XV. Input, Output, and the Stock of Capital in Value Terms

CHAPTER XV INPUT, OUTPUT, AND THE STOCK OF CAPITAL IN VALUE TERMS 1 IN the last two chapters we saw that wherever it is possible, on purely technological grounds, to attribute a definite quantity of output to the application of a definite quantity of input at a particular Tho rolaUooahlp bemoment of time, the values of these tween Input and outt ·t· t be 1 t· hi t pulln value lerml quan lieS mus ar a re a IOns p 0 one another corresponding to a compound rate of interest which (in the special sense defined) is uniform throughout the system. It will be remembered that it is possible to establish such a technological or causal connection between individual units of input and individual units of output not only in the simplest "point input - point output" cases but also in some "continuous input point output" cases. This is possible in these latter cases provided the shape of the input function can be continu ously varied so that the physical marginal product of units of input applied at successive stages of the process can be isolated. In the next chapter we shall have to investigate the cases where no such unambiguous physical relationship exists between individual units of input and individual units of output, and where all that we know is that a certain aggregate of output is due to a certain aggregate of input, either or both of which may be spread over a period of time. We shall then have to seek a solution of the problems in these cases on the principle of an "im putation " of a marginal value product.

1 The substance of this and the following chapter, although in a different fonn, was the subject of an article, by the present author, which appeared some time ago in the Economic Journal (1934b). 193 14 194 Investment in a Simple Economy PT. II Before proceeding, however, to these more complicated cases, it will be useful if we try to give a general descrip tion of the relations between aggregllltes of input and aggregates of output in the simpler cases where these relations can be built up, so to speak, from the known relations between the elements of which the aggregates are composed. Our task consists essentially in devising a suitable method of adapting our earlier representations of the complete structure of production, so as to show the new G hi factor of the growth of value of every unit rap c represenlalion of changes of of input invested, during the time for value In lIme h· h ·t .. t d F thO w lC 1 remams mves e. or IS purpose we must go back to the three-dimensional diagram which we used earlier (Fig. 6, p. 117) to represent the complete structure of production in terms of units of input. In that diagram all quantities of intermediate and final products had to be measured in terms of such quantities as were the product of equal units of invest ment. We neglected any change in value which the results of the investment of a unit· of input underwent in the course of the process. It is, however, just this growth of value in which we are now mainly interested, and in order to be able to show it in the original diagram it would evidently be necessary to introduce an additional dimen sion. But since the usual diagrammatic methods do not allow us to show variations in more than three dimensions in one diagram, we shall have to make room for the additional dimension by leaving out one of the variables that wer~ shown in that earlier diagram. The one that can be dispensed with with the least loss is that repre sented by the vertical or 8-axis of the said diagram (i.e. the axis which indicates the" stages" to which the different quantities of intermediate products measured along the r-axis belong). The relationship which was indicated along this axis can be shown in another way.

CH. xv The Stock of Oapital in Value Terms 195 To explain this alternative device for representing the phenomenon of " synchronisation" shown by means of the third axis in the earlier diagram, we must return for a moment to the very first diagram intro-h I I I hi h T e pr DC peon w C duced in this book (p. 101) and now re-the earlier diagram Is produced, with some additions, in Fig. 15. mocWled That diagram shows the part of the expected income stream which can be said already to exist at any moment in the "inchoate" form of nonpermanent resources. The curve T mR bounding t this part of the future in-Tp come stream (the output -To curve) represents the dis-Tit tribution in time of the product of the different T m units of current pure input invested at the mo ment O. The area under this curve may be taken to represent also all the T 3 R 3 successive stages through T2 R2 which the pure input in-T1 R 1 . vested at 0 will have to pass before it matures into its final . product. l o R FIG. 15 ,.

But since under stationary conditions similar investments will be made at every successive moment (i.e. at Tl , T 2 , Ts, and every other point on the t-axis), we may visualise an infinite' number of similar triangles telescoped one into another. And at every point on the t-axis all these suc~ cessive stages will be simultaneously present. In other words, if we conceive of this continuous series of triangles telescoped into one another, each of them will also re present the position which, under' stationary conditions, 1 Thus interpreted the curvilinear triangle of the present figure corresponds to the plane P 3RR3 in the earlier three· dimensional diagram (Fig. 6 above), the meaning of which was not explicitly dis cussed at that previous juncture.

