Karl Marx’s Das Kapital

Abstract

Karl Heinrich Marx (1818–1883) was born in Trier, Germany, in a Jewish family that converted to Protestantism during his childhood. He studied philosophy in the Universities of Bonn, Berlin and Jena from where he earned his doctorate in philosophy at the age of 23. As a student, he was involved in circles of young philosophers known as the Young-Hegelians. He worked as a journalist and editor for the influential newspaper Rheinische Zeitung of Cologne. The radical perspective of the newspaper led the Prussian authorities initially to censorship and later to the closing of the newspaper and to the exile of Marx. He took refuge in France and settled in Paris, where he had the opportunity to study French utopian socialism and English Political Economy, while at the same time, he was involved in the socialist movement in Prussia.

Appendix

5.1.1 A1: The Mathematics of the Falling Rate of Profit

We argued that Marx’s thesis for the falling rate of profit is that the organic composition of capital as a result of mechanisation of the production process and that this increase is higher than that of the rate of surplus-value (s/v) and necessarily leads to a long run falling tendency in the rate of profit. This result can be shown starting from the formula of the rate of profit which can be rewritten as:

$$ r = \frac{s}{C} = \frac{{s/l}}{{C/l}} $$

where l=s+v, that is, the total labour time (l) is equal to the surplus (s) and necessary (v) labour time. The advantage of this formula is that it sets limits to the variation of the rate of profit. For example, we derive that regardless of the rate of increase in the rate of surplus value (s/v), the numerator of the above formula has as an upper limit the one, and the rate of profit for v→0 (i.e., “workers leave on thin air”) is equal to the l/C, that is, the maximum rate of profit (the rate of profit for s=l). The mechanisation process leads to a rising C/l ratio or what amounts to the same thing a falling maximum rate of profit. The latter implies that the general rate of profit (whose magnitude depends on the level of v) fluctuates with an interval with a falling upper limit. In short, the general rate of profit with the passage of time starts to display a falling tendency, for it is depressed from above from the falling maximum rate of profit. However, this in itself is not an adequate proof of the falling rate of profit, and one must show that the limit of the rate of profit is zero (Kurz 1993, p. 113).

For the proof of this proposition, let C΄=C/l, s΄=s/l, v΄=v/l or v΄=1–s΄ and the rate of profit can be rewritten as:

$$ r = \frac{{s'}}{{C'}} $$

Assume now that C΄ increases at a rate equal to α, whereas the variable capital decreases at a constant rate equal to β. By using time subscripts, we can write for the evolution of each of these variables as follows:

$$ {C'_t} = {C'_0}{(1 + \alpha )^t}\,{\text{and}}\,{v'_t} = {v'_0}{(1 - \beta )^t} $$

and the evolution of s΄ is residually determined, that is, ${s'_t} = 1 - {v'_0}{(1 - \beta )^t}$. The rate of profit therefore can be rewritten as follows:

$$ {r_t} = \frac{{1 - {{v'}_0}{{(1 - \beta )}^t}}}{{{{C'}_0}{{(1 + \alpha )}^t}}} = \frac{{{{(1 - \beta )}^{ - t}} - {{v'}_0}}}{{{{C'}_0}{{(1 + \alpha )}^t}{{(1 - \beta )}^{ - t}}}} $$

as t increases without bounds the numerator and the denominator of the rate of profit increase to infinity, so we end up with an indeterminate form, ∞/∞. Thus, we can write,

$$ \mathop {{\lim }}\limits_{t \to \infty } \frac{{{{(1 - \beta )}^{ - t}} - {{v'}_0}}}{{{{C'}_0}{{(1 + \alpha )}^t}{{(1 - \beta )}^{ - t}}}} = \frac{\infty }{\infty } $$

We apply L’Hôpital’s rule, which gives:

$$ \mathop {{\lim }}\limits_{t \to \infty } \frac{{ - {{(1 - \beta )}^{ - t}}\ln (1 - \beta )}}{{{{C'}_0}{{(1 + \alpha )}^t}\ln (1 - \alpha ){{(1 - \beta )}^{ - t}} - {{C'}_0}{{(1 + \alpha )}^t}\ln {{(1 - \beta )}^{ - t}}\ln (1 + \beta )}} $$

which simplifies to the following:

$$ \mathop {{\lim }}\limits_{t \to \infty } \frac{{ - \ln (1 - \beta )}}{{{{C'}_0}{{(1 + \alpha )}^t}\ln \left[ {(1 + \alpha )/(1 - \beta )} \right]}} = 0 $$

5.1.2 A2: The Incremental Rate of Profit and its Components

In the analysis of competition in Marx’s capital, we faced the following paradox. On the one hand, the rates of profit were equalised across industries and on the other hand, there was a stratification of the rates of profit between firms within. How can these contradictory observations be reconciled? The idea is that the average rate of profit is the average of all firms that comprise the industry. And an industry consists of firms that use very advanced technology and excellent location and firms whose technology is old. Certainly, investment flows would be directed neither towards the old type of capitals because of low profits nor towards the very new, precisely because they have not been tested adequately, so there is too much risk involved. Besides, there are problems in investing in these kind of capitals simply because these capitals are not easily reproducible, for example, things such as patents, location near a source of raw materials and the like.

