# Cost Models

## Cost Efficiency Model with Radial Technical Efficiency

Let us denote by $C\left(\mathbf{y},\mathbf{w}\right)$ the minimum cost of producing the output level $\mathbf{y}$ given the input price vector $\mathbf{w}$: $C\left(\mathbf{y},\mathbf{w}\right)=\min \left\{ \sum\limits_{i=1}^{m}{{{w}_{i}}{{x}_{i}}} | {\mathbf{x}} \geqslant X\mathbf{\lambda} {\mathbf{y}_{o}} \leqslant Y{\mathbf{\lambda },\;{\mathbf{\lambda }} \geqslant {\mathbf{0}}} \right\}$, which considers the input possibility set capable of producing $\mathbf{y}_{o}$. For the observed outputs levels we can calculate minimum cost and the associated optimal quantities of inputs $\mathbf{x^{*}}$ consistent with the production technology by solving the following program:

\begin{aligned} & \underset{\mathbf{x} ,\mathbf{\lambda }}{\mathop{\min }}\,\quad \quad \quad \;\ C\left(\mathbf{y}_{},\mathbf{w}\right)=\mathbf{wx^{*}} \\ & \text{subject}\ \text{to} \\ & \quad \quad \quad \quad \quad \ {{\mathbf{x}}}\ge X\mathbf{\lambda } \\ & \quad \quad \quad \quad \quad \;Y\mathbf{\lambda }\ \ge {{\mathbf{y}}_{o}} \\ & \quad \quad \quad \quad \quad \ \mathbf{\lambda }\ge \mathbf{0}. \end{aligned}

The measurement of cost efficiency assuming variable returns to scale, VRS, adds the following condition:

$$$\sum\nolimits_{j=1}^{n}\lambda_j=1$$$

Cost efficiency defines as the ratio of minimum cost to observed cost: $CE=C\left(\mathbf{y},\mathbf{w}\right)/\mathbf{wx_{o}}$. Thanks to duality results presented by Shephard (1953) , and following Farrell (1957), cost efficiency can be decomposed into the radially input oriented technical efficiency measure and the residual difference corresponding to allocative cost efficiency. Allocative efficiency defines as the ratio between minimum cost to production cost at the technically efficient projection of the unit under evaluation.

In this example we compute the cost efficiency measure under variable returns to scale:

using DataEnvelopmentAnalysis

X = [5 3; 2 4; 4 2; 4 8; 7 9.0];

Y = [7 4; 10 8; 8 10; 5 4; 3 6.0];

W = [2 1; 2 1; 2 1; 2 1; 2 1.0];

deacost(X, Y, W)
Cost DEA Model
DMUs = 5; Inputs = 2; Outputs = 2
Orientation = Input; Returns to Scale = VRS
──────────────────────────────────
Cost  Technical  Allocative
──────────────────────────────────
1  0.615385      0.75     0.820513
2  1.0           1.0      1.0
3  1.0           1.0      1.0
4  0.5           0.5      1.0
5  0.347826      0.375    0.927536
──────────────────────────────────

### 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.