4—Capital-interest and wages in the stationary economy A. Mathematical presentation

Let us therefore assume that a group of workers wish to start a productive undertaking on their own account, in which one commodity or a number of goods is produced once. They themselves possess no capital. They can, however, within certain limits, obtain any amount of money on loan at a rate of interest which for the time being we shall think of as given. In order to make the matter as simple as possible, we shall assume that they do all the necessary preparatory work themselves, make the tools, and so on. However, once the production process is complete and the goods are ready, these tools are assumed to be worn out and valueless. The more labour they devote to these preparations for production, the lengthier will be the production process. But, as compensation for this, the quantity of goods produced, or rather their value, will be greater, according to our assumptions; and, what is more, this value must here be assumed to be growing in a greater proportion than the length of the period of production itself; so that the value of the average (i.e. annual or daily) production of a worker is also to be thought of as growing with the length of the period of production (in which case, however, the scale of the surplus profits is necessarily a decreasing one).

If now we ask what method of production or—what is here the same thing—how long a period of production these workers are to choose with most advantage to themselves, this problem, it is clear, remains vague, since the workers can obviously pursue two different aims: on the one hand, they may strive to attain the greatest possible ultimate profit; on the other, they may desire to procure for themselves a subsistence as abundant as possible while the work lasts. But since we still wish to keep the hypothesis of the stationary condition and must consequently regard the sum of the capitals as an invariable magnitude, we can disregard completely the gain which will ultimately result and which would obviously be a new capital.¹We therefore assume that the workers, even when they themselves are the entrepreneurs, merely strive to attain the second of these two aims, the greatest possible subsistence or wages. Then the problem is quite definite and very easy to solve.

Let the value of the final product be s. According to our last assumption, we shall find that this value comprises the whole capital engaged in production plus interest on this capital, and no more. But the capital consists here merely of the cost of maintaining the workers and will consequently amount to t. l for each worker, if l stands for the annual subsistence or annual wage of one worker, still to be determined, and t for the length of the period of production expressed in years (and fractions of years). If now the whole capital was borrowed already at the beginning of the production process, then, on the assumption of simple interest and if z stands for the rate of interest, t .l. z . t (or t² .l. z) must consequently be paid as interest. But if the capital is only invested by instalments, this sum has to be multiplied by some proper fraction, which in the case of a constant taking-up of capital can, it is evident, become as small as ½, and no smaller. We therefore write

(12)

The value can be taken as the average length of the investment of capital, which therefore need only amount to half the length of the process of production, if the production is constant.

If both sides are divided by t, we have, since obviously stands for the average annual production of one worker which we shall call p:

(13)

s and p are here, as has already been said, to be understood as functions and, what is more, as known functions of t ; z is assumed to be a known value; and the task is now to determine t in such a way that l becomes as great as possible. This is done, of course, by means of differentiating on both sides in respect of t, as if l were a constant; since, in the case of a maximum, dl = 0.¹ We consequently obtain

(14)

and this equation gives us, together with (13), the values of t and l, expressed in terms of z, which we require to know.

In order to make the understanding of this problem easier, we shall also illustrate this result geometrically. We assume that t and p are abscissa and ordinate of a curve that, according to the known attributes of p, must follow a rising course which, however, is concave in respect of the axis of abscissae and (since something can always be produced, even in production for immediate use which is carried out without any capital) intersects the axis of ordinates at a certain distance from the zero-point. If we take any one point on this curve and connect it by a straight line to a fixed point which lies on the negative side of the axis of abscissae at a distance of from the zero-point, then this straight line will cut off a section of the axis of ordinates which is equal to l, as follows from equation (13) if it is written in the form

The greatest value of l can consequently be attained, if from the fixed point mentioned a tangent is drawn to the curve. This is just what equation (14) expresses.

Let us now deal with the contrary question. Let us suppose that the wages are given and that an entrepreneur who is himself a capitalist wishes to direct his production in such a way that the greatest possible profit accrues to himself from the capital which he has expended on each of the workers employed and consequently on the whole production. This problem (the only one which Böhm-Bawerk has dealt with) seems at first sight to be quite different from the former, but leads to precisely the same expressions. That is to say, when p and l stand for annual production and annual wages of a worker, we obtain in this case also

Here, however, l is understood as a known value and our task is to determine t in such a way that z becomes a maximum. But the differentiation in respect of t takes place in both cases as if l as well as z were a constant, and we obtain as before

Using these equations, t and z can now be expressed in terms of l.

The geometrical solution is arrived at in this case by taking a point at a distance l from the zero-point on the axis of ordinates and drawing from this point a tangent to the curve. This tangent now determines on the negative side of the axis of abscissae the length , which in turn determines the value of z. (If, for instance, the length so determined is 40, z becomes equal to or 5 per cent.)

