From Marx's fundamental equalities to the solving of the transformation problem -- Coherence of the model

TL;DR

This paper presents an exact algorithmic solution to Marx's transformation problem, demonstrating its coherence and addressing longstanding debates about the rate of profit and capital reallocation across multiple branches.

Contribution

It introduces a novel, exact, and computationally efficient method to solve the transformation problem for systems with any number of branches, confirming Marx's model coherence.

Findings

The transformation problem is solvable with exact solutions using the proposed algorithm.

The solution set for three or more branches is infinite, reflecting market competition.

The approach shows that fixed capital is not essential for solving the transformation problem.

Abstract

Recently, V. Laure van Bambeke used an original approach to solve the famous problem of transformation of values into production prices by considering that capital reallocation to each department (branch) was part of the problem. Here, we confirm the validity of this consideration in relation with the satisfaction of demand (social need which is able to pay for the given product). In contrast to V. Laure van Bambeke's method of solving an overdetermined system of equations (implying that compliance with Marx's fundamental equalities could only be approached), we show that the transformation problem is solvable from a determined (two-branch models) or an underdetermined system of equations enabling to obtain exact solutions through an algorithm we provide, with no approximation needed. For systems with three branches or more, the solution of the transformation problem belongs to an infinite ensemble, accounting for the observed high competition-driven market fluidity. Furthermore, we show that the transformation problem is solvable in the absence of fixed capital, supporting that dealing with the latter is not essential and cannot be seen as a potential flaw of the approach. Our algorithm enables simulations illustrating how the transient rise in the rate of profit predicted by the Okishio theorem is consistent with the tendency of the rate of profit to fall (TRPF) subsequent to capital reallocation, and how the TRPF is governed by the increase of organic composition, in value. We establish that the long-standing transformation problem is not such a problem since it is easily solved through our algorithm, whatever the number of branches considered. This emphasizes the high coherence of Marx's conception, and its impressive relevance regarding issues such as the TRPF, which have remained intensely debated.