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Demographics and the long-run rate of profit
Paul Cockshott and Allin Cottrell∗
‘... the difficulty which has hitherto troubled the economists, namely to explain the falling rate of profit’ (Marx, 1971: 230)
The rate of profit tells us something about the potential rate of expansion of capital stock. It sets an upper limit on the rate of expansion – the rate of capital growth that will be achieved if all profit is reinvested. The focus of the following analysis is be on how this rate of expansion will change over time, if capital actually is reinvested. We also examine the circumstances under which capital might be reinvested, and the consequences of capital not being reinvested.
1 The mathematics of profit and labour time
If we approach the time-evolution of the rate of profit from the standpoint of capital accumulation, the issue is basically quite simple. Initially we will assume that all measurements are performed either in labour hours, or – what amounts to the same thing – in a monetary unit whose labour-time equivalent does not change from year to year. Using this approach we will derive an equation for the time-evolution of the rate of profit and show that the rate of profit tracks towards a long-run value which depends on the rate of growth of the working population along with the fraction of profit that is reinvested. To begin with, consider the implications of taking a labour-time perspective on the question of profit. Profit can be measured as a flow of labour value, in which case its units are person hours per annum, which in dimensional terms is just persons since the division hours/annum gives a scalar. Thus the annual flow of profit when measured in labour terms corresponds to a certain number of people – the number of people whose direct and indirect output is materialized in the goods purchased out of profits. The capital stock of a nation is, in these terms, a quantity expressed in millions of person years. And the rate of profit is then:
R =
Millions of workers whose product is bought by profits Millions of worker years represented by the capital stock
The evolution of R then depends on how rapidly the capital stock is built up compared to the growth rate of the number of workers producing the surplus that corresponds to profit. Let us represent the total profit or surplus value as a given share of the economy’s net output (we consider the effects of a change in this share later).¹ We will write this share as ( 1 − w), where w is the share of wages and salaries. Now, the concept of ‘net output’ requires some clarification. First, since we are working in terms of labour hours, the gross output of the economy is measured by the total hours worked, which we’ll write as L. To get the net output we need to subtract from the total hours worked the number of hours required to maintain the capital stock. This includes ‘physical’ depreciation, whereby part of the existing capital stock, K , wears out each year. We assume that depreciation occurs at a constant rate, δ, so that δK hours must be spent in maintaining the capital stock in physical terms. But, since we are measuring capital stock in terms of labour-hours embodied, there is an additional factor to be taken into account. Namely, if the productivity
∗This document is an extract from Chapter 14 of the book Classical Econophysics by Paul Cockshott, Allin Cottrell, Greg Michaelson, Ian Wright and Victor Yakovenko, which is due to be published by Routledge in 2009. ¹ At the level of abstraction of this argument, we are not distinguishing the components of surplus value – profit of enterprise, rent and interest – but rather treating the entire surplus as ‘profit’.
of labour is increasing over time, at some proportional rate g, then a given physical collection of commodities will come to embody a declining number of labour-hours. Call this effect devaluation of the capital stock. To maintain the capital stock in value terms (worker years embodied), the physical capital stock must be expanded. We can then write the following expression for net output:
L − (g + δ)K
That is, net output equals gross output minus the sum of depreciation and devaluation of the capital stock. Writing S for total profit, we then have
S = ( 1 − w) (L − (g + δ)K ) (1)
We next examine the growth of the capital stock. This is given by gross investment (‘source’) minus the sum of depreciation and devaluation (‘sink’). Gross investment we take to be a proportion, λ, of total profit. That is, K˙ = λS − (g + δ)K (2)
where K˙ is shorthand notation for the time-derivative of the capital stock, d K /dt.² Now the rate of profit (in labour-time terms) is the ratio of the surplus, S, to the capital stock, K :
R =
S
K
Using equation (1) to substitute for S, we get
R =
( 1 − w)(L − (g + δ)K ) K
= ( 1 − w)
L
K
− (g + δ)
If the wage-share in net output, w, remains constant (along with g and δ), then the time-derivative of the rate of profit is given by
R^ ˙ = ( 1 − w) d(L/K^ ) dt
That is, the change in the profit rate over time is a fraction of the change in the ratio of labour to capital stock. Via basic calculus, d(L/K ) dt
K
L˙ − L
K 2
K˙ = L
K
L
L
K˙
K
We will assume that the total labour performed per year changes at a proportional rate n (that is, L˙/L = n). In addition we infer from (2) that K˙ K
λS − (g + δ)K K
= λR − (g + δ)
It follows that
R^ ˙ = ( 1 − w) L K
(n − λR + (g + δ)) (6)
Here we have an expression for the time-evolution of the value-rate of profit in terms of the basic parameters of the system. Under what condition is the rate of profit unchanging (i.e. R˙ = 0)? Given that the wage share, w, must lie between 0 and 1, and that total hours worked, L, must be positive, the required condition is that n − λR + (g + δ) = 0. That is,
R?^ =
n + g + δ λ
where R?^ is the value of R that yields R˙ = 0. This is the steady-state rate of profit – the rate which, once attained, will persist over time.