Here it is assumed, however, that the capital is employed only successively. Temporarily, therefore, a use must be found for it outside the business. In order to avoid this difficulty, we could imagine that the entrepreneur carries on not merely one, but several businesses of the same kind at the same time, in all of which the period of production is the same, but which are at different stages of progress, so that the entrepreneur can consequently market finished goods once a month, say, or once a week. The proceeds from these provide him with necessary money for the next payment of wages. Since in this case each of the workers employed has, on an average, half of the production process behind him,¹ the average capital invested in each worker obviously amounts to . But the average monthly production of each worker is and the monthly wage and their difference can be regarded as monthly interest on the capital invested in each worker; so that the monthly rate of interest amounts to and the yearly rate of interest consequently amounts to and we obtain

as above.

But it is not necessary to suppose such a rigorously conducted gradation of production within the separate businesses. It is sufficient if this phenomenon appears as the result of the total production This is the same as saying that the different products and half-finished products are produced precisely when consumption and production require them. That is to say, the capital, too, can then find employment through the mechanism of the loan-market just when it becomes free.¹

This is more or less the actual state of affairs, or rather it is the ideal towards which production continually strives. But this ideal, for several reasons, can only be partly attained.

It was, by the way, assumed in what has just been said, that production is itself constant, so that at each moment of production the same number of workers is employed. This, too, is of course not the case. At certain stages of production there is perhaps room for very few workers or for no workers at all—when, for instance, the goods in process of production are simply exposed to the action of natural forces; for example, when ripening grain continues standing in the fields throughout the summer, or when, in the production of wine, after the completion of the actual production, the new wine remains lying in the cellar, perhaps for years. Finally, the period of production ought actually to be thought of as lasting until the finished goods are in fact sold.

Still, we shall allow for all these facts if we put the general expression €. t (where € is a proper fraction) instead of for the length of the investment of capital.

It is clear that in this case the gradation of production within the particular economy must at least be carried to the point at which the workers employed find uninterrupted occupation. Our equation (13) can then take the form

p = l(1+ € . t .z)

The distribution of labour over the period of production can itself be altered, however, and e is therefore in reality a variable quantity. The product € . t, that is to say, the investment period of the capital, can here, however, be conceived as a single variable, so that the expressions undergo no essential alteration, at least when calculating simple interest.

If now, within the branch of the business in question, the total existing capital and the total number of workers employed were each a constant magnitude, we could find out not only the above relations between wage, level of interest and length of the period of production (which, to be sure, can be assumed to be equal practically everywhere within this branch of the business), but even these magnitudes themselves. That is to say, since the capital invested in each worker is, on an average, (or more generally €. t.l), we obtain

(15)

when K stands for the total capital and A for the number of workers employed. Using this equation in combination with the equations (13) and (14), we can express l, t and z in terms of K and A. We obtain, in fact, from (13) and (14), by eliminating z,

(16)

and if the value of l obtained from the above equation is substituted in (15), we get

(17)

This equation can be solved for t, since p and must be thought of as known functions of t; and so forth.

This assumption, however, will not do. Capital and labour which are to-day employed in the manufacture of goods of a certain kind, can to-morrow have been partly transferred to other branches of business. Within the whole economy, however, the number of available workers and the total capital can be regarded as approximately given magnitudes. If, therefore, following Böhm-Bawerk’s precedent, we may assume as a first approximation within all branches of the business the same productivity and the same increase in productivity when the length of the period of production is increased, then, obviously, our equations set forth above can be regarded as valid for the whole economy, since t stands for the length of the period of production, p the yearly production of one worker, and l and z the wage and the level of interest. According to our assumption, these values must be the same in all the businesses.

Indeed, in practical life the fulfilment of these equations would take place in the following way. At each level of wages a period of production of a certain length proves to be the most advantageous to the entrepreneur-capitalists, since it promises the greatest possible interest (makes z a maximum). If in this case all the workers find employment and the whole of the existing capital is invested, these proportions will undergo no further change: equilibrium on the capital-labour market has been reached. But if more labour is demanded than is available, wages must rise. At the new level of wages a new and, what is more, a longer period of production proves at once to be the most profitable, as is evident, and the superfluous capital is absorbed partly by the rise in wages, and partly by the lengthening of the period of production.

If, on the other hand, more labour is available than can be employed during a period of production of the length in question, wages must fall, owing to competition of the workers. At this lower level of wages a new and, what is more, a shorter period of production recommends itself as the one which is now most profitable to the capitalists. This is adopted, and the capital which was before insufficient is now able to give employment to all workers, partly owing to the decrease in wages, but partly also to the shortened period of production.

In both cases equilibrium is finally achieved, although only after several oscillations in this and that direction; and in the case of equilibrium all our above equations are fulfilled.

