Imposing and Testing Curvature Conditions on a Box–Cox Cost Function

Abstract

Die Discussion Papers dienen einer möglichst schnellen Verbreitung von neueren Forschungsarbeiten des ZEW. Die Beiträge liegen in alleiniger Verantwortung der Autoren und stellen nicht notwendigerweise die Meinung des ZEW dar.Discussion Papers are intended to make results of ZEW research promptly available to other economists in order to encourage discussion and suggestions for revisions. The authors are solely responsible for the contents which do not necessarily represent the opinion of the ZEW.Download this ZEW Discussion Papers from our ftp server:ftp://ftp.zew.de/pub/zew-docs/dp/dp0070.pdf Abstract. We present a new method for imposing and testing concavity of a cost function using asymptotic least squares, which can easily be implemented even for cost functions which are nonlinear in parameters. We provide an illustration on the basis of a (generalized) Box-Cox cost function with six inputs: capital, labor disaggregated in three skill levels, energy, and intermediate materials. A parametric test of the concavity of the cost function in prices is presented, and price elasticities are compared when curvature conditions are imposed and when they are not. The results show that, although concavity is statistically rejected, the estimates are not very sensitive to its imposition. We find that substitution is stronger between the different types of labor than between any other pair of inputs.

Exports and Labor Demand

“…In earlier work on similar data, Koebel et al (2003) found that the datatransformation parameters γ also varied across groups of industries; thus here we allow these parameters to vary across industries (hence the notation γ 1n and γ 2n ). This means that the functional form of the cost function is allowed to differ across the industries; it may, for example, be translog for some industries (those with γ 1n → 0 and γ 2n → 0), whereas for others, a normalized quadratic specification (γ 1n = γ 2n = 1) is more adequate.…”

“…For the special case in which γ 1 = 1 and γ 2 = 1, the normalized quadratic functional form is obtained, whereas for γ 1 → 0 and γ 2 → 0, (33) coincides with the translog. Other usual functional forms nested within (33) have been discussed by Koebel et al (2003).…”

“…A version of Berndt and Khaled's (1979) Box-Cox cost function, extended to nest both the normalized quadratic and the translog functional forms by Koebel, Falk, and Laisney (2003), is considered here. Let p = (w , q ) be the price vector.…”

“…The assumption that γ 1n = γ 2n , leading to a Berndt and Khaled (1979) point, γ = (.71, .30), a test for the hypothesis that the functional family is the same across industries is rejected at the 1% threshold. Several usual functional specifications that are nested within the Box-Cox are all rejected (see Koebel et al 2003).…”

Increasing Pressure on Unregulated Species Due to Changes in Individual Vessel Quotas: An Empirical Application to Trawler Fishing in the Baltic Sea

“…Symmetry requires α ik = α ki and α jh = α hj , imposing homogeneity implies Σ iI α ij = 0, whereas convexity is imposed locally at average values according to methodology described by Koebel, Falk, and Laisney (2003). The procedure implies a correction of unrestricted parameters according to: α Ã ¼ αþ Ω α ∂g′ðαÞ ∂α ∂gðαÞ ∂α′ Ω α ∂g′ðαÞ ∂α −1 ðη Ã H −∂gðαÞÞ, where α Ã is corrected for the convexity coefficient, α is the original regression coefficient, Ω α is the original regression variance-covariance matrix, gðαÞ is vector of linear independent values of the Hessian matrix for average prices and quantities of fixed factors, and η Ã H −∂gðαÞ is the difference between the parameters for the convexity restricted and unrestricted Hessians.…”