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.