If, on the contrary, we had set out from the assumption that the workers are themselves entrepreneurs, the result would have been the same—with this difference, however, that supply and demand now occur on the loan market, so that the rising or falling rate of interest now takes the place of increasing and decreasing wages.

In both cases the equations of equilibrium will be the same, and, what is more, a large amount of capital and a comparatively small number of workers will always be connected with a longer period of production, high wages and a low rate of interest—and vice versa. That is to say, when the capitalists are entrepreneurs, the lengthening of the period of production is seen to be a reaction on the part of the capitalists against the increase in wages which has taken place and the low rate of interest which results therefrom. But as a result of this lengthening of the period, interest can again be raised to some extent, but cannot reach the level achieved in the case of the previous lower level of wages.

If, on the contrary, the workers are entrepreneurs, and the rate of interest, due to increased demand for capital, has risen, the workers will shorten the period of production, and by this means once again be able to improve to some extent their incomes (i.e. wages), diminished by the rise in interest. But neither wages nor interest can in this case return to quite the former position.¹

The question could be asked, how far the above result is affected by the existence of people who work with their own capital. This question is, however, easily answered. If such a worker has enough capital to observe (in the case of steadily-flowing production)² the usual period of production, then he will select just this period (always supposing that his purpose is merely to conserve his capital and not to increase it). If he has less capital, he must adopt a shorter period of production; if he has more, then he can, if he so desires, adopt a longer period. In both cases, however, he will obtain a greater income if he chooses the customary period of production. In order to do so, he will, in the first case, procure the capital which he still requires by means of a loan at the ordinary rate of interest, and in the second case he will lend the superfluous capital or use it to employ other workers. For the validity of our formulae it is therefore of no consequence whatever who possesses the capital, provided only that the latter is employed as capital.

Here, however, I must draw attention to a certain ambiguity in the problem, which was not taken into consideration by Böhm-Bawerk and which I in my criticism of his presentation (in Conrads Jahrbücher, December 1892) had not yet noticed.

One could imagine capitalists adopting longer and longer periods of production in a quite haphazard way, wages being in this case determined every time by the competition of capitalists and workers according to equation (15),

The interest attained is still given by (13),

but since l is now no longer regarded as a constant but depends, according to (15), on t, it follows that when we try to determine t in such a way that z becomes a maximum, we are led (as can easily be seen) not to equation (14) but to the quite different equation

Since l and t are essentially positive, would have to be negative here ; that is to say, supposing the length of production is increased more and more, the greatest possible interest will only be attained when the scale of productivity (the annual production of one worker) has changed into a decreasing one. Practically speaking, no real maximum of the rate of interest consequently exists here, but each lengthening of the period of production will be advantageous to the capitalists.

This result may seem strange, but is not difficult to understand. What we have been considering above was the case of free competition, where everyone pursues his own advantage. But our last assumption presupposes that capitalists combine to depress wages and that the workers can do nothing about this. But then each lengthening of production will prove in the end to be remunerative, provided it is undertaken simultaneously in all businesses; since the wage-capital available for each year is diminished and wages must consequently fall. Even if the productivity of one worker remains unaltered or even undergoes a slight decline, it will still be remunerative. In this case, of course, the fall in wages will sooner or later cause, somehow or other, a drop in the number of workers within the economy, or the workers must be partly supported by charity. But if this point has not yet been reached, it will always be in the interest of the capitalists as a class to extend the period of production.

But the situation is different if there is free competition between the capitalists, because in this case the low level of wages will be a temptation to every individual capitalist to shorten the period of production and to use his capital for the employment of a greater number of workers. But if several capitalists do this, wages will, of course, rise.

On the other hand, by sticking together, workers can, within certain limits, undoubtedly enforce a shorter period of production if, for instance, they refuse to work with the new ‘labour-saving’ machines. As a result of this, wages will rise—if, of course, we assume that the capital remains undiminished in spite of the lower rate of interest. But if there is free competition amongst workers, this reduced rate of interest will, for some workers, be a temptation to become entrepreneurs themselves—and, what is more, according to the lower level of interest—by adopting longer periods of production. And so the demand for capital would again become greater, etc. We cannot pursue this subject further here. However, what has been said will suffice to show that the new concept ‘period of production’ seems destined to bring order and clarity to some of the most complicated problems of political economy, problems which are far from being explained.

B. Böhm-Bawerk’s presentation and his ‘positive’ law of interest. His criticism of Jevons’s theory of interest

The above-mentioned presentation is substantially identical with the theory to which Böhm-Bawerk has devoted the last chapter of his book. But this theory obviously contains merely an element of a complete theory of interest, because, on the one hand, the services of the land (actually the services of all rent-goods) were left unconsidered, and because, on the other hand, the theory assumes that there is an identical productivity and scale of productiveness for all branches of production—which is very far from reality. In what follows I shall try to replace this theory by another, which is complete in both these respects; and in this way I hope, in the end, to be able to take up again our problem of determining the exchange value, which was not brought to a conclusion in the previous chapter.

