XVII. Time Preference and its Effects with Constant Returns on Investment

CHAPTER XVII TIME PREFERENCE AND ITS EFFECTS WITH CONSTANT RETURNS ON INVESTMENT 1 THE assumption employed up to this point that the dic tator of our economic system will under all circumstances aim at a constant income stream was only an expository The assumption that device, adopted provisionally. Its purpose a constant Income was to enable us to study the significance stream Is desired under all circumstances of certain factors in isolation; but it has Is abandoned little relevance to the conditions existing in real life. It is extremely unlikely that anyone would want a constant income in time whatever the circum stances, and regardless in particular of the prevailing opportunities of varying the size of the returns obtainable by investment and disinvestment respectively. Our next task is therefore to introduce a technique by which we can adequately describe the possible attitudes a person may hold towards income streams of different time shapes, and which will enable us to use more realistic assumptions in this connection.

It will, however, not be expedient to drop at once all the simplifying assumptions hitherto employed. It will be advisable at first to study the effects of the dispositions In all other respects over time on the assumption that, apart tbe assumption 01 from the effects of these dispositions themstationary conditions I •• tlll retained selves, everything else remains unchanged. This assumption of constant data will in particular include the assumption that the tastes and the knowledge of the economic subject and the flow of services from the per manent resources which he commands remain the same. 1 Parts of this and the following chapter are taken verbatim from an article which appeared some time ago in the Economic Journal (1936b). 216 CR. xvn Time Preference with Constant Returns 217 The meaning of these assumptions, especially of that of " constant tastes ", will require careful definition. It will soon become evident that the existence of constant data as just defined does not necessarily imply the existence of a stationary state in the traditional sense, since these conditions are fully compatible with actions on the part of any individual who will change his position from period to period and will therefore alter the conditions which affect the decisions he will make at successive points of time, even though we assume his tastes to remain constant.

Before· we proceed, however, to define the critical concept of constant tastes, it will be useful to say a few words about the implications of the other main assump tion, i.e. that the dictator of our communist TIM e.oted !low of society expects to command a constant pun laput fa assum" . f· t Th· t' to be OODltaat stream 0 mpu. IS assump Ion provisionally rules out one of the main factors which in the real world lead people to accumulate capital. If, as would follow from this assumption, people planned for ~n eternal life during which they would invariably command the same quantity of permanent resources, some of the main motives which govern their actions in the world as it is would be absent. Yet it is only on this assumption, together with that of constant tastes, that we are able to obtain a clear view of the significance of time ,preference for the decisions to save and invest, or to take advantage of opportunities for obtaining larger additions to future income by giving up a given amount of present income.

In later sections of this chapter we shall supplement this preliminary analysis by considering the factors which we are here ignoring. The idea of tastes which remain constant in time is of course again not something which is supposed to exist in reality; it is merely an expository device and is closely connected with the concept of the stationary state. This concept is of particular importance in the theory of 218 I nvestment in a Simple Economy PT. II interest because all the familiar theories concerning the psychological factors affecting interest imply the exis tence of constant tastes at least in this particular sense. Th i . h Indeed the main reason why it is so im-e s gOlftcance of t e assumption of con-portant to have a clear grasp of this constant tastes . h . 11 11 h d'ffi It' cept IS t at practlca y ate 1 cu les which arise in this connection are due not to any special complexity of the problem, but to the fact that the path is strewn "rith the effects of various confusions on this point. The blame for this rests not with the individual authors \vho have treated these problems in the past but ,vith the state of the general theory of value of which they had to make use. It is only in comparatively recent times that the development of the" substitution" tech nique (and the connected method of representation by in difference curves) has provided us \vith a method which effectively avoids the dangers of the earlier utility analysis.

