Incorporating monotonicity and concavity conditions in flexible functional forms

Abstract

SUMMARY Empirical economists using flexible functional forms often face the disturbing choice of drawing inferences from an approximation violating properties dictated by theory or imposing global restrictions that greatly restrict the flexibility of the functional form. Focusing on the cost function, this paper presents an alternative approach which imposes monotonicity and concavity properties only over the set of prices where inferences will be drawn. An application investigating elasticities for Bemdt-Wood data set using the translog, generalized Leontief, and symmetric generalized McFadden flexible functional forms illustrates the technique. Violations of theoretical properties pose one of the most difficult challenges faced by empirical economists. The introduction of flexible functional forms provided specifications capable of approximating many technologies and of violating monotonicity and concavity conditions implied by theory. Imposing these restrictions at all non-negative prices eliminates this problem, but with a significant loss in flexibility. This paper assesses the benefits of imposing monotonicity and concavity on cost functions over a range of prices through appropriate choice of prior distribution. The evolution of flexible functional forms approximating cost functions over the last twenty years supplies a clear indication of the importance of incorporating properties from theory into empirical analysis. Diewert's (1971) introduction of the generalized Leontief and Christensen, Jorgenson, and Lau's (1973) translog supplied attractive alternatives to earlier Cobb-Douglass and CES forms. These fexible functional forms are capable of attaining arbitrary price elasticities at a point, unlike other forms which imposed strong restrictions on elasticities between inputs. Unfortunately, such advances leading to gains in flexibility also produced functional forms that often violate monotonicity and concavity properties implied by theory. Caves and Christensen (1980) and Barnett and Lee (1985) find small regular regions for the translog and generalized Leontief, and concavity violations often appear in empirical applications using these forms.1 For example, Diewert and Wales (1987) report violations of