196 Investment in a Simple Economy PT. n would exist at anyone moment. And the same will apply if, instead of two-dimensional triangles, we use a corre sponding series of three-dimensional solids which are similarly telescoped into each other. In what follows we shall start, not from the output curve, but from the input curve. Before proceeding to the construction of a diagram, however, it is necessary to LlmUationslo the DIe recall once more the exact meaning of the ofaslnglelnputourve input functions that are involved. They describe the range of periods for which different parts of the input applied at one moment of time are invested. These parts can be stated in physical terms only on condi tion that there is only one kind of homogeneous input. The corresponding information for a number of different kinds of input can be combined into a single input curve only on the assumption that the relative values of these different kinds of input are given. This means that; we ought really to start with as many separate input curves as there are different kinds of input; and that, although it will be possible to describe the conditions of equilibrium by means of a single composite input curve, this curve can be constructed only on the basis of given relative values of the different kinds of input, which are themselves determined by that equilibrium.

The three-dimensional diagram (Fig. 16) below depicts the growth of value in time. It is constructed by using the last diagram (Fig. 15) but interpreted as representing The proce .. In time the input function as base, and erecting on In value Iorms it a perpendicular v-axis along which are measured the changing values which the products of the various units of input obtain at successive moments of time. 1 The units of input invested at 0 (or at any later 1 "Value" is here measured in terms of anyone commodity on the assumption that identical quantities of this commodity available at different dates will be identical in value. The" growth of value in time" means that at successive dates the product of the investment of a given quantity of input at a particular date will be equal in value to increasing quantities of the commodity chosen. Under the CR. XV The Stock of Oapital in Value Terms 197 moment) may, of course (and if more than one kind of in put is involved must), be expressed in terms of value, and the fact that values are measured along two different axes in this diagram may at first appear confusing. The explanation is simple, however, when we remember that the r-axis measures the quantities of intermediate and final products in such units as are the product of a unit of input (in factor units) and not in units of their own (commodity units). And, as we have seen, the value of FIG. 16 the product of a given unit of input will vary in time.

While therefore the value of the input necessarily remains the same throughout, and it would be superfluous to show it a second time in addition to its measurement along the r-axis, the values of the products of this input have to be indicated separately. In the diagram, then, the total value of the input invested at any moment is represented by the rectangle OR1Ql V 1 (and T 2R2Q2 V'2 and all the similar rectangles which can be imagined at all other points on the t-axis). stationary conditions here postulated, it is immaterial what com· modity is used as the " standard of value" .

198 Investment in a Simple Economy PT. II The investment periods of these units of input vary between zero and OT l' as is indicated by the input curve R 1T 1• Over the same range of periods the value of the product of any unit of input invested will grow at a uniform compound rate of interest as indicated by the curve V IT' 1. Any perpendicular cross-section of the resulting solid, parallel to the plane vOt, corresponds to Fig. 13 above; that is, it shows how the value of the product of an infinitesimally small unit of input applied at 0 grows at compound interest during the period for which it remains invested. Anyone of the parallel curves, which have been drawn in the upward-curving interest surface V1QIT'I, corresponds to the compound interest curve of the earlier diagram, and the height of the curved perpendicular surface QIRITIT'1 at the point where any one of these vertical planes ends shows the value of the product due to the investment of a small unit of input for the corresponding period.

Each of the two solids shown in the diagram - and all the others that are not shown but may be conceived to be telescoped into each other in an infinite series - can accordingly be regarded as being made up by adding all the thin slices which, in the manner of Fig. 13, show the gradual growth of the value of a single unit of input invested. 1 The difference is that while in the two dimensional diagram it was possible to show only the growth of a single s;mall unit invested for a particular period, the present diagram gives a simultaneous view of the growth of all the units of input invested at a moment of time for the continuous range of periods described by the input curve R 1T 1• 1 If it did not make the diagram too complicated, we might also make it show the condition of equilibrium in the same way as Fig. 13. The condition is that, at the point where the product of every unit of input matures, the segment of the productivity curve of that unit of input (which shows the change in the size of its marginal product when its investment period is slightly changed while all other invest.

ment periods remain constant) must just touch the interest surface from below.

