Abstract
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.
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Notes
- 1.
Interestingly, a precursor of the input distance function was also introduced independently by Malmquist in the same year but in a consumer context (Malmquist, 1953).
- 2.
- 3.
General notation: For a vector v of dimension D, \( v\in {\mathrm{\mathbb{R}}}_{++}^D \) means that each element of v is positive; \( v\in {\mathrm{\mathbb{R}}}_{+}^D \) means that each element of v is nonnegative; v > 0D means v ≥ 0D but v ≠ 0D; and, finally, 0D denotes a zero vector of dimension D. Given two vectors v and u, v ≤ u ⇔ vd ≤ ud, d ∈ {1, …, D}, v < u ⇔ vd < ud, d ∈ {1,…, D}, and v <* u ⇔ vd < ud or vd = ud = 0 and d ∈ {1, …, D}. The inner—or scalar—product of two D dimensional vectors v ≡ [v1, …,vD] and u ≡ [u1, …, uD] is denoted as \( v\cdot u\equiv {\sum}_{d=1}^D{v}_d{u}_d \).
- 4.
In these definitions, at least one element of the comparison vectors, x′ and y′, could be equal or, respectively, smaller or greater than those in the strongly efficient reference vectors x and y.
- 5.
This is one of the regularity conditions imposed on the production function or, more generally, the transformation function, so they are well behaved, i.e., they are characterized by a decreasing (increasing) marginal of substitution (transformation) between inputs (outputs) resulting from strictly convex (concave) isoquants.
- 6.
In these definitions, all the elements of the comparison vectors, x´ and y´, are, respectively, smaller or greater than those in the weakly efficient reference vectors x and y.
- 7.
- 8.
Again, this issue is not of concern in the parametric approach based on well-behaved production functions satisfying the desirable regularity conditions. The usual functional forms present equal strong, weak, and isoquant efficiency subsets. An example is the Cobb-Douglas function, whose input set corresponds to \( L(y)=\left\{{\prod}_{m=1}^M{x}_m^{\alpha_m}\ge y,\kern0.5em {\alpha}_m>0\right\} \).
- 9.
It is relevant to recall that the weakly efficient subsets ∂WLS(y) and ∂WPS(x), generated under the DEA assumption of strong disposability of inputs and outputs, are not to be confused with the production possibility sets LW(y) and PW(x), defined under the assumption of weakly disposable inputs and outputs.
- 10.
DEA started with the seminal paper of Charnes et al. (1978) where the CRS versions of the input-oriented and output-oriented radial models where proposed. Having been formulated as linear programs, they had, from the beginning on, two alternative dual formulations: the envelopment form, or primal linear program, and the multiplier form, or dual linear program.
- 11.
The second-stage additive model is a particular case of the third oldest DEA model (Charnes et al., 1985). It corresponds to its primal or envelopment form where the original right-hand side term of the first M restrictions, xmο, m = 1, …, M, have been replaced by θ∗xmο, m = 1, …, M, being θ∗ the optimal value of the first-stage model (3.23). It can be refined searching for the strong efficient projection that minimizes the same objective function, at the expense of solving a more complex nonlinear program (see Aparicio et al., 2007).
- 12.
Program (3.24) can be refined and substituted by a more complex nonlinear program that has the advantage of identifying the closest projection (minimizing the objective function) instead of the “farthest” projection, which means that the new optimal slacks can be much smaller; see Aparicio et al. (2007, 2017c).
- 13.
This procedure can also be refined, by reducing the set of possible peers from the subset of strong efficient firms to the subset of extreme strong efficient firms, those that cannot be deleted without modifying its frontier facet. For VRS technologies, the last subset guarantees that any projection accepts a single representation as convex linear combination of extreme strong efficient firms.
- 14.
The subset of efficient peers are all the efficient firms that belong to the mentioned optimal hyperplane. For large problems, it is interesting to consider as peers only the subset of “extreme efficient firms,” that is, firms that cannot be expressed as a convex linear combination of other efficient firms.
- 15.
This results was named after Shephard by Diewert; see Fox (2018: 514).
- 16.
Shephard himself did not introduce the concept of cost or revenue efficiency, neither proposed their decompositions.
- 17.
The proofs of the propositions included in this chapter can be found in these references.
- 18.
This is a standard assumption in the parametric approach to economic efficiency measurement where the preferred functional forms impose homotheticity, for example, the case of the Cobb-Douglas specification. Even when relying on flexible functional forms such as the translog or the quadratic specifications, homotheticity is routinely imposed by restricting the value of the associated parameters, normally without testing whether this restriction holds empirically or not. For a discussion on how to decompose cost efficiency relying on the parametric approach under non-homotheticity, we refer the reader to Aparicio and Zofío (2017).
- 19.
Färe et al. (2019) rely on this property in their proposal to decompose profit efficiency in a multiplicative way.
- 20.
The allocative efficiency term corresponding to the reverse approach, \( {AE}_{R(I)}^R\left(x,\hat{y},w\right) \), can be expressed in terms of the disparity between shadow prices and market prices at \( {\partial}^WL\left(\hat{y}\right) \), as in (3.46).
- 21.
Now, the reverse allocative efficiency term corresponding \( {AE}_{R(O)}^R\left(\hat{x},y,p\right) \) can be expressed in terms of the disparity between shadow prices and market prices at \( {\partial}^WP\left(\hat{x}\right) \), as in (3.58).
- 22.
Both options can be easily changed to constant returns and weak disposability. The syntax necessary to calculate both options simultaneously is the following:
- 23.
We refer the reader to Sect. 2.6.1 in Chap. 2 for the installation of the “Benchmarking Economic Efficiency” package. All Jupyter Notebooks implementing the different economic models in this book can be downloaded from the reference site: http://www.benchmarkingeconomicefficiency.com.
- 24.
When calculating the decomposition of revenue efficiency as presented in Table 3.5, the function
reports the inverse of these efficiency scores.
reports the inverse of these efficiency scores.Bibliography
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Pastor, J.T., Aparicio, J., Zofío, J.L. (2022). Shephard’s Input and Output Distance Functions: Cost and Revenue Efficiency Decompositions. In: Benchmarking Economic Efficiency. International Series in Operations Research & Management Science, vol 315. Springer, Cham. https://doi.org/10.1007/978-3-030-84397-7_3
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