Exponential distance function and duality theory

Charnes, Cooper, Seiford, and Stutz (1982, 1983) and Banker and Maindiratta (1986) suggested multiplicative radial measures for efficiency gauging. More recently, Peyrache and Coelli (2009) presented a multiplicative directional distance function that was elaborated by Mehdiloozad, Sahoo, and Roshdi (2014). In this paper, we extend these studies and propose an exponential approach based upon a new exponential distance function endowed with a multiplicative production structure. The main purposes of this paper are twofold: one is to provide a general production theoretic basis for the approach and the other is to extend it to a nonparametric framework. The first purpose is accomplished as follows: (1) the exponential distance function is formally defined and its properties are established; (2) it is shown how the exponential distance function is characterized under a Napierian technology; (3) a duality relationship between Napierian profit and the exponential distance functions is established; (4) shadow prices of inputs and outputs are derived based on the Napierian technology. The second purpose is accomplished by providing nonparametric programming extensions, which include data envelopment analysis (DEA) models, productivity indexes and returns to scale models. Here, the efficacy of our nonparametric theoretical results is demonstrated by applying DEA to the data on accommodation establishments in OECD.