Is the theory of a falling prot rate valid? Paul Cockshott Contents 1 The theory in Volume 1 . . . . . . . . . . . . . . . . . . . . . . . 1 2 The theory in Volume 3 . . . . . . . . . . . . . . . . . . . . . . . 3 3 Criticisms of the theory . . . . . . . . . . . . . . . . . . . . . . . 6 3.1 A counter example to Okishio . . . . . . . . . . . . . . . . . . . . 9 4 A dynamic solution . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5 Comparing the theory with reality . . . . . . . . . . . . . . . . . 14 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Abstract Marx's theory of the falling rate of prot makes two main appearances in his work. The rst is in Chapter 25 of Capital Volume 1, entitled: The General Law of Capitalist Accumulation. It is further developed in Part III of Volume 3 of Capital, entitled The Law of the Tendency of the Rate of Prot to Fall. In this paper I will outline the structure of the theory presented in these two volumes of Capital. Following that I will look at some criticisms that have been levelled at it. I will go on to argue that the criticisms are based on a misunderstanding of some of the dynamic causal mechanisms that Marx assumed. Following on from this I shall present a dynamic solution to the equations of accumulation and show under what circumstances these lead to a falling rate of prot. The dynamic model will then be used to analyse the trajectories of some contemporary capitalist economies and to help understand the current structure of the world economy. 1 The theory in Volume 1 Marx presents both of his discussions in the context of capital accumulation. In volume I he is concerned primarily about the interaction of accumulation with the working population. Although this is not so evident in Volume 3 I believe that the same concerns are present there too. According to Marx a key factor in understanding the impact of accumulation is the composition of capital. The composition of capital is to be understood in a two-fold sense. On the side of value, it is determined by the proportion in which it is divided into constant capital or value of the means of produc- tion, and variable capital or value of labour-power, the sum total of wages. On the side of material, as it functions in the process of production, all capital is divided into means of production and living labour-power. This latter composition is determined by the relation 1 1 The theory in Volume 1 2 between the mass of the means of production employed, on the one hand, and the mass of labour necessary for their employment on the other. I call the former the value-composition, the latter the technical composition of capital.( [Mar87], page 387 ) If the value composition of capital remains the same, an increase in the stock of capital necessarily implies an increase in employment. Accumulation of capital is, therefore, increase of the proletariat.( [Mar87], page 388 ) However, if the growth of the labour supply is slow, the demand for labour may exceed the supply allowing wages to rise. This in turn can tend to reduce the rate of prot. Whilst this condition was the one most favourable to the labouring classes, Marx believed it to be temporary and self limiting. a rise in the price of labour resulting from accumulation of capital implies the following alternative: Either the price of labour keeps on rising, because its rise does not interfere with the progress of accumulation. In this there is nothing wonderful, for, says Adam Smith, "after these (prots) are dimin- ished, stock may not only continue to increase, but to increase much faster than before.... A great stock, though with small prots, gen- erally increases faster than a small stock with great prots." (l. c., ii, p. 189.) In this case it is evident that a diminution in the unpaid labour in no way interferes with the extension of the domain of capi- tal.  Or, on the other hand, accumulation slackens in consequence of the rise in the price of labour, because the stimulus of gain is blunted. The rate of accumulation lessens; but with its lessening, the primary cause of that lessening vanishes, i.e., the disproportion between capital and exploitable labour-power. The mechanism of the process of capitalist production removes the very obstacles that it temporarily creates.