Classical economists were aware of these limitations in the flows of capital; perhaps the best example is the case in agriculture where the most productive pieces of land are already cultivated and they are not available to new entrants, so the new entrants enter not to the average quality of land since it is not available but rather to the worst type of land because only that is available. Classical economists therefore considered as the average rate of profit not simply the arithmetical average but rather the type of capital where expansion or contraction of accumulation takes place.

Turning to manufacturing, the regulating conditions of each sector are determined by exactly the same method; that is, by the type of capital where expansion or contraction of accumulation takes place. This concept is similar to what business people call the capital, which embodies “the best generally available method of production”, and is often called “the best-practise method of production”. This should not lead to the conclusion that all firms adopt this method of production immediately, since firms operate fixed capitals of different vintages and managers have different expectations about the direction of demand and profitability. Consequently, firms do not easily switch from one method of production to another. However, new capitals are expected to enter into the method of production, which can be duplicated and, furthermore, the expected rate of profit is attractive enough. The production method which is targeted by the new entrants is usually the most recent in the industry and not the older or the most profitable. The older methods of production ceteris paribus, will have a rate of profit lower than the average, whereas the most profitable, methods of production may not be easily reproducible or their reproduction is associated with certain degree of risk, which new entrants may not wish to undertake. However, over “a cycle of fat and lean years”, that is, a complete business cycle, there is a tendency for the rate of profit to equalise among regulating capitals between industries. The profit rates of the regulating capitals across each industry are those that will be tendentially equalised.

The rate of profit earned on regulating capital is, therefore, the measure of new investment’s return and determines the rhythm of accumulation in industries. If two regulating capitals have different rates of profit, the investment will flow differentially and will not just stop flowing in the industry with the lowest rate of profit because of uncertainty and differences in expectations. It is important to point out that the regulating conditions of production do not necessarily specify a single rate of profit, but rather a narrow spectrum of rates of profit. This is true even in the case of a single regulating condition of production, because there are still differences in management, demand, etc., which may give rise to profit rate dispersions. Consequently, at any given moment in time, the rates of profit between regulating capitals across industries are not equal, and only in the long run, there is a tendential equalisation of the regulating rates of profit to an average. Anwar Shaikh (1995, 2008) argued that the rate of profit that tends to be equalised between industries is not necessarily the average rate of profit, but rather the rate of profit that corresponds to the regulating conditions of production within an industry. “incremental rate of return on capital” (henceforth IROR) and he approximated it by taking into account the following considerations: investment flows are conditioned more by short-run rate of return such as the incremental rate of profit than the rate of profit over the lifetime of investment. Hence, he expresses current profits (S _t ) that accrue to a firm as the sum of profit from the most recent investment (ρI _t–1) and profits that accrue to the firm from all the previous investments (S*), which is equivalent to saying “the current profits in the absence of new investment”. Consequently, we write:

$$ {S_t} = \rho {I_t}_{--1} + S* $$

If we subtract profits of the past period from both sides of the this equation, we get:

$$ {S_t}--{S_t}_{--1} = \rho {{\rm I}_t}_{--1} + (S*--{S_t}_{--1})\,{\text{or}}\, \Delta {S_t} = \rho {{\rm I}_t}_{--1} + (S*--{S_t}_{--1}) $$

The term in parenthesis is expected to be very small in comparison with the term ρΙ _t–1 and for practical purposes it can be ignored. The justification is the view that the shorter the evaluation horizon, the closer the current profit will be on carried-over vintages S* to the last period’s profit on the same capital goods (S _t–1). Moreover, since uncertainty and ignorance increase with the passage of time, it is reasonable to assume that the short-run (up to a year) is the relevant time horizon. After all, current profits are fraught with many ephemeral factors, and we know that abnormally high or low profits direct investment accordingly, which in turn gives rise to new uncertainty and thus profits or losses, and so forth. With these considerations in mind, it is reasonable to assume that expectations about future returns to investment are nearsighted; that is, expectations depend on the short-run rate of return. Consequently, the current rate of return on new investment will be

$$ {\rho_t} = \Delta {S_t}/{I_t}_{--{1}} $$

that is, the change in profits of each industry divided by the investment in the previous period. The above configuration provides a practical way to identify the IROR in the case that we do not have data on the best practise technique and the firm that utilises it over the years. Consequently, the motion of the IROR determines whether or not there is a tendential equalisation of profit rates for the regulating capitals.

