XIV. The Marginal Productivity of Investment and the Rate of Interest

CHAPTER XIV THE MARGINAL PRODUCTIVITY OF INVESTMENT AND THE RATE OF INTEREST THE analysis of the last chapter has provided us with a convenient means of giving numerical expression to the magnitude which in the chapter preceding it we called, rather clumsily, the " rate of increase of the product due to the extension of the investment period". But to identify this magnitude, as we did in the last chapter, with the rate of interest was somewhat premature - even though by the rate of interest we mean here merely the general rate of return on real capital and disregard the problems of the relation between this rate and the rate at which money is lent and borrowed. Strictly speaking we can call it the rate of interest only in an equilibrium situation where the "rates of increase" have been equalised for all the different investments. So long as we are investigating the conditions of equilibrium and are talking about the rates of increase due to particular investments it will be convenient to use the expression "the marginal productivity of an investment". This phrase is here introduced as a technical term with the specific meaning of the ratio of increase of output from a particular unit of input, due to an extension of the investment period of that unit of input, and expressed as 80 time rate.

We shall now proceed to apply the general rules governing the relationships between the marginal produc tivity of investments for different periods The dlstrlbutloD of to the problem of determining the choice IDvutmeDts over perlodl of dIlrereDt of the period for which any individual unit leDgth of input will be invested. We shall first confine our atten179 180 Investment in a Simple Economy PT. II tion to the case where the investment period of the particu lar unit of input can be continuously varied, and where the product due to the investment of this unit of input can be clearly isolated. This does not mean that we shall be exclusively con cerned with the rather exceptional " point input - point output" case. Our conditions will be satisfied equally Cases where the phy-well whenever it is possible continuously slcal marginal pro-to vary (at least within certain limits) the duct 01 units 01 Input can be isolated amounts of input invested at different stages of the process of production. In" continuous input - point output" cases of this kind where the shape of the input curve can be continuously varied, it is possible to change either the rate at which input is applied in particular stages of a process yielding a product at a particular date, or (what really amounts to the same thing) the periods for which units of input are invested. In such circumstances it will always be possible to observe what changes in the quantity of the product are caused by changes in the investment periods of particular units of input. On the assumption that the use to be made of all other input is given, we can then represent the size of the product as a function of the investment period of the unit of input concerned.

The cases which we shall have to exclude from con sideration for the present are all cases where no such connection between particular units of input and par This Is Impossible ticular increments of output can be estab where the Input lunc-lished on purely technological grounds, and tion Is rigid or where It can only be derived where all that can be regarded as a techIn value terms I . I d t' . b t no oglCa a um IS a connectIOn e ween certain aggregates of input and aggregates of output. Included under this head there are first all those cases of time-consuming processes (i.e. "continuous input point output" cases) where the rates at which input is invested during the process are not continuously variable, or, that is, where the input function is more or less rigid.

CR. XIV Productivity and Rate of Interest 181 Secondly, there are all cases of durable goods (i.e. " point input - continuous output" and "continuous input continuous output" cases) where the variability of the individual investment periods, which is required to estab lish a technological link between units of input and units of output, is, if it exists at all, much more limited. These cases will have to be reserved for separate discussion in the next chapter. Among the cases that we shall consider here, the simplest one of "point input - point output" has one great advantage for purposes of exposition which makes it advisable to consider it first. In The" point Input many instances the intermediate products point output" case which arise in the different stages of the process are similar in character to the final product; in conse quence, what is being reinvested at every stage can be directly compared, in terms of physical quantities, with the final product. It is to this circums.tance that such instances as the growing of timber and the maturing of wine owe the great popularity which they have long enjoyed with writers on the subject. In such instances, the addition to the product which is attributable to the extension of the period for which the input applied at the beginning of the process remains invested can be calcu lated by a direct comparison of the quantities of the product which result from one and the same process according as it is terminated at an earlier or a later date.

