Shephard’s Input and Output Distance Functions: Cost and Revenue Efficiency Decompositions

In this chapter, we present the classic approach to calculate and decompose cost and revenue efficiency based on Shephard’s radial input and output distance functions. These decompositions follow closely the presentation done in Chap. 2 where both economic efficiency measures can be separated into technical and allocative components, i.e., expressions ( 2.33 ) and ( 2.44 ). However, rather than resorting to Farrell’s input and output technical efficiency measures, the decomposition is formalized in terms of the distance function and duality theory. At the time of publishing his seminal paper in 1957, Farrell did not seem to be aware of the work by Shephard, printed initially in his 1953 book titled Theory of Cost and Production Functions. There, he formalized the duality between the cost function and the input distance function, constituting the theoretical base for the decomposition of economic efficiency. Farrell cites Debreu’s (1951) “coefficient of resource utilization” as a source of inspiration, although, had he been aware of it, he could have relied equally on Shephard’s contribution, which is specific to production theory. Nevertheless, Shephard never introduced the concept of overall economic efficiency nor that of allocative efficiency, being one step short of proposing this decomposition explicitly.