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Notes on the Structure and Evolution of Societies

Doctoral thesis, Complutense University of Madrid, 2011. 

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Abstract

We propose two models: maximum proportional growth (VN) and the steady state with the greatest reduction in working hours (TE). We use VN as a theory of capitalism and some biological societies, and although we explain that it is only a rough first approximation, we demonstrate that this model fulfills many mathematical laws observed in these real systems. TE is used as a sociology of allocation. We also analyze optimal economies, the role that value and accounting play in the economic mechanisms necessary to establish allocation, and contrast VN with other possible optimal economies. We point out that the evolutionary processes that have designed societies allow us to understand their similarity to VN, and more specifically natural selection in the case of biological societies, but that the structure of human societies cannot be explained solely by a Darwinian hypothesis. Finally, we provide brief notes on the history of economic thought.

Part One. Model Approach

We study how to write production processes and the conditions that simple behaviors must meet. We postulate two models: proportional growth with the largest expansion factor, or the Von Neumann model (VN), and the steady state with the largest reduction in working hours, or the Theory of Exploitation (TE). Both are restricted maximization problems, and for them to be solvable, certain maximum conditions must be met, involving auxiliary variables, the Lagrange multipliers. These multipliers and maximum conditions can be interpreted in VN as prices and the law of profitability, and in TE as labor values and the Ricardian law. The solutions of these models inform us about the economic structure and values in each case, so we have precise theories of two different economies. We use VN as an approximation of systems that behave according to production for production's sake, such as capitalism and certain biological societies, and TE as a sociology of allocation, a study of workers' control of the economy. We point out that limiting ourselves to simple behaviors allows us to build very simple and powerful theories, but at the cost of not being able to study non-simple behaviors.

Part Two. Model Development

We analyze the links between our models and theories and concepts of past economists, as well as some of their difficulties. We explain how to deal with different aspects: spatial position, durable materials, demographic and labor structure, durable processes, production functions, cyclical behaviors, types of processes and materials, external and internal exchanges, or material flows imposed on the system. We deduce from VN a series of precise quantitative laws that are empirically observed in capitalisms, such as:

We also deduce the corresponding laws in TE, which do not always coincide with those observed in capitalisms and VN. We also demonstrate that if any economic mechanism were able to coordinate subsystems through exchanges, it could only do so according to the VN solution if each subsystem grows proportionally as much as possible, or according to the TE solution if the subsystems reduce working hours as much as possible while maintaining a steady state. We also study the relationship between a change in the material flows imposed on the system and value.

Part Three. Analysis of the Models

We demonstrate that under certain assumptions, VN and TE are solvable. We also demonstrate how this solution changes when we modify the starting data, and thus, by slightly perturbing the recipes, the VN factor increases at a rate equal to the perturbation price and the TE factor at a rate equal to its labor value. We develop algorithms to solve our models, which allow us to find all relative maxima in small problems and also to tackle very large problems. We also understand the relationship between our models and other problems such as eigenvalue problems and linear programs. We explain a simplex-like algorithm and its similarity to certain aspects of the market mechanism.

Part Four. Allocation, Value, and Accounting

Optimal economies are those in which the allocation maximizes an objective function while fulfilling material balances; they are constrained maximization problems. For these to be solvable, it is necessary that, in addition to the restrictions, certain maximum conditions be met, involving auxiliary variables, the Lagrange multipliers. In these problems, a mathematical calculation parallel to economic calculus is defined, where maximum conditions play the role of accounting and where Lagrange multipliers play the role of values.

However, the problem of finding the solution to the equations that correspond to an optimal economy with a mathematical algorithm is very difficult, and the problem of establishing an allocation in a real economy is even more difficult. In Robinson's economy, direct allocation can be used, but not in an economy with advanced technology. Under these conditions, allocation mechanisms such as the market mechanism are essential because they act as real algorithms. Therefore, one way for the category of value to take shape in reality is for a mechanism to establish the allocation using accounting, imitating the way some mathematical algorithms operate.

Furthermore, accounting and values depend on the objective function. In an economy oriented toward the satisfaction of needs, commodities are valued by how much the objective function increases with the introduction of a small amount of the corresponding material, and the accounting balances also take this objective into account. But in VN, as in capitalism, allocation maximizes final weighted output, materials are valued at their expected yield, and accounting balances consider only marginal benefits. The objective function does not depend on human needs, and workers are inputs on an equal footing with others, to the point that VN has a solution for an automatic economy without humans. Furthermore, we can attempt to mathematically reconstruct the objective function that is maximized in a real economy if we understand the accounting that operates within it. Since accounting in capitalism obeys the law of profitability, VN is its mathematical counterpart because its maximum conditions correspond to this law.

We also detail some equations that can be understood as a generalization of VN for non-simple behaviors.

Part Five. The Evolution of Societies

We consider why VN resembles capitalism and beehives, and why humans and bees live in such systems.