First of all, however, I shall go a little more deeply into Böhm-Bawerk’s treatment of this problem, in order to emphasize once again the great importance of this theory, but also because of several remarks which he makes, as it seems to me that his reasoning there does not hold good in all points.

Böhm-Bawerk lays down as an hypothesis an invariable pattern, which is supposed to represent the productiveness of production when, for instance, a period of production of one, two, three, etc., years is adopted. After this he shows how, assuming different levels of wages, now this and now that period of production yields the highest interest on the capital which has to be invested in each worker. I reproduce here one of the relevant tables. This corresponds to just the level of wages, 500 fl., which would prove absolutely right for the number of workers and amount of capital chosen in this example.

Level of wage 500 fl.
Period of production	Product of one working-year	Annual profit per worker	Number of employed	Total annual profit on each 10,000 fl.
1 year	350 fl.	– 150 fl.	40	(Loss)
2 years	450 „	– 50 „	20	„
3 „	530 „	30 „	13.33	400.00 fl.
4 „	580 „	80 „	10	800.00 „
5 „	620 „	120 „	8	960.00 „
6 „	650 „	150 „	6.66	1,000.00 „
7 „	670 „	170 „	5.71	970.70 fl.
8 „	685 „	185 „	5	925.00 „
9 „	695 „	195 „	4.44	866.66 fl.
10 „	700 „	200 „	4	800.00 „

The first three columns require no explanation. The fourth column shows the number of workers that can be employed with a capital of 10,000 fl. in a period of production of 1, 2, 3, etc., years respectively; in which case it is assumed that the advance of capital amounts to only half the sum of wages paid during the period of production—as will really be the case if there is an appropriate ‘gradation’ of production and payment of wages. If, therefore, the period of production is χ years, the figures in this column are determined by the formula

The fifth column can now be obtained by multiplying the appropriate figures of the third and fourth columns. Its figures give, therefore, for each year, the profit on 10,000 fl., or, divided by 100, the level of interest, expressed in percentages.¹

From this table we see that, when the rate of wage is 500 fl., the adoption of a period of production of six years will yield the highest interest on the invested capital, i.e. 10 per cent, whilst a period of five years would yield only 9-6 per cent, and a period of seven years only 9-7 per cent. This depends entirely, however, upon the level of wages. In the same way we see that, at a rate of wage of only 300 fl., and under otherwise identical circumstances, a period of production of only three years would prove the most profitable, and the capital would even yield interest at the rate of 51 per cent. At a rate of wage of 600 fl., on the other hand, a production period of eight years must be selected, ‘which will yield the modest, but still advantageous interest of 3 · 54 per cent.’²

If now—as the author for the sake of argument supposes—a national capital of 15,000 million gulden and 10 million workers are available, then, at a rate of wage of 500 fl. and with the correspondingly most advantageous production period of six years, the market will be in equilibrium. In other words, the existing capital will be just sufficient to keep all these workers fully occupied (and vice versa), since

And this state of equilibrium will necessarily also arise of its own accord through the competition of workers and capitalists. If, that is to say, wages were somewhat higher, i.e. 510 fl., then the six-year production period would still be the most remunerative. However, with the existing capital of 15,000 million fl., only 9,800,000 workers could be employed, ‘and the unemployed remainder, by creating a situation in which supply far exceeds demand, would exert pressure on the price of labour, until such time as they themselves can be, and are, employed’; which can only happen at a rate of wage of 500 fl. (This superfluity of workers shows itself, of course, in a much more marked degree when the rise in the rate of wages leads also to a lengthening of the period of production, which in our example will only be the case when the rate of wages is 530 fl. or more. It will, however, always by the case if we assume a continuously variable period of production.) If, on the contrary, the wage were a little lower, say 490 fl., then only 14,700 million fl. capital would be taken up by the employment of the existing 10 million workers. The unemployed remainder would then obtain employment through overbidding, and the result would again be a rise in wages which would continue until the point was finally reached at which everything can and does come into equilibrium.

So far everything seems to be correct.¹ The agreement with our formulae set forth above will be clear to every mathematically-trained reader. Strange to say, however, Böhm-Bawerk believes that he has found in the series of numbers which he has set down ‘other relationships as well, which in a positive (?) way point to the resulting rate of interest of 10 per cent and which can provide the material for a positive law of the level of interest.’ I reproduce here literally what he has to say on this subject.

