Abstract
This paper introduces a series of new concepts under the name of Economic Cross-Efficiency, which is rendered operational through Data Envelopment Analysis (DEA) techniques. To achieve this goal, from a theoretical perspective, we connect two key topics in the efficiency literature that have been unrelated until now: economic efficiency and cross-efficiency. In particular, it is shown that, under input (output) homotheticity, the traditional bilateral notion of input (output) cross-efficiency for unit l, when the weights of an alternative counterpart k are used in the evaluation, coincides with the well-known Farrell notion of cost (revenue) efficiency for evaluated unit l when the weights of k are used as market prices. This motivates the introduction of the concept of Farrell Cross-Efficiency (FCE) based upon Farrell's notion of cost (revenue) efficiency. One advantage of the FCE is that it is well defined under Variable Returns to Scale (VRS), yielding scores between zero and one in a natural way, and thereby improving upon its standard cross-efficiency counterpart. To complete the analysis we extend the FCE to the notion of Nerlovian cross-inefficiency (NCI), based on the dual relationship between profit inefficiency and the directional distance function. Finally, we illustrate the new models with a recently compiled dataset of European warehouses.
Highlights
Data Envelopment Analysis (DEA) is a data-driven approach for estimating a piece-wise linear frontier enveloping from above a cloud of points in a space with dimensions associated with variables categorized as inputs and outputs
We show that under the customary assumption of input homotheticity, the traditional bilateral notion of input cross-efficiency for unit l, when the weights of unit k are used in the evaluation, coincides with the Farrell notion of cost efficiency for unit l when the weights of unit k are used as market prices
Despite the capability of cross-efficiency to yield a suitable ranking of observations based on the prices associated with all the sample units when evaluating each observation, these techniques have developed without establishing any connection with the literature devoted to measuring economic efficiency when prices are present; i.e., relying on microeconomic theory
Summary
Data Envelopment Analysis (DEA) is a data-driven approach for estimating a piece-wise linear frontier enveloping from above a cloud of points in a space with dimensions associated with variables categorized as inputs and outputs. Other important property is P6 since it means that, assuming for example perfect competition, the new approach collapses to the well-known Farrell measure of cost efficiency, which should be the standard reference to be used for evaluating performance and ranking units when information on a common set of prices, in this case market prices, is available This property is not satisfied by the traditional notion of cross input technical efficiency in the literature, as s n pr yrl pr yrl r 1. We consider initially the case of variable returns to scale DEA technologies and, subsequently, constant returns to scale production possibility sets In this way, and inspired in the Farrell cross-efficiency notion introduced in the previous section when dealing with input-oriented models, we suggest to consider the shadow prices for inputs and outputs of each unit k ,.
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