Beehives operate with great reproductive efficiency, and we can explain this fact as a consequence of natural selection designing societies that reproduce as well as possible. But in VN, we choose precisely the "fittest" proportional growth; we act by imitating natural selection. This is why this model is a chapter of Darwinian evolutionary theory, and the properties of VN (the law of profitability, the law of compound interest, depreciation formulas, etc.) are laws that societies resulting from natural selection approximately fulfill (if overall growth is maximized and if they behave in a simple manner). We have another way for the category of value to take shape in a real system if it has emerged as a consequence of a Darwinian-type process, and in fact, this category has already been used by biologists in relation to natural selection. But not all biological systems obey VN, and systems that do not tend toward overall growth (such as predator-prey), those that do not behave simply (such as many organisms), or those that are the result of artificial selection, can greatly deviate from maximum growth. VN is not an inescapable physical law but rather the result of selection processes that can be altered.

In the case of human civilizations, we observe that they operate very differently from each other and with clear distances from VN, and if civilizations had been designed solely by natural selection, this could not be the case. Furthermore, humans have intelligence and have shown that they are capable of escaping spontaneous processes like natural selection, which raises the question of why humans have not artificially selected their own societies. Therefore, although capitalisms resemble VN, a Darwinian hypothesis alone cannot explain the history of humanity, and a specific study is essential.

Part Six. Historical Notes

We detail different theories on the relationship between labor values and prices and critically analyze them. We describe the history of economic models and how they have been the result of many progressive improvements. We point out that one of the tasks of our time is to develop models of real economies, but also of alternative economies, so that humanity can make an informed choice about its preferred society.

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Sraffa, Piero

[1] Production of Commodities by Means of Commodities. Prelude to a critique of economic theory, Cambridge University Press, London 1960, http://pombo.free.fr/sraffa60.pdf 

[1b] Producción de mercancías por medio de mercancías, Oikos-Tau, Barcelona 1966, https://es.scribd.com/document/375927835/Produccion-de-mercancias-por-medio-de-mercancias-Piero-Sraffa-pdf 


Thünen, Johann-Heinrich von

[1] Der isolierte Staat in Beziehung auf Landwirtschaft und Nationalökonomie, several editions in http://books.google.es/books?id=Z1lGAAAAYAAJ and http://www.archive.org/search.php?query=Der%20Isolierte%20Staat

[1b] Recherches sur l'influence que le prix des grains, la richesse du sol et les impôts exercent sur les systèmes de culture, (primera parte de [1]), Guillaumin et Cie., Paris 1851, http://books.google.com/books?vid=UCM5320561546

[1c] Le salaire naturel et son rapport au taux de l'intérêt, (second part of [1]), Guillaumin et Cíe., Paris 1857, http://books.google.com/books?id=y7VLmR_DeJMC

[1d] Von Thünen's Isolated State (parts 1 and 2 of [1]), Pergamon Press, 1966, https://archive.org/details/isolatedstateeng0000thun

[1e] The Isolated State in Relation to Agriculture and Political Economy. Part III (part 3 of [1]), Palgrave McMillan, 2009, https://es1lib.org/book/2680919/3a205f 


Tugan-Baranovskyi, Mijailo I.

[1] Theoretische Gundlagen des Marxismus, Duncker & Humblot, Leipzig 1905, https://archive.org/details/theoretischegru00tugagoog 

[1b] Los fundamentos teóricos del marxismo, Hijos de Reus, Madrid 1915, https://web.archive.org/web/20120414200840/http://www.ucm.es/info/bas/es/tugan/fundamentos.pdf 

[2] Historia de las crisis industriales en Inglaterra, La España Moderna, Madrid 1914, https://web.archive.org/web/20120206084532/http://www.ucm.es/info/bas/es/tugan/crisis600.pdf  


Vegara, Josep María

[1] Economía política y modelos multisectoriales, Tecnos, Madrid 1979, https://es.scribd.com/document/401533622/Vegara-Economia-politica-y-modelos-multisectoriales-1979-EXCELENTE-MATRIZ-E-TEORIA-MARXISTA-pdf 


Walras, Léon

[1] Éléments d'économie politique pure ou théorie de la richesse sociale, R. Pichon et R. Durand-Auzias, Paris 1926, https://gallica.bnf.fr/ark:/12148/bpt6k1181431v 

[1b] Elementos de economía política pura, Alianza, Madrid 1987, https://archive.org/details/leon-walras-principios-de-economia-politica-pura-teoria-matematica-de-la-riqueza-social

Wilkinson, James H.

[1] The algebraic eigenvalue problem, Oxford University Press, 1988, https://archive.org/details/algebraiceigenva0000wilk_c5b6 


Wilson, Edward O.

[1] Sociobiology: the new synthesis, Harvard University Press, 2000, https://archive.org/details/edwardo.wilson_sociobiology_the-new_synthesis 

[1b] Sociobiología: La nueva síntesis, Omega, Barcelona 1980.


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