Distance Functions and Generalized Means: Duality and Taxonomy

1. Problem Statement

The paper addresses a fragmentation in production economics between two major theoretical frameworks:

• Consumer Theory: Relies heavily on utility functions (specifically the Stone–Geary utility function) to model preferences and decision-making.
• Producer Theory: Focuses on efficiency measures, productivity indices, and distance functions (e.g., Shephard's, Directional Distance Functions) to evaluate technical efficiency.

While these fields share deep interdependencies (duality, efficiency frontiers), they have largely developed independently. Existing efficiency measures (radial, non-radial, directional, slack-based) often lack a unified formal foundation. Furthermore, many duality results linking efficiency measures to profit functions rely on strong convexity assumptions, limiting their applicability to non-parametric or non-convex technologies.

The Core Objective: To establish a formal, unified link between utility functions and distance functions. The author aims to demonstrate that a broad class of known efficiency measures can be derived from a specific type of "producer utility function" inspired by the Stone–Geary concept, and to establish duality theorems connecting these new measures to the profit function without requiring convexity.

2. Methodology

The paper employs a rigorous mathematical approach combining production theory, functional analysis, and duality theory.

• Stone–Geary Utility Framework: The author adapts the Stone–Geary utility function (originally used for subsistence consumption levels) to the context of production. Instead of minimizing the distance to a subsistence level, the producer maximizes the "distance" (improvement) from a current netput vector $z$ to the efficient frontier.
• Generalized Means (Aczel/Atkinson): The core innovation is the introduction of a Generalized Mean Directional Distance Function. This function utilizes generalized means (parameterized by $p$) to aggregate netput expansions.
  • The utility function $W_p$ is defined using $\phi_p$-generalized sums (power means, geometric means, min/max functions depending on $p$).
  • The distance function $D_{W_{p,g}}(z)$ is defined as the supremum of $W_p(u-z)$ over all feasible netput vectors $u$ that dominate $z$.
• Duality via Norm Maximization and Lagrangian Duality:
  • For $p \geq 1$, the author utilizes a "farthest distance" theorem (related to the maximization of a norm over a compact set) to derive duality results. This approach does not require the production set to be convex.
  • For $p < 1$, the utility function is quasi-concave. The author uses standard Lagrangian duality and the concept of indirect utility functions to establish dual relationships with the profit function.
• Limit Analysis: The paper systematically analyzes the behavior of the generalized distance function as the parameter $p$ approaches specific limits ($-\infty, 0, 1, +\infty$) to recover known measures.

3. Key Contributions

A. The Generalized Mean Directional Distance Function

The paper defines a new class of distance functions, $D_{W_{p,g}}(z)$, which unifies various existing measures under a single Stone–Geary-inspired utility formalism.

• Definition: It measures the maximum generalized sum of netput expansions required to reach an efficient point in a given direction $g$.
• Unification: It encompasses the Directional Distance Function (DDF), the Färe-Lovell measure, the Asymmetric Directional Distance Function, and multiplicative measures as special cases or limiting cases.

B. Duality Without Convexity

A significant theoretical contribution is the establishment of duality theorems linking the generalized distance functions to the profit function ($\Pi$) without assuming convexity (T5) of the technology set.

• Case $p \geq 1$: The distance function is interpreted as maximizing a weighted norm. The dual problem involves maximizing the difference between the profit function and the value of the current netput, subject to a dual norm constraint on prices.
• Case $p < 1$: The distance function is linked to quasi-concave utility. The dual problem involves minimizing the profit gap subject to constraints derived from the indirect utility function.

C. Taxonomy of Efficiency Measures

The paper provides a comprehensive taxonomy showing how different values of the parameter $p$ correspond to specific, well-known efficiency measures:

• $p = -\infty$: Recovers the standard Directional Distance Function (DDF).
• $p = 0$: Recovers the Multiplicative Directional Färe-Lovell measure.
• $p = 1$: Recovers the Directional Färe-Lovell measure.
• $p = +\infty$: Recovers the Asymmetric Directional Distance Function.

D. Axiomatic Properties

The author characterizes the axiomatic properties of the new class of functions:

• Efficiency Characterization: The function characterizes the strong efficient subset ($E(T)$) when $p > 0$ and the weak efficient subset ($F(T)$) when $p \leq 0$.
• Translation Invariance: The functions satisfy translation invariance properties consistent with directional distance functions.
• Monotonicity: The functions are non-increasing, with strong/weak decreasing properties depending on $p$.

4. Key Results

1. Representation Theorem: Any distance function satisfying specific conditions (upper semi-continuity, non-decreasing) can be represented as the supremum of a utility function over the dominating set $T_z$.
2. Convergence: The sequence of generalized mean distance functions converges uniformly to the limiting cases (e.g., as $p \to -\infty$, it converges to the standard DDF).
3. Duality Formulas:
   • For $p \geq 1$: $D(z) = \sup_{w} \{ \Pi_z(w) - w \cdot z : \|w\|_q = 1 \}$.
   • For $p < 1$: $D(z) = \inf_{w} \{ \Pi_z(w) - w \cdot z : W^*(w) = 1 \}$.
   • These results generalize previous duality theorems (e.g., Shephard, Luenberger, Chambers et al.) and show that convexity is not a prerequisite for duality in this framework.
4. Ranking of Measures: The paper establishes an ordering of the distance functions based on $p$:
   $$D(z; g) \leq D_{W_{q,g}}(z) \leq D_{W_{p,g}}(z) \leq AD(z; g)$$
   for $p \geq q$.

5. Significance

• Theoretical Unification: The paper bridges the gap between consumer utility theory and producer efficiency analysis, demonstrating that diverse efficiency measures are manifestations of a single underlying principle (Stone–Geary utility maximization).
• Robustness to Non-Convexity: By proving duality results without convexity assumptions, the framework is applicable to a wider range of production technologies, including those analyzed via Data Envelopment Analysis (DEA) with non-convex frontiers or free-disposal hulls.
• Flexibility in Efficiency Measurement: The parameter $p$ allows researchers to choose an efficiency measure that balances the trade-off between characterizing strong efficiency (requiring $p>0$) and maintaining continuity or specific axiomatic properties (often associated with $p \leq 0$).
• Connection to Bargaining Theory: The approach links production efficiency to bargaining theory (Nash and Kalai-Smorodinsky solutions) and multicriteria decision analysis (compromise programming), offering new interpretative lenses for efficiency scores.
• Practical Application: The inclusion of a numerical example and the clear taxonomy (Tables 1-3) provide a roadmap for researchers to select appropriate distance functions and their corresponding dual profit formulations for empirical studies.

In conclusion, Briec's work provides a powerful, generalized mathematical framework that not only recovers existing efficiency measures as special cases but also extends the theoretical toolkit of production economics to handle non-convex technologies and diverse efficiency criteria through the lens of generalized means and utility duality.