Capital Accumulation and Structural Impoverishment: Toward a Mathematical Marxism

This paper provides a mathematical formalization of the thesis that peripheral relative impoverishment is a necessary consequence of the internal logic of capitalist accumulation. Surplus is defined as structural subtraction: the difference between the value produced and the value returned to labor. It arises from the competitive constraint that governs the survival of agents within the system. This constraint constitutes the primitive condition on which the entire dynamics of accumulation rests. From this foundation, two main results are derived through a system of dynamic equations. First, the rate of return on capital structurally exceeds the rate of income growth, producing a necessary and persistent divergence in the distribution of wealth. This divergence operates across multiple scales (local, national, and global) through a network model in which transfers of value from subordinate nodes to dominant nodes emerge as an endogenous property of the system. The magnitude of these transfers is proportional to the asymmetry in capital stocks. Reformist interventions that remain internal to capitalist logic encounter a formally identifiable structural ceiling: beyond a certain threshold, meaningful correction requires a modification of control over the surplus. The contribution relative to the existing literature is twofold. First, the fundamental inequality is derived from surplus theory rather than observed empirically. Second, the core–periphery structure is formalized as a stable asymmetric equilibrium with endogenous attractors. The paper identifies three directions for partial exit: economic democracy, selective withdrawal from the market, and reform of international institutions. Each is necessary, but none is sufficient in isolation.