5. Completion of Böhm-Bawerk’s Theory. Capital-Interest, Wage and Rent in their Relationship to each other

Böhm-Bawerk’s theory forms, as was remarked above, only one element in the complete determination of the level of interest. The main reason for this is that the operation of natural forces, i.e. the services of the land, are not taken into consideration or, rather, are regarded as free. However, it would not be impossible to consider this factor also,² particularly as the services of the land with regard to capital behave, in several respects, exactly like labour. The landowners, too, get their rent in advance, before the products are ready for the market. We can even assume, for the sake of simplicity, that ground-rent is paid by instalments, just as wages are; so that here also the necessary advance of capital comprises, on an average, half the length of the period of production.

In what has been said above we have assumed with Böhm-Bawerk, as the simplest hypothesis, that all labour is paid at the same rate and that in all branches of production the scale of surplus returns is the same, so that one and the same period of production is adopted everywhere. In the same way we can assume as a first approximation, that landed property everywhere is of the same quality and that in all branches of production an equally large area of land is required for each worker. The problem is then susceptible of exact treatment in its broadened form also, and we can generalize our equations, laid down above, in such a way that they also include the factor which has now been added.

Let us express the yearly wage by l, as before, and the ground-rent per hectare by r. If now h hectares of land are required for each worker, it is obvious that the capital advanced, calculated for a single worker, amounts in a t-year production to . This is analogous to what has been said before.

Here the yearly production of one worker depends not only on the length of the period of production, but also, obviously, on the size of the area of land which falls to him. In other words, this magnitude becomes here a function of two variables which are independent of each other, namely t and h, and must be expressed by p = F(t, h). We notice at once that this function possesses, with regard to h, attributes which are quite analagous to those which it possesses in respect of t ; it increases when h increases, but the surplus return from one worker for every hectare of land added is as certainly a decreasing magnitude as the surplus return from every new extension of production.

The yearly expenditure of capital, calculated for each worker, is here consequently l + h. r, and equation (13) is now replaced by the equation

(20)

which changes into (13) as soon as r = 0, that is to say, as soon as the use of land is supposed to be free.

Now thrift requires that at each level of wages and ground-rent the greatest possible capital interest should be attained. z must therefore become a maximum (l and r being assumed to be constant). As is well known, this is done by making its partial derivatives in respect of t and h, each separately, equal to zero. (That in this case a maximum is actually reached, can easily be proved by reference to the attributes of the function p indicated above.) Or we simply differentiate the above equation partially with regard to t and h, as if z, too, were a constant, and we obtain thereby the two new equations

(21)

(22)

and

If, therefore, l and r were known, t, h and z could be determined from these three equations; so that we should obtain the most advantageous length of the period of production and the most profitable proportion of the use of land per worker, as well as the rate of interest itself, expressed in terms of wages and ground-rent.

But l and r, too, belong to the unknowns of the problem. To be able to solve it completely, we consequently need two independent equations as well. One of these is modelled on our previous equation (15). The existing capital of the community K must just suffice, in the case of the period of production and proportion of use of land in question, to employ fully all the available workers, and at the same time pay the necessary ground-rent. We therefore obtain, if the number of workers is A,

(23)

But just as all the available workers must here be employed by the capital, so, too, must the whole of the available area of land. If this is not the case, or if, on the contrary, more land is demanded than is available, the present level of ground-rent cannot be maintained; it must rise or fall, respectively. In other words, when equilibrium is to be attained, the most advantageous proportion of the use of land per worker, found above, must be equal to the proportion in which the number of hectares of land existing within the whole economy stands to the existing number of workers. If we express the former magnitude by B, we consequently obtain as the required fourth equation simply

(24)

The problem is now solved in its entirety.

Equation (23) can in this case, of course, also be replaced by

(23^*)

The existing capital must suffice to pay all the workers during the period of production adopted, and must at the same time be sufficient to rent the whole of the land.

