Why no mathematical models of Marxist economics?

It's been commonplace in recent decades to observe, even decry the extreme mathematization of orthodox micro- and macroeconomics. But Marxist economics - still an active area of research - remains resolutely descriptive and verbal.

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If, nevertheless, you search you come across "Mathematical Model of Marxist Economics" (2015) by Wu Yifeng - but that's in Chinese.

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In English we have "Analytical Foundations of Marxian Economic Theory" by John E. Roemer (1989) which is available, partially, as a PDF. Roemer says some sensible things in the introduction about his approach:

"the microfoundations approach consists of deriving the aggregate behavior of an economy as a consequence of the actions of individuals, who are postulated to behave in some specified way. I have taken this approach throughout the book.

For example, in the chapters on the falling rate of profit, it is postulated that a technical innovation is introduced only if it increases profits for a capitalist. This micro approach is different from a macro approach, which might say: We postulate that technical change takes the form of an increasing aggregate organic composition of capital.

From the micro vantage point, one is not allowed to postulate an increasing organic composition of capital unless one can show what individual entrepreneurial mechanism leads to it.

Marxists might question the microfoundations methodology because one of the forceful points of Marx's theory is that the individual is not the relevant unit to examine - it is the class. This might lead one to try and build a model in which classes are the atoms of the system.

I think it should be possible to produce such a model, but I do not believe that model would be contradictory to the ones I have described in this book. The reason is this: that individuals act as members of a class, rather than as individuals, should be a theorem in Marxian economics, not a postulate.

Marx's point is that despite the capitalist's incarnation as a human being, he or she is forced by the system to act as an agent for the self-expansion of capital. Workers, similarly, may have their individual yearnings and habits, but conditions of life force them to acquire a class consciousness and to act, at times, as agents of the working class as a whole and not as their own agent. (This might be the situation, for instance, in a strike, where the striker takes great chances for the good of the strike, which are not personally worthwhile.)

In each case, Marx has claimed that although people exist organically as individuals, we can conclude that they act as members of classes. It is in this sense that class behavior is a theorem and not a postulate of the Marxian theory.

Taking the argument a step further, I would say that it is not only admissible, but important, to take a microfoundations approach in Marxian theory. A common error in Marxian discussions is functionalism: to assume that a mechanism necessarily exists to perform actions that must be performed to reproduce the system.

Put more simply, if the occurrence of X will further the reproduction of capitalist relations, then X occurs. For example, if racist attitudes exist among the working class, then capital will be strong. Therefore, capitalism foments racism.

What is missing here is a description of the mechanism by which this is accomplished. It may be in the interest of capital as a whole to maintain discriminatory wage differentials for black and white workers of equal productivity, but why should the individual, profit-maximizing capitalist respect this differential when he or she can increase profits by unraveling the differential - that is, by hiring only black workers at a slightly higher wage than they are receiving under the racist regime? [For one answer to this question, see Roemer (1979c).]

If we postulate capitalism as a system of anarchic, competitive capitals, each bent on its own expansion, we must face this sort of contradiction from functionalist arguments.

Another example comes from some Marxist-radical theories of education. Capitalism does not require a highly educated working class, so the theory goes, but it does require a well-socialized and docile working class. Schools, then, will serve the role of socializing and channeling people into capitalist society, but not of educating them.

Now, this conclusion may be true, but the functionalist nature of the argument eclipses the mysterious and difficult part of the phenomenon - how does capitalism ensure this role for schools, when teachers try to teach, students try to learn, and so on?

A third example is the role of the state. The capitalist state acts in the interests of the capitalist class - that is the theory. But the theory cannot be convincing unless one can demonstrate the mechanism by which this occurs, especially because capitals do not have a habit of cooperating with each other, as the primary aspect of each capital's existence is self-expansion and competition against other capitals.

What one requires, then, are microfoundations for the role of racism, education, and the state under capitalism. Other examples abound. In the cases when capitalism is guided as if by an invisible hand to coordinate its preservation in the ways mentioned, one requires an explanation of how anarchic capitals produce such a result.

 A second form of functionalism that exists among Marxists is the converse of the first form: If X has occurred, then X must be in the interests of capital. We again can take an example from education:

Because we observe compulsory education, it must be that capital requires such for its reproduction. But this may not be the case: compulsory education may exist because the working class fought for it.

One can find many examples of this form of functionalism in Marxian work: The general consequence of the error is to attribute an omnipotence to the capitalist class that it does not possess in Marxian theory. The capitalist class is pushed along by historical developments: Not everything that happens under capitalism was planned by it, nor is in its best interests. In fact, according to Marx, the general tide of historical development favors the working class. Again, a defense against this form of functionalism is a microfoundations approach.

There is a third form of functionalism among Marxists that, strangely, seems diametrically opposed to the first two forms: If the occurrence of X is necessary for the demise of capitalism, then X will come to pass. We can see the general rule of which these different functionalist forms are special cases if we phrase the general functionalist position this way. We postulate a certain outcome for the social system; functionalism then takes the form of claiming that only events occur which lead to that outcome.

In the first two forms of functionalism discussed, the outcome is the reproduction of capitalist relations; in the third form, the outcome is the transformation of capitalism into socialism. Perhaps the first two forms of functionalism are short-run variants, and the third form is a long-run variant of the general functionalist interpretation of Marxism.

Examples of the third form of functionalism in Marxian economics are prevalent in crisis theory. The system must have crises, because crisis is necessary for capitalist demise. The rate of profit must fall, because only in this way can crisis be brought about. The working class must become impoverished, because otherwise it will never perform its revolutionary task. Bourgeois democracy must transform itself into fascism, because only fascism will heighten the contradictions of capitalism sufficiently to produce revolutionary transformation, which must occur. These arguments are less than convincing; the form of functionalism they involve is similar to that of the utopian socialism of Marx's time, which postulated socialist transformation without a mechanism.

