Israfil Roshdi | The University of Auckland Business School 

We develop a piecewise log-convex environmental technology by introducing a multiplicative variant of the weak disposability assumption of outputs. Using several numerical and real-world data sets, we elaborate the structural differences between the piecewise log-convex environmental technology and the traditional piecewise linear one. Our analysis demonstrates that, despite the piecewise linear technology which requires marginal products to be increasing and an everywhere concave frontier, the envelopments of the piecewise log-convex technology being of Cobb-Douglas type allow for all three possible types of marginal products - increasing, constant and decreasing - and thus capture all three types of production structures - concavity, linearity and convexity. To gauge the environmental performance, we develop new radial/non-radial directional hyperbolic distance functions (DHDFs) that satisfy several desirable properties and encompass the existing measures as special cases. While the formulation of the hyperbolic distance function with respect to a piecewise linear environmental technology leads to a non-linear optimization problem, the value of DHDF with reference to the proposed piecewise log-convex one can be simply computed via the directional distance function, using a logarithmic transformation. Moreover, our models provide useful information about the returns to scale and shadow prices of the inputs and outputs quantities.

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