² Note that since we have defined λ as the ratio of gross investment to S, and S is defined as a fraction of net output, it is possible in principle to have λ > 1. (This could happen if the capitalists fully cover depreciation and devaluation, and at the same time plough all of the surplus into new investment.)
0
0 10 20 30 40 50 60 70 80 90 100 years
rate of profit population growth
Figure 1: Evolution of the profit rate under constant population growth.
0
0 10 20 30 40 50 60 70 80 90 100 years
rate of profit population growth
Figure 2: Evolution of the profit rate given declining population growth. The rate of profit declines further than in Figure 1.
0
1
0 10 20 30 40 50 60 70 80 90 100 years
profit share rate of profit population growth
Figure 3: Evolution of the profit rate given declining population growth and constant real wages. The rate of profit declines despite the increasing share of profit in output (the top curve).
at 3 per cent a year, while real wages remain constant. Under these circumstances the wage share will fall by 3 per cent a year. (That is, w in year t equals w in year t − 1 divided by 1.03.) Observe that the rate of profit still falls. Investment in new plant and equipment is often associated with improvement in production techniques and hence a reduction in the labour-content and price of capital goods. Under these circumstances the existing of stock of capital capital will be devalued. This has two contrary effects on the rate of profit. On the one hand the devaluation of capital slows down the growth of the capital stock, which tends to mitigate any decline in the profit rate – see equation (2). On the other hand, devaluation produces a loss on the capital account which directly reduces profits (equation 1). The simulation shown in Figure 4 shows the relatively complex effects which emerge when a change oc- curs in the pace of technical improvement. During years 1–14 and 55 onwards there are no improvements in productivity, but over years 15 to 54 labour productivity rises by 5 per cent a year.³ Real wages are assumed to be held constant, so the wage share falls and the profit share rises during the period of growth in productivity. In year 15 the rate of profit drops sharply because of induced devaluation of capital. But for the following 15 years the rate of profit rises. This is due to the combined effect of slower growth of the capital stock and a rise in the profit share. The rate of profit subsequently settles into a declining trend because of the slowdown in population growth. As we showed earlier (equations 5 and 7), the rate of profit R?, towards which the actual rate of profit tends in the Figures above, is the only one at which the ratio of current labour to the labour embodied in the capital stock is stable. If the rate of profit is higher then the organic composition is driven up; if it is lower, then the organic composition falls. This has the interesting implication that if an economy were to have a declining population – which, given trends in birth rates, is quite plausible for many capitalist nations – then the long- term profit rate might be zero or negative. To retain a positive profit rate with a declining workforce, the rate of improvement in labour productivity must be greater than the rate of shrinkage of the workforce.
³ To make the effect on the graph more apparent we have also set the depreciation rate, δ, to 2 per cent over this period. Other than that, physical depreciation of the capital stock is ignored in these simplified examples.
