Measurement and decomposition of profit efficiency under alternative definitions in nonparametric models

A competitive firm is considered to be inefficient when its observed profit falls short of the maximum possible profit at the applicable prices of inputs and outputs. Two alternative measures of performance of the firm are the ratio of the actual to the maximum profit and the difference between the maximum and the actual profit. The ratio measures its profit efficiency and is naturally bounded between 0 and 1 so long as the actual profit is strictly positive. However, the possibility of negative actual profit and zero profit at the maximum has prompted many researchers to opt for the difference measure of profit inefficiency, which is necessarily non-negative. For a meaningful comparison of performance across firms, however, the difference needs to be appropriately normalized to take account of differences in the scale of operation of firms. Three common variables used for normalization are the observed cost, the observed revenue, and the sum of revenue and cost. It is a common practice to measure the separate contributions of technical and allocative efficiencies to the overall profit efficiency of the firm. When the firm is not operating on the frontier of the production possibility set, there are many ways to project it on to the frontier. This leads to different decomposition of profit efficiency into technical and allocative components. In this paper, we consider McFadden’s gauge function and an endogenous projection based on the overall efficiency of the firm along with the usual input- or output-oriented distance functions, the graph hyperbolic distance function, and a slack-based Pareto Koopmans efficiency measure for technical efficiency. We show that in a multiplicative decomposition of profit efficiency, the technical efficiency component of profit efficiency is independent of input-output prices only from the projection based on the gauge function. Also, an empirical application using data from Indian banks shows that the alternative normalized difference measures of inefficiency generate different performance ranking of the firms. Thus, comparative evaluation of performance of firms depends on the analyst’s preference for a specific type of normalization.