XVIII. Time Preference and Productivity: Their Relative Importance

CHAPTER XVIII TIME PREFERENCE AND PRODUCTIVITY: THEIR RELATIVE IMPORTANCE THE assumption of constant productivity of successive investments had to be introduced, it will be remembered, as a consequence of two other simplifying assumptions which were necessary in order to render The assumption of a diagrammatic treatment of our problem conslanl relurns on Investments abanpracticable. These assumptions were, doned first, that only one commodity (or" income" conceived as a sort of composite commodity consisting of constant proportions of the different components) was produced, and, secondly, that there was only one possible period of investment. l As soon as we drop either of these assump tions we come to the more realistic case where successive investments will yield decreasing returns. It will be seen, however, that the results obtained under the less realistic assumptions are not altogether useless or irrelevant, since, as compared with time preference, the productivity of investment is likely to be fairly constant.

There is, however, the serious difficulty of exposition that as soon as we drop either of the two assumptions concerned, the diagrammatic method so far used is no longer strictly applicable. The reason why C I dim I ," onsequen eu the existence of different commodities lies of diagrammatic makes the returns on successive invest-representalion ments decrease is twofold. In the first place the advantage of time-consuming. processes will be different in the pro duction of different commodities, and in the second place if the production of some commodities is increased more 1 Probably a third condition is also required in order to obtain constant productivity, i.e. that input should be homogeneous. 229 230 Investment in a Simple Economy PT. II than that of others the relative values of the former will fall. But in our diagram only decreases in physical returns can be properly represented. The reason why the possibility of varying the investment period implies decreasing returns is that the existence of different kinds of possible methods of production will always mean that the most profitable ones will be selected first, and the less productive ones will be taken up only gradually as the supply of capital increases. The difficulty of representing this case diagrammatically is due to the fact that if the periods of investment vary, it is no longer possible to assume that the results of the investments made at any one date will all become available after a single and con stant interval of time. But while it would be difficult to take account of this factor in the diagram, the in accuracy caused by neglecting it is not so serious as to make the simpler form of the diagram useless for the elucidation of this case. We shall therefore simply postulate that decreasing returns occur as a result of the variety of investment periods, but shall neglect the particular complication mentioned in order to be able to use the same sort of diagram as before.

The way in which the diagram (Fig. 22, p. 233) will ex press the fact that the returns on investment decrease is that the transformation lines will be curved and concave The shape 01 the towards the axes instead of being straight transformation curves as in the former case. Since the return on further investment must be assumed to depend only on the amount of investments already made (i.e. only on the provision made for the future) and to be independent of the size of the income left for the present, it cannot be presumed that the shapes of the different members of the complete family of transformation curves will be inde pendent of each other. At any point corresponding to a given value on the abscissa the slope of all the transforma tion curves will have to be the same, and they will there fore all be similar in shape. But those further up and to CR. XVIII Time Preference and Productivity 231 the right will possess additional parts at their lower end which are missing from those further down and to the left.

Which member of this family of transformation curves will be relevant at any particular moment will depend on the size of present income. But this affects only the variety of the choice and not the size of It Is practically Inde the returns obtainable for the sacrifice of a pendent of the length of the period over given amount of income. What propor-which the Investtional share of current income a given ments are made quantity to be invested represents will have little if any effect on the size of the returns that will be obtained. The latter will depend almost exclusively on the aggre gate of the investments already made, no matter whether the individual investments making up that aggregate have followed each other at long or at short intervals. But while the size of the present income, out of which we save a given amount, will have no influence on the return on the investment of a unit of that income, it will have a very considerable effect on our The willingness to willingness to save any given amount. save a given amount . depends on the length And since the aggregate of income out of of period during which which a given amount is saved will vary Ii Is to be saved in proportion to the length of the period during which that amount is saved, it follows that the shape of the indifference curve will vary with the length of the period which we regard as " the present". As the period during which a given amount is to be saved and invested becomes shorter, this given amount becomes a proportionately larger share of the income of that period. For a person with an annual income of £600, for example, saving £25 in the course of a year is an altogether different proposi tion from savin~ the same amount in the course of a month. The sacrifice in the second case amounts to reducing one's expenditure to one-half one's income.

