Stochastic Efficiency

Efficiency measurement in data envelopment analysis (DEA) has mostly used deterministic models, where the input-output data D = (X,Y) are assumed to be known. Here the input and output matrices (X,Y) are deterministic. If a particular decision-making unit (DMU) e.g., DMU_k is found to be efficient by a certain type of DEA model, one could aggregate these efficient units into a number N₁, where N2 = N - N₁ would then be the total number of inefficient units in the total industry comprising N units. The proportion p = N1/N of efficient units provides in this framework a natural measure of efficiency in the whole industry. When one considers time series data D_t =(X_t,Y_t), two additional dimensions are introduced. One is due to the wider choice of DEA formulations e.g., one may specify a DEA model for each t and then observe how p_t = N_1t/N_t changes over time. Alternatively, one may take a cumulative volume of input and output Dc =(Xc,Yc) over a certain period and then apply a DEA model based on the data set Dc to measure efficiency of a DMU_k. A second problem is due to the nonstationary nature of input-output data particularly for growing firms or DMUs. In such a case the steady state (t -→ ∞) version of the DEA model may not be valid.