([Mar87], page 390) The system goes through a cycle. Rapid accumulation uses up the supply of labour. This allows wages to rise. This in turn reduces prots and reduces accumulation. So we return to the starting point. Here we have the basic mech- anism by which unemployment acts as a break on wages, and fullemployment as a break on accumulation. The business cycles of the 19th century illustrate this process very clearly. Figures 1 and 2 show how from 1881 to 1883 the rate of accumulation quickened, rising to over 7% of prots being accumulated. This kept unemployment low. But the process reached a crisis in 1883 as competion for labour caused accumulation to slacken. For the next ve years accumula- tion fell and unemployment rose. Once unemployment had reached a peak of 10%, falling wages stimulated a gradual acceleration of accumulation causing the cycle to start again. But Marx then argues that even if the natural increase in population is inadequate to the needs of capital accumulation, a further mechanism comes into play. At times of labour shortage, rms try to replace workers by machinery. This creates a new reserve army of labour that opens up further opportunities for accumulation. In the process, the value composition of capital changes vc rises and each additional increment to capital requires fewer workers. 2 The theory in Volume 3 3 Fig. 1: The basic cycle of accumulation is well shown in 19th century British trade cycles. Relationship between accumulation and unemployment. Reproduced from [Coc77]. So the theory in Volume 1 is of a cyclical process by which capital accumu- lation adjusts its pace to the supply of labour, and in turn regulates the supply of labour. If the quantity of unpaid labour supplied by the working-class, and accumulated by the capitalist class, increases so rapidly that its conversion into capital requires an extraordinary addition of paid labour, then wages rise, and, all other circumstances remaining equal, the unpaid labour diminishes in proportion. But. as soon as this diminution touches the point at which the surplus-labour that nour- ishes capital is no longer supplied in normal quantity, a reaction sets in: a smaller part of revenue is capitalised accumulation lags, and the movement of rise in wages receives a check. The rise of wages therefore is conned within limits that not only leave intact the foun- dations of the capitalistic system, but also secure its reproduction on a progressive scale. .([Mar87], page 390) 2 The theory in Volume 3 In Volume 3 Marx concerned himself with another implication of the change in the value composition of capital. He had previously been concerned with how this aected the demand for labour power, now he looks at its implication for the rate of prot p0 = c+v s . It is clear from this formula that if there is a change in c or in v , let us call them 4c, 4v , then with the mass of surplus value s unchanged, the rate of prot will fall or remain the same if 4c + 4v ≥ 0 and otherwise the rate of prot will rise. On the other hand, if s rises with c, v remaining unchanged, then the rate of prot will rise. 2 The theory in Volume 3 4 Fig. 2: The relationship between accumulation, unemployment and rate of change of wages. 2 The theory in Volume 3 5 If the labour theory of value is correct, then the sum daily added value per worker s + v is bounded by the length of the working day. If we assume that the maximum practical working day is say 12 hrs then s + v ≤ 12. On the other hand there is no equivalent limit to c the value of constant capital equipment used by each worker. Since wages can never fall to zero, it follows that the maximum daily rate of prot rb would be limited by the relation rb < 12c . Marx asserted that over time the value of constant capital used per worker tended to rise, and that for any given rate of exploitation this rise tended to reduce the rate of prot. He gives the following example The rate of surplus-value is 100%: If c = 50, and v = 100, then p0 = 100/150 = 66 23 %; c = 100, and v = 100, then p0 = 100/200 = 50%; c = 200, and v = 100, then p0 = 100/300 = 33 31 %; c = 300, and v = 100, then p0 = 100/400 = 25%; c = 400, and v = 100, then p0 = 100/500= 20%. This is how the same rate of surplus-value would express itself under the same degree of labour exploitation in a falling rate of prot, because the material growth of the constant capital implies also a growth  albeit not in the same proportion  in its value, and consequently in that of the total capital. If it is further assumed that this gradual change in the compo- sition of capital is not conned only to individual spheres of pro- duction, but that it occurs more or less in all, or at least in the key spheres of production, so that it involves changes in the average organic composition of the total capital of a certain society, then the gradual growth of constant capital in relation to variable capital must necessarily lead to a gradual fall of the general rate of prot, so long as the rate of surplus-value, or the intensity of exploitation of labour by capital, remain the same.