Statistical Mechanics of Resource Allocation

We sketch the usefulness of spin-glass techniques for the analysis of the macroscopic properties of certain resource allocation problems, focusing in particular on a simple linear production model and on the competitive equilibrium model. The efficiency with which resources are allocated is a central issue in economics. The concept of ‘allocation’ is to be intended here in a broad sense that includes both the exchange of commodities among agents and the production of commodities by means of commodities whereas ‘efficiency’ is usually connected to the solutions of maximum or minimum problems, as for instance firms try to meet demands at minimum costs. 1) A question that naturally arises concerns the collective properties of efficient states, or the aggregation of the microeconomic behavior of producers, consumers, etc. into macroeconomic laws — like the distribution of prices, consumption levels or firm sizes — that can eventually be checked against empirical observations. To this aim, it is important to take into account the existence of heterogeneities across agents, that is of different budgets, endowments, technologies and preferences. Traditional economic methods like the so-called representative-agent approach are inadequate in this respect. 2) Mean-field spin-glass theory offers an alternative toolbox whose usefulness in many cases goes beyond the merely technical level. To begin with, we consider the classical linear model of production to meet demand at minimum cost. Let there be N processes (or technologies) labeled by i and P commodities labeled by µ. Each process allows the transformation of some commodities (inputs) into others (outputs) and is characterized by an input-output vector ξ i = {ξ µ } where negative (positive) components represent quantities of inputs (outputs). Moreover it can be operated at any scale si ≥ 0 and the cost of operating it at scale si =1i spi. One wants to choose the si’s so as to minimize the total cost i sipi subject to the requirements that i siξ µ i = κ µ for all µ, so that if κ µ > 0 (κ µ