In what follows an attempt will be made to give a straightforward statement of the relevant relationships as they appear in the light of the modern theory of value. All discussion of the relations between this positive state ment and the traditional concepts or theories is relegated to Appendix I. The assumption of constant tastes was introduced into equilibrium analysis to indicate that a person will act in the same way at different points of time if faced The meaning of con-with the same circumstances. "Acting" stant tastes here includes, of course, the distribution of resources between the -present and the future,! and the 1 There is a further difficulty arising out of the question what are to be regarded as the same time intervals; and according as we give different answers to this question we obtain a wider or a narrower concept of constant tastes. We can define constant time preferences, as is done in the text, as a state of affairs in which at any point of time the relative values attached to a unit of present income and a unit of incOlne a year hence are the same. But we might also include under that description the case where the relative values attached to incomes at the times tm, tm to ... are the same at t1 , t2 , ta •.. without postulating that the attitude between any pair of successive points of time tm, tm CR. XVII Time Preference with Oonstant Returns 219 assumption of constant tastes therefore means that, faced with the same possibilities, a person will at different moments of time distribute his resources in the same way between present and future. The same distribution in this sense need not be an even distribution through time, i.e. a distribution which provides for the same income in the present and in the future. The attitude would still be the same if at both dates the person were equally willing to increase his future income at the expense of the present or vice versa. But there arises the further difficulty that if he provides for an increasing or decreasing income stream, the circumstances in which he will have to act at different moments will necessarily vary. And in order to say whether his tastes have remained the same or have changed we shall have to compare his decisions at the later moment not with the decisions he actually made at the earlier moment, but with the decisions he would have made if the choice before him at the earlier moment had been the same as it is at the later moment.

Or, in other words, the fact that the tastes of a person have remained constant cannot be established merely by comparing his actual decisions at successive moments but only by comparing his (hypothetical) attitude to all possible sets of circumstances at each moment, as expressed by his complete " indifference map" to ••. (which ar~ assumed to be equidistant) must be the same. As wIll be readily seen, this question is closely connected with the assumptions we make regarding foresight. If we assume that the person in question plans investments at the initial date tl for all the future dates t., t3 •.• tm' tn' to ... , then his tastes will have to be regarded as constant in the strict sense only if the relative importance he attaches to marginal increments at tm and tn (or any other pair of future points of time) is the same whether he is at tl , t., t., or any other point, irrespective of whether at any given moment his attitude as between different pairs of future moments is the same or not. This would be the wider concept of constant tastes, and although it would satisfy the requirements of equilibrium analysis, it would complicate matters in the present con text. For this reason the narrower concept iii used in the text, which involves identity of the attitude as between any two moments and which, as will be easily seen, is a special case of the wider definition.

220 Investment in a Simple Economy PT. n Complete" indifference maps" as we should require them for an adequate treatment of the question cannot, however, be shown by graphical methods. Each of the The use 01 the In- "indifference maps", showing the attitude durerenoe curve me-of a person at any moment of time, would thod makes oonslderable slmplllloations have to show his relative preferences not necessary only for all the different commodities, but also for each of these commodities at all future points of time which he considers. This means that the complete "indifference map" would have to be drawn in as many dimensions as correspond to the number of different com modities plus the number of different dates for which the person plans, or, strictly speaking, since time is continuous a.nd infinite, in an infinite number of dimensions. But since the tastes in which we are primarily interested are the relative preferences for present and future goods, it is possible to show the essential points on a drastica.lly simplified model.

In the first place we can provisionally ignore the existence of a variety of different commodities and speak of income as if it consisted of a single commodity. I I d or, what for our purposes amounts to the neome COON va as a single (oomposlte) same thing, as if it were a composite comoommodlty d' I mo lty, a ways made up of exactly the same proportions of the different goods. This involves either or both of the assumptions that for technological reasons the different commodities can be produced only in fixed proportions, and/or that there exists such a peculiar kind of psychical complementarity between them that they are only wanted in fixed proportions. And it means that, whether the income stream is expanded or contracted, the quantity of every single commodity con tained in it will be increased or decreased in the same proportion. The possibility that changes in the size of the income stream will lead to changes in its composition will be considered explicitly in later sections.

What we have to consider, then, is the attitude of the OR. XVII Time Preference with Oonstant Returns 221 person at successive moments of time when he has to choose between an addition to his income now and an addition at a later date. But since it will usually be possible by a given sacrifice of present in-I •• .... . nves.men. usum_ come to increase future income at anyone to be pouibl' only for f I . f f d one deftnlte period o a ong contmuous range 0 uture ates (or vice versa), we should still have to consider many more variables than can be shown in any diagram. Our indifference map would, strictly speaking, still have to have an infinite number of dimensions - that is, as many as there are future moments of time to be considered. We can, however, obviate this difficulty by assuming that there is only one possible period of investment, say of one year, as might be the case in a purely grain-growing community, and that there will periodically occur the opportunity of deciding what part of the current input should be devoted to current consumption and what part of it should be invested for, one year.