([Mar94]page 148) He believed the tendency to exist, rst because of the mechanism described in the rst Volume, but also because he thought that the constantly growing mass of capital that was thrown into accumulation will in the long term outstrip the growth of thee proletariat. There would be absolute over-production of capital as soon as additional capital for purposes of capitalist production = 0. The purpose of capitalist production, however, is self-expansion of capi- tal, i.e., appropriation of surplus-labour, production of surplus-value, of prot. As soon as capital would, therefore, have grown in such a ratio to the labouring population that neither the absolute working- time supplied by this population, nor the relative surplus working- time, could be expanded any further (this last would not be feasible at any rate in the case when the demand for labour were so strong that there were a tendency for wages to rise); at a point, there- fore, when the increased capital produced just as much, or even less, surplus-value than it did before its increase, there would be absolute over-production of capital; i.e., the increased capitalC + 4C would produce no more, or even less, prot than capital C before its expan- sion by 4C . In both cases there would be a steep and sudden fall in 3 Criticisms of the theory 6 Tab. 1: Initial technology table, chosen to give equal value compositions of cap- ital. Industry capital goods labour used output luxuries 1 2 3 subsistence goods 1 2 3 capital goods 1 2 3 the general rate of prot, but this time due to a change in the com- position of capital not caused by the development of the productive forces, but rather by a rise in the money-value of the variable capital (because of increased wages) and the corresponding reduction in the proportion of surplus-labour to necessary labour.([Mar94]page 172) Marx was careful with the way he specied this tendancy of the rate of prot to fall. He saw it as a 'law', but one with counteracting inuences. In particular, if constant capital goods became cheaper this would tend to oset a fall in prots, and if the rate of exploitation that would also counter a fall in the rate of prot. These might act as partial osets to a long term tendancy for prot rates to decline. 3 Criticisms of the theory These undened elements : changes in the rate of exploitation and cheapening of the elements of constant capital, left Marx's theory open to criticism. The most serious challenge to the theory has come from the Okishio who in a landmark paper [Oki61] showed that any technical invention that is costsaving to the individual capitalist must raise the overall rate of prot in the economy. This paper combined into a single mathematical framework both processes of cheaping constant capital and raising exploitation. It purports to show that any economically rational investment by capitalists will tend to raise the rate of prot. Similar arguments have been made by the inuential economist Roemer [Roe86]. The maths used by Okishio is quite complex so rather than present his algebra we will take some worked examples that illustrate his points. Suppose we have a capitalist setup with the structure in terms of technology shown in Table 1. We have three industries, the rst produces luxury goods, the second subsis- tence goods, and the last capital goods. In order to produce 3 units of luxuries, two units of labour and one unit of capital goods are required. Similar condi- tions of production apply in all three industries. We can view the outputs if we like in concrete terms. Thus we might think of the subsistence output in terms of tons of food, the labour inputs as person years, and assume that the capital goods are measured in terms of numbers of machines. But this is just an aid to the imagination. In a real economy each of these industries would produce a large number of dierent types of machines, dierent types of food and clothing etc. We will also assume as a starting point that the rate of surplus value is 100%, so that a worker is paid half a working year of value in wages for each 3 Criticisms of the theory 7 Tab. 2: Economy in initial state. whole economy Mpers.yr value of value cost C v s output per unit per unit luxuries 1 1 1 3 1 0.67 subsistence goods 1 1 1 3 1 0.67 capital goods 1 1 1 3 1 0.67 totals 3 3 3 9 wage in value terms 0.5 wage in real terms 0.5 rate of prot 0.5 value composition of capital 1 year worked. From these assumptions about technology wages we can obtain Table 2 that describes the whole economy. Since we are talking about the whole economy we can, if we want, imagine that the units are now in millions of person years rather than individual person years. Since the wage is 0.5, the original 2 (million person years) of labour per industry translated into 1 (million person years) of variable capital per industry. The total surplus value is then equal to 3 (million person years). The 3 million person years of surplus value, correspond to the value of out- put produced by the luxuries industry  also 3 million person years. Which in turn tells us, that of the 6 million person years work done in the hypothetical economy, 3 million person years are spent producing luxuries for the upper class. Calculations like this in terms of labour value tell us how the population is dis- tributed. It means that half the population are working directly and indirectly to meet the needs of the upper class: two million working in the luxury goods