CH. xv The Stock of Capital in Value Terms 199 The three-dimensional diagram gives us a clear picture of two important relationships which, up to this point, have been left unexplained. First, it gives us a descrip tion of the total value, at a given rate of interest, of the stock of capital which corresponds to a given input function or output function. Secondly, it shows the relationship between the range of periods for which the input is invested (i.e. the input function) and the shape of the income stream derived from it (i.e. the output function). These two relationships are clearly inter connected. The value of the stock of intermediate products exist ing at any moment, under stationary conditions, is repre sented in the diagram by the volume of each of the solids V10R1Q1T\T 1, etc. This becomes evi-R 1 h epresentatlon 0 t e dent at once if we think of these solids value 01 the stock 01 b . d f 'nfi't . f capital as emg compose 0 an 1 me senes 0 perpendicular planes, parallel to the plane vOr, each of which represents the value of the intermediate goods belonging to the corresponding " stage " of production.

The only point which needs further emphasis is, as will be seen from the diagram, that the value of the capital stock, as r6presented by the volume of the solid, depends not only on the shape of the input function, or the size of the area which it. encloses, but also on the rate of interest. The reason is, of course, that what is being invested further at every stage is not merely the value of the original input; it includes the interest already accrued (or the value of the product which could have been obtained if the input in question had been in vested only for the shorter period). This representation of the factors which determine the value of the stock of capital shows that there is no simple or unique correlation between the shape of the input curve and this value. It shows also that, even given the rate of interest, we could not determine the value of the stock of capital if, instead of having a full description 200 Investment in a Simple Economy PT. II of the range of investment periods such as is provided by the input function, we knew only the aggregate or average of these periods. If such an aggregate or average were Its value can be deter mined only If we have a full description of the range of Invest ment periods and the rate of Interest all that were given, it would mean that we should know only the size of the area enclosed by the input function, and not its shape. There will be any number of different input functions which enclose areas of the same size, i.e. which correspond to the same aggregate or average of investment periods. And if we take two such input curves (as for example those t T o a T o R r b FIG. 17 shown in Fig. 17), one of which extends over a shorter investment period and is less curved, and the other of which extends over a longe~ period and is more curved, it is clearly possible that at one rate of interest the first, and at another rate of interest the second, will correspond to the greater quantity of capital. Similarly, at any given rate of interest, two input curves of different shapes, but enclosing the dame area (and therefore representing the same average or aggregate investment period), will correspond to different quantities of capital.

It will now be easy to derive the shape of the output stream, in terms of units of the product (or the output curve) which corresponds to any given input curve. In CH. xv The Stock of Oapital in Value Terms 201 fact, the output function in its non-cumulative form (as shown earlier in Fig. 4) is implicitly represented by the curved vertical surface RIQIT'lTl in Fig. 16. The height of this surface indicates the D rl" (.h ._ e va .. on 0 • e ou ... value of the product of a small (strictly put OlU'Ye from the speaking infinitesimal) unit of input ma-Input elU'Ye turing at the moment concerned. But in order to find the actual magnitude of the product maturing during any interval of time of finite length, we must also take account of the rate at which the product of such a small unit of input will be maturing at the relevant moment. This time rate is shown by the slope of the input curve RIT!> which forms the base of the vertical surface. Conse quently the shape of the output stream is obtained by multiplying the height of the surface at each point by the slope at that point of the curve which forms its base.

In this way we arrive at a non-cumulative description of the output stream as represented by the strips shown alongside the solid in Figs. 18 and 19 below. It is im portant to observe that the representation of the shape of the output stream thus obtained is not the usual (cumu lative) form of the output function: the strips represent the non-cumulative curves, or, that is, the first derivatives of the output functions proper.l We shall return to the discussion of the relationship between the input and output curves, as shown in these two diagrams, in Chapter XVI below, in connection with the" point input - con tinuous output" cases. 1 In this respect the earlier exposition of these relationships given by the author in the article quoted before (19Mb) was somewhat confused.