If a capitalist wants to go up in the world, he should watch his value rate of profit. If he does not want his absolute standard of living to decline, he should ensure that his money rate of profit is greater than price inflation. Equation (7) showed how, for a representative capitalist, the growth of his power comes to be constrained by investment (λ), population growth (n), technical progress (g) and the rate of depreciation (δ). We can transform the attractor of the value rate of profit, R?, into an attractor for the monetary rate of profit, R? M, by adding the rate of inflation in value terms, πv , i.e., R? M = R?^ + πv. This πv measures the annual price increase for a commodity bundle containing, say, 100 hours of labour. As labour productivity rises, the 100-hour bundle will get physically bigger at the rate g. Suppose that the cost of living, measured in the usual way by the aggregate price of a representative bundle of goods, is constant. This means we have zero inflation in the ordinary sense of the word, but the rate of inflation in value terms is not zero: it is equal to g. The 100 hours corresponds to a larger bundle of goods, with a higher aggregate price. With ordinary price-inflation, π, at zero, πv = g. If ordinary inflation is positive, this augments value-inflation, hence in general πv = g + π. It follows that the attractor for the monetary rate of profit will be
R? M = R?^ + πv =
n + g + δ λ
Table 1 shows five possible scenarios to get a feel for the behaviour of the long-term value- and money- rates of profit. The scenarios are labeled by historical periods that have similar general features. Note that in the table R?^ and R? M are not the actual rates of profit, but rather the limits towards which the rates of profit evolve. Scenario 1 represents a period of high inflation, high accumulation and slow population growth – for example,
Table 1: The rate of profit: five scenarios. A depreciation rate of δ = 0 .03 is assumed in all cases. Scenario n g λ π R?^ R? M Comment 1 0.5% 2.25% 100% 8% 2.78% 13.03% UK 1970 2 1.0% 2.00% 20% −1% 15.15% 16.15% UK 1870 3 −1.0% 3.00% 30% 0% 6.77% 9.77% Europe 2020 4 −1.0% 3.00% 70% 0% 2.90% 5.90% Europe 2020, high accum. 5 5.0% 10.00% 100% 0% 15.03% 25.03% China 2000
the UK in the late 1960s and early 1970s. The long-run money rate of profit, R? M, is high but discounting the effect of inflation the underlying value-rate of profit is very close to g, the improvement in labour productivity. Going back a century (Scenario 2), we have faster population growth, but a much lower rate of accumulation out of profits and, because of the deflationary effect of the gold standard, slightly declining money prices. The value- and money-rates of profit are close to each other and stand considerably higher than in Scenario 1 due to the slow pace of accumulation. Scenarios 3 and 4 envisage a future European economy with a declining population, with the Euro managed so as to maintain a zero rate of price inflation, and assuming 3 per cent growth in labour productivity. Given relatively rapid accumulation, the value-rate of profit is approximately equal to the growth of labour productiv- ity; if accumulation is slower the value-rate is higher. In both cases the money-rate exceeds the value rate by the growth of labour productivity. Scenario 5 pertains to a rapidly emergent capitalist economy such as China in the early twenty-first century: the population is growing fast and the rate of investment is high. At the same time, the importation of advanced technology allows for much faster growth of labour productivity than can be attained in a mature capitalist economy. The long-run profit rate is therefore high despite the rapid pace of accumulation.
4 A tendency for the rate of profit to fall?
Smith, Ricardo and Marx all held the view that the rate of profit would tend to fall in the course of capitalist development. However, each offered a different reason for this thesis.
Smith thought that as capital accumulated, and the supply of commodities expanded, there would be in- creased competition between the capitalists to sell their wares, which would drive down both prices and the rate of profit. Ricardo rejected this argument. In modern terminology, he accused Smith of confusing microeco- nomics with macroeconomics. It’s true, at the microeconomic level, that if capitalists expand the production of some particular commodity, while the demand for that commodity remains unchanged, then its price will fall and so will the rate of profit in that line of production. But if the capitalists collectively expand the production of all commodities, at the macroeconomic level, one cannot assume that demand remains unchanged. With higher employment and higher wages (due to the increased demand for labour), accompanied by greater orders for capital goods, demand must expand along with supply. But then there’s no necessity for prices or profits to fall. Ricardo’s own case for a falling rate of profit hinged on increasing costs in agriculture. As population expanded it would be necessary to bring less fertile land into cultivation, and/or to farm the best land more intensively. The result would be a rise in the labour-time required to produce basic foods (‘corn’ or wheat, in Ricardo’s day), and hence a rise in the price of food. Profits would be put under a double squeeze: on the one hand wages would have to rise to cover the increased cost of subsistence for the workers; on the other, rent would increase since the rent on any given piece of land is equal to the wedge between the cost of production at the margin (i.e. on the worst land in use) and the cost of production on the land in question. This wedge will grow as progressively less fertile land comes to be cultivated. Ricardo’s argument is logically valid, but one of his factual premises has proved false. He thought that improvement in agricultural productivity, due to the application of science and technology, would be at best a mitigating factor, slowing the inevitable rise in the cost of food. In fact, to date we have managed to do better than that. The most basic index of the huge increase in agricultural productivity is the declining percentage of the workforce employed in agriculture. In the UK this fell from a large majority at the beginning of the nine- teenth century to 22 per cent at mid-century, to 10.7 per cent in 1891, and to 7.8 per cent in 1911 (Thompson, 1993). That leaves Marx. Writing at a time when the ongoing increase in the productivity of labour in agriculture had become readily apparent, Marx did not subscribe to Ricardo’s prognosis. His basic argument concerned the balance between the organic composition of capital (i.e. roughly speaking, the capital–labour ratio) and the rate of surplus value (i.e. the division of the working day between the ‘necessary labour’ required to maintain and reproduce the workforce and the ‘surplus labour’ that produces the goods purchased out of profit, rent and interest). Marx thought that (a) the organic composition tends to rise as capital accumulates, which of itself reduces the rate of profit (basically, by increasing the denominator), but also (b) the rate of surplus value tends to rise as the productivity of labour increases, which of itself raises the rate of profit. It might then seem that the net outcome for the rate of profit is indeterminate, but Marx had the intuition that the rising organic composition would ultimately win out. This intuition is borne out by the ‘demographic’ model we developed above. Unfortunately, Marx presented two arguments regarding the falling rate of profit, one correct and one fal- lacious. The fallacious one attracted a good deal of attention from the 1960s onward, hence obscuring the real issue. So it is necessary to disentangle the two. The problematic argument of Marx’s runs as follows.