This assumption makes it easy to see the main point. It is clear that what is being invested for the further period by which the original period is extended is not simply the amount of input that was E II h qual s ng t e marinvested in the beginning, but the product ginal producUvity 01 . t h· h th t . t h t th d different investments In 0 w lC a Inpu as grown a e en of the shorter period and which could have been consumed at that time. If any such extension of the investment period is to be profitable, the proportional increase 182 Investment in a Simple Economy !'T. n in the product due to it must be at least equal to the proportional increase in the product due to any other input which is invested for exactly the same (additional) period. So long as the'proportion in which the product already obtained will continue to grow by further invest ment is greater than the proportional increase for any other input invested over the same period, it will evi dently be advantageous to invest the product further for that period. And all the input will be invested in the most profitable way only if none of the products maturing at any moment would increase in a greater proportion than any other maturing at the same moment if its investment were continued for a further short period.

(At the limit this rate of increase during a short period of time again becomes, of course, a rate of proportional increase at a moment of time.) Thus, if we take a particular unit of input whose invest ment period can be continuously varied while that of all other input remains constant, the particular investment period which will be most advantageous will be the one where the proportional rate of increase of the product is equal to the rates of increase for all other products maturing at the same moment of time. And under stationary conditions it will also have to be equal to the rates of increase for the products maturing at all other moments of time. So much for the considerations which determine the choice of the investment period for a particular unit of input. The same considerations will, of course, also de DlstrlbulioD 01 In-cide how a number of units of one kind of vestments between input or of different kinds of input have dilierent .. point Input - point output" pro-to be distributed between different uses.

cesses When input of any particular kind can be used in a variety of processes, the additional returns that are obtainable by lengthening the investment periods will presumably decrease at different rates in the different processes. The proportional rates of final increase will OlI. XlV Productivity and Rate of Interest 183 consequently be equalised if the different units of input are invested for periods of different lengths. For the simple "point input - point output" cases which we are still considering, and in which the increase of the product during anyone process can be directly measured in terms of quantity or value, the conditions of equilibrium as between processes of different lengths can easily be shown in a simple diagram (Fig. 12). In this diagram the ordinate measures the value of the product obtained from the investment of a unit of v input for the different periods which are mea sured along the ab scissa. The curves Pll Pi' and P a represent the value of the pro duct obtained after different intervals from three different processes starting at the moments 0, Tv and Tz respec-0 T1 T2 T3 t tively. Equilibrium re-FIG. 12 quires that for all such processes which terminate at a given date, say T3 , the ratio between the rate of final increase of the product (due to the last extension of the investment period) and the product obtained must be the same. In the case shown in the diagram, in which the product obtained from each of the three processes in question at T3 will be the same, namely,T 3P, this condition will be satisfied if the slope of the three productivity curves at P is the same.

(For the purpose of the diagram we have made the size of the product maturing from the three processes at T3 equal by assuming that the quantities of input invested in each of the three processes will be such as to produce the same output at that date. As will appear presently, 184 Investment in a Simple Economy PT. II this means that more input will have to be invested in the process beginning at Tl than in the process beginning at 0, and that still more input will be invested in the process beginning at T2 than in the process beginning at T 1 • If instead we had chosen to represent the curves describing the amount of output resulting from investing equal quantities of input at the three dates so that the output obtained at Ta from the processes beginning at T2 and Tl would be smaller than the output obtained from the process beginning at 0, the condition of equilibrium would not be that the slopes of the curves above Ta should be identical but that they should stand in the same proportion to the height of the curve at this point.) The same condition will of course have to be satisfied at any moment, that is, for all processes maturing at the v same moment of time the ratio between the rate of increase of the product and the size of the product itself will have to be the same.

