When profit-seeking firms find an efficient destination—but not the best route

An economy can have an efficient equilibrium without an obvious explanation of how it gets there. Firms face different returns, resources take time to move, and nobody necessarily knows what future prices will be. Can decisions based only on present profits nevertheless produce an orderly adjustment?

In The invisible hand as an emergent property, Giorgio Fabbri, Davide Fiaschi and Cristiano Ricci construct a model in which they do. This article examines their preprint’s revised version, arXiv:2508.14498v2, dated 10 November 2026; the preprint was first submitted on 20 August 2025. It establishes a conditional mathematical result, not a general claim that actual markets always work efficiently. Its central distinction is between reaching the best allocation eventually and taking the best route once the costs of getting there count.

An economy that adjusts at two speeds

The model contains a fixed total mass of firms spread across a continuous range of sectors. Consumers supply labour and receive income from wages and firm ownership. Sectors differ in productivity, while consumer preferences specify how readily one good can substitute for another.

In the baseline economy, prices and wages adjust instantly, and workers move freely between sectors. Firms change sectors more slowly. Each follows current profit differences rather than forecasting the full future economy.

Movement carries a quadratic cost: doubling the speed of adjustment quadruples its instantaneous cost. These costs represent implicit frictions, rather than an explicit destruction of productive capital. A firm therefore moves faster when nearby profit differences are steeper and slower when movement is more expensive. As firms enter a sector, profit per firm there falls, providing a counterweight to further inflows.

For mathematical convenience, sectors sit on a circle, eliminating endpoints. This is an assumed structure for the model’s sectoral distances, not an empirical map of industries.

From individual incentives to collective improvement

The authors translate these decisions into an equation tracking how the distribution of firms changes. They then identify a mathematical structure called a Wasserstein gradient flow.

Here, Wasserstein distance measures how costly it is to rearrange one distribution of firms into another using squared movement distances. A gradient flow describes movement in the direction of steepest improvement, measured with that geometry.

The striking result is what improves: the logarithm of aggregate consumption. Although individual firms pursue profits, their combined movement makes this economy’s consumption aggregate non-decreasing. Equivalently, the continuous adjustment can be represented as the limit of successive aggregate optimization steps that balance consumption gains against movement costs. No firm needs to solve that collective problem.

Under the stated smoothness and strict-positivity assumptions, the authors prove convergence to a unique long-run equilibrium, with an exponential bound on the remaining distance. Profit rates become equal across sectors, and the resulting firm distribution maximizes aggregate consumption.

Higher productivity does not automatically mean a larger equilibrium sector. That relationship depends on substitution: more productive sectors attract a greater share of firms when the elasticity of substitution exceeds one. Consumer demand is therefore integral to the result.

The destination is not the journey

Consumption increasing along a path does not establish that the path maximizes welfare after adjustment costs are deducted.

To examine that distinction, the authors introduce a planner who chooses movement speeds to maximize discounted consumption utility minus reallocation costs. The efficient stationary allocation is shared, but the transition generally differs.

Their comparison concerns linear approximations near equilibrium, under additional regularity assumptions. Within that scope, the planner can converge faster or slower than decentralized firms. The outcome depends on the value placed on consumption, discounting and adjustment frictions. Faster arrival alone is therefore not a sufficient welfare test.

Evidence, extensions and limits

The empirical application uses more than 750,000 firms in 14 EU countries, aggregated into 680 sectors, comparing 2018 with 2023. Sectoral returns on equity tend to converge, whereas labour productivity shows weak divergence rather than comparable convergence. The authors interpret this contrast as consistent with capital adjusting more readily than labour.

However, returns on equity are a proxy for model profits. The dataset does not directly track sectoral capital reallocation or provide a reliable measure of the model’s firm-capital distribution. These patterns motivate the mechanism; they do not establish it causally. Parameter estimates also depend on adopting the extreme assumption of immobile labour. Reported consumption losses of roughly 2% from labour immobility belong to numerical experiments, not measured economy-wide losses.

The extensions expose important boundaries. Unequal consumer preferences change sector sizes without removing the underlying structure. Immobile labour can preserve wage and productivity differences and reduce long-run consumption relative to the mobile-labour benchmark. Fixed movement costs can halt adjustment with unequal profits. Sectoral spillovers preserve favourable results only under restrictions; sufficiently strong negative externalities can make decentralized adjustment reduce consumption instead.

The contribution is thus a precise account of when an invisible-hand mechanism emerges—and why neither its efficient endpoint nor its rising consumption path guarantees an optimal transition.