Imposing and Testing Curvature conditions on a Box-Cox Function

Many empirical studies in applied production analysis are based on functional forms that have to satisfy some curvature conditions in order to be compatible with microeconomic theory. Curvature conditions means that the cost function must be concave in factor prices, which implies that the factor demand curves are downward-sloping (i.e. a rise in the price of a factor will lead to a decline in the demand for it). Tests of the concavity property assumption appear interesting both from the economic and statistical perspective. When the curvature conditions are statistically rejected a concavity unrestricted specification should be preferred. Furthermore, price elasticities can be used with more confidence in applied policy simulations when curvature conditions are tested rather than checked. Since violations of price concavity may call into the question the applicability of the neoclassical theory, tests of the concavity assumption appear also interesting from the economic perspective. For instance positive own-price elasticities may indicate that the producer does not minimize costs or that heterogeneity over the observations has been neglected. Until recently very few contributions test whether concavity is statistically rejected. Furthermore, few contributions experimented with different functional forms of the cost function.
The aim of the paper is to present a new method for imposing and testing concavity of a cost function. The advantage of our framework over existing methods is that it can be applied for imposing curvature conditions on a great variety of functional specifications. We provide an empirical application with a new variant of a Box-Cox function nesting six different functional forms. We compare the empirical performance as well as the elasticity estimates of the alternative functional forms. In addition, we compare price elasticities when curvature conditions are imposed and when they are not.
A main result is that concavity is rejected for all functional forms considered. However, the estimates are not very sensitive to whether concavity is imposed or not. In contrast, the elasticities are sensitive on the choice of the functional form. In particular, the Translog overestimates the extent of the substitution, while the normalized quadratic underestimates it.

Koebel, Bertrand, Martin Falk und François Laisney (2000), Imposing and Testing Curvature conditions on a Box-Cox Function, ZEW Discussion Paper Nr. 00-70, Mannheim.