- Each individual capitalist has an incentive to introduce new technology to cut his costs (true).
- The bias in such new technology will tend to be to increase the capital–labour ratio (probably true).
- In the short run, this will increase the profits of the innovating capitalist, who is able to sell his product at the ‘old’ price while having a reduced cost of production (true).
- When the new technique becomes generalized, the effect is to raise the economy-wide organic composi- tion of capital (true, if point 2 is correct).
- And this will lower the general rate of profit (false).
Marx’s argument has an air of paradox: a policy which raises the rate of profit for each capitalist, considered individually, nonetheless ends up having the effect of lowering the overall rate of profit. That a proposition of economic theory seems paradoxical is not in itself an argument against that proposition, but in this case the paradoxical claim is not sustainable; it can be shown on the basis of Sraffian input–output analysis that Marx’s
0
10
20
30
40
50
60
70
1860 1880 1900 1920 1940 1960
Figure 5: Accumulation of capital as a percentage of profit in the UK, 1855–
proved possible to organize unions among the the bulk of the working class, not just an aristocracy of skilled artisans. The dynamic interaction of industrial capital and the banks polarizes capital and precipitates out a class of rentiers. By the late Victorian era this process was well underway. A capitalist class whose grandfathers had been the pioneering cotton masters or iron masters of the industrial revolution had been transformed into a rentier class. Where frugality and accumulation had once been their watchwords, they now increasingly aped the lifestyles of their former political enemies, the landed aristocracy. Fortunes were spent constructing stately homes in the country and on employing retinues of domestic servants. With so much going on luxury consumption, less was left for investment. The late Victorian rate of investment was low: typically, less than 15 per cent of profit was reinvested in new plant and equipment within the UK (see Figure 5). In consequence the organic composition of capital remained low, and rate of profit did not decline significantly, in the late Victorian period (Figure 6).⁴ Again during the inter-war period, recession meant that capital accumulation was relatively slow. It was not until the period after 1945 that rapid capital accumulation, along with slow population growth, produced a substantial decline in profitability. David Zachariah has computed what the steady-state rate of profit in the UK and Japan should have been if the theory in this chapter is correct. As in section 1, the profit rate is defined as R = S/K , where S is the total net profit, computed by subtracting wages, W, and depreciation, δK , from the Gross Domestic Product, Y. That is, S = Y − W − δK. The steady-state rate of profit is given by equation (7) above. The actual and steady-state rates are compared in Figure 7. It can be seen the the actual rate of profit closely follows the steady-state rate – with a time lag, as one would expect. Edvinsson shows data for Sweden indicating that over a prolonged period there had been a significant rise in the organic composition of capital and a fall in the rate of profit. Dum´enil and L´evy (2002) show that there was a prolonged decline in profit rates in the USA in the postwar period (see Figure 8).
6 The Okishio critique revisited
Many people have interpreted Okishio’s critique of Marx to mean that the sort of historical trends discussed above ought not to have occurred. Since they did occur, we are led to believe that there must be some premises in his argument that are not an accurate reflection of the way capitalist economies actually work.
⁴ Figures 5 and 6 are constructed from Tables 3.1 and 3.2 in Cockshott et al. (1995).
2
4
6
8
10
12
14
16
1860 1880 1900 1920 1940 1960
profit rate organic comp.
Figure 6: Organic composition of capital and profit rate in the UK, 1855–
Table 2: Rising organic composition of capital, manufacturing and mining in Sweden (Edvinsson, 2003: table 7.5)
average average 1871–1900 1971–2000 change c/(s + v) 184% 305% 66% s/(s + v) 34% 21% −38% s/c 19% 7% −61%
Figure 8: Evolution of the profit rate in the USA (Dum´enil and L´evy, 2002).
