The invisible hand as an emergent property: a gradient flow approach

TL;DR

This paper models the economy as a gradient flow in a Wasserstein space, showing how decentralized firm reallocation leads to efficient long-run equilibrium with emergent properties, supported by empirical analysis of EU firms.

Contribution

It introduces a novel gradient flow framework for general equilibrium dynamics, linking decentralized decisions to global optimization and efficiency, with extensions and empirical validation.

Findings

Long-run equilibrium maximizes aggregate consumption.

Decentralized decisions can be viewed as global optimization.

Empirical evidence of sectoral profit convergence and limited reallocation costs.

Abstract

We develop a general equilibrium model in which, at each instant, a short-run competitive equilibrium arises. Heterogeneity in factor allocation generates differential profit rates across sectors, prompting firms to move between them under myopic profit-seeking behaviour, subject to quadratic reallocation costs. The aggregate dynamics of the economy can be formalised as a gradient flow in a Wasserstein space, starting from a partial differential equation that describes the reallocation of firms across sectors. Two key emergent properties arise: (i) decentralised and uncoordinated decisions can be reinterpreted as the solution to a sequence of global optimisation problems, involving a function of aggregate consumption, which increases monotonically along the dynamic path; (ii) the long-run competitive equilibrium is efficient, as the distribution of firms maximises aggregate consumption and profit rates are equalised across sectors. We extend the baseline model to incorporate non-symmetric preferences, intrasectoral externalities, a fixed cost of reallocation, and labour immobility. These extensions reveal conditions under which the efficiency and uniqueness of the long-run equilibrium may fail, but also highlight the surprising result that the decentralised equilibrium can remain efficient even in the presence of externalities. Finally, using a large sample of EU firms from the period 2018-2023, we empirically document convergence in sectoral profit rates, but not in labour productivity, pointing to a certain degree of labour immobility. We also find evidence suggesting the absence of significant fixed costs of reallocation at the sectoral level, the presence of positive but limited intrasectoral externalities, and a moderate degree of substitutability among goods.