“To arrive at the position of equilibrium, the capital of the community had to be withdrawn from the shorter processes of production, in which full employment could not have been found for it with the existing stock of labour, and employed in gradually lengthening processes, until it was fully occupied. This happened in a six-year period of production. On the other hand, the adoption of still longer processes, for which the capital would not have been sufficient, had, economically, to be prevented. In these circumstances the producers who adopt the six-year period of production are the last buyers, the ‘marginal buyers’; the producers who would like to adopt a seven-year period of production are the most capable excluded suitors for means of subsistence; and, according to the well-known law, the price that results must fall between the subjective valuations of these two. How does it stand with the valuations?

“What we have to consider is simply this: What is the utility which, for these two sets of buyers, depends on the disposal over a definite sum of means of subsistence? First of all, the general assertion must be made, that on the disposal over each half-year’s wage—in the present case 250 fl.—depends one year’s extension of the production period per worker.¹ Thus the ability to embark on or continue in the six-year, instead of the shorter five-year period of production, employing one labourer, depends, especially for the producers who adopt the six-year period, on the possession or non-possession of 250fl.; and since, according to our scheme of productivity, the year’s product from one worker in a five-year production period amounts to only 620 fl., whereas in a six-year period it amounts to 650 fl., the attainment of an annual surplus product of 30 fl. depends, for the marginal buyers, on their having at their disposal 250 fl. On the other hand, those would-be producers who try to take means of subsistence out of the market, in order to extend the production period to seven years even, could gain by this extension only a surplus return of 20 fl. (670 − 650 fl.) . . .

“If, therefore—and this is indispensable to the attainment of equilibrium—the extension of the production period is to halt at the limit of six years, the agio established by the fixing of the price (i.e. the interest) must lie between the rate that corresponds to the valuation of the last buyers (30 fl. on 250 fl., or 12 percent) as upper limit, and the rate of 8 per cent, corresponding to the valuation of the competitors first excluded, as lower limit. . . . The fact that, within these bounds, a rate of interest of 10 per cent was precisely indicated, is, of course, no longer due to the limiting effect of the valuations of the marginal pairs, but, as described on p. 226 ff., simply to the quantitative effect of supply and demand.”

All this sounds very clear and convincing, to be sure. But, when we look at it more closely, it unfortunately no longer seems clear. How could a surplus return of 30 fl., i.e. a net profit of 12 per cent, depend, for the producers who have adopted the six-year period, on their having the disposal over 250 fl. ? we are obliged to ask ; since at the assumed rate of wage of 500 fl. the capital can, at most, yield an interest of 10 per cent! And vice versa: if they can obtain this net profit, why should ‘supply and demand’ depress the interest which has to be paid to only 10 per cent? This could only occur if the capital sufficed for more than a six-year period, which, however, was not the case. But as a matter of fact a net profit of only 25 fl., or just 10 per cent, depends, for the producers who wish to go over, at the rate of wage mentioned, from the five-year to the six-year period, on having at their disposal 250 fl. ; the remaining 5 fl. of the surplus return are due to the fact that their capital, which was already employed before, and which in the five-year period amounted to 5 x 250 fl. per worker, is now employed in a six-year period of production where it now yields 10 per cent instead of only 9·6 per cent. For 5 x 250 fl. it consequently yields 125 fl. instead of 120 fl.

And this increase of profits they could obtain in any case, even without having new capital at their disposal, if they only decreased the number of their workers in a corresponding proportion.

Likewise, an added capital of 250 fl. would yield, when changing over to a seven-year period, not only 20 fl., but more than 24 fl. But at the same time the capital which was previously employed in the six-year period will have to be content with an interest of only 9-7 per cent instead of 10 per cent.

It is, therefore, certainly true that interest, calculated for half the level of wage, comes to lie ‘between the surplus return of the last permissible extension of production and that of the no longer permissible extension of production’; but between these limits its definite level is not determined by supply and demand but simply by the productiveness of the most profitable period of production. Whether in this case wages will really remain at the assumed rate or can be kept there, will depend on the supply and demand situation with regard to labour. This, however, is quite a different question.¹

The idea of regarding the ‘producers who adopt the six-year period of production’ as ‘the last buyers,’ etc., must be regarded as altogether wrong, for, at the rate of wage in question, everybody will choose this period, and neither a longer nor a shorter one. It would, indeed, not be impossible to conceive the present problem also as one which involves an exchange between present and future goods—with this reservation, however, that the exchange is an alternative one, in that the length of the period of production to be chosen influences the quantity of the future commodity (the average annual production) as well as that of the present commodity (namely the wage-capital to be employed in the present year). But we shall not dwell longer upon this.