One source of weakness in the Okishio theory is the assumption of an equalized rate of profit. This rate of profit is used as a benchmark against which possible improvements in productivity are measured. We have argued that this assumption is unrealistic. Actual profit rates show a wide dispersion (Farjoun and Machover (1983)), wider than the dispersion in the rate of surplus value for instance. The general rate of profit is not a given datum for an individual firm. A firm knows what its own rate of profit last year was, and it knows what the interest rate is, but the ‘general rate of profit’ is difficult to measure and is of interest mainly to economic statisticians. The process by which equilibration of profit rates is supposed to come about was originally invoked by classical economists in the context of comparing processes such as the maturation of wine and forestry, which had turnover times of several years, with corn-growing which had an annual turnover period. The argument was that capital would be invested in low-turnover activities only if it yielded the same return as in normal agri- culture. This argument may have some plausibility when applied to activities such as the production of vintage wine, where the rate of technical change is low and decades or centuries can be allowed for the establishment of relative prices, but it is less clear that it can be invoked where there is rapid technical change. In this case the time taken to establish equilibrium can be longer than the lifespan of the technology. This is especially true in certain industries with a very high capital–labour ratio, ones which are particularly relevant to the question at hand. Consider the Victorian railways. Here was an entirely new technology requiring huge capital investment. The lifetime of the capital in the form of bridges, embankments and stations would be a century or more. The railway booms resulted in over-capacity, which led to line closures by the early twentieth century. But before the capital invested in railways could depreciate to the level at which it would yield an equilibrium rate of return, the whole technology was superseded by road transport. The sort of equilibrium that is required for the Okishio theorem can be so long in coming that the industry has died before it is relevant. One can distinguish three rates that might act as benchmarks for a capitalist contemplating new investment:
- the statistical average rate of profit;
- the average rate of return on equities; and
- the rate of interest available from the banks.
We have ruled out item 1 as a practical guide, what about the rate of return on equities? This rate is certainly more accessible, since there are well developed stock markets that make such data available. This makes it a
more plausible investment benchmark. Suppose we have a static working population, and the rate of return on equities is equal to the general rate of profit. And suppose that any net investment in fixed capital would lead to a lower return on capital than allowed by the equity market. That is, in Okishio’s terms, no ‘viable’ labour-saving techniques are available.⁵ Firms will thus tend to select only capital-saving technical innovations. This implies that firms, taken as a whole, will need no net infusion of capital, so there should be zero net issue of new equities. If the rentier class as a whole is content to carry out no net saving then the situation is stable. But this is unlikely. If the rentiers attempt to accumulate capital by investing in shares, the net effect will be to bid up the price of equities, given that no new equities are being issued. The effect of this is to depress the rate of return on equities. This will affect the discount rate used in assessing investment projects. Previously unprofitable ones will seem profitable. New equities will be issued and the proceeds invested. Given that the population is static, this will raise the organic composition of capital and depress the real rate of profit. Similar considerations apply to the interest rate. If rentiers wish to accumulate assets in the form of bonds, and if capitalist enterprises are not issuing bonds to finance new investment on the grounds that such investment will lower the rate of profit, then the price of bonds will be bid upwards, which corresponds to a reduction in the interest rate. Just as with equities, this will make previously unprofitable investments seem profitable.
References
Cockshott, W. P., A. F. Cottrell and G. Michaelson (1995) ‘Testing Marx: some new results from UK data’, Capital and Class 55: 103–129.
Dum´enil, G. and D. L´evy (2002) ‘The profit rate: where and how much did it fall? Did it recover? (USA 1948–2000)’, Review of Radical Political Economics 34(4): 437–461.
Edvinsson, R. (2003) ‘A tendency for the rate of profit to fall’. Presented at the economic-historical meeting in Lund.
——— (2005) Growth, Accumulation, Crisis: With New Macroeconomic Data for Sweden 1800–2000. Ph.D. thesis, Stockholm University.
Farjoun, E. and M. Machover (1983) Laws of Chaos, a Probabilistic Approach to Political Economy, London: Verso.
Marx, K. (1971) Capital, vol. 3, Moscow: Progress Publishers.
Okishio, N. (1961) ‘Technical change and the rate of profit’, Kobe University Economic Review, 7: 86–99.
Thompson, F. M. L. (1993) The Cambridge Social History of Britain, 1750–1950, Cambridge: Cambridge University Press.
Zachariah, D. (2008) ‘Determinants of the average profit rate and the trajectory of capitalist economies’. Pre- sented at the conference on Probabilistic Political Economy, Kingston University, July 2008.
⁵ A ‘viable’ technique is one which raises the profitability of the firm that introduces it, taking the current vector of prices and wages as given. As noted above, Okishio argues that if a technique is viable in this sense, its generalization must raise the economy-wide rate of profit so long as the real wage remains constant.