industy, and 1 million in the capital goods industy. The value composition of capital in the example is unity, since the total variable capital and the total constant capital are the same. What does a value composition of capital of 1 mean in real terms? It means, at the current rate of exploitation, for every two workers, the plant and machinery and raw material they use required one years work to make. Table 2 is set out so that certain constraints required for reproduction are met. These were analysed by Marx when he looked at simple reproduction in Chapter 20 of Capital volume 2. In particular we need to ensure that the total surplus value equals the total output of luxuries, that total wages equal the total output of the subsistence goods, and that the capital goods bought as inputs equal the capital goods sold as outputs. Suppose now that capitalists reorganise production and nd that they can produce 3 units of subsistence goods using only 1 worker and 1 machine instead of 2 workers and one machine. At the then prevailing exchange values this appears a very protable innovation. The rms in this sector expect to make a saving of 25% in their total costs and expect to sell the output at the same price. They expect the situation to be as shown in Table 3. But this situation would not be sustainable. The capitalists making the 3 Criticisms of the theory 8 Tab. 3: What capitalists producing wage goods hope will happen after they cut their labour costs. whole economy value of value per C v s output unit output luxuries 1.00 1.00 1.00 3.00 1.00 subsistence goods 1.00 0.50 1.50 3.00 1.00 capital goods 1.00 1.00 1.00 3.00 1.00 totals 3.00 2.50 3.50 9.00 wage in value terms 0.50 wage in real terms 0.50 rate of prot 0.64 value composition of capital 1.20 Tab. 4: The economy after stabilisng following prot raising technical change. whole economy value of value per C v s output unit output luxuries 1.00 0.80 1.20 3.00 1.00 subsistence goods 1.00 0.40 0.60 2.00 0.67 capital goods 1.00 0.80 1.20 3.00 1.00 totals 3.00 2.00 3.00 8.00 wage in value terms 0.40 wage in real terms 0.60 rate of prot 0.60 value composition of capital 1.50 employment 5.00 3 Criticisms of the theory 9 change in productive technique do it on the assumption that prices will not change, and that they will get 3 million units of money for the 3 million units of subsistence goods they still produce. But can they sell all of it at the old price? They have laid o 1 million workers. If we are talking 19th century capitalism these workers may well have been driven by poverty to emigrate. The advance in the productivity that saved them labour costs has deprived them of a market. There is no way that the workers can buy 3 million worth of consumer goods out of an income of 2.5 million. So their actual sales will be at most 2.5 million and the price of consumption goods must fall. Once we allow consumption goods prices to fall to the level attainable in the shrunken market we get the situation shown in Table 4. The rate of prot in the whole economy has risen, because the rate of surplus value is now higher at 120% and this has oset the rise in the value composition of capital. But it has not risen as much as the capitalists in the consumer goods industry might have hoped. So this is what happens if there is an improvement in the productivity of labour with no net accumulation of constant capital ( it is 3 million person years before and after the change ). 1. The rate of exploitation rises. 2. The number of jobs shrinks. 3. The value composition of capital rises. 4. The real wage rises ( from 0.5 to 0.6 ). 5. The rate of prot across the whole economy is higher. This is one of the kinds of technical change that Okishio had in mind  one which would increase both the value composition of capital and the rate of prot. A word on method Drawing up tables like this has a certain arbitrary char- acter to it unless one follows clear rules. How did I obtain Table 4 from Table 3? It was done using the linear equation solver package within a spreadsheet. The linear programme package was set to maximise total prot under the new technical conditions and subject to the constraints that: 1. Total sales of consumer goods ≤total wages. 2. Total sales of luxuries ≤total surplus value. 3. Total sales of capital goods ≤ total capital goods consumed. The solver was allowed to vary the wage and the relative scales of the dierent industries. 