And as we make the period still shorter the reluctance to save and invest a given amount will increase further.

232 Investment in a Sirnple Economy PT. II The faet that of the two relevant eurves, the trans formation eurve and the indifference curve, the first is independent of the length of the period considered while The relevant period the second is dependent on it, means, in terms of the diagram, that as we make the income period shorter the indifference curve will become more and more curved while the transformation curve will remain practically unchanged. Par our purposes, however, the relevant periods will clearly be the com paratively short intervals at which income accrues and at which the decisions have to be made as to what part of income to consume and what part to save,1 Compared with the totals that may be profitably invested, even the total income for a year, and still more the total income for a month, are relatively small magnitudes. But it is these relatively short periods which we have to consider if we want to understand the position of a person at different points on the path of saving.

It is, then, the ratio of the curvatures of (or the ratio of the elasticities of substitution expressed by) the two groups of curves which is of fundamental importance for At every step In the process 01 saving the variable rate 01 time preference adapts it self to the relatively constant rate of return what follows. Perhaps it is necessary to dwell somewhat longer on this point. The above statement about the ratio of the curvatures is an expression of the fact that, if we consider a relatively short time interval, the sacrifice of successive parts of the income of 1 New decisions of this sort are required for each date at which, in consequence of past investment, current income increases. It is irrelevant for our purpose whether we assume that the" new decision" is not made until that moment or whether we assume that the altered disposition after that moment has been decided upon from the begin ning. The difficulty, as mentioned previously, concerns the date or dates when the successive fruits of investment become available. Con trary to the first case discussed, these investment periods are no longer identical with the income period. Thus, strictly speaking, it is no longer true that the decision to save made during any income period is directly affected by the result of the investment in the immediately preceding period. This is a complication which, as mentioned previously, will be deliberately neglected.

CR. XVIII Time Preference and Productivity 233 this interval of time in the interests of the future will meet with" a rapidly increasing resistance, while, on the . other hand, the investment of successive fractions of a small total income will have quite an insignificant effect on the rate of return obtainable from investment. The situation will still be similar to the case discussed before to the extent that during any short income-period the rate of time preference (as represented by the indifference curve) will have to adapt itself to a relatively constant rate efreturn. In terms of the diagram: If the Y process starts at P, then, during the first time interval the indi vidual will move along the transformation curve until he reaches R, where the rate of time preference on fur ther doses of present income is equal to the rate of return, which will have been little affected by the como w L x FIG. 22 paratively small amount invested. In consequence of this investment he will some time later find himself at P', and will again save until his time preference has been raised to equal the comparatively constant rate of return. That is, he will move to R'. And when, in consequence of this, he finds himself in the position indicated by P", he will again save, and so on.

The imaginary curve R, R', R" . .. thus again repre sents the path along which the saver moves in time. The main points that are brought out by this method of repre sentation are these: First, saving must necessarily be treated as a process in time the effects of which continu ally change the actual position of the saver and thereby 234 Investment in a Simple Economy PT. II his willingness to perform further saving. Secondly, at any point in this process the rate of saving is practically determined by the productivity of investment alone, and Time preference dl-time preference is important only in deter reotly dects only tbe mining how fast the saver will move towards rate of saving : Its elreot on rate of In-a position in which the productivity of urest Is Indirect only further investment, and therefore the rate of interest, is lower. 1 It will also be seen that, so long as saving continues, the time preference of the indi vidual as represented by his marginal rate of substitution between equal present and future income streams (the slope of the indifference curve where it crosses the line Ow) may be zero, or even negative, and there will nevertheless be a positive rate of interest.