From the assumption already introduced (i.e. that the tastes of the person considered remain unchanged) it follows that any investment, once it has been made, is intended to be permanent: that is, to Any Investment once be repeated in every successive year For madeJsusumecltobe • Intended as pennaour assumption implies that in so far as Dent he has to choose at successive moments of time between the same alternatives, he will always decide in the same way. Any question as to what he will do at successive moments arises only to the extent that, in consequence of past decisions, the opportunities now open to him have changed. This means that all that is to be regarded as the addition to next year's income due to a given invest ment is the net return: that is, the amount which is left for consumption after provision has been made for the same quantities to be invested (or reinvested) as in the previous year. In such circumstances any investment can be expected to make equal additions to the income of all future years.

222 Investment in a Simple Economy PT. II The choice which is described by the "indifference maps" below is, then, one between additions to this year's income and additions to the income of all future years. In consequence, net It is important to remember this through (and not gross) re-out the exposition that follows, since the turns of the Investments have to be maps differ in this respect from the othercompared wise similar diagrams made familiar by Professor Irving Fisher's discussion of the same problems. 1 The advantage of the method adopted here, as will be seen, is that it enables us to use the diagrams to describe the successive decisions of an individual, or the process of saving over time. In the following diagram (Fig. 21) this year's income is measured along the ordinate Oy and the income of the next and all future years along the abscissa Ox. The line The construction of Ow is drawn across the quadrant at an the diagram angle of 45 degrees to the two axes to indicate the locus of all points representing an equal y w distribution of income between this year and all future years (corre sponding to stationary conditions). It will be convenient to start the discussion from a point where' no investments have yet been made, no nonpermanent re sources exist, and 3 constant stream of cur rent input is expected.

x The income which could permanently be expected from the direct use of the input may be repre sented by the distance OLe If no investments were to be o L N FIG. 21 1 cr. The Theory of Interest (1930), p. 237 et seq. The" Willing ness" and "Market" lines there employed refer to gross (and not CR. XVII Time Preference with Constant Returns 223 made, and, in consequence, the expected future incomes were equal to the present income, the initial position would be represented by the point P (with the ordinates x=y=OL).l This may serve as our starting point. The choice to be made in this initial situation is between retaining a given amount of present income and obtaining a permanent addition to the income of each period in the future. It may, perhaps, not be immedi-Only cases where In ately obvious that in this situation saving vestments are expected to yIeld a post will occur only if such a permanent net live return Deed be return is obtainable, and we must be care-cODsidered ful not to beg this question. But it can easily be showll that on our present assumptions this will be so. As we have already seen, it follows from our assumption of constant tastes and constant external data (including in particular constant technical possibilities) that the invest ment which is made at the first moment must be repeated at every successive moment. But if no net return were obtainable (of if the gross product obtainable from the investment were actually smaller than the amount in vested), the sacrifice of present income would not be balanced by any addition to future income (or might even lead to an actual decrease in future income). And since we are here excluding the possibility of a desire to provide for uncertain contingencies (i.e. unforeseen changes in the net) returns. In consequence, in Professor Fisher's diagrams, " a given rate of interest is represented by the algebraic difference between the slope of a given Market line, and the 100 per cent slope of the 45° zero interest line," while in the diagrams that follow the rate of interest is directly represented by the slope of the curves.

1 As we start out from a position in which only permanent resources are supposed to exist, an increase of present income at the expense of future income is evidently impossible, either for the isolated individual or for the community as a whole. For this reason the part of the plane yOx (Fig. 21) which lies to the left of the line LP can be neglected, since it contains no points which represent possible combinations of present and future income. Even after capital has been accumulated and, in consequence, it becomes in principle possible to increase present income at the expense of future income, it will never be possible to reduce future income below OL, since by their nature the services of the per. manent factors cannot be consumed before they mature.

224 Investment in a Simple Economy PT. II data), a permanent investment without the prospect of a net return would imply a desire to bring about a net decrease in total income, and may for this reason be excluded from our considerations. The possible attitudes as between alternative marginal increments of present and future income which we shall have . to consider will therefore range from one limiting P lbl d b bl case where every addition, however small, osa ean pro a e rate~ of time prefer-to the permanent income stream will inenee duce the person to give up part of his present income, to the other limiting case where no addi tion, however large, to the permanent income stream will induce him to give up part of his present income. It is impossible to decide on a priori grounds what the actual attitude of a person in any given situation will be. But it is a reasonable assumption, borne out by common experience, that in order to induce a person who, apart from this decision, would be certain of a constant income, to give up part of his present income in order to increase his future income, this permanent addition to future income would have to reach a definite minimum magni tude. And it is even more certain that in order to induce him to make larger and larger sacrifices of present income at a given moment, more than proportionately larger additions to future income will be required. Since the preservation of life requires a certain minimum of present income, no addition to future income, however large, will induce him to reduce present income below this minimum.

