Entropy Geometry and Condensation in Wealth Allocation

TL;DR

This paper addresses how wealth concentrates across agents by counting admissible configurations in a value--wealth space. It introduces a primitive value--wealth convertibility function $V_i(w)$ and derives an entropy functional from microstate counting, yielding a stationarity condition based on the curvature of $V_i(w)$. Two condensation pathways emerge: an equilibrium, capacity-driven condensation when the regular sector's absorptive capacity is finite, and an instability-driven condensation that arises from the geometry of the mapping even without a stable interior maximum. Extending to open systems shows exponential weighting in wealth and agent number without optimization, revealing that condensation reflects a balance between growth and absorptive capacity; together, the framework provides a structural baseline for inequality arising from configuration space.

Abstract

We develop a statistical framework for wealth allocation in which agents hold discrete units of wealth and macrostates are defined by how wealth is distributed across agents. The structure of the economic state space is characterized through a value convertibility function, which captures how effectively additional wealth can be transformed into productive or meaningful value. The derivative of this function determines the effective number of internally distinct configurations available to an agent at a given wealth level. In a closed setting with fixed total wealth and a fixed number of agents, we show that equilibrium wealth distributions follow directly from unbiased counting of admissible configurations and may display a condensation phenomenon, where a finite fraction of total wealth accumulates onto a single agent once the remaining agents can no longer absorb additional wealth. We then extend the framework to open systems in which both total wealth and the number of agents may vary. By embedding the system within a larger closed environment and analyzing a finite subsystem, we show that exponential weighting in wealth and agent number emerges naturally from counting arguments alone, without invoking explicit optimization or entropy maximization principles. This extension leads to a richer interpretation of wealth concentration: accumulation is no longer driven solely by excess wealth, but by a balance between wealth growth and the system's capacity to accommodate new agents. Condensation arises when this capacity is limited, forcing surplus wealth to concentrate onto a few agents. The framework thus provides a minimal and structurally grounded description of wealth concentration in both closed and open economic settings.