3.1 A counter example to Okishio My example was of a technical change which used the same amount of constant capital and less labour to produce the same amount of output. What happens if we introduce a technical change that requires more constant capital and less 3 Criticisms of the theory 10 Tab. 5: An example of cost saving technical change lowering the rate of prot. whole economy sales of price per C v s output unit output prot rate luxuries 0.935 0.95 0.92 2.81 0.94 0.49 subsistence goods 1.065 0.81 0.84 2.71 0.90 0.45 capital goods 1 0.95 1.05 3.00 1.00 0.54 totals 3 2.71 2.81 8.52 wage in value terms 0.48 wage in real terms 0.53 rate of prot 0.49 value composition 1.11 employment 5.52 labour to produce the same amount of output. Suppose that instead of reducing the labour required to produce 3 units of consumer goods by 50%, we reduce it by only 15%, and increase the constant capital required by 10%? This more modest change will still be cost saving from the standpoint of the capitalists in the consumer goods industry. The eect on the whole economy when we put it into the solver is shown in Table 5. The overall rate of prot has now fallen, despite the initial change apparently being cost saving. Okishio's maths was sound, so how have I been able to produce a counter example? I have been able to do it because Okishio imposed additional very stringent constraints in his models : that the rate of prot must be identical in all in- dustries and a constant wage. In Table 5 the rates of prot are not identical and wages change. If one imposes extra constraints on a mathematical model, one restricts the possible outcomes. The question one has to ask is whether the constraints used in Okishio's model are an accurate reection of reality? Okishio himself recognised in a later paper [Oki90] that his argument against the tendancy of the rate of prot to fall could be invalidated if a rapid rise in capital stock allowed wages to rise. I believe that Okishio's constraint of equal rates of prot is also unrealistic for a capitalist economy. What production price theorists like Roemer and Okishio do not answer is by what dynamic mechanism is the rate of prot supposed to equilibrate? There have been a couple of attempts to address this recently but they arrive at radically dierent conclusions [SD09, Wri11] claiming respectively to have proven that such convergence can not occur or that it can occur. In the second case, the author argues that the equlibriation of prot rates will come about via the interest rate. In Farjoun and Machover's work [FM83] they argued that the chaotic character of capitalist economies must be taken into account when modeling them. This chaotic character meant that it is misleading to assume tight equilibrium conditions like an equal rate of prot in all sectors. Their objection has been well born out by empirical studies [CC03, Zac06] which show that not only are prot rates quite dispersed accross sectors but also, that industries with a high vc value composition tend to have a lower rate of prot than those with low vc compositions. These results not only cast doubt 4 A dynamic solution 11 on Okishio's assumptions, but also support Marx's basic theory in that they provide empirical evidence for the determining role of the value composition of capital in prot rates. This determining role is incomprehensible outside the labour theory of value. 4 A dynamic solution There is, I think, a more fundamental objection to the work of Okishio and Roe- mer than these rather technical points. It is that they are addressing a dierent theoretical problem from that of Marx and the classical political economists. Marx was concerned with the overall dynamics of accumulation in the whole economy. He was asking what happens if capital is accumulating at a certain rate, if say 25% of all prots are ploughed back as new capital. This accumu- lated capital, in his picture, crowds in to dierent elds of business and competes with existing capital already there. In the end too much capital accumulates undermining the very purpose of capital accumulation itself - the growth of prot. Okishio shifts the debate to a dierent question: the optimal choice of tech- nique by individual capitalists. In the example I gave above of cost saving technical change producing a fall in the rate of prot, the costings were done in terms of values not production prices. The essential dierence is that I costed constant capital just in terms of what its purchase price. If you use production price theory as Okishio did, then you have to cost constant capital in terms of capital plus expected average prot. This tends to make capital intensive innovations appear less protable. If rms reckon, using Okishio's calculus, that there are no prot opportuni- ties for investment in their own line of business at how instead will they attempt to accumulate their prot? If they simply deposit it with the banks and don't reinvest it, then the lack of investment demand brings on a recession and the rate of prot will fall because of the slackness of trade. Alternatively, as individual rms, they can put their prots into the stock market. This will tend to drive up the price of equities and reduce their yield. This yield on equities is the main indicator that rms have of a general rate of return, so a falling equity yield, brought about by purely nancial considerations, will cause rms to mark down their cost of constant capital. Investments that under