Laws of Chaos

A Probabilistic Approach to Political Economy

Emmanuel Farjoun, Moshe Machover

• Publisher: Verso
• Year: 2020
• ISBN:
  9781788737036
• Topic: Économie
• Subtopic: Économie politique

A defining work of Econophysics, republished for the first time since 1983, Laws of Chaos is an attempt to construct a non-deterministic theoretical framework for the foundations of political economy. It relies on probabilistic and statistical methods of the kind used in the modern foundations of several other sciences, introducing scientific modelling into economics for the first time.

Chapter One
Non-Uniformity of the Rate of Profit

We have seen that the assumption of a uniform rate of profit, as an idealized expression of equilibrium under perfect competition and of equality among the various portions of capital, is basic to all classical theories of prices and profit, including the traditional Marxist theory. The same is also true of neo-classical theories. Before we offer an alternative theorization of the rate of profit as a random variable, we would like to examine and criticize the uniformity assumption from both the mathematical and the economic points of view.

Ever since Adam Smith, economists have put forward various explanations for the observed deviation from uniformity. But most of those explanations associate differences in the rates of profit accruing to different firms or branches of production with some important but contingent structural inequalities between them, such as inequality of ‘risk’, or differences in the degree of monopolization. Explanations of this sort are of no interest to us here. Rather, we are interested in those random but necessary differences in rates of profit that must arise even if all firms are assumed to compete on an equal footing, in the sense that no lasting structural bias of the system favours any branch or firm.

We shall try to explain why the uniformity assumption is in principle incompatible with a theorization of the capitalist system as a system of free competition and private property in the means of production. Our objections are of two principal kinds—mathematical and economic. Each of these, by itself, would be sufficient to undermine the uniformity assumption as a legitimate reasonable abstraction from reality.

Mathematical Objections

In the following two chapters we shall see that the mathematical tools for dealing with a large and disorderly collection of moving objects are well developed, and that these methods can be adapted to the study of an economy comprising a multitude of economic units, acting in an uncoordinated way to secure their survival and to improve their relative position. In that connection, we shall illustrate by means of examples from physical science the fact that, in general, the short and long-term behaviour of such a system cannot be captured correctly by assuming that the movement of its constituent parts has reached a uniform average.

Here we shall raise a few preliminary elementary objections to the uniformity assumption. Let us assume that the long-term average rates of profit in different branches of production are equal or approximately equal. Does it then follow that at any particular time the rates of profit in different branches are clustered close together? By no means.

To illustrate this simple mathematical point, consider the following—admittedly, drastically simplified—example. Suppose there are just two branches, A and B, each with a capital of £1,000. Let us assume, moreover, that the amounts of capital remain the same over a period of ten years. During five years (not necessarily consecutive) out of the ten, branch A yields profit at 5% per annum, while branch B yields 35%. In the remaining five years the position is reversed: branch A yields 35% and branch B only 5%. The long-term average rate of profit (over the whole ten-year period) is the same for both branches: 20%, which is also the yearly average rate of profit of both branches, taken together. Yet, each year the rates of profit in the two branches are wide apart.

Is it, nevertheless, legitimate in a theoretical calculation to replace the different rates of profit by a single figure (their average) and pretend that there is just one uniform rate? The answer is that it depends on the precise use to which such a simplifying assumption is put. If—to continue our example—we want to calculate the total yearly profit, we get the same result, £400, whether we assume a uniform rate of 20% for the total capital of £2,000, or whether we take £1,000 at 5% plus another £1,000 at 35%. The result in this case is the same, because the total profit is a magnitude that depends solely on the average rate of profit and on the total amount of capital, but does not depend on the dispersion of the different rates of profit among the various portions of capital.

However, certain quantities are very sensitive to the dispersion of the rate of profit. Suppose that our A and B are not branches but individual firms. Suppose also that a firm must pay tax at the rate of 50% on all its annual profits, except on the first £150, which are not taxed. Therefore, a firm that makes a profit of 5% on its capital of £1,000 will pay no tax; while the other firm, which makes a profit of 35% in the same year, will pay £100 tax. (The firm’s profits for the year amount to £350, of which £150 are tax-free and the remaining £200 are taxable at 50%.) The total tax paid each year is therefore £100. But if we assume that both firms make profits at a uniform rate of 20% per annum, then the yearly profit of each is £200, of which only £50 are taxable, so that each firm would pay £25 in tax, and the total tax paid (by both together) would be £50. We see that in this case the final result is drastically altered by assuming, contrary to fact, that the average rate of profit actually prevails as a uniform rate.

