# Revenue Russell model

The revenue Russell model is computed by solving am output-oriented Russell DEA model for the technical efficiency.

Taking the Russell measure as reference, Pastor, Aparicio and Zofío (2022, Ch. 5) present the Fenchel-Mahler inequality obtained from the dual correspondence between the revenue function and this measure of output technical inefficiency. The Russell output-oriented measure quantifying the technical inefficiency of a firm can be calculated through DEA methods by solving the following program:

$$$\begin{split} & T{{E}_{RM(O)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}} \right)= \underset{\begin{smallmatrix} \pmb{\phi}, \lambda \end{smallmatrix}}{\mathop{\min }}\, \frac{1}{N} \sum\limits_{n=1}^{N}{{{\phi }_{n}}} \\ & s.t. \quad \sum\limits_{j=1}^{J}{{{\lambda }_{j}}{{x}_{jm}}}={{x}_{om}}, \quad m=1,...,M \\ & \quad \quad \sum\limits_{j=1}^{J}{{{\lambda }_{j}}{{y}_{jn}}}={{\phi }_{n}}{{y}_{on}}, \quad n=1,...,N \\ & \quad \quad \sum\limits_{j=1}^{J}{{{\lambda }_{j}}}=1, \\ & \quad \quad {{\phi }_{n}}\ge 1, \quad n=1,...,N \\ & \quad \quad {{\lambda }_{j}}\ge 0, \quad j=1,...,J \\ \end{split}$$$

In this program $\phi^*_{n}$ evaluates the relative proportional increase of output $n, n=1,...,N$. The objective function averages these proportional rates of output expansion. Contrary to the Russell graph DEA model, this program is linear and therefore easy to calculate through the simplex method.

Considering $T{{E}_{RM(O)}}({{\textbf{x}}_{o}},{{\textbf{y}}_{o}})$ we can define its technical inefficiency counterparts as $T{I}_{RM(O)}( {\textbf{x}_{o}},{\textbf{y}_{o}})=1-T{{E}_{RM(O)}} ({\textbf{x}_{o}},{\textbf{y}_{o}})$. It is then possible to decompose revenue inefficiency into technical and allocative components: $R{{I}_{RM\left( O \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},{\tilde{\textbf{p}}} \right)$ = $T{{I}_{RM\left( O \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}} \right)$ + $A{{I}_{RM\left( O \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}}, ,\tilde{\textbf{p}} \right)$. That is

$$$\underbrace{\frac{R \left( \mathbf{x}_o,\mathbf{p} \right) - \sum\limits_{n=1}^{N}{{{p}_{n}}{{y}_{om}}} }{ N \min \left\{ {{p}_{1}}{{y}_{o1}},...,{{p}_{M}}{{y}_{oM}}\right\}}}_{\text{Norm. Revenue Inefficiency}}= \underbrace{1-\frac{1}{M}\sum\limits_{n=1}^{N}{\phi _{n}^{*}} }_{\text{Output Technical Inefficiency}}+\underbrace{A{{I}_{RM\left( O \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\tilde{\textbf{p}} \right)}_{\text{Norm. Allocative Inefficiency}}\ge 0. \\$$$

Reference

Chapter 5 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 Russell measure under variable returns to scale:

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];

revenuerussell = dearevenuerussell(X, Y, P)
Russell Revenue DEA Model
DMUs = 8; Inputs = 1; Outputs = 2
Orientation = Output; Returns to Scale = VRS
──────────────────────────────────
Revenue  Technical   Allocative
──────────────────────────────────
1  0.0       0.0       0.0
2  0.25      0.0       0.25
3  0.25      0.0       0.25
4  1.0       0.866667  0.133333
5  1.33333   1.33333   2.22045e-16
6  1.0       0.5       0.5
7  0.5       0.458333  0.0416667
8  2.5       2.05556   0.444444
──────────────────────────────────

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

efficiency(revenuerussell, :Economic)
8-element Vector{Float64}:
0.0
0.25
0.25
1.0
1.3333333333333333
1.0
0.5
2.5
efficiency(revenuerussell, :Technical)
8-element Vector{Float64}:
0.0
0.0
0.0
0.8666666666666665
1.333333333333333
0.5
0.4583333333333335
2.0555555555555554
efficiency(revenuerussell, :Allocative)
8-element Vector{Float64}:
0.0
0.25
0.25
0.13333333333333353
2.220446049250313e-16
0.5
0.04166666666666652
0.44444444444444464

### dearevenuerussell Function Documentation

BenchmarkingEconomicEfficiency.dearevenuerussellFunction
dearevenuerussell(X, Y, P)

Compute revenue efficiency using Russell data envelopment analysis for inputs X, outputs Y and price of outputs P.

Optional Arguments

• rts=:VRS: chooses variable returns to scale. For constant returns to scale choose :CRS.
• monetary=false: decomposition in normalized terms. Monetary terms if true.
• names: a vector of strings with the names of the decision making units.
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