The exact direction of the path of saving will evidently depend on the shape and position of the two sets of curves. Whether and when a stationary position will be Positive time prefer-reached depends on the rate at which the ence a oondltlon for productivity of investment decreases as the the existence of Interest under stationary amount invested increases, compared with conditions the change in the rate of time preference consequent upon the increase in income. It is conceivable that saving might not cease until all opportunities for investment had been exhausted, because any rate of return, however small, would lead to some saving. This would mean that at the point where they cross the con stant-income line Ow, the indifference curves, at least in the region of the higher incomes, would be perpendicular. In this case stationary conditions would be reached only at a zero rate of interest. But if the indifference curves have a definite slope along that line (i.e. if there is, in the traditional meaning, a positive time preference), saving will cease, and a stationary state will be reached, with a positive rate of interest corresponding to the rate repre1 The credit for having made this point clear belongs to Professor F. H. Knight, with whose more recent statements on this point I find myself in complete agreement. ·Cf. particularly his articles 1932a and b.

CK. XVln Time Preference and Productivity 235 sented by that slope. And in this sense it can be said that the rate of time preference determines the rate of interest existing in the final stationary equilibrium, and there only.l Lastly, there is also the possibility that as the return on investment decreases, the rate of saving may become so small that the investment opportunities will not be exhausted within any time in which we are at all interested. In this case saving will go on indefinitely (i.e. the path of saving will end at the line Ow only after an indefinitely long time), and while it continues the rate of interest will be determined not by time preference but by the productivity of investment. The shape of the curve describing the path of saving depends on the rate at which the willingness to save changes with a change in income, and the rate at which the returns on further investment decrease Factors determining in consequence of past investment. -Only the path of saving in the case of constant returns from successive invest ments (see Fig. 21) will its shape depend entirely on, and be a property of, the structure of the "indifference map".

In this case the curve becomes an instance of the" ex penditure" curve, which is familiar from the modern expositions of the theory of value, 2 and which represents the change in the distribution of " expenditure" (in our case between present and future income) consequent upon an increase of income. In the other case we considered, shown in our second diagram (Fig. 22), this effect of an increase in income is mixed up with the effects of the decreasing returns from further investment. It will be 1 "The" rate of time preferenoe is here the rate at whioh a person would just be indifferent towards giving up a marginal quantity of his present income in return for a oorresponding addition to his otherwi.se equal future income. Since, a.s will be clear from what has been said before, a positive rate of time preference in this sense is entirely com· patible with the assumption of constant tastes, we have here the case of a stationary equilibrium with a positive rate of interest, which Professor Schumpeter regards a.s impossible (cf. Appendix I).

• Cf. J. R. Hicks and R. G. D. Allen, "A Reconsideration of the Theory of Value ", Economica, N.S., vol. i (1934).

236 Investment in a Simple Economy PT. II useful to isolate this influence also, and to show how the person in question would react if, while his income remained unchanged, the returns on investment were gradually altered by extraneous circumstances. This can be done by deriving from the "indifference map" a curve showing the demand for future income, in terms of present income, at every possible rate of return on The elleet of the rate investment. The phenomenon which this 01 Interest on saving curve describes is, of course, identical with that which has been widely discussed under the heading of how the individual saver will react, ceteris paribus, to changes in the rate of interest. It will immediately appear, perhaps contrary to the first impression, that the assump tions so far made do not in any way prejudice this question. They are Y compatible with the w saver's either increas ing or decreasing the amount saved in re sponse to a rise in the rate of interest, and vice versa.

The demand curve which we are now con sidering sh ows th e effect of a change in price on the distribution of outFIG, 23 x lay between present and future income as distinguished from the effect of a change in income (shown tl 1 by the expenditure curve). It is constructed The construe ou 0 the demand curve for (Fig. 23) by rotating the price line PQ future Income d h fi d . P ( h' h roun t e xe pomt w lC represents present income on the assumption that no investments have yet been made), and connecting up all the points (R, R', R", R''', RiV) where this line becomes tangential to the successive indifference curves. The slopes of these CR. XVIII Time Preference and Productivity 237 lines represent the different rates at which present income can be transformed into future income, i.e. the rates of return or the rates of interest. As they become less and less steep, the permanent addition to future income which can be obtained for a given sacrifice of present income becomes greater and greater.