Alternatively, we can derive the IROR from the simple definition of the rate of profit r=S/K or S=rK, whose total differential gives: dS = rdK + Kdr. We divide by dK and we get

$\frac{{{\text{d}}S}}{{{\text{d}}K}} = r + K\frac{{{\text{d}}r}}{{{\text{d}}K}} = r\left( {1 + \frac{{{\text{d}}r}}{{{\text{d}}K}}\frac{K}{r}} \right) = IROR$ ³³

The term (drK/dKr) in the parenthesis is the elasticity of profit rate for which the following holds:

$$ {\text{if}}\,\left( {\frac{{{\text{d}}r}}{{{\text{d}}K}}\frac{K}{r}} \right)\frac{ > }{ < }0\,{\text{then}}\,IROR\frac{ > }{ < }r $$

It can be shown that the ΙROR is a variable that encapsulates the operation of a series of other variables such as the profit and wage shares, productivity of labour, capacity utilisation and capital–output ratio. In order to show the operation of all these variables, we start from the definition of total income (Y) as

$$ Y = rK + wL $$

whose total differential gives:

$$ dY = rdK + Kdr + wdL + dwL $$

We divide throughout by dK and get:

$$ \frac{{{\text{d}}Y}}{{{\text{d}}K}} = r + K\frac{{{\text{d}}r}}{{{\text{d}}K}} + w\frac{{{\text{d}}L}}{{{\text{d}}K}} + \frac{{{\text{d}}w}}{{{\text{d}}K}}L = r\left( {1 + \frac{{{\text{d}}r}}{{{\text{d}}K}}\frac{K}{r}} \right) + w\left( {\frac{{{\text{d}}L}}{{{\text{d}}K}} + \frac{{{\text{d}}w}}{{{\text{d}}K}}\frac{L}{w}} \right) $$

$$ = IROR + w\left( {\frac{{{\text{d}}L}}{{{\text{d}}K}} + \frac{{{\text{d}}w}}{{{\text{d}}K}}\frac{L}{w}} \right) $$

After some mathematical manipulation, we get:

$$ IROR{ = }\frac{{{\text{d}}Y}}{{{\text{d}}K}} - w\left( {\frac{{{\text{d}}L}}{{{\text{d}}K}} + \frac{{{\text{d}}w}}{{{\text{d}}K}}\frac{L}{w}} \right) = \frac{{{\text{d}}Y}}{{{\text{d}}K}}\left[ {1 - w\left( {\frac{{{\text{d}}K}}{{{\text{d}}Y}}\frac{{{\text{d}}L}}{{{\text{d}}K}} + \frac{{{\text{d}}K}}{{{\text{d}}Y}}\frac{{{\text{d}}w}}{{{\text{d}}K}}\frac{L}{w}} \right)} \right] $$

or

$$ IROR{ = }\frac{{{\text{d}}Y}}{{{\text{d}}K}}\left[ {1 - w\left( {\frac{{{\text{d}}L}}{{{\text{d}}Y}} + \frac{{{\text{d}}w}}{{{\text{d}}Y}}\frac{L}{w}} \right)} \right]{ = }\frac{{{\text{d}}Y}}{{{\text{d}}K}}\frac{K}{Y}\left[ {1 - w\left( {\frac{{{\text{d}}L}}{{{\text{d}}Y}} + \frac{{{\text{d}}w}}{{{\text{d}}Y}}\frac{L}{w}} \right)} \right]\frac{Y}{K} $$

or

$$ IROR{ = }\frac{S}{Y}\left[ {1 - w\frac{L}{Y}\left( {\frac{{{\text{d}}L}}{{{\text{d}}Y}}\frac{Y}{L} + \frac{{{\text{d}}w}}{{{\text{d}}Y}}\frac{Y}{w}} \right)} \right]u{\left( {\frac{Y}{K}} \right)^*} $$

From the foregoing analysis, we observe that the IROR is directly related to the profit share (S/Y), the rate of capacity utilisation (u=(Y/K)/(Y/K)*), the growth rate of productivity of labour (dY/dL/Y/L) and to the normal capacity output–capital ratio (Y/K)*. In addition, the IROR is inversely related to the share of labour income (wL/Y) and the elasticity of wage with respect to income (dwY/dYw).