This last example merely highlights a simple mathematical fact, ignorance of which is a common source of fallacy. The fact is this: a mathematical relation that holds among variable quantities does not, in general, hold between their respective averages.¹

The moral of this is that one must exercise extreme care in considering an ideal ‘average’ state as though it were a real functioning state, with the usual relations between various quantities. Without such care, one can fall into the same error as the poor statistician who drowned in a lake whose average depth was six inches.² To sum up this point: Even on the hypothesis that the long-term average rates of profit in different branches of production are equal,³ it does not follow that at any given time there is a uniform, or nearly uniform, rate of profit. The uniformity assumption is an additional assumption, and a very drastic one at that, because it distorts those phenomena and relations that are sensitive to the dispersion of the rate of profit.

Our second objection is closely related to the first. Technically speaking, it concerns the question of stability—a question that mathematicians have been studying, in various forms and contexts, at least since the time of Laplace, who investigated it in connection with Newtonian celestial mechanics. In very general terms, it is a question about the behaviour of a system that is perturbed away from a state of equilibrium (as every real system usually is).

Suppose it is proved that some variable that describes the behaviour of a given theoretical system (model) assumes a particular numerical value (or, more generally, that a particular relation between several such variables holds) when the system is in a state of equilibrium. Does this result remain at least approximately correct when the system is slightly perturbed away from equilibrium? It turns out that—perhaps contrary to naive expectation—the answer is often negative.

Therefore, even if it were reasonable to assume the uniformity of the rate of profit for a state of equilibrium of a system with perfect competition, one must not jump to the conclusion that results deduced from this assumption remain approximately true for a nonequilibrium state. This caveat is especially pertinent in view of the undeniable fact that in reality the rates of profit of different firms, and even of different branches of a capitalist economy, are always quite far from uniformity.

Yet, as far as we know, none of the input-output theorists, for example, who use the uniformity assumption to deduce the prices of commodities ‘at equilibrium’, has ever attempted to show that the resulting prices are not strongly sensitive to slight variations in the rate of profit between branches. Had they raised this question, they would have found that in general their models may be quite sensitive to such variations, so that the results they prove have doubtful validity, even as first approximations, for the real world, in which rates of profit are not uniform.

Economic Objections

We must now raise another question, essentially economic rather than mathematical: is it theoretically sound to suppose that in an ideal state of equilibrium rates of profit would be uniform?

The concept of economic equilibrium is, of course, a construct of economic theory. What we are questioning is not the usefulness of such a construct in general, but a particular way of theorizing it. Whatever conditions are postulated for a state of equilibrium, any disparity between them and empirical reality must be explainable by the intervention of disequilibrating forces. Yet, even a cursory glance at the detailed economic statistics of an advanced capitalist country reveals that at any moment in time the disparity between rates of profit of different firms, or in different branches of production, is so large, that one begins to suspect that it cannot be explained by external constraints on free competition, or by mere deviation from equilibrium. Surely, the external inhibitions upon the mobility of capital cannot be so strong as to produce such an enormous ‘deformation’. And what is the meaning of a putative state of ‘equilibrium’ if the real economy is always so very far from it?

One’s suspicions are aroused still further upon closer examination of the real data.⁴ It transpires that in reality rates of profit are at least as widely dispersed as certain other important economic parameters, such as the rate of labour costs (as measured by the ratio between a firm’s total annual wage bill and its invested capital). Moreover, at least one parameter—the ratio of profits to labour costs (= the ratio between a firm’s annual gross profit and its annual wage bill)—is much more narrowly distributed, and therefore much closer to uniformity, than the rate of profit. Yet, while economists since Adam Smith have repeatedly argued that the rate of profit must tend to uniformity, and at equilibrium must actually be uniform, their theories do not impose any limit on the dispersion of rates of labour-costs; nor do they explain why the ratio of profits to wages should be so close to uniformity.⁵ What happens in reality is not explained by theory, while what theory tells us to expect does not actually occur.

Leaving empirical observations aside for the moment, let us take a closer look at the traditional theoretical argument that purports to show that in a state of equilibrium (under perfect competition) the rate of profit must be uniform. The argument consists of two parts, a premiss and a conclusion.

The premiss is that if the production of a particular type of commodity yields an abnormally high rate of profit, then competition will set in motion countervailing forces: capital will tend to crowd into the production of that type of commodity, leading eventually to its being over-produced; competition among its producers will then become fiercer, and its price will be forced down, bringing down also the rate of profit. Exactly the opposite process will operate if the initial rate of profit is abnormally low. The conclusion that is supposed to follow from this is that, as these processes play themselves out, the economy will tend towards (or oscillate around) a state of equilibrium in which all rates of profit are equalized.