This does not mean that these ratios must also be equal for processes terminating at different moments. That is true only under stationary o Tl T2 T3 t conditions. It is, howFIG. 13 ever, useful briefly to follow the usual practice and to discuss this stationary case. We can then consider the relation between processes resulting from the invest ment of equal quantities of input at a given moment for different periods. If in this case the ratio of the final in crease of output to the absolute size of this output is also to be the same for all the processes, the condition of equilibrium will be determined by all the various productivity curves CR. XIV Productivity and Rate of Interest 185 touching the same compound interest curves at different points. This situation is represented in Fig. 13, where the process represented by the productivity curve P 1 is terminated at TI> the process described by the productivity curve P2 at T2 , etc.

The condition of equilibrium thus described is a neces sary but not yet a sufficient condition, for there will always be not merely one but many different compound interest curves, which can be made to touch all the productivity curves and which therefore will give us different sets of equilibrium points. We shall see later that the explana tion lies in the circulllstance that the rela Equalisation of mar ginal productlvltles of Inveslmenl B neces sary but nol a .um cienl condition of equilibrium tive values of the different kinds of input and the different kinds of output will depend on the amounts invested in different processes and for different periods. But before we can go on to deal with th~s point we must first general ise the conclusionE' so far obtained by applying the same argument to other cases than the simplest" point input point output" one. We must now consider the conditions governing the investment period of input which cannot be regarded as an isolated point input, but which is applied at one point of, and as part of, a "continuous input Conditions of equlll. t t t " Wh th' t brlum In a ... on-- pOln ou pu process. ere e Inpu IInuou.lnput _ point function describing such a process is con-output .. process tinuously variable, the question which arises is whether particular units of input should be invested at an earlier or a later point of the process (or whether more or less units should be invested at a particular point). Before we can answer this question, two modifications of our argument are necessary.

The first of these modifications is made necessary by the fact that in the case we are now considering the extension of the investment period of particular units of input cannot be brought about simply by continuing the same process somewhat longer; the units of input 186 Investment in a Simple Economy PT. n have to be invested from the beginning in an altogether different process. There will not in this case be any consumable product available after the interval at the The marginal pro-end of which the product would have be ductlvlty or Invest-come available in the original process. ment In this ease Is not the Increase In We have to deal, therefore, not with produot obtained by f h' h ld continuing the same different qua.ntities 0 output w 10 wou process - emerge from the same process at different dates, but with quantities of output which would emerge at different dates from alternative processes in anyone of which the input can be invested. And we shall have - but the Increase to compare not the total size of the proobtained by choosing d t f th diff t btl th an alternative, slightly UC SO e eren processes u on y e longer, process size of the contributions to those products which are due to the co-operation of the particular units of input concerned. That is, we have to compare the marginal addition to the product which can be obtained by investing some small quantity of input at one point in one process with the addition which can be obtained by investing the same quantity at the same moment at a somewhat earlier stage of a similar process which will give forth its product a little later.

But although it would be impossible in this case to prolong the investment period by just continuing for a little longer a process already started, the initial decision about which of the alternative investments to undertake would have to be made on exactly the same principles as if we were dealing with products which could be obtained from one and the same process at different dates. This means that the result of an extension of the investment period of a particular unit of input would have to be judged as if what was being invested for the addi tional interval were the marginal addition which could have been obtained from that unit of input if it had been invested for the shorter period. 1 And the concept of the proportional rate of final increase of the product - the 1 I.e. at a later stage of the same process.

OR. XlV Producti'Vity and Rate of Interest 187 magnitude which in equilibrium must be equal for all investments - would in this case refer to the difference between the marginal products which could be alter natively obtained by applying the quantity of input earlier or later in the process. 'The second modification which has to be introduced at this point is a qualification which was implicit in the discussion of the earlier, simpler case, but which now becomes more, obviously necessary and The return from the must therefore 'be made quite explicit. It lnvenment of a nnlt of Inpnt ean here DO was pointed out in that earlier discussion 10D,er be regarded h h f h d uafuncUoDofthe t at t e size 0 t e pro uct which is Investment period of obtainable from a particular use of a given that nn1t onIJunit of input, and the variations in the size of the product which are due to changes in the investment period of that unit of input, can be regarded as given on the assumption that the investment periods of all' other units of input are determined.