Okishio's criterion would have appeared unprotable now seem worthwhile and accumulation can resume. But this then gives rise to a new question. Just how low does Marx's theory predict that prot rates will be driven? Can we use Marx's theory to predict what the rate of prot will be in the immediate future? There is no ready answer to be found in Capital, but one can relatively easily extend the theoretical framework laid out by Marx into a dynamic model that does give answers to these questions. The variables of interest, given Marx's treatment in the sections of Capital that I have mentiioned are: the division of surplus value between revenue and accumulation, the cheapening of constant capital and the rate of exploitation. I will look at how these aect the endpoint to which the rate of prot will decline1 . 1 What follows is a condensed version of the analysis in [CCM+ 08]. 4 A dynamic solution 12 0.6 0.5 0.4 0 pop growth 0.3 5% pop growth 5% pop growth 25% acc 0.2 0.1 0 0 5 10 15 20 25 30 Fig. 3: The way the rate of prot declines with dierent population growth rates and dierent accumulation fractions. In all cases the starting position is one where c = v = s. 4 A dynamic solution 13 Start with a simple scenario, an economy where there is no population growth and all prots are accumulated. According to Marx's logic, in this economy the rate of prot will tend towards 0 because the capital stock grows without limit whilst the surplus value has a xed upper bound. This is shown in the bottom line of Figure 3 which gives the result of numerical simulations of the result of capital accumulation in dierent scenarios. Next consider the situation where the working population grows by 5% a year, and again all prots are accumulated. In this case the rate of prot will decline until it reaches 5%. Why? Because at a 5% rate of prot, all reinvested, the capital stock grows at the same rate as the working population, at which point the value composition of capital stabilises. This scenario is shown as the middle line in Figure 3. Finally consider the scenario where only 25% of prots are reinvested, the rest being unproductively consumed. What is the nal rate of prot in this case? Clear it will be 20%, because at a 20% rate of prot, with a quarter being reinvested, capital stock will again grow at 5% to keep up with the growth of the working population. It follows that the basic equation for dening the equilibrium rate of prot re is re = g/α (1) where g is the growth rate of the employed workforce and α is the share of prot that is accumulated. There is one element of Marx's argument that this leaves out - the cheapening of the elements of constant capital. We can assume that this will progress at the same rate as the rise in labour productivity, let us call this t for technical progress. The eect of a cheapening of constant capital is to devalue the existing capital stock. A 5% annual growth in labour productivity will reduce the value of existing plant and machinery etc, by 5% a year. Its eect on the rate of prot is thus the same as that of population growth. Suppose there is no population growth but a 5% rate of technical progress, and assume that all prots are accumulated. Clearly the rate of prot will stabilise at 5% because at that rate of prot the reinvestment is just sucient to oset the technical devalorisation of the capital stock. So at that rate the value composition of capital must stabilise. The nal equation for the long term rate of prot, on Marx's assumptions must be: g+t re = (2) α This rate of prot re is the level to which the law of the falling rate of prot drives the actual rate of prot. What does it tell us: 1. That the equilibrium prot rates rise with the growth of the workforce. This is important because in developed capitalist countries with a low birthrate the population has tended to stabilise. The theory shows that if the population starts to fall, and if the rate of improvement in technology stagnates, then the rate of prot will tend to become negative. 2. That equilibrium prot rates rise with technical progress. This acts through its eect on cheapening constant capital. 5 Comparing the theory with reality 14 3. That the equilibrium prot rate does not depend on the rate of exploita- tion. Rises in the rate of exploitation can produce short term rises in the rate of prot, but they do not aect the nal level to which it will decline, they can only postpone the decline. 