Counting Equations and Unknown Variables
Marx’s Economic Contribution
In his letter to Friedrich Engels dated August 24, 1867, Karl Marx identified what he considered the two best points in his newly published Capital: first, the two-fold character of labor input (use-value and exchange-value); and second, “the treatment of surplus-value regardless of its particular forms as profit, interest, ground rent, etc.” Marx noted that “the treatment of the particular forms in classical political economy, where they are forever being jumbled up together with the general form, is an olla potrida” (a rotten pot of stew).
A few months later, in a letter to Engels on January 8, 1868, Marx stated the point with even greater precision: “That in contrast to all former political economy, which from the very outset treats the different fragments of surplus value with their fixed forms of rent, profit, and interest as already given, I first deal with the general form of surplus value, in which all these fragments are still undifferentiated—in solution, as it were.”
This claim has been understood in philosophical terms—as an insight into the “essence” behind “appearances,” or as a dialectical move from the abstract to the concrete. But there is a more precise, indeed algebraic, way to understand Marx’s contribution: it concerns the counting of equations and unknowns in the classical theory of price and income distribution.
Adam Smith’s Price Equation: The Problem of Indeterminacy
Adam Smith posited the fundamental price equation of classical economics:
p = w + π + ρ
where p is unit price, w is wages, π is profits (including interest on loans), and ρ is rent. This gives us one equation with four unknowns (p, w, π, ρ)—or three unknowns if we measure price in terms of some numeraire.
This price system is underdetermined. Smith employed several subterfuges to close it:
First subterfuge: For many commodities, assume ρ = 0 (no rent). This eliminates one unknown but still leaves one equation with three unknowns (p, w, π).
Second subterfuge: Assume w = constant, which Smith called “subsistence wages.”
This phrase “subsistence wages” has confused generations of readers into thinking Smith was making an empirical claim about miserable wages. But the analytical function of the assumption is to treat wages as a parameter—a datum determined outside the price equation by custom, bargaining, or other factors—rather than as an unknown to be solved within the price system.
With rent eliminated and wages given exogenously, Smith’s equation becomes:
p = w̄ + π
But even here, we have one equation and two unknowns (p and π). The system remains underdetermined. Smith could not derive the division between wages and profits from his price theory alone; he needed additional, often ad hoc, assumptions. David Ricardo inherited this problem and, despite his greater analytical rigor, could not solve it either.
Marx’s Simplification: From Three Unknowns to One
Marx’s analytical move was simple. Instead of treating profit, interest, and rent as separate unknowns to be determined simultaneously with price, he reduced them into a single category: surplus value (s):
s = π + i + ρ
where i represents interest (now separated from profit of enterprise). This reduces three unknowns to one. The price equation becomes:
p = w + s
(In Marx’s notation, w = v, variable capital, leaving aside constant capital c for simplicity.)
Like Smith and Ricardo, Marx treats wages as given—determined by the value of labor-power. The terminology of “subsistence wages” remains misleading, but the analytical function is identical: w is a parameter, not an unknown.
With w given, we now have:
• 1 equation
• 1 unknown (s)
The system is determinate. Given p (from market prices or labor-time calculations) and w (given exogenously), surplus value is determined as the residual:
s = p − w
This is the analytical payoff of treating surplus value in its “general form.” The subsequent division of s into its component forms—profit of enterprise, interest, and rent—becomes a distribution problem, not a value creation problem. That division depends on competition among capitals (equalizing profit rates), bargaining between industrial and finance capital (determining the interest rate), and ownership of land (ground rent). But these distributional struggles occur over a magnitude already determined at the point of production. Equalization of the profit rate is another mathematical convenience because profit rates per industry can be modeled as a matrix.
A Class Conflict Theory of Distribution
Marx’s simplification yields not only a determinate theory of price but also an explicit theory of income distribution. Divide the price equation by p:
1 = w/p + s/p
Let ω = w/p (the wage share) and σ = s/p (the profit share). Then:
1 = ω + σ
Therefore:
σ = 1 − ω
This class conflict theory of distribution has a simple algebraic form: the profit share and the wage share are inversely related, summing to unity. One class’s gain is the other’s loss. The income distributional conflict between capital and labor is zero-sum.
This insight was present in Smith and Ricardo, but their treatment of the three “particular forms” of surplus as separate categories from the outset obscured it. By reducing profit, interest, and rent into surplus value, Marx made the zero-sum nature of the income distribution between capital and labor explicit and transparent.
Fractions of Capital and Shifting Alliances: Poulantzas’s Extension
Nicos Poulantzas, the Greek-French political theorist, was to my knowledge the first to systematically analyze (albeit without algebra) the conflicts among what he called fractions of capital: industrial, financial (banking), commercial, and landed property. In Political Power and Social Classes (1968) and Classes in Contemporary Capitalism (1974), Poulantzas developed a theory of shifting alliances among these fractions, analyzing how their coalitions shape state policy and political outcomes.
