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On Debreu-Koopmans Theorem: Bridging Neoclassical and Behavioral Economics via Star Quasiconvexity

Felipe Lara

Abstract

The Debreu Koopmans theorem restricts separable aggregation to at most one nonconvex component. We solve this by proving that a separable, additive or multiplicative, function is star quasiconvex, those with star shaped sublevel sets about minimizers, if and only if each component is star quasiconvex. This immediately yields star quasiconvexity of separable sums of quasiconvex functions, formally bridging diversification theory with the S shaped value functions of Prospect Theory. Furthermore, we develop a complete calculus, monotonic composition, pointwise minima, quasi-arithmetic means, and we apply it to Cobb-Douglas functions, multifactor risk models, and constant function market makers in decentralized finance. Star quasiconvexity thus provides a unified framework for economic modeling beyond the classical Debreu Koopmans constraint.

On Debreu-Koopmans Theorem: Bridging Neoclassical and Behavioral Economics via Star Quasiconvexity

Abstract

The Debreu Koopmans theorem restricts separable aggregation to at most one nonconvex component. We solve this by proving that a separable, additive or multiplicative, function is star quasiconvex, those with star shaped sublevel sets about minimizers, if and only if each component is star quasiconvex. This immediately yields star quasiconvexity of separable sums of quasiconvex functions, formally bridging diversification theory with the S shaped value functions of Prospect Theory. Furthermore, we develop a complete calculus, monotonic composition, pointwise minima, quasi-arithmetic means, and we apply it to Cobb-Douglas functions, multifactor risk models, and constant function market makers in decentralized finance. Star quasiconvexity thus provides a unified framework for economic modeling beyond the classical Debreu Koopmans constraint.
Paper Structure (14 sections, 12 theorems, 53 equations, 2 figures)

This paper contains 14 sections, 12 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries and Basic Definitions
  3. Main Results
  4. Additive Separable Star Quasiconvexity
  5. Monotonic Transformations
  6. Product Separable Star Quasiconvexity
  7. Applications
  8. Weighted Quasi-Arithmetic Means
  9. Ratio Models with Multiple Risk Factors
  10. Conclusions and Future Work
  11. Declarations
  12. Availability of Supporting Data
  13. Competing Interests
  14. Funding

Key Result

Theorem 3

(Additive Separable Star Quasiconvexity) Let $X = \prod^{m}_{i=1} X_{i}$, $h: X \rightarrow \mathbb{R}$ be defined as in separable, and $\overline{x} := (\overline{x}_{1}, \ldots, \overline{x}_{m}) \in {\rm argmin}_{X}\,h$. Then, $h$ is (strongly) star quasiconvex with modulus $\gamma \geq 0$ with r

Figures (2)

  • Figure 1: Function $V$ in \ref{['pvalue:func']}. A 3D plot of $V$ (left), and 2D plots for different parameters (right).
  • Figure 2: An illustration of the sublevel sets at height $\delta = -4, 0, 2$ of function $V$ des-cri-bed in \ref{['pvalue:func']}. All sublevel sets are star shaped w.r.t. $(\overline{x}_{1}, \overline{x}_{2}) = (-5, -5)$.

Theorems & Definitions (34)

  • Definition 1
  • Remark 2
  • Theorem 3
  • proof
  • Corollary 4
  • proof
  • Example 5
  • Corollary 6
  • Example 7
  • Corollary 8
  • ...and 24 more