Economists Reveal How Prices Find Their Equilibrium Through Hidden Eigenvalue Dynamics

More than six decades after the Italian economist Piero Sraffa published his landmark work on the production of commodities by means of commodities, a pair of researchers has breathed new mathematical life into his framework. In a study published in the open-access journal Heliyon, Emiliano Álvarez and Pilar Lorenzo extend the classical Sraffa system into discrete time and, crucially, allow wages to move endogenously rather than treating them as a fixed parameter handed down from outside the model. The result is a dynamic portrait of an economy in which prices and wages chase each other through time, yet the ratios between prices settle with remarkable speed onto a trajectory dictated entirely by the structure of production itself.

The original Sraffa model describes a multi-sector economy conceived as a circular flow, in which goods are produced by means of other goods. Each industry uses inputs, labor, and produced means of production to generate its output, and the central analytical question concerns how the social surplus is distributed between profits and wages. In the static formulation, owners of the means of production target a uniform rate of profit across all sectors, and prices adjust to make that target consistent with the given technology and the real wage. Prices, in this classical view, are not signals of marginal utility but means of distributing the social surplus, reflecting both technical conditions of production and social arrangements surrounding income distribution.

What has remained underexplored, the authors argue, is the temporal dimension of this system. Mainstream dynamic models frequently treat prices and wages as exogenous or predetermined variables, overlooking the feedback loops that arise when each is determined within the same economy it helps to describe. Earlier work by Nikaido and Kobayashi added dynamics to Sraffa’s equations using continuous time, capturing a wage-price spiral in which wage growth responds to inflation with a bargaining-power parameter. But a continuous-time formulation has a subtle drawback: without explicit lags, production delays vanish, and with them the very possibility of capital advances and a meaningful rate of profit. As the economist Luigi Boggio observed, without stocks and production lags there is no place for a rate of profit at all. Discrete time, by contrast, imposes a time-to-build structure that mirrors the sequential nature of real production cycles.

Building on a discrete-time extension developed by Brida and colleagues, Álvarez and Lorenzo modify the price equation so that the wage paid in each period is not a constant but the cost of a standard subsistence basket valued at the previous period’s prices. The dynamics then collapse into a single linear system: prices today equal a matrix applied to prices yesterday, where that matrix combines the technology coefficients scaled by the profit rate with a term capturing labor’s consumption of the subsistence basket. This compact formulation admits an exact analytical solution, allowing the researchers to trace price paths from any initial configuration of prices forward through time without approximation.

The mathematics that emerges is strikingly elegant. Because the system’s matrix contains only non-negative entries, the Perron-Frobenius theorem guarantees that its largest eigenvalue is a real number associated with an eigenvector whose coordinates are also real. When the matrix is diagonalizable, which occurs with probability one when coefficients are drawn from continuous distributions, the individual price levels may grow or shrink without bound, but the ratios between any two prices converge to the ratio of the corresponding coordinates of the dominant eigenvector. In a two-sector economy, the ratio of the second price to the first inexorably approaches the quotient of the second and first components of that eigenvector, and the same logic generalizes to economies with any number of sectors.

Numerical simulations confirm the analytical results with almost disorienting speed. In a two-sector economy with initial prices set at equal values and randomly generated coefficients constrained to keep the system economically reasonable, relative prices become essentially constant by roughly the fifth period of a thirty-period horizon, settling near 0.33, precisely the ratio predicted by the dominant eigenvector. A three-sector simulation tells the same story: relative prices converge rapidly to the eigenvector ratio of approximately 2.57, even though the dominant eigenvalue in that case exceeds one and absolute price levels are exploding. The economy, in other words, can experience unbounded inflation in all prices simultaneously while the structure of relative values locks into place within a handful of periods.

The speed of that convergence depends on a quantity economists have long recognized as decisive: the subdominant eigenvalue, the second-largest in absolute value. The smaller the subdominant eigenvalue relative to the dominant one, the faster the transitory components of the price dynamics die away and the faster the system settles onto its eigenvector trajectory. This echoes earlier work by András Bródy, who showed that for Leontief matrices the second eigenvalue is the crucial quantity for calculating convergence speed. Random-matrix reasoning suggests that in larger systems convergence should be even faster, though recent empirical studies caution that multiple subdominant eigenvalues in real input-output structures can slow the adjustment, leaving open important questions about price stickiness in high-dimensional industrial economies.

Perhaps the most philosophically significant finding concerns what determines the equilibrium relative prices. They do not depend on the profit rate at all, nor on preferences or bargaining strength, but solely on the technological coefficients of the production system. This resonates with Sraffa’s own insistence that relative prices are completely constrained by the system of production and the conditions of its reproduction, and it aligns with the interpretation, advanced by Steenge and others, that Sraffa’s famous standard commodity is nothing more than an eigenvector of the system. It also departs from comparative statics treatments and from models in which individual preferences drive the dynamics. In this framework, the properties of the whole determine the properties of the parts: uniform profit rates emerge from an interconnected production system rather than being imposed as an assumption.

Allowing wages to adjust endogenously does change one headline conclusion relative to earlier discrete-time work. There is no fixed long-run equilibrium price level; prices change from period to period, with the rate of price growth governed by technological factors. What stabilizes is not the level but the structure, the web of relative prices that classical economists identified with natural prices. The wage-price spiral, driven by workers’ purchasing power over a subsistence basket, generates persistent nominal movement while leaving the deep architecture of value untouched, a result consistent with Nikaido and Kobayashi’s continuous-time analysis and robust to the shift into discrete time.

The authors are candid about the model’s limits. Technology is held fixed, the profit rate is assumed uniform and decided in advance, and the wage adjustment rule, in which wages rise exactly in proportion to prices, leaves little room for institutional bargaining power or the nonlinearities of the wage-profit curve. The simulations, moreover, are confined to low-dimensional systems. Future extensions could allow sectoral profit rates to differ and adjust toward a global rate, let technological coefficients evolve over time, or redefine the wage basket in terms of Sraffa’s standard commodity, which would freeze distribution entirely. The policy implications, the authors suggest, point toward regulating corporate profit margins, strengthening antitrust enforcement against clustered oligopolies, and bolstering collective bargaining, so that the distributive struggle this framework is designed to illuminate can be observed and shaped in the real economies it describes.

Subject of Research: Dynamic price and wage adjustment in a discrete-time Sraffa-Leontief production system

Article Title: Dynamic price and wage setting in a discrete-time Sraffa-Leontief system

Article References: Dynamic price and wage setting in a discrete-time Sraffa-Leontief system. (n.d.). https://doi.org/10.1016/j.heliyon.2026.e45505

Image Credits: AI Generated

DOI: 10.1016/j.heliyon.2026.e45505

Keywords: Sraffa system, Leontief model, price dynamics, wage-price spiral, eigenvalues, Perron-Frobenius theorem, discrete time, income distribution, classical economics, relative prices, input-output analysis, production prices