When the length of the process of production can be changed by indefinitely small steps, as is for the most part really the case in practical life, the productiveness of the last small step that can actually be taken, and the productiveness of the step which is just out of reach, approach each other closely. Böhm-Bawerk therefore believes that he is able ‘to formulate the law of the level of interest in such a way that this level is determined by the surplus return of the last still permitted extension of production’; and in his controversy with Jevons (p. 427, note) he remarks that ‘the level of the rate of interest is to be deduced from the relation of the last surplus return to the sum of subsistence which allows the last extension of production.’

Without further qualification, however, the last statement is misleading. The words ‘at an unchanged rate of wage’ need to be added to it, and the word ‘allows’ should be replaced by ‘brings about’ or some such phrase. But then this statement simply expresses a consequence of the fact that the highest possible level of interest is already reached, and throws no further light on the nature of interest. One could, however, be led by the wording of the sentence to believe that, if an increase in the national capital leads to an extension of the period of production, the number of workers remaining the same, then the surplus return obtained through this extension, divided by the capital increase in question, will give us approximately the level of interest. This would be decidedly wrong. The result of this division sum is, as we shall see, always smaller than the interest and, what is more, it is smaller by a finite amount, even when it is a question of a minimum change. This is connected with the fact that this increase in the national capital is accompanied by an increase in wages which partially swallows it up, with the result that the lengthening of production actually achieved always falls short of the lengthening of production possible when the rate of wage remains unchanged.

With the help of the equations which we used before, this can be shown quite easily, and further relationships between the values occurring here, which might not be without interest, can be stated.

If p is replaced by F(t) and by F'(t), then, generally speaking

F(t) − F(t − Δt) > F'(t)Δt > F(t + Δt) − F(t)

since F(t) is an increasing, and F'(t), on the contrary, a decreasing function of t. Here Δt stands for a small quantity of time. Now, according to (14), when the level of interest reaches a maximum,

We therefore obtain for the corresponding value of t

In this inequality Böhm-Bawerk’s rules stated above find expression, since an extension of the period of production amounting to Δt requires a new capital investment per worker of .¹

In the case of a given national capital and a given number of workers, the length of the period of production and the wage are found, as we have seen, by means of the equations

(15)

(16)

and

in which p' replaces dp: dt. The rate of interest proper to them is then given by one or other of the identical expressions

If, however, the total capital is slightly increased, whilst the number of workers remains the same, a new state of equilibrium is reached, with a change in the level of wage and in the length of the period of production; with the result that, when K becomes K + dK, l is changed to l + dl and t to t + dt. The relationships between the quantities dK, dl and dt are found simply by differentiation of the above equations (15) and (16), namely

(18)

(19)

and

where p" is written for . We shall now apply these equations in various ways.

The annual production p of one worker undergoes, when t becomes t + dt, the increase dp or p'dt; the total surplus return is consequently A. p'dt. If we want to find out the proportion of this quantity to the increase in the national capital, we obtain from (18) and (19)

Since p" is always negative, the latter expression will always be smaller than —that is to say, smaller than the rate of interest, as I have remarked above.

In the case of a relative increase of the national capital the wage increases and the level of interest decreases. This circumstance is generally explained by the fact that, with increasingly capitalistic production, the workers’ share in the result of the production becomes greater and greater, whilst that of the capital becomes smaller and smaller. This, however, is not unconditionally true. It might very well happen that the workers, although they now have higher wages, nevertheless obtain a smaller share in the production, since its productiveness has in the meantime increased; or—which is the same thing—the share of the capitalists might be greater, although this share amounts to a smaller interest on the capital, which in the meantime has increased. In order to be able to decide whether this is really the case or not, we must see whether the expression increases or decreases when t increases, that is to say, whether

is positive or negative.

Taking into account the equations (19) and (16), this expression becomes

-tp" . p + t(p')² − p' . p

The first two terms of the expression are positive (since p" < 0); the third term, on the contrary, is negative. In certain circumstances, therefore, the sum of the three terms can be positive or negative.

For example, at a rate of wage of 280 fl. a two-year period of production would be the most remunerative (if we base our calculations upon Böhm-Bawerk’s figures). At a rate of wage of 300 fl., on the other hand, a three-year period would be the most remunerative. The annual production of one worker in the two-year period was 450 fl., in the three-year period, on the other hand, 530 fl. Now 280 : 450 > 300 : 530. If, consequently, the period of production is here extended from two to three years through a corresponding increase in capital, the share of the capitalists in the production increases and the share of the workers decreases, in spite of the fact that the wages have risen and the capital-interest has decreased. If, on the contrary, it is a question of periods of production of greater length, every new extension of the period of production will, in general, diminish the share of the capitalists and increase that of the workers.