The landowners who work with their own means are here conceived in the double role of capitalists and landowners, just as, in the foregoing, we have treated the workers who are themselves capitalists. All three functions can, of course, be united in one person.

Discussion of the equations set forth above would now reveal the true relationship between capital-interest, wage and ground-rent—in so far as the assumptions which we have made are in approximate agreement with reality.¹

Just as the equations set forth above constitute a completion of Böhm-Bawerk’s theory of interest, so they also include, as I shall now show, the older (Ricardo-Thünen) theory of ground-rent as a special case.

Our conditional equations obviously remain unchanged if, assuming in the first place any two of the three magnitudes l, r and z to be constant, we try to determine t and h in such a way that the third of these quantities becomes a maximum. If, therefore, we assume that z is constant and, in the meantime, for the sake of simplicity, = zero (or, which is the same, if we assume that its amount is already included in l and r), and if, moreover, we make the assumption that the length of the period of production is unchangeable, then equation (21) drops out and instead of (20) and (22) we obtain simply

The former equation means that the yearly production of one worker must replace his yearly wage and, in addition, the ground-rent of the area of land which he has used. The latter equation, in its turn, expresses the fact that production will develop in the most advantageous way when each worker disposes of just so many hectares of land that the addition of a further hectare would increase his yearly production merely by the amount of the ground-rent of this hectare; since the wage reaches its highest possible level if the ground-rent is unchanged, and, vice versa, if the wage is unchanged, the ground-rent per hectare reaches the highest possible level.

In order to show that this is nothing else but the ordinary theory of ground-rent, we choose as unit for the used area of land, instead of one hectare only, an area so great that on each of these area-units a large number of workers can be employed. Our h then becomes a proper fraction; indeed, , if n stands for the number of workers employed per unit of area. In the same way, , when q stands for the yearly production attained by the unit of land. Although n is here a whole number according to the nature of the matter, it can be treated approximately as a continuous magnitude. Thus we obtain, according to the rules of the differential calculus,

and the above-mentioned system of equations turns into

or into

which is the same thing.

What these two equations provide is precisely the mathematical expression of Ricardo’s theory of rent in the form given to it by Thünen. The significance of the first equation is self-evident (here, of course, r stands for the ground-rent of the present area-unit). But the second equation expresses the fact that the most advantageous production is attained if on each area-unit just so many workers are employed that the employment of a further worker would yield merely his annual wage and no more; which agrees with Thünen’s well-known law, mentioned above.

If we wish to take into consideration capital-interest as well here, we have simply to multiply the right side of the equations by . But t must here be assumed to be a constant, otherwise a third relation is necessary, namely equation (21)¹, which now turns into

Now in the older theory of ground-rent the last relation was missing—quite naturally, since the length of the period of production has never been laid down as an independent concept. For this reason, however, the whole theory remained a very incomplete one. Without more exact definitions, there was talk of different quantities of ‘labour and capital’ or of different ‘doses’ of capital which are added to the land successively. But labour and capital can be used in various ways, and in particular it makes an important difference whether the capital is used simply to employ several workers in direct production, or for preparatory work, production of machines, breeding of draught-animals and food-producing animals, etc., as well—in other words, whether a longer or shorter period of production is adopted. Altogether, one could never arrive at the necessary factors which determine the level of capital-interest without considering this circumstance, and for the relationship between capital and wages there was, after all, only the completely insufficient wage fund theory. In all these respects Böhm-Bawerk’s theory forms, so to speak, the corner-stone which before was missing. Once this corner-stone had been laid, the science of economics could be looked on as something complete in itself.

All rent-goods (buildings, railways, etc.) which form, each group by itself, an unvarying sum of goods (assuming a stationary economy), would, in my opinion, have to be treated in the same way as landed property. In this case, of course, a special unit would have to be chosen for each group. However, I will not dwell on this matter, but will at once proceed to show how, with the help of the theory of capital-interest and ground-rent which we have obtained, our problem of the exchange values of goods, which we left for the time being at the end of the previous chapter, can now be treated in an exact way.