The diagram is constructed in such a way as to show a case where, up to a point, the sacrifice of present income which the person is willing to make in order to obtain additions to future income increases as the The elasticity of de returns on investment increase. But even mand for future Inif, as the assumptions underlying the dia-come gram imply, the additions to future income which will be demanded continue to increase indefinitely with increases in the rate of return on investment, this does not mean that the demand for future income in terms of present income will continue to increase. As the additions to future income that are obtainable for given sacrifices of present income increase, the demand for such additions to future income may well increase less than in inverse proportion to the decrease of their price in terms of present income. That is, the amount of present income given up to obtain additions to future income may actually decrease. This is shown in the diagram by the fact that the demand curve turns upwards at some point. At this point the elasticity of the demand for future income becomes less than unity, and the supply of savings will henceforth decrease as the rate of interest increases.

We can say nothing a priori about the point at which in concrete cases the rate of saving will begin to decrease with further increases' in the rate of interest. It will be different with different individuals, and it may conceivably be so low (although this seems very unlikely) as to make a fall in the supply of savings in response to in creasing rates of interest the rule rather 110 general rule as to whetber the rate of saving will move in the same or in tbe opposite dIrection to the rate of Interest than the exception, at least so far as concerns those rates 238 Investment in a Simple Economy PT. II of interest with which we have to deal in actual practice. But for the theoretical questions with which we are dealing here our lack of certain knowledge as to which case is most likely to occur is of small consequence. All that we want to stress is that our treatment implies no particular assumptions in this respect: either of the two cases fits into our scheme. In practice we ought probably to assume that the rate of saving will increase with in creasing rates of interest up to a point and decrease beyond that point. And the position of this point will be different for different individuals and at different times.

In the case of the economic dictator of a communist society on which our attention has so far been centred, it is probably most reasonable to assume that the dictator Elleet of IImllalion plans for an indefinite future and expects of period over which to command a constant stream of output plan exleniIJ d· th t f t B t·t·· t t· urlng a u ure. u 1 IS lmpor an , In view of later applications of our argument, to drop these assumptions now, and to consider the case where the future period in which the person is actively interested is limited, and where also, perhaps, he cannot expect a stream of permanent services which will last throughout the period in which he is interested. The latter condition means, of course, that initially he does not command any really" permanent" 1 resources: all that he has are relatively durable wasting resources. Thus even if he does not want to increase (by investment) the stream of final services which he can obtain from the direct use of these resources while they last, he will be obliged to make provision for their replacement by producing resources for the time when they are exhausted.

The more important of the two assumptions is, how ever, the first, i.e. that the person in question plans only 1 But since the services from these resources cannot in any way be utilised before they accrue in the course of nature, i.e. since they are non-anticipatable services, they may, so far as his initial plan is con cerned, to a large extent possess the essential characteristics of" per manent" resources.

CR. XVIII Time Preference and Productivity 239 for a limited period beyond which he is not interested. This means, of course, that we can no longer assume that he will make provision for a permanent income stream : we have therefore to allow for the possibility of his re converting capital that has been accumulated during the earlier part of the period into income during the later part of the period. If we make the further assumption that he has no permanent resources whatever, but is from the beginning only equipped with a stock of wasting assets of varying minimum durability, the problem be comes one of how to use this stock of capital so as to produce an income stretching over the desired period and possessing the most preferred shape. But even on this assumption we shall have to take into account that the services of some of the wasting assets are more like those of permanent resources in that they cannot be consumed before a certain date. This means that even if none of the resources are strictly permanent, the essential fact remains that some of them cannot be used before a certain date.