But, finally, the question could be raised, to what extent the net profit of the capitalists—in the absolute sense—will, in fact, increase when the capital is increased and the period of production is extended. This is obviously a question of the greatest practical significance. If, that is to say, an increase in capital merely helped to diminish the profit on the capital, then such a capital increase would conflict with the interests of the capitalists as a class and would probably be prevented in some way or other. On the other hand, every increase in capital is, of course, advantageous to the workers. The result would be that the interests of the capitalists and the workers, which in this respect hitherto went hand in hand to some extent, would now clash.

The yearly profit on each worker was p' − l. When t becomes t + dt, this quantity undergoes the change

d(p − l) = p'dt − dl

or, taking into account (19),

= (p' + tp")dt

The solution of our problem consequently depends on whether the latter expression is positive or negative, p' is positive; p", on the contrary, is negative. If now p" (taken positively) is very small, that is to say, if p' is approximately constant, so that each extension of the period of production yields nearly the same surplus return, then the expression becomes positive. Every extension of the period of production and every increase of the national capital will then increase the net profit also (although, of course, not in the same proportion as the capital itself increases). If, on the other hand, p" is relatively big, that is to say, if p' decreases rapidly, then the expression becomes finally negative: the surplus return of the extended period of production is more than counterbalanced by the increase of wages.

If we suppose that p increases with t in a logarithmic proportion, so that p = α + β log nat t, where α and β are constants, then and ; we therefore now have for every value of t

p' + tp" = 0

The net profit, then, remains constant, even if the period of production is lengthened to a very great extent by continuous formation of capital: a national capital of 15,000 million fl. does not yield more than a capital of 1,500 or even of 150 million fl.—provided the number of workers is always assumed to be unchanged. But if ρ increases in a greater proportion, then, in the case of an extended period of production, the net profit increases also. If, on the other hand, ρ increases in a smaller proportion,¹ then the absolute net profit decreases with every new increase of capital and lengthening of production. If we base our calculations on the figures of productiveness given in the table, we see, for instance, that if the capital increases from 15 milliards fl. to 19¼ milliards fl., then, at the new rate of wage of 550 fl., the seven-year period would prove to be the most profitable one. But the annual profit from each worker would then amount to only (670 − 550) = 120 fl. instead of the 150 fl. obtained before, and the total net profit would, of course, diminish in the same proportion.

The figures in the table are, to be sure, only examples, but the decreasing scale of surplus returns which characterizes them may be regarded as a well-established fact or, rather, a matter of course. Sooner or later, if the formation of capital is continued and if the population remains relatively unchanged, the point must therefore be reached, at which the increasing capital is not only accompanied by a fall in the rate of interest, and not only has to be content with a smaller quota of the total production, but even leads to a smaller amount of the total profit; so that every new accumulation of capital directly damages the capitalists—always assuming, of course, completely free competition of capitalists.

As is well known, Thünen had already laid down a law of the level of interest, analogous to his familiar proposition which stated that the average wage² depended on the ‘yield of the last worker.’ According to this law, the level of the rate of interest depends on the productiveness of the ‘last invested particle of capital.’ The agreement of this theorem with Böhm-Bawerk’s own is obvious and is rightly emphasized by the latter. Only it must be remembered that here it is always a question of the capital investments of the individual entrepreneurs only, in which case the wage can and must be assumed to be given.¹ This theorem can by no means be applied to the increase in the national capital itself and to the surplus return brought about thereby.

Jevons in his Theory of Political Economy (2nd edition, p. 266) sets out from somewhat different considerations, in order to arrive at a general formula for the level of the rate of interest. Jevons supposes that, when the actual production is completed, the value of the product goes on rising for a while (for example, through its being exposed to the influence of the free forces of nature, as wine lying in the cellar; or because the sale conditions have improved in the meantime). So then the increase in value, taking place at each moment of time, can be thought of as the natural interest on the value which the product possessed at the beginning of this moment of time. If, therefore, F(t) denotes the value of the product after a certain length of time t has elapsed, and F'(t) stands for its derivative, the level of this interest is expressed by the following equation:

Under the assumptions which Jevons makes, this formula is not incorrect, but it is still rather meaningless, for it says nothing about the way in which this natural, continuously variable rate of interest becomes the decisive factor for the interest actually gained. In Jevons’s works the problem of the increase of the rate of interest to a maximum, and the relationships between interest and wages, are nowhere discussed.

However, the above-mentioned formula could also quite well be chosen as a point of departure, and is even the most natural starting point if we wish to take compound interest into consideration. But in this case, if it is a question of a continuous production, the labour element and wage element which have been added in each case must be taken into consideration too.¹

Böhm-Bawerk, as can be seen from his criticism of Jevons’s theory (Positive Theorie, p. 427, footnote), has completely misunderstood the latter’s train of thought, and reproaches him without reason for an ‘error’ or an Oversight in principle.’