There are two reasons for this in the present case. The first reason, which is the' more general one and also applies, although perhaps less obviously, to the former case, is that the size of the product ob- -parUyowlngtothe tained at a particular date can be desqribed elreat of changes In the relative qunlltles only in terms of value, and this value of d1fterent products will be determiI,Ied only if the quantities on theIr valnesof all other' commodities available at this and all other ,dates are given. The second reason, which applies ex clusively to the present case but is very conspicuous here, is that even the physical size of the contribution due to a particular unit of input which co-operates with many other units in anyone process, is depend - but mainly owing to the teohnlcal com plementarity between Investment periods of dIIIerent units of In put ent, because of technical complementarity, not merely on its own inv>estment period, but also on the investment periods of all the units of input used in that process.

This means that we are not entitled to regard the productivity curves of the investment of different units of 188 I nvestmeni in a Simple Economy PT. II input, such as we drew previously, as independently and simultaneously valid. The shape of each of them is liable to change with any, change in the use that is made of The productivity any other unit of input, and anyone curve curves of dlft'erent will have a determinate shape only on units of Input are not independent the assumption that the use of all other units of input is determined. In other words, it is not really possible to start out from the notion that the pro duct of each unit of input is a function solely of the period for which that unit is invested. We shall have to take as our' initial datum a description of the way in which the total income stream and the relative values of its com ponent items are dependent on the investment periods of all the units of input used. The value of any part of this total income stream will depend on, or will be a function of, the investment periods of all the units of input used.

And the contribution due to a particular unit of input can be determined only by observing and comparing the effects, first of taking it out of a particular use, and then of applying it in a way in which it will yield its product at a slightly later date, the use made of all other units of input remaining the same. Fundamental as is the importance of this modification, it does not deprive our earlier construction of its value as a description of the conditions of equilibrium. It still remains true that in a state of equilibrium there must be a uniform ratio, for all units of input, between the rate of increase of the marginal product of the unit, due to a slight increase in its investment period, and this marginal product. The only modification which we have to make in order that the diagram used before may still be a correct description of this equilibrium condition is that we must not regard the productivity curves as being simultaneously valid, but must consider each only as describing the change caused by a change in a particular investment period when all the other investment periods are such as to correspond to an equilibrium position.

CR. XIV Productivity and Rate of Interest 189 Perhaps it would be less misleading if, instead of showing complete productivity curves, we drew only short seg ments in the immediate vicinity of the point of tangency with the compound interest curve. The segment may then be looked upon as indicating the rate of change of the product of a unit.of input consequent upon a small change in its investment period while assuming that all the other investment periods remain unchanged (Fig. 14). v o T t FIG. 14 The consequences of this last modification are very far-reaching. The concept of the ratio between the rate of increase of the product and the product (the ratio which must be equal to the instantaneous Jevons'" rate or In rate or force of interest) is of course the crease of lb. produ .. divided by the whole same as that on which W. S. Jevons' ex-produce .. planation of interest was based. Jevons described it as " the rate of increase of the produce divided by the whole produce", and defined it in mathematical terms as IJ;g;, where the function F(t) describes the size of the product as a function of the investment period of the input. 1 And both Bohm-Bawerk and Wicksell followed Jevons 1 W. S. Jevons, The Theory of Political Economy, 1st ed. (1871), p. 237; 4th ed. (1911), p. 246.

190 Investment in a Simple Economy PT. II in the method he used to determine this rate. The method all three authors employed was to assume that the aggregate of waiting, or the sum of the periods for which the different units of input could be invested, was unequivocally determined by the size of the" subsistence fund". They then concluded that.the marginal product ivity of waiting could be determined by so distributing the total waiting between the different units of input that the "rate of increase of the produce divided by the produce" became everywhere the same. We have already observed, however, that the supply . of capital cannot ever be assumed to be given in a " free" form as an actual subsistence fund, and that the actual The Investment period stock of nonpermanent resources cannot not one of the data be identified in any definite and unam but one of the unknowns 01 the prob-biguous way with quantities of future lem consumers' goods or determinate waiting periods. It is therefore impossible to regard the average or aggregate investment period as a datum from which we can derive the marginal productivity of investment in the same manner as we determine the marginal pro ductivity of any other factor of which there is a given quantity available for distribution among the various uses.