It is hardly possible to discuss these questions in a satisfactory manner without making allowance for one further important element. I refer to the uncertainty which will usually exist concerning both Slgnlll.cance of un the length of the period for which provision certainty will be required and the time for which the resources are likely to last. Any attempt to go beyond what has already been said would therefore mean entering upon a systematic analysis of the role played by uncertainty in this connection. But for a number of reasons it has seemed advisable to stop short of this discussion here. One reason is that there appears to be little possibility of saying much that is worth while on the subject without making more concrete assumptions about the institutional background than are required for the more general con siderations to which the bulk of this study is devoted. It seems in particular that the significance of uncertainty may be more properly discussed in connection with the 240 Investment in a Simple Economy PT. II monetary aspects of our problem, which we are not dis cussing here. Furthermore, most of what can usefully be said about the significance of uncertainty on the general plane of abstraction on which this discussion proceeds, seems to ,have been said by others, especially by Professor Irving Fisher. 1 Therefore, instead of adding to the length of this discussion by going over what to most readers will be familiar ground, it is perhaps preferable to refer to these other works and to confine ourselves here to the little that can be said about the topic which is our im mediate concern without introducing the factor of un certainty.

If, then, we assume that an income stream is wanted for a limited (but definite) period only, this will in general mean that any act of saving and investment will increase Effect of anticipated income during the remainder of the period length of life on wlll-not only by net interest, as the term has Ingness to save b d b bib f h een use a ove, ut a so y a urt er amount corresponding to the rate at which it will be possible gradually to use up the capital previously accumulated. Or, in other words, any given increase of one's income for the rest of one's life will be obtainable at a smaller pre sent sacrifice than a similar increase of a perpetual in come. This will probably mean that a person will be ready during the earlier years of his life to provide year by year for some addition to his future income. But since given additions to future income can in this case be obtained at a smaller present sacrifice, the effect is likely to be that he will not save as big a portion of his present income as would seem advisable if he were providing for 'a perpetual income. And he will certainly stop saving at an earlier age. On the whole, then, we should expect that a limita tion on the time for which a person wants to provide an income - and any shortening of this time - will make him save less than he otherwise would.

1 See also an unpublished thesis by Dr. Helen Makower on The Theory oj Value on the Capital Market (University of London, 1937).

CR. XVIII Time Preference and Productivity 241 The effect of the second of the two factors mentioned above, an expected decrease in the stream of "per =-anent. " services that would be available without saving, works in the opposite direction. Additions Err f I I d e.t 0 ant c pate to a future income which, in the absence decrease of non-anUf . ld b I h cipatable services o present savmg, wou e ower t an present income are likely to appear more urgent than increases of future income beyond the level of present income. An expected decrease in the stream of services of the "permanent" resources is therefore likely to in crease saving and so to counteract (though not necessarily to offset, or not more than to offset) the effect of the limita tion of the period for which an income has to be provided. Since we are not concerned here with the concrete magni tude of time preference, but only with the direction in which various factors are likely to affect it, we may leave this problem here.

The last point which needs to be touched on briefly in this chapter is the possibility of the existence of different rates of time preference in respect to different commodities. Up to now we have looked upon I Rates of t me prefer" income" as if it were a composite com-ence for dlrrerent d· h' h '1 d f commodities mo Ity w lC was necessarl y ma e up 0 constant proportions of the different goods. This in volved either or both of the assumptions that the different commodities could, for technical reasons, be produced only in fixed proportions, and that there existed such a peculiar degree of complementarity between them that they were desired only in fixed proportions. If tastes remained constant and conditions were in all other respects stationary, we should expect the rate of interest to be the same no matter in which commodity it was expressed. But, even with constant tastes, there is no reason why people, when their total incomes increase, should not want to spend these incomes in different proportions on the various commodities. And, in addition, we shall have to take into account that at different dates 242 I nve8tment in a Simple Economy PT. II the relative preferences for the various commodities may be different. This need not mean a " change in tastes"

in the sense that a person has altered his intentions. He may from the beginning have anticipated that at different periods he will want his income in different forms (and made his plans accordingly), just as everyone will foresee that he will have different needs at different seasons of the year and different hours of the day. But we may begin with the simpler case where tastes are constant in the sense that, with equal opportunities before him, the individual will always decide in the same Eltecls 01 accumu-way at the different dates. All that we lallon 01 capItal h t ·d· thO . th ff t f on relative values 01 ave 0 conSI er In IS case IS e e ec 0 commodities the gradual accumulation of capital on the relative values of commodities. This works in two ways. On the one hand, with given relative costs (i.e. given dis placement curves) for the various commodities, relative preferences, and therefore relative values, may change in response to increases in income. On the other hand, the fall in the rate of interest accompanying the accumulation of capital will affect the relative costs of the different commodities in varying degrees according to the propor tional amounts of capital! that can be profitably used in their production at the different rates of interest.