4. That rapid accumulation tends to reduce the long term rate of prot. This is also signicant because it shows the antagonistic and in the long term reactionary relation that the capitalist class has with the develop- ment of the productive forces. The long term rate of prot is higher if the capitalists consume the surplus unproductively rather than investing it. Their economic interest becomes directly counter to the further devel- opment of the productive forces, and they become increasingly concerned with rent seeking activity: securing monopolies via intellectual property rights, trademarks, acquisition of landed property etc. 5 Comparing the theory with reality A scientic theory is only as good as the predictions that it produces. If the basic model of accumulation put forward by Marx is right, we should be able to use it to predict the evolution of the rate of prot in real capitalist economies. Equation (2) has three variables on the right hand side: the accumulation share, the rate of increase in labour productivity, and the rate of growth of the working population. Given these three variables for a country one can easily calculate what re should be. If the theory is correct then the actual rate of prot will move towards that given by re . The short term, but not long term, movement of the actual rate can also be aected by changes in vs , so we should not expect the predictions to be 100% right, but they should be right most of the time, since sudden changes in vs are not that common. In Figure 4 we show for 4 countries the time seriess for re and the actual rate of prot. Note that re almost perfectly predicts what the actual rate will be a few years later. The equilibrium rate itself changes, primarily due to two causes. In the rst period we see a decline in re brought about by the exhaustion of labour reserves, which in turn was the eect of declining birth rates and the ending of migration from the land. In Japan, where the natural rate of population growth is low and where immigration is heavily restricted, the prot rate continued an apparently remorseless decline. In the other 3 countries it stabilises at a slightly higher level probably because of the combined eect of inward migration and a fall in the accumulation fraction. 6 Conclusion The theory of the declining rate of prot that Marx developed was remarkably insightful and fruitful. It captures the key features of accumulation in capitalist economies. It can be cast in a mathematical form that allows one to compute the future evolution of the rate of prot in a capitalist country, and the predictions it gives are remarkably good. 6 Conclusion 15 0.26 0.28 CAN equilibrium filtered USA equilibrium filtered CAN real profit filtered USA real profit filtered 0.24 0.26 0.22 0.24 0.2 0.22 0.18 0.2 0.16 0.18 0.14 0.16 0.12 0.14 1965 1970 1975 1980 1985 1990 1995 2000 2005 1965 1970 1975 1980 1985 1990 1995 2000 2005 0.4 0.3 JPN equilibrium filtered FRA equilibrium filtered JPN real profit filtered FRA real profit filtered 0.28 0.35 0.26 0.3 0.24 0.25 0.22 0.2 0.2 0.15 0.18 0.1 0.16 0.05 0.14 1965 1970 1975 1980 1985 1990 1995 2000 2005 1965 1970 1975 1980 1985 1990 1995 2000 2005 Fig. 4: Evolution of prot rates in Canada, USA, Japan and France. Solid lines are re dashed lines the actual rate. Note how the theory predicts the actual rate two or three years in advance. Data taken from the Penn World Tables[Mar09] and processed by Tamerlan Tadjadinov. 6 Conclusion 16 References [CC03] Paul Cockshott and Allin Cottrell, A note on the organic composi- tion of capital and prot rates, Cambridge Journal of Economics 27 (2003), 749754. [CCM+ 08] P. Cockshott, A. Cottrell, G. Michaelson, I. Wright, and V. Yakovenko, Classical econophysics: Essays on classical political economy, thermodynamics and information theory, Routledge, 2008. [Coc77] W.P. Cockshott, The Recession and Socialist Strategy, Conference of Socialist Economists, 1977. [FM83] Emmanuel Farjoun and Moshe Machover, Laws of chaos, a proba- bilistic approach to political economy, Verso, London, 1983. [Mar87] K. Marx, Capital, vol. 1. the process of production of capital, Trans. S. Moore and E. Aveling, Ed. F. Engels. Moscow: Progress Publish- ers. URL (accessed December 2007): Marx/Engels Internet Archive http://www. marxists. org/archive/marx/works/1867-c1, 1887. [Mar94] , Capital: Critique of political economy, vol. 3, the process of capitalist production as a whole,, The Marx/Engels Internet Archive, 1894. [Mar09] Adalmir Marquetti,Extended Penn World Tables: Economic Growth Data on 118 Countries, http://homepage.newschool. edu/~foleyd/epwt/, 2009. [Oki61] N. Okishio, Technical change and the rate of prot, Kobe University Economic Review, 7 (1961), 8699. [Oki90] , Constant and variable capital, The New Palgrave  Marx- ian Economics (John Eatwell, Murray Milgate, and Peter Newman, eds.), W. W. Norton and Company, New York and London, 1990, pp. 91103. [Roe86] J. Roemer, Should marxists be interested in exploitation?, Cam- bridge University Press, Cambridge, 1986. [SD09] A. Sinha and M.S. Dupertuis, Sraa's system: Equal rate of prots and the notion of center of gravitation, Journal of Economic Behav- ior & Organization 71 (2009), no. 2, 495501. [Wri11] I. Wright, Classical macrodynamics and the labor theory of value, Open Discussion Papers in Economics (2011). [Zac06] David Zachariah, Labour value and equalisation of prot rates, In- dian Development Review 4 (2006), no. 1, 121.