Marginal Rates and Two-dimensional Level Curves in DEA

Of great importance to management, the computation of trade-offs presents particular difficulties within DEA since the piecewise linear nature of the envelopment surfaces does not allow for unique derivatives at every point. We present a comprehensive framework for analyzing marginal rates, and directional derivatives in general, on DEA frontiers. A useful characterization of these derivatives at given points can be provided in terms of the ranges they can take; equivalently, the bounds of these ranges correspond to derivatives to the rightand to the left at these points. We present two approaches for their computation: first, the dual equivalents calculation of minimum and maximum multiplier ratios / finite differences, and then a modified simplex tableau method. The simplex tableau method provides a more general application of the method introduced by Hackman et al. (1994) to generate any two-dimensional section of the isoquant and is a practical tool to generate level plots of the frontier. By giving a complete picture of trade-offs and allowing a better visualization of high dimensional production possibility sets, these tools can be very useful for managerial applications.     

Combining DEA Window Analysis with the Malmquist Index Approach in a Study of the Canadian Banking Industry

The banking industry in Canada is essentially an oligopoly with five large participants controlling about 90% of the market. To evaluate the industry's performance over time, we need to deal with the problem of a small number of DMU's compared to the number of relevant inputs and outputs. To overcome this problem we use data envelopment analysis (DEA) window analysis, whereby efficiency scores for the 20 year period 1981–2000 are obtained. To measure productivity changes over time, Malmquist indices can be calculated from DEA scores. Using DEA window analysis scores, however, raise the question of how to define the same period frontier in a DEA window analysis. We show that for both the adjacent and the base period Malmquist index and for all suggested definitions of same period frontier, the standard decomposition into frontier shift and catching up effects gives inappropriate results when Malmquist indices are based on DEA window analysis scores.