Unlocking Efficiency: How Data Envelopment Analysis Evaluates Production Functions

Introduction

Background and Motivation

Efficiency concerns the relationship between input and output; with equal input, higher output means higher efficiency, making efficiency evaluation crucial for managing production. In practice the true production function is rarely observable, yet without it efficiency cannot be assessed, so methods to estimate production functions are needed.

What is Data Envelopment Analysis?

Data Envelopment Analysis (DEA) is a benchmark‑based, non‑parametric method that estimates production functions from existing data to evaluate the efficiency of comparable entities (decision‑making units, DMUs).

Why Use DEA?

In practice production is often a multi‑input multi‑output (MIMO) system with high‑dimensional data, making it almost impossible to derive or even guess the true production function. DEA’s advantage is that it does not require a specific functional form for the frontier, yet it can approximate the production function using observed data.

Limitations of DEA

Input and output data must be precise; categorical or dummy variables can introduce bias.

DMUs being compared need to be homogeneous; units of different nature or scale should not be compared.

DEA yields relative efficiency among DMUs, not absolute efficiency, so using relative scores as absolute values is inappropriate.

DEA is highly sensitive to data noise; the data should be as accurate as possible.

Essential Knowledge

Production Function

A production function represents the production frontier and indicates the maximum output achievable from a given input vector. Moreover, because different industries may have different input‑output ratios, data should ideally come from the same industry.

Properties of Production Functions

Nonnegativity: Output is a finite, non‑negative real number.

Weak Essentiality: At least one input must be used to produce output.

Monotonicity: An additional unit of any input does not reduce output (non‑decreasing).

Concavity: For any two input vectors, the output from their convex combination is at least the convex combination of their individual outputs, reflecting diminishing marginal returns.

Exceptions

The above properties are not exhaustive nor universally applicable.

When inputs are abundant, the monotonicity assumption can be relaxed; e.g., adding many machines may initially raise productivity but eventually reduce it due to external factors.

For S‑shaped production functions, the concavity assumption may be relaxed.

Variable Returns to Scale

In economics, returns to scale describe how long‑run output changes as all inputs are proportionally varied. The concept stems from the production function, linking output growth rates to changes in input levels. Over the long term, all inputs are variable and adjust with scale.

Three Possible Types of Scale Returns

Constant Returns to Scale (CRS): Output increases proportionally with inputs, representing the optimal production state.

Decreasing Returns to Scale (DRS): Output increases less than proportionally with inputs, indicating that the firm is too large and should not increase inputs.

Increasing Returns to Scale (IRS): Output increases more than proportionally with inputs, suggesting the firm can benefit from expanding scale.

Mathematical Comparison of Return Types

References

[1] Coelli, T. J., Rao, D. S. P., O'Donnell, C. J., & Battese, G. E. (2005). An introduction to efficiency and productivity analysis. Springer. [2] Data Envelopment Analysis. (2020, June 9). Wikipedia. Retrieved June 20, 2020, from https://en.wikipedia.org/wiki/Data_envelopment_analysis [3] Lee, C. Y., & Johnson, A. L. (2012). Two‑dimensional efficiency decomposition to measure the demand effect in productivity analysis. European Journal of Operational Research, 216(3), 584‑593. [4] Returns to scale. (2020, April 16). Wikipedia. Retrieved June 22, 2020, from https://en.wikipedia.org/wiki/Returns_to_scale [5] Vörösmarty, G., & Dobos, I. (2020). A literature review of sustainable supplier evaluation with Data Envelopment Analysis. Journal of Cleaner Production, 121672.