A LONG ANSWER TO A SHORT QUESTION

Marco Aurelio Denegri comments, “I am curious about the tools from the mathematical sciences that have been most useful to you in exploring Marx's thought. What are books that survey these applications?”

 

The simple answer would be Linear Algebra, followed by a list of titles, but there is a good deal more to be said about the subject and it might be of interest to some of you for me to elaborate.

 

Modern economic theory began in France in the 18^th century with the work of a small group of scholars called Physiocrats who put forward a tabular analysis of the French agricultural economy that emphasized the fact that the inputs into any cycle of production – the seed, tools, fertilizer, and so forth – are the outputs of previous cycles of production, so that reproduction rather than production is the proper analytical concept to be employed. This important and foundational idea was taken up by Adam Smith, David Ricardo, and other classical political economists in a series of books that constituted the core of economic theory up through the beginning of the 1870s.

 

It was first Smith and then Ricardo who developed the idea that the natural price or “value” of a commodity brought to market is determined by the amount of labor directly and indirectly required for its production. Hence the phrase “labor theory of natural price” or “labor theory of value,” which captured in summary form their central analytical insight into the newly developing capitalist economy.  Although their work was powerful and sophisticated, Smith, Ricardo, and the other classical political economists used virtually no mathematics in the exposition of their theories.

 

In the 1870s, three economic theorists – Stanley Jevons, Leon Walras, and Carl Menger – more or less independently of one another introduced into economic theory the notion of marginal product, expounding their theories with the use of calculus. This so-called “triple revolution” completely transformed academic economic theory and quickly gave rise to the characteristic mathematical analyses of capitalist economies with which any student of modern economics is familiar.

 

Somewhat obscured from view by the flashy mathematics of the new economic theories was the fact that the classical economists and the new economists, the so-called neoclassicals, asked fundamentally different questions about capitalist economies and in their analyses furthermore adopted alternative and opposed simplifications for purposes of analysis. The classical political economists sought answers to three fundamental questions: first, what is the source of economic wealth (hence the full title of Adam Smith’s great work, An Inquiry into the Nature and Causes of the Wealth of Nations)?; second, in what way is economic wealth divided among the three great classes of society, the landed gentry, the entrepreneurs, and the workers?; And third, what factors promote or place obstacles in the path of economic growth? The neoclassical economists, in contrast, sought to understand, in the famous characterization of economic theory put forward by Lionel Robbins, the efficient allocation of scarce resources with alternative uses.

 

In addition to asking different questions, the classical economists and the neoclassical economists adopted opposite simplifications to assist them in their analyses. The classical economists assumed that each commodity had only one established mode of production. From time to time, new techniques or modes of production would be introduced by entrepreneurs of an experimental turn of mind and these would either win out and become the new established mode or prove unprofitable and die away. This was of course not strictly speaking true. At any given time there were probably several established alternative modes of production for a given commodity – involving different productive techniques and different combinations of inputs – but the classical economists made the simplifying assumption that there was only one established technique for each commodity.

 

The neoclassicals made the opposite simplifying assumption. They assumed that there were in fact an infinite number of alternative ways of producing a given commodity, involving different combinations of inputs, the quantity of each one being capable of being varied by very small increments. This assumption permitted them to use the powerful mathematical tool of the calculus and so one eventually got the standard analyses of marginal product and marginal cost that fill modern economics texts and bedevil poor business students with incipient math phobia.

 

In the 1960s and 70s and 80s, a number of quite gifted mathematical economists around the world took a new look at the classical school of political economy, using modern mathematical techniques to analyze and test the truth of the propositions advanced by Ricardo, Marx and the others. They found that the appropriate tool for translating the theories of the classicals into rigorous mathematical form is not calculus but rather linear algebra. The reason, quite simply, is that if each commodity has a single mode of production, then an entire economy can be represented by a system of linear equations, each of which relates quantities of inputs of a given commodity to the quantity of its output.

 

This formal reconsideration of the classical school began with the work of the great Italian–English economist Piero Sraffa, who not insignificantly was the general editor of the magnificent 10 volume edition of the works of David Ricardo.  Sraffa himself in his seminal monograph Production of Commodities by Means of Commodities, published in 1960, did not in fact make use of linear algebra but the economists inspired by his work did.

 

Michio Morishima in Japan, Luigi Pasinetti and Pierangelo Garegnani in Italy, Gilbert Abraham-Frois and Edward Berrebi in France, and Andras Brody in Hungary, among others recast the central ideas of Ricardo and Marx in mathematical form and demonstrated that many of the most important claims that they had advanced in their works could in fact be given rigorous mathematical demonstrations.

 

That, as briefly and simply as I can state it, is the answer to Marco Aurelio Denegri’s question.