A consequential assumption sits behind Marxian analyses of the average rate of profit whenever differences in the degree of labor exploitation across productive sectors are not explicitly modeled: sectoral rates of surplus value may be treated as tending toward uniformity. Adam Smith supplied the original economic argument for such a tendency, and within Marxian political economy the proposition acquired central importance. But can that tendency be defended when actual sectors are neither independent nor statistically identical? A 2022 paper by José Mauricio Gómez Julián in Ciencia Económica approaches the question from three directions at once: probability theory, the concrete labor-market mechanisms discussed by Smith and Marx, and statistical evidence from the United States between 1960 and 2020.

The Problem, in Plain Language

In Marxian theory, surplus value is the value produced by labor beyond the value represented by labor-power. Empirically, the paper approximates the rate of surplus value using modern national accounts: gross operating surplus is taken as a representation of surplus labor time and employee compensation as a representation of necessary labor time. Their ratio provides the sectoral rate used in the statistical analysis.

The issue matters because the tendency toward uniform sectoral rates of surplus value is part of the foundation on which the formation of an average rate of profit and, consequently, Marxian prices of production rests. But an important nuance is easily lost here. The paper does not treat prices of production as pre-existing equilibrium prices around which market prices merely oscillate. Drawing on Carchedi and de Haan, it emphasizes a dynamic interpretation: market prices exist first, and capitalist competition — including technological competition and movements of capital — pressures them toward a systemic average. Prices of production exist through this very process of convergence rather than as a fixed center established in advance.

The empirical question is therefore not whether every industry displays exactly the same exploitation rate at every instant. The proposition under examination is a tendency toward asymptotic uniformity: as the analysis approaches a sufficiently long period and/or a sufficiently large collection of relevant sectors, do sectoral rates behave in a way consistent with convergence toward a global expected value?

Why the Asymptotic Perspective Matters

The paper connects this question to Marx’s theory of knowledge. Its argument is that Marx’s method cannot be reduced to the passive acceptance of isolated empirical facts. Through a materialist appropriation of Hegelian dialectics, the analysis privileges totality: the phenomenon must be understood through its relations, its development, and the larger system of which it forms a part. In the paper, the term asymptotic regime condenses this idea statistically — a sufficiently long time horizon, a sufficiently large number of units, or both.

This matters because a sector observed in isolation may differ substantially from another sector. Uniformity, if it exists, need not appear as point-by-point equality. It may instead be a property that becomes visible only when the system is considered on a sufficiently large scale.

Enter the Law of Large Numbers

The mathematical framework is the Law of Large Numbers (LLN). In broad terms, laws of large numbers describe conditions under which averages stabilize around an expected value as the amount of information grows. The strong version concerns almost-sure convergence; the weak version concerns convergence in probability.

The difficulty is immediate. Classical formulations typically rely on strong assumptions such as independence and identical distribution. Those conditions are not realistic for capitalist sectors. Industries interact through production chains, competition, wages, technology, demand, and capital movements; and their productive structures differ. The paper’s own statistical results confirm that the sectoral variables are neither identically distributed nor, in general, linearly independent.

That does not make the LLN irrelevant. Rather, the theoretical part of the paper asks whether more general versions of the law can accommodate the structure of the economic problem. The answer is qualified: several mathematical results relax different parts of the classical assumptions, but they do not all do so in the same way.

The word certain is essential. The paper does not claim that simply invoking a non-classical LLN automatically proves the Marxian proposition. Formal consistency depends on satisfying the conditions of the particular theorem being used. In some cases this may require transformations of the dataset — for example, grouping or disaggregating sectors, arranging observations in structures such as triangular arrays, or estimating missing periods where appropriate. Whether such requirements can actually be met depends on the concrete dataset.

What Counts as a Productive Sector?

Before performing the statistical analysis, the paper faces a specifically Marxian classification problem: not every activity appearing in national accounts necessarily belongs in the calculation of the average rate of profit. The relevant sectors are those treated as productive in relation to the circuit of capital and the production of surplus value.