The ‘concrete example’ which Böhm-Bawerk uses to illustrate the ‘bearing of this oversight’ is badly devised and shows that Böhm-Bawerk, as was pointed out before, has himself not arrived at a perfectly clear understanding of the necessary conditions of the problem. He says: ‘Let us suppose the case of an entrepreneur whose means would allow him to carry through an eight-year production period with a yearly return of 685 fl., who, by a loan of 300 fl., which would guarantee him subsistence for a ninth ( ?), is put in a position to go over to a nine-year production period with a return of 695, or a surplus return of 10 fl. According to Jevons, the rate of interest here should be 10 : 685, or 1-46 per cent. But clearly there is no reason whatever why the suitor for the loan should be ready to offer 10 fl. per year and no more as interest for a sum of 685 fl. It is not the sum of 685 fl., but that of 300 fi., acquisition of which makes the extension of production possible,’ etc. According to Böhm-Bawerk, ‘an interest of 10 fl. on 300 fl., i.e. 3⅓ per cent—or even, assuming a steadily-flowing production, a rate of 10 fl. on 150 fl., i.e. 6⅔ per cent—would be economically possible.’

It is obvious that Jevons has been misunderstood here. But, what is more, where does Böhm-Bawerk get his figure of 300 fl. from? How does he know that the entrepreneur, who before used to earn 685 fl. a year, will be content for a whole year with the very small subsistence of 300 fl. ?

In fact, the problem is unsolved so long as it is not known how much of his income the entrepreneur in question is accustomed to save. The simplest hypothesis is, however, that he does not save anything, but merely preserves his existing capital, that is to say, creates it afresh from period to period. But then his yearly subsistence and the average yearly return from his production (when he only works with his own means) are simply identical magnitudes; for his investment of capital would then merely consist in the fact that he supplies himself with his own subsistence while the work lasts; and in the final product he gets back the value of this amount of means of subsistence, neither more nor less. In the case of steadily-flowing production only half the sum of subsistence is necessary as capital. Consequently, for a one-year extension of production, an increase of capital of 685 : 2 = 342½ fl. is necessary. But for this sum he will be able to pay at most 10 fl. per year as interest ; so that at the very best a rate of interest of 2·92 per cent is ‘economically possible’ under the assumptions here made.

Instead of this simplest hypothesis, we could, of course, make any other assumption about the dispositions of this entrepreneur in general. But if no definite assumption of this kind is made at all, the whole problem obviously lacks a solid basis.

C. Böhm-Bawerk’s theory and the wage fund theory

After my efforts to give to Böhm-Bawerk’s presentation greater precision and to clarify what is obscure in it, I should like to draw attention once again to the great importance of his theory. As the author himself has explained, this importance consists partly in the fact that in this theory for the first time a real substitute is provided for the obsolete wage fund theory, which several writers have tried to overthrow by cheap criticism without being able to replace it by a better.

The wage fund theory, as is well known, represented the wage as equal to the results of dividing the capital destined for the payment of wages by the number of workers. Now it was pointed out with good reason by the opponents of this theory, that the first of these magnitudes is from the very start completely undetermined. For from the very first it is uncertain how much of the existing national capital will be used productively; nor will the whole of the capital used productively be paid out as wages. Rather, it is more or less ‘permanently’ invested in buildings, machines, tools, raw materials and half-finished products of all kinds.

The first objection applies equally well to Böhm-Bawerk’s theory and can only be removed by a comprehensive theory of savings and capital formation. As for the latter objection, it was clear from the beginning that the actual division of productive capital into means of labour and means of subsistence (into, shall we say, fixed and variable capital) is not arbitrary, but takes place according to the principle of the greatest possible profit ; but no one has been able to say anything more definite on this subject. This gap has now been brilliantly bridged by Böhm-Bawerk’s theory, which introduces the length of the period of production as one of the factors of the problem and replaces the vague ‘wage capital’ by the whole national capital, which is relatively definite.

Let us now return to mathematical language. While, according to the wage fund theory, the relationship between wage, number of workers and ‘capital’ is expressed by the equation

which leaves nothing to be desired in the matter of simplicity but has this drawback, that it gives only a single relation for two quantities which have to be determined, the new theory expresses these relationships by the equation

in which, however, K is now the relatively known magnitude of the total national capital productively used. Here, too, in order to determine the new unknown t, the further relation

or, which is the same thing,

is added.

The boundary between fixed and variable capital is in this case really abolished.¹ The whole capital, at least in so far as it is ‘turned over’ during the period of production, will subsequently appear in the form of money and means of subsistence, and, when no account is taken of ground-rent and the like, will be paid out in wages up to the last penny, but, as Böhm-Bawerk rightly remarks, not in one year, but during a period of time which, incidentally, amounts to half the length of the period of production.