This maps directly onto the algebraic framework developed above:
First stage (production): The bilateral conflict between labor and capital determines the primary division:
1 = ω + σ
Second stage (distribution of surplus): The conflict among property-owning fractions determines how the surplus share σ is subdivided:
σ = π + i + ρ
Poulantzas analyzed how, for example, finance capital might ally with industrial capital against landed interests in one historical conjuncture; or how finance and real estate might combine against industrial capital in another; or how industrial and landed interests might align against finance—depending on which fraction dominates what Poulantzas called the “power bloc” and controls state policy.
Marx’s analytical simplification bears direct political fruit here: by showing surplus value as a unified magnitude prior to its division, it becomes clear that struggles among capitalist fractions are over the distribution of already-extracted surplus—a secondary contradiction, nested within the primary one between capital and labor.
The Working Classes as Political Pivot
In my reading of Poulantzas, there is a crucial further dimension: the working classes, themselves fractioned between wage earners (manual, industrial workers) and salary earners (white collar, professional, managerial employees), become the pivot—the swing vote—manipulated by the shifting alliances among capitalist fractions.
The structure thus becomes:
Capital fractions: Industrial (π), Financial (i), Landed (ρ)—competing over the division of σ.
Labor fractions: Wage earners, Salary earners—competing over the composition and level of ω, but also disponible (available) as political allies for capitalist coalitions.
The dominant capital fraction, or coalition of fractions, secures its hegemonic position not merely by winning the intra-capitalist distributional struggle, but by recruiting segments of the working classes to its political project. This recruitment occurs through multiple mechanisms:
Ideological appeals: Homeownership ideology aligns workers with landed/real estate interests; “shareholder democracy” aligns them with finance capital; nationalist industrial policy aligns them with domestic manufacturing capital.
Material concessions: Credit access and mortgage availability align worker-debtors with finance capital; pension fund investments create “fictitious” alignment with capital markets; sector-specific wage gains tie workers to “their” industry’s fortunes.
Employment dependence: Workers in a given sector often support policy favorable to that sector’s capital fraction, perceiving (sometimes correctly) their immediate employment interests as aligned with their employers’ profitability.
The working classes are thus pivotal but not autonomous in Poulantzas’s framework—mobilized, divided, and realigned according to the strategic needs of shifting capitalist coalitions. The fracturing of labor along multiple dimensions (wage/salary, public/private, industrial/service, homeowner/renter, creditor/debtor, native/immigrant) provides numerous cleavage points for such manipulation.
Contemporary Relevance
This framework illuminates contemporary political economy. Consider the shifting alliances of recent decades:
The neoliberal period (1980s–2008): Finance capital achieved hegemony within the power bloc, allied with real estate interests (the “FIRE” sector: Finance, Insurance, Real Estate). Industrial capital was subordinated or offshored. Segments of the working class were recruited through homeownership, 401(k) retirement accounts, and credit-fueled consumption. The wage share (ω) declined; the profit share (σ) rose; and within σ, the interest and rent components (i + ρ) expanded relative to industrial profit (π).
Post-2008 instability: The financial crisis fractured this coalition. Segments of the working class, devastated by foreclosures and stagnant wages, became available for realignment. Populist movements of both left and right emerged, variously targeting finance capital (Occupy Wall Street, Sanders), globalized industrial capital (Trump’s trade wars), or immigrants (as a displaced target for economic anxiety).
Contemporary realignments: We observe competing attempts to construct new hegemonic coalitions. Tech capital seeks alliance with finance; industrial capital in some sectors seeks protection from both globalization and financialization; real estate interests navigate between finance dependence and populist homeowner politics. In each case, success depends on recruiting sufficient segments of the fractured working classes as the swing vote.
Conclusion: The Analytical Power of Simplification
Marx’s claim to have discovered the “general form” of surplus value, in contrast to the “particular forms” with which classical economists began, is not merely a philosophical or methodological assertion. It represents a genuine analytical advance: the reduction of an underdetermined system (one equation, multiple unknowns) to a determinate one (one equation, one unknown).
This simplification yields:
1. A determinate theory of price: p = w + s, with w given and s as the residual.
2. An explicit theory of distribution: σ = 1 − ω, the zero-sum class conflict between capital and labor.
3. A framework for analyzing intra-capitalist conflict: the division of σ into π, i, and ρ as a secondary distributional struggle.
4. A political theory of hegemony: Poulantzas’s analysis of shifting alliances among capital fractions, with the fractured working classes as the pivot, the swing vote to be courted and captured.
The algebra is simple. Its political implications are revealing.
References
Marx, Karl. Capital: A Critique of Political Economy, Volume I (1867).
Marx, Karl. Letter to Friedrich Engels, August 24, 1867. In Marx-Engels Collected Works, Volume 42.
Marx, Karl. Letter to Friedrich Engels, January 8, 1868. In Marx-Engels Collected Works, Volume 42.
Marx, Karl. Letters to Dr. Kugelmann. London: Lawrence & Wishart, 1936.
Poulantzas, Nicos. Political Power and Social Classes (1968).
Poulantzas, Nicos. Classes in Contemporary Capitalism (1974).
Ricardo, David. On the Principles of Political Economy and Taxation (1817).
Smith, Adam. An Inquiry into the Nature and Causes of the Wealth of Nations (1776).