But we see now that we do not need a description of the total time dimension of investment, an aggregate investment period, or a definite fund of capital, as an The Inyestment initial datum; and that these are more in periods are not glYen the nature of results of the forces which by a determinate IUp- . ply of free capital determine the equilibrium. The factor which limits the possible extensions of the investment periods is that as one unit of input is invested for a longer period, the output stream at the earlier date is reduced and the value of the products maturing at this earlier date is consequently raised. This means that the value of the marginal products of units of input invested for that earlier date increases, with the result that it becomes OB. XIV Productivity and Rate of Interest 191 profitable to invest more for that date. We have postu lated that the available input must be used in suoh a way that the resulting income stream will be of constant size, i.e. that every gap caused by investing . some units of input for longer periods must be filled by investing other units .of input for c.orresp.ondingly shorter periods.

Therefore the c.ondition that all input must be invested in such a way that the ratio -between the marginal rate of increase of the pr.oduct and the size .of the whole pr.oduct is the same for all units of input , als.o determines the period f.or wp.ich each .of the units .of input has t.o be invested. The nature of our present assumption about the desired shape of the income stream, i.e. that under all c.onditions it must remain c.onstant in time, makes it impossible for the moment to give this "A ftnal solution oan c.onclusion m.ore exact expression In be given only after the • Introduotlon of time terms of utility analysis this ass~mption, preference which has so far been stated in only very general terms, would mean that IjLny addition to the .output at a date when the t.otal output is smaller than at .other dates W.ould . have a greater value than any addition, however large, to the output at those other dates. Apart fr.om the obvious lack of reality of such an assumption, it is exceedingly inconvenient. We shall therefore postpone further discussion .of this point until after we have made a m.ore careful study .of the p.ossible psych.ol.ogical attitudes t.owards income streams .of different sh~pes (cf.

Chapter XVII). But even without this more· exact formulation, it will be evident by now· that if. we start .out with a given stock ofn.onpermanent res.ources, the factor which will determine the investment periods of the various items in that stock will be the condition of maximising the result ing stream, and the consequent condition of equalising the proporti.onal rates of final increase of all the different investments. This st.ock .of n.onpermanent res.ources in the f.orm in which it exists as a datum is n.ot some definite 192 Investment in a Simple Economy PT. II quantity of capital; for it can be expressed as a single magnitude only after the relative values of the items of which it is composed have been determined. And these values are clearly a resultant of the same equilibrating forces as determine the investment periods. The initial datum from which we have to start is simply an enumera tion of all the items of which this stock of nonpermanent resources is composed, and of all their technical attributes.

As will appear later in more detail, the quantity 9f capital as a value magnitude, no less than the different invest ment periods, are not data, but are among the unknowns which have to be determined. 1 1 Wicksell saw this quite clearly, although he proceeded in his ex position as though the investment periods or the quantity of capital were given magnitudes. It is evident that he realised that this was not so from a passage in his Lectures (vol. i, p. 202) already quoted, where he emphasises that "it would clearly be meaningless - if not altogether inconceivable - to maintain that the amount of capital is fixed before equilibrium between production and consumption has been achieved". Wicksell did not, however, consistently follow this up. It seems that he discovered this point rather late and neve~ fully incorporated it in his system. This is borne out by the fact that in the German translation of his Lectures, which was prepared from an earlier Swedish edition, the passage quoted above is much less emphatic (cf.

Vorlesungen uber National6konomie, vol. i,1913, pp. 272-273).