This is particularly important for services. Drawing on the literature on productive and unproductive labor, the study distinguishes activities that directly produce surplus value, activities that facilitate its production elsewhere, and activities outside the relevant circuit of capital. After harmonizing sector classifications across the historical period and applying those theoretical criteria, the empirical analysis works with 36 productive sectors.

What the U.S. Data Show

The empirical component uses data from the U.S. Bureau of Economic Analysis for the period 1960–2020. Sectoral rates of surplus value are constructed from gross operating surplus and employee compensation, and the study then examines probability distributions, pairwise Pearson correlations, and the differences between sectoral location measures and their global counterparts.

This pattern is important precisely because the empirical data do not reproduce the assumptions of the classical LLN. The sectors remain heterogeneous and interconnected. Yet the descriptive behavior of their deviations provides evidence, in the author’s interpretation, in favor of a tendency toward reciprocal nullification when the system is considered as a whole.

Why Conventional Significance Tests Were Set Aside

The paper originally considers tests for differences between sectoral means or medians and their global counterparts, including Student’s t procedures and Wilcoxon procedures, together with bootstrapping. It ultimately argues that the available inferential tests are not appropriate for drawing valid conclusions from this particular structure of data.

The problem is not simply non-normality. Depending on the procedure, assumptions concerning distributional form, linear independence, pairing, and equal sample sizes become relevant. The economic variables are interdependent by construction, and bootstrapping does not solve the entire difficulty: although resampling can break the relation between the immediately compared groups, it does not eliminate dependence among the sectoral components that make up the global group.

An especially important nuance is that the discarded tests did indicate statistically significant differences. The paper nevertheless refuses to treat those results as valid inferential evidence because the required conditions are not adequately satisfied. It therefore replaces that inferential route with a descriptive — or, as the paper itself suggests, perhaps more accurately semi-descriptive — analysis based on the behavior and probability distributions of the differences.

Smith, Marx, and the Economic Mechanism

The probability argument is only one dimension of the paper. A second dimension returns to Adam Smith’s account of labor-market adjustment. Smith argues that the total advantages and disadvantages of different employments in the same locality must either be equal or continually tend toward equality; otherwise workers and capital would move in ways that erode unusually favorable or unfavorable conditions.

The paper organizes Smith’s discussion around seven factors affecting compensation: the simplicity or difficulty of the work; whether it is pleasant or unpleasant; danger and effects on health; regularity or temporariness of employment; the degree of trust placed in the worker; the probability of professional success; and subjective considerations such as passion for an occupation, reputation, confidence in one’s abilities, and confidence in one’s luck.

These factors matter because they do not imply isolated sectors. Quite the opposite: sectoral rates can be linked directly through the organic interdependence of capitalist production and indirectly through common forces affecting wages and working conditions. In this sense, the economic mechanism described by Smith and Marx is more naturally compatible with generalized probabilistic frameworks that permit dependence than with a classical model requiring reciprocal independence.

So What Does the Paper Actually Establish?

The conclusion is favorable to the uniformity hypothesis, but it is carefully delimited. At the theoretical level, the logic of Smith’s tendency toward uniform sectoral rates of surplus value is consistent with certain strong and weak versions of the Law of Large Numbers that allow correlated variables and/or do not require identical distributions. This formal consistency is conditional on the concrete requirements of the relevant mathematical results being satisfiable.

At the empirical level, the U.S. data show precisely the heterogeneity and interdependence that rule out a simplistic classical-LLN argument. At the same time, the descriptive analysis of sectoral deviations produces results that point in a direction favorable to Smith and Marx’s proposition that sectoral rates tend toward uniformity.

This is therefore not a demonstration that every capitalist economy, in every historical period, must exhibit a uniform rate of surplus value. Nor does the study claim that all sectors possess the same rate at each moment. Its narrower and more defensible conclusion is that the assumption used in long-run Marxian analyses of the average rate of profit has both a plausible probabilistic formulation and empirical evidence in its favor for the U.S. case examined.

That distinction matters. The paper’s contribution is not to eliminate sectoral diversity, but to ask whether diversity at the level of the parts can coexist with an asymptotic regularity at the level of the whole. Its answer is cautiously affirmative — and it is precisely the combination of heterogeneity, interdependence, and systemic regularity that makes the problem mathematically and economically interesting.