) suggest that the functional form may affect elasticity estimates in the primal and dual methods. Lusk et al (2002) underscored the need of high-quality databases to estimate dual relationships in the presence of measurement error and low variability of relative prices for different sample sizes.…”