Revenue Directional Distance Function model

The revenue directional distance function model is computed by solving a directional distance function DEA model for the technical efficiency.

The directional distance function DDF, projects firm $\left( {{\mathbf{x}_o,\mathbf{y}_{o}}} \right)$ to the production frontier in the pre-assigned direction $\mathbf{g}= {\left({{\mathbf{g_{x}^-},\mathbf{g^{+}_y}}} \right)\neq\mathbf{0}_{M+N}}$, $\mathbf{g^{-}_{x}}\mathbb{\in R}^M$ and $\mathbf{g^+_{y}}\mathbb{\in R}^N$. Decomposing revenue inefficiency requires defining the output-oriented directional distance functions. Departing from the general definition this implies setting the output or input directional vectors to zero; i.e., $\mathbf{g}= \left({{\mathbf{g_{x}^-}},{\mathbf{g_{y}^+}}} \right)=\:$$\left({{\textbf{0},\textbf{g}_{\textbf{y}}^+}} \right)$, respectively. Therefore the output-oriented directional distance functions defines as follows:

\[T{{I}_{DDF(O)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}}, \textbf{0},\textbf{g}_{\textbf{y}}^{{+}} \right)=\,\max \,\left\{ \beta_O :\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}}+{{\beta_O}}\textbf{g}_{\textbf{y}}^{{+}} \right)\in P(\textbf{x}_o),\ {{\beta_O}}\ge 0 \right\}.\]

The linear programs that allows calculating this measure is:

\[\begin{split} & T{{I}_{DDF(O)}}\left( {{\mathbf{x}}_{o}},{{\mathbf{y}}_{o}} \right)=\underset{{{\beta_O }_{{}}},\lambda }{\mathop{\text{max}}}\,\beta_O \,\, \\ & s.t. \quad \sum\limits_{j\in J}^{{}}{{{\lambda }_{j}}{{x}_{jm}}}\le {{x}_{om}},\,\,\,m=1,...,M \\ & \quad \quad \sum\limits_{j\in J}^{{}}{{{\lambda }_{j}}{{y}_{jn}}}\ge {{y}_{on}}+\beta g_{{{y}_{n}}}^{{+}},\,\,n=1,...,N \\ & \quad \quad \sum\limits_{j\in J}^{{}}{{{\lambda }_{j}}}=1 \\ & \quad \quad {{\lambda }_{j}}\ge 0,\,\,j\in J. \end{split}\]

Once again, as in the graph case already presented for the Profit Directional Distance Function model, the choice of directional vector corresponds to the researcher. Customarily, to keep consistency with the radial models, the observed amounts of outputs set the direction: $\mathbf{g}= \left({{\mathbf{g_{x}^-}},{\mathbf{g_{y}^+}}} \right)=\:$$\left({\textbf{0},{\mathbf{y}_o}}\right)$. In this case it can be shown that the directional model nests Radial Output Oriented Model. Indeed, if $\left({{\mathbf{g_{x}^-},\mathbf{g^+_y}}} \right)=\:\left( {\mathbf{0},{\mathbf{y}_o}} \right)$, then $\beta_O^{*}=\phi^*-1$ (in the Revenue Radial model. However, other choices are available, which are included as options in BenchmarkingEconomicEfficiency.jl–see the documentation below accompanying this function.

The notion of Nerlovian revenue inefficiency corresponds to the decompostion of economic efficiency based on the output-oriented directional distance function: $R{{I}_{DDF\left( O \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\mathbf{g_{y}^+},{\tilde{\textbf{p}}} \right)$ = $T{{I}_{DDF\left( O \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o},\mathbf{g_{y}^+}} \right)$ + $A{{I}_{DDF\left( O \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\mathbf{g_{y}^+}, {\tilde{\textbf{p}}} \right)$. This results in the following expression:

\[\underbrace{\frac{R \left( \mathbf{x}_o,\mathbf{p} \right) - \sum\limits_{n=1}^{N}{{{p}_{n}}{{y}_{om}}} }{\sum\limits_{n=1}^{N}{p_{n}^{{}}g_{on}^{+}}}}_{\text{Norm. Revenue Inefficiency}}=\underbrace{\beta_{{I}}^{*}}_{\text{Technical Inefficiency}}+\underbrace{A{{I}_{DDF\left( I \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\mathbf{g_{y}^+},\tilde{\textbf{p}} \right)}_{\text{Norm. Allocative Inefficiency}} \ge 0.\]

Reference

Chapter 8 in Pastor, J.T., Aparicio, J. and Zofío, J.L. (2022) Benchmarking Economic Efficiency: Technical and Allocative Fundamentals, International Series in Operations Research and Management Science, Vol. 315, Springer, Cham.

Example

In this example we compute the revenue efficiency directional distance function measure using the option Gy=:Monetary.

using BenchmarkingEconomicEfficiency

X = [1; 1; 1; 1; 1; 1; 1; 1];

Y = [7 7; 4 8; 8 4; 3 5; 3 3; 8 2; 6 4; 1.5 5];

P = [1 1; 1 1; 1 1; 1 1; 1 1; 1 1; 1 1; 1 1];

revenueddf = dearevenueddf(X, Y, P, Gy = :Monetary)

Revenue DDF DEA Model 
DMUs = 8; Inputs = 1; Outputs = 2
Orientation = Output; Returns to Scale = VRS
Gy = Monetary
─────────────────────────────────
   Revenue  Technical  Allocative
─────────────────────────────────
1      0.0       0.0         0.0
2      2.0       0.0         2.0
3      2.0       0.0         2.0
4      6.0       5.0         1.0
5      8.0       8.0         0.0
6      4.0       0.0         4.0
7      4.0       3.0         1.0
8      7.5       5.75        1.75
─────────────────────────────────

Estimated economic, technical and allocative efficiency scores are returned with the efficiency function:

efficiency(revenueddf, :Economic)

8-element Vector{Float64}:
 0.0
 2.0
 2.0
 6.0
 8.0
 4.0
 4.0
 7.5

efficiency(revenueddf, :Technical)

8-element Vector{Float64}:
 0.0
 0.0
 0.0
 5.0
 8.0
 0.0
 3.0000000000000004
 5.749999999999999

efficiency(revenueddf, :Allocative)

8-element Vector{Float64}:
 0.0
 2.0
 2.0
 1.0
 0.0
 4.0
 0.9999999999999996
 1.7500000000000009

dearevenueddf Function Documentation

dearevenueddf(X, Y, P; Gy)