# Profit Models

## Profit Efficiency Model with Directional Distance Function Technical Efficiency

The profit function defines as $\Pi\left(\mathbf{w},\mathbf{p}\right)=\max \Big\{ \sum\limits_{i=1}^{s}{{p}_{i}}{{y}_{i}}-\sum\limits_{i=1}^{m}{{w}_{i}}{{x}_{i}} \,|$ ${\mathbf{x}} \geqslant X\mathbf{\lambda},\;{\mathbf{y}} \leqslant Y{\mathbf{\lambda },\;{\mathbf{\mathbf{e\lambda=1}, \lambda }} \geqslant {\mathbf{0}}} \Big\}$. Calculating maximum profit along with the optimal output and input quantities $\mathbf{y^{*}}$and $\mathbf{x^{*}}$ requires solving:

\begin{aligned} & \underset{\mathbf{x,y,\lambda} }{\mathop{\max }}\,\quad \quad \quad \;\ \Pi\left(\mathbf{w},\mathbf{p}\right)=\mathbf{py^{*}-wx^{*}} \\ & \text{subject}\ \text{to} \\ & \quad \quad \quad \quad \quad \ {{\mathbf{x}}}\ge X\mathbf{\lambda=x } \\ & \quad \quad \quad \quad \quad \; {{\mathbf{y}}} \le Y\mathbf{\lambda =y} \\ & \quad \quad \quad \quad \quad \; \mathbf{e\lambda=1} \\ & \quad \quad \quad \quad \quad \ \mathbf{\lambda }\ge \mathbf{0}. \end{aligned}

Profit efficiency defines as the difference between maximum profit and observed profit. Following the duality results introduced by Chambers, Chung and Färe (1998) it is possible to decompose it into technical and allocative efficiencies under variable returns to scale. Profit efficiency can be then decomposed into the directional distance fucntion and the residual difference corresponding to the allocative profit efficiency. Allocative efficiency defines then as the difference between maximum profit and profit at the technically efficient projection on the frontier. The approach relies on the directional vector to normalize these components, thereby ensuring that their values can be compared across DMUs.

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

using DataEnvelopmentAnalysis

X = [1 1; 1 1; 0.75 1.5; 0.5 2; 0.5 2; 2 2; 2.75 3.5; 1.375 1.75];

Y = [1 11; 5 3; 5 5; 2 9; 4 5; 4 2; 3 3; 4.5 3.5];

P = [2 1; 2 1; 2 1; 2 1; 2 1; 2 1; 2 1; 2 1];

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

deaprofit(X, Y, W, P, Gx = :Monetary, Gy = :Monetary)
Profit DEA Model
DMUs = 8; Inputs = 2; Outputs = 2
Returns to Scale = VRS
Gx = Monetary; Gy = Monetary
─────────────────────────────────────
Profit     Technical    Allocative
─────────────────────────────────────
1     2.0   0.0           2.0
2     2.0  -5.41234e-16   2.0
3     0.0   0.0           0.0
4     2.0   0.0           2.0
5     2.0   0.0           2.0
6     8.0   6.0           2.0
7    12.0  12.0          -1.77636e-15
8     4.0   3.0           1.0
─────────────────────────────────────

### deaprofit Function Documentation

DataEnvelopmentAnalysis.deaprofitFunction
deaprofit(X, Y, W, P; Gx, Gy)

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

Direction specification:

The directions Gx and Gy can be one of the following symbols.

• :Zeros: use zeros.
• :Ones: use ones.
• :Observed: use observed values.
• :Mean: use column means.
• :Monetary: use direction so that profit inefficiency is expressed in monetary values.

Alternatively, a vector or matrix with the desired directions can be supplied.

Optional Arguments

• 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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