While we do not wish to dispute the premiss of this argument, we claim that the supposed conclusion does not follow from it at all, and is in fact false. To see this, we must consider what is meant by a state of equilibrium of any system whatsoever. Stated in very general terms, a system is in a state of equilibrium when all its internal forces neutralize each other, so that if left to its own devices the system will continue in the same state, and will be perturbed away from it only under the influence of external forces.⁶ If the state of equilibrium is stable, and the system is subjected to a small perturbation by external forces, the internal forces of the system create a negative feedback effect, pulling the system back towards equilibrium. The system will then either converge to that state of equilibrium or oscillate around it.

A simple and familiar mechanical example of such a system is provided by the pendulum. When at equilibrium, a pendulum hangs vertically downwards; and it remains at rest in this position unless it is subjected to external perturbation.⁷ If externally perturbed, it will start to oscillate around its state of equilibrium. In the absence of friction, it would continue its oscillation for ever; but due to friction its oscillation is gradually dampened and the state of equilibrium is eventually restored. In the case of the pendulum, this state of equilibrium makes good sense not because such a state is necessarily ever reached—for, on the contrary, a real pendulum may perhaps be continually subjected to perturbations, and may therefore never come to rest—but because any departure from this state can always be ascribed to the action of external forces, different from the internal forces that tend to pull the pendulum towards equilibrium. The point is that this concept of equilibrium, ideal though it may be, does not violate the fundamental laws of motion of the pendulum itself.

Let us now return to a capitalist system in conditions of perfect competition. At first sight, this case may seem analogous to that of the pendulum, with a state of equilibrium here characterized by a uniform rate of profit. And so it has seemed to economic theorists since Adam Smith. The fact that in reality rates of profit are not uniform does not, in itself, seem to refute the assumption that in an ideal state of equilibrium they would be uniform—just as the oscillations of a continually perturbed pendulum do not refute the physical law that, in an ideal state of equilibrium, it would hang down motionless.⁸

But in fact there is a crucial difference between the two cases. For in a capitalist economy the very forces of competition, which are internal to the system, are responsible not only for pulling an abnormally high or low rate of profit back towards normality, but also for creating such ‘abnormal’ rates of profit in the first place. To make this crucial difference clearer, let us consider the following two ‘thought experiments’.

The first thought experiment is concerned with a pendulum. Suppose that the pendulum is pinned down, by an external constraint, to its vertical position. Then imagine that the constraint is removed and the pendulum is left to its own devices, free from the intervention of external forces. What will happen? Clearly, the pendulum will persist at rest in its vertical position. The persistence of this state is guaranteed by its being a state of equilibrium.

Now consider an analogous thought experiment with a perfectly competitive capitalist economy. Suppose that, due to the intervention of some all-powerful planning authority, rates of profit are forced to be absolutely uniform throughout the economy for a couple of years; suppose also that other conditions which are traditionally thought to characterize a state of equilibrium are enforced. Then imagine that the external constraint is removed, and the economy is left to its own devices, shielded from external intervention. Perfect competition will then resume its unfettered operation. Will rates of profit then remain uniform for any length of time, or will the uniformity be rapidly scrambled by competition itself? Clearly, the latter; but then it follows that the initially enforced state could not possibly have been a state of equilibrium!

Indeed, under any reasonable theorization of the concept of competition, the competitive forces that tend to scramble rates of profit away from uniformity are at least as real and powerful as those that pull towards uniformity. For one thing, even if rates of profit were to start from an initial uniform level, this would not prevent the flow of new investment capital from one branch of production to another. This flow is motivated not only by past differences in rates of profit in different branches, but at least as much by conjectures about future demand for various products. For example, firms in the coffin-making business may decide to invest their profits in another branch, say in furniture-making, rather than expand the manufacture of coffins, not because that other branch is at present more profitable, but because they do not anticipate a growing demand for coffins.

But even leaving such considerations aside, there are various competitive strategies that have a ‘scrambling’ effect on the rate of profit. For example, a large motor-car manufacturing firm, wishing to maximize its profits in the long run, may actually price its products down in order to encourage demand, or in order to drive its competitors into bank...

Table of contents

• Cover Page
• Halftitle Page
• Title Page
• Copyright Page
• Contents
• Dedication
• Foreword
• Introduction
• Chapter One: Non-Uniformity of the Rate of Profit
• Chapter Two: A Paradigm: Statistical Mechanics
• Chapter Three: The Rate of Profit as a Random Variable
• Chapter Four: Labour-content as a Measure of Commodities
• Chapter Five: Price and Wage as Random Variables
• Chapter Six: Dissolution of the Transformation Problem
• Chapter Seven: Elements of Dynamics
• Chapter Eight: Empirical Data and Open Problems
• Appendix I: Probability Theory
• Appendix II: The Determination of Labour-content
• Appendix III: The Value Controversy
• Notes
• Index