Abstract
This chapter provides an overview of the dual measurement of efficiency by means of distance functions and their value duals, the profit, revenue and cost functions. We start by showing how the Shephard (input) distance function in quantity space is a cost function in price space and how the cost function in quantity space is a distance function in price space. We then proceed to formulate a more unifying structure that allows for the simultaneous adjustment of inputs and outputs via establishing duality between the profit function and the directional (technology) distance function which also enables us to derive duality results for the revenue and cost functions as special cases. We complete our exposition by explaining how we can implement empirically dual forms of these efficiency measures either via activity analysis accounting for environmental technologies, slack-based measures and endogenous directional vectors or via parametric methods.
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Notes
- 1.
See Färe and Primont (1995) for a discussion of these axioms.
- 2.
We think of \(x \in \Re^N_+\) as belonging to the primal (quantity) space and \(w \in (\Re^N_+)^*\) as belonging to its dual (price) space. Since x is a vector of real numbers its dual space \((\Re^N)^*\) equals ℜN. Hence in this case we need not distinguish between the primal and dual spaces. See Luenberger (2001).
- 3.
For the existence of the minimum see Färe and Primont (1995).
- 4.
This requires convexity.
- 5.
This was first noted by Shephard (1953), in less detail.
- 6.
Note that the input directional vector g x is taken here to be non-negative, however we ‘deduct’ it from the input vector x, analogous to the subtraction of cost from revenue to obtain profit.
- 7.
This was first introduced by Luenberger (1992) in the form of a shortage function.
- 8.
This paper is the consequence of many conversations with Professor R.W. Shephard.
- 9.
A ‘must read’ on this topic is W. Schworm and R. R. Russell: Axiomatic Foundations of Technical Efficiency Measurement (in progress).
- 10.
This is often the case in the DEA/Activity Analysis case, and always when inputs are strictly positive as in Charnes et al. (1978).
- 11.
Färe and Grosskopf (2010) develop their model based on technology T rather than as we do here with the input set \({L(y)}\).
- 12.
See Färe et al. (2013b) for the output orientation.
- 13.
An alternative form of nonparametric estimators which is considered to be econometric is discussed in Martins-Filho in Färe et al. (2013a).
- 14.
This appendix builds on Chambers et al. (2013).
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Färe, R., Grosskopf, S., Margaritis, D. (2015). Distance Functions in Primal and Dual Spaces. In: Zhu, J. (eds) Data Envelopment Analysis. International Series in Operations Research & Management Science, vol 221. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7553-9_1
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