The general implications of such a situation were dis cussed above (Chapter XII, p. 167), when we dealt with the meaning of the concept of a uniform rate of interest in value terms. We know that equilibrium in these intertemporal value relationships is quite compatible with different intertemporal marginal rates of substitution for different commodities. All that need be added here is an explanation of how this situation fits into our representa tion of time preferences. And although the present case falls outside the field in which the graphical representa1 The decisive magnitude in this connection is the "elasticity of substitution" of new capital for the concrete resources already in existence.

CH. XVIII TimePrejerence and Productivity 243 tions used previously are applicable, a conceptual exten sion of our scheme to cover this case meets with no serious difficulties. We shall have to conceive of a structure of n-dimen sional indifference surfaces where n includes the number of commodities considered plus the number of different points of time (or periods) for which the Conditions of Inter person plans. Similarly the different rates temporal equilibrium of transformation between the different of values commodities, or between commodities at different points of time, can be represented by a corresponding system of n-dimensional displacement surfaces. The relative values of the commodities, and their individual time rates of increase, will then be formally determined on exactly the same principles as those determining the relative values of a number of commodities that are assumed to be simultaneously available. Every increase or decrease in the value of anyone commodity, relative to that of others, which takes place in response to an increase in total income will mean a corresponding rate of interest during the period in terms of that commodity. Suppose, e.g., we take as our standard of reference the rate of interest in terms of commodity a, and consider a commodity b, the value of which increases in terms of-a from one moment of time to another, so that at the second moment a smaller quantity of b corresponds to a given quantity of a than at the first moment. Then equilibrium requires that the rate of interest in terms of b should be smaller than the rate in terms of a. And the same applies, mutatis mutandis, to a fall in the value of one commodity in terms of another: here the rate of interest in terms of the former will have to be greater than in terms of the latter.

Fundamentally the same considerat~ons also apply where the cause of the change in relative values is not a change in total income but a foreseen" change" in tastes. If - to add just a few words on this case - it 244 Investment in a Simple Econo1ny PT. II is known in advance that at a later date relatively more of commodity a will be demanded and relatively less of commodity b, and an increase in the output of a at the Etlect of foreseen expense of that of b is obtainable only at changes In relative increasing cost then the rate of interest preferences for dlt- , ferent commodities in terms of a (i.e. the price of present a's in terms of future a's) will be smaller than the rate of interest in terms of b. It will be seen that this description of int.ertemporal relationships covers not only the case of constant tastes in the narrower sense but also the case where the actual distribution of the resources is different at different dates but where it has been correctly foreseen from the initial moment when the planning for the whole period was done.

This means that it covers as much as can be covered by general equilibrium analysis - i.e. all actions based on the knowledge possessed at one moment of time.

PART III CAPITALISTIC PRODUCTION IN A COMPETITIVE COMMUNITY CHAPTER XIX THE GENERAL CONDITIONS OF EQUILIBRIUM WE are now ready to drop some of the special assumptions on which we have so far discussed the problems of capital and interest, and to apply the results obtained to the phenomena of the market. The case of a centrally directed communist society which we have been consider ing . has shown us the role that is played by two basic factors (i.e. the various opportunities to invest and the time preferences of the persons concerned) under the (analytically) simplest conditions, that is when all resources are under the control of a single mind which uses them in the service of a coherent system of ends. Here we shall move one step nearer to reality by considering the case where the command over the existing resources i3 distributed between a multitude of independent persons, each of whom uses his share of them in the service of his individual system of ends and all of whom are in a position to exchange on a market.