# Cost Radial model

Pastor, Aparicio and Zofío (2022, Ch. 3) summarize the duality results that allow to relate numerically the value of cost efficiency with that of Farrell's radial input measure $R(I)$–whose inverse is Shephard's input distance function. Farrell's measure represents the maximum equiproportional reduction in the observed input vector necessary to the reach the production frontier. For the firm under evaluation $(\mathbf{x}_o,\mathbf{y}_o)$ it can be calculated by solving the following DEA model.

$$$\begin{split} & TE_{R(I)}( {{\mathbf{x}}_{o}},{{\mathbf{y}}_{o}}) = \underset{\theta ,\mathbf{\pmb{\lambda} }}{\mathop{\min }}\,\quad \theta \\ & \text{s}\text{.t}\text{.}\quad \,\sum\limits_{j=1}^{J}{\lambda _{j}^{{}}x_{jm}^{{}}}\le \theta{{x}_{om}},\ m=1,...,M\text{,}\ \\ & \quad \quad \sum\limits_{j=1}^{J}{\lambda _{j}^{{}}y_{jn}^{{}}}\ge y_{n}^{{}},\ n=1,...,N\text{, } \\ & \quad \quad \sum\limits_{j=1}^{J}{\lambda _{j}^{{}}}=1, \\ & \quad \quad \lambda \ge 0. \\ \end{split}$$$

Denoting the optimal solution to this program by $\theta^*$, the constraints require the projection $\left( {\theta^{*}\mathbf{x}_o,\mathbf{y}_o} \right)$ to belong to the technology $L(\textbf{y}_o)$, while the objective functions seeks the minimum value of $\theta$ that projects radially the input vector $\mathbf{x}_o$ to its frontier benchmark represented by $\hat{\textbf{x}}_o=\theta^*\mathbf{x}_o$. A feasible solution signaling technical efficiency is $\theta^*=1$. Therefore if $\theta^*<1$, the observation is technically inefficient and $(\pmb{\lambda} X,\pmb{\lambda} Y)$ outperforms $\left( {{\mathbf{x}_o,\mathbf{y}_o}} \right)$.

We can now show the main duality result relating cost efficiency and Farrell's radial input distance function $R(I)$, allowing its decomposition: $CE^{{}}_{R(I)}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\textbf{w} \right)=TE_{R(I)}\left( {{\textbf{x}}_{o}},{{\textbf{\textbf{y}}}_{o}} \right)+AE_{R(I)}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\textbf{w} \right)$, i.e.,

$$$\begin{split} \underbrace{\frac{C(\textbf{y}_o,\textbf{w})}{\sum\limits_{m=1}^{M}{{{w}_{m}}{{x}_{om}}}}}_{\text{Cost Efficiency}}=\underbrace{\theta^{*}}_{\text{Technical Efficiency}}\times \underbrace{A{{E}_{R\left( I \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},{\textbf{w}} \right)}_{\text{Allocative Efficiency}} \le 1. \end{split}$$$

Example

In this example we compute the cost efficiency measure:

using BenchmarkingEconomicEfficiency

X = [2 2; 1 4; 4 1; 4 3; 5 5; 6 1; 2 5; 1.6 8];

Y = [1; 1; 1; 1; 1; 1; 1; 1];

W = [1 1; 1 1; 1 1; 1 1; 1 1; 1 1; 1 1; 1 1];

costradial = deacost(X, Y, W)
Cost DEA Model
DMUs = 8; Inputs = 2; Outputs = 1
Orientation = Input; Returns to Scale = VRS
──────────────────────────────────
Cost  Technical  Allocative
──────────────────────────────────
1  1.0        1.0         1.0
2  0.8        1.0         0.8
3  0.8        1.0         0.8
4  0.571429   0.6         0.952381
5  0.4        0.4         1.0
6  0.571429   1.0         0.571429
7  0.571429   0.666667    0.857143
8  0.416667   0.625       0.666667
──────────────────────────────────

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

efficiency(costradial, :Economic)
8-element Vector{Float64}:
1.0
0.8
0.8
0.5714285714285714
0.4
0.5714285714285714
0.5714285714285714
0.4166666666666667
efficiency(costradial, :Technical)
8-element Vector{Float64}:
1.0
1.0
1.0
0.6000000000000001
0.4
1.0
0.6666666666666667
0.625
efficiency(costradial, :Allocative)
8-element Vector{Float64}:
1.0
0.8
0.8
0.9523809523809522
1.0
0.5714285714285714
0.857142857142857
0.6666666666666667

Reference

Chapter 3 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.

### deacost Function Documentation

DataEnvelopmentAnalysis.deacostFunction
deacost(X, Y, W)

Compute cost efficiency using data envelopment analysis for inputs X, outputs Y and price of inputs W.

Optional Arguments

• rts=:VRS: chooses variable returns to scale. For constant returns to scale choose :CRS.
• dispos=:Strong: chooses strong disposability of outputs. For weak disposability choose :Weak.
• names: a vector of strings with the names of the decision making units.