Marx's method was to counter utopian thinking by trying to expose the mechanism that would bring the socialist transformation about. ...

Finally, the equilibrium method has been used in the models in this book. While I have defended the approach of mathematical modeling, and the microfoundations approach, I have less confidence about the equilibrium method. Like many economists of my generation, I am strongly influenced by the power of the equilibrium method: of examining a model when it is at rest, so to speak, in the sense that all the rules that describe how its parts work are simultaneously fulfilled.

What is disturbing about the equilibrium method is that it pictures the typical position of the system as a position the system rarely or never enjoys. Of course, no sophisticated economic model builder would claim that economies are in equilibrium in the sense of a static equilibrium model. A model is only an ideal type.

However, there seems to be a deep contradiction between using models whose main analytical trick is to postulate a position that is precisely at variance with the most interesting and important aspect of capitalist economy as described by Marxian theory - its incessant, contradictory motion.

There is, therefore, the danger that if this intuition is correct, the equilibrium method will prevent one from seeing the most important aspects of the Marxian theory of capital. Knowing no other method, I use the equilibrium method, with the vague thought that, when rereading these pages in twenty years, its obsolescence as a modeling tool for Marxian theory may be clear.

(I might add that there is plenty of precedent in Marx's modeling of his theory for the equilibrium method: Consider, for example, the notion of equalization of profit rates among capitals, or the models of balanced growth designed to show that capitalism was capable of reproducing itself.) ..."

After this thoughtful introduction we encounter chapter 1.

"1 Equilibrium and reproducibility: the linear model

1.1 A brief review

We begin with a discussion of what constitutes an equilibrium, in Marxian terms, of a capitalist economy, and for this chapter we study a linear, Leontief specification of production.

Capitalists all have access to the same Leontief production system {A, L}, where

   A is an n x n input matrix of commodities

   L is a 1 x n vector of direct labor inputs.

If p is a l x n commodity price vector and the wage is normalized at unity so that prices p are interpreted as wage prices, then the usual notion of an equilibrium-price vector in a Marxian system is a vector p for which there exists a nonnegative number π such that

   p = (1 + π)(pA + L)           (1.1)

The vector pA + L is the vector of unit costs of production (assuming that there are no fixed capital costs), and so π may be interpreted as the uniform profit rate for the economy.

There may be, of course, many pairs (p, π) satisfying Equation 1.1.

We add more information to the system by requiring that workers each receive a subsistence vector of commodities b, an n x 1 vector. By subsistence, we mean simply that the wage must be precisely sufficient for b to be affordable, thus

   pb = 1                                 (1.2)

Here, the unit of time that denominates L, which is the same unit of time for which the wage of unity is paid, is to be thought of as one working day.

Combining (1.1) and (1.2), we see immediately that

   p = (1 + π)p(A + bL)         (1.3)

Thus, for a vector p satisfying (1.1) and (1.2) to exist, it is necessary and sufficient that the augmented input coefficient matrix M = A + bL possess an eigenvalue 1/(1 + π) that does not exceed one, to which can be associated a nonnegative eigenvector p.

According to the Frobenius—Perron theorem, if M is an indecomposable matrix, then M has a unique eigenvalue that can qualify; that is, M possesses a unique positive eigenvalue to which can be associated a nonnegative eigenvector. It is, furthermore, the content of Morishima-Okishio's fundamental Marxian theorem (FMT) that if the rate of exploitation e is positive, then the Frobenius eigenvalue of M will be less than one, and so the associated profit rate will be strictly positive. ..."

and so it goes on. One reviewer writes:

"The analytical rigor in treating the subject is among the things that make the book worth buying for anyone interested in mathematical economics: sitting down with paper and pencil for a couple of months in reading the book will be a beautiful intellectual experience, and it will add a whole set of "weapons" to the neoclassically trained economist."

But perhaps we just don't yet have the right (agent-based?) mathematics for Marx.

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I'm been reading Duncan Foley's excellent primer on Capital.

PDF link

Foley is a clear and creative thinker who is prepared to bring some simple mathematics to the table. He's interested in presenting the conceptual apparatus of Marxism in an orderly manner rather than simply providing a slimmed-down exegesis of the books themselves. His approach is very much rooted in applying Marxism to the present day economy, which informs both his appraisal and his examples.

He reconstructs in mathematical form the circulation of capital and Marx's simple and expanded reproduction equations before moving on to the 'transformation problem', crisis theories and Marx's concept of socialism. Here's the table of contents:

Contents

1 On Reading Marx: Method                            1
2 The Commodity: Labor, Value, Money        12
3 The Theory of Capital and Surplus Value    31
4 Production under Capitalism                       49
5 The Reproduction of Capital                       62
6 The Equalization of the Rate of Profit         91
7 The Division of Surplus Value                   105
8 The Falling Rate of Profit                          125
9 The Theory of Capitalist Crisis                  141
10 Socialism                                                 158
Suggested Readings                                    173
References                                                   177
Index                                                             181

In addition as you can see, this introductory book is quite short.
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When I did my PhD research - the formal theory of intentional agents - I explicitly rejected the framework of differential equations used by the control theorists and the cyberneticists (cf Ross Ashby). For the same reason I am somewhat skeptical about aspects of Peter Turchin's Cliodynamics.

My approach was in the classical AI tradition of modal logics and generalised automata theory where intentional agents (those with perceptions, beliefs, goals, plans, actions) are modelled explicitly. I still think this is the best approach to a twenty-first century formalised Marxist psychohistory where we're interested in history at the simulation level.