Differential Characteristics of Efficient Frontiers in Data Envelopment Analysis

The implicit definition and nondifferentiability of efficient frontiers used in data envelopment analysis are two major obstacles to obtaining their differential characteristics, including various elasticity measures and marginal rates of substitution. In this paper we invoke the theorem of the directional derivative of the optimal value function and show how this can be used to define and calculate the required elasticities without any simplifying assumptions. This approach allows us to extend the known elasticity measures and introduce new ones to the entire efficient frontier, including all its extreme points, in one single development. We also construct linear programs that are required for the calculation of elasticity measures. Our main development is undertaken in the variable returns-to-scale technology but extends to other polyhedral technologies of efficiency analysis.