If we now describe this attitude by means of indiffer ence curves, the first limiting case - that in which every permanent addition to future income is considered worth The slope of the In-som~ sacrifice of present income - will be durerence curves represented by a curve with a perpendicular slope at the point P. If, however, some definite addition to future income is necessary in order to induce the person to give up even the smallest quantity of present income, the curve at this point will show a (negative) slope correCR. XVII Time Preference with Oonstant Returns 225 sponding to the rate of (net) return which will just leave the person indifferent as between saving and not saving. In either case any point to the right of this curve will represent a combination of present and future income which will be preferred to the constant income represented hyPo In all cases the slope of the curve will be as indicated only at the point where it crosses. the line Ow, and, as we move along the curve further to the right, the slope ,vill gradually diminish. Whether the slope at The curvature of the P is perpendicular or slightly inclined, it lndUlerenee curves will gradually turn more and more to the right with decreasing values of the ordinate: ultimately it will become horizontal at some point before present income (measured by the ordinate) has dwindled to zero. And the same will be true of all the other mem bers of the complete family of indifference curves which may be drawn through all other points on the line O'w.

This set of curves describing the psychical attitude of the person at any moment will enable us to derive his actual behaviour if we combine it with a corresponding representation of the technical possibilities Investment opportunof investment. The simplest assumption iUes represented by to make in this respect - and the only transformation lines assumption compatible with the case where there is only one commodity and only one possible investnlent period. is that up to a definite limit 1 investments yield constant returns, and that at this limit the net return falls suddenly 1 In this case successive investments will be made by investing successive doses of the pure input for the given investnlent period instead of using them for the satisfaction of current wants. And all opportunities for investment will be exhausted when all the available input is invested for that period. Up to this point we Inay expect constant physical returns unless we assume that some kinds of the pure input are more suitable for investment than other kinds. The point at which all input has been invested represents the absolute limit beyond which future income cannot be increased. In the diagranl this is expressed by the fact that all transformation lines end (or become perpendicular) at the points corresponding to this maximmn incOlne.

16 226 Investment in a Simple Economy PT. II to zero. This constant rate of transformation of present income into permanent additions to future income can be expressed by a family of straight transformation (or " displacement ") lines of corresponding slope, as shown in the diagram. With this representation of the technical possibilities added to the indifference curves, the action of the in dividual can immediately be deduced. If he starts at the position represented by P, he can and The Drst act of saving.. • • wIll move In the manner IndIcated by the transformation line going through that point. From among the possible positions represented by all the points on this line the one that is most preferred will evidently be the position indicated by the point R where the trans formation line just touches an indifference curve. This means that at the first date the person will save, out of his income LP, the amount MP, in order to increase next year's and all future years' income from OL to ON.

But this is only the beginning of a process that will continue for some time. The diagram, it will be re membered, is supposed to represent the tastes of the person not only at the initial moment but The path of saving at every moment, since his tastes are assumed to remain constant. Thus, as we can derive from the diagram the position in which the person will be at the second moment in consequence of his decision at the first moment, we can also derive from it what his decision will be at the second moment. As a result of the investment at the first date, the person will later find himself in command of an increased present income with an assured future income of equal magnitude, i.e. he will be in the position represented by P'. And he will again find it to his advantage to move along the transformation line from P' to R' on to a higher indifference curve, and so on. If, as we have assumed, the decisions to save and invest take place discontinuously at definite intervals, the person will be in successive years at the points R, R', R".

ClIo XVII Time Preference with Constant Return8 227 There are two questions to which we must give at least provisional answers before we can go on to make more realistic assumptions concerning the shape of the trans formation curve. The first is: What is the relative im portance of the productivity element and the psycho logical attitude respectively in determining the rate of interest while the process of saving continues? The second is: At what point will that process come to an end, and on what will the rate of interest depend in that final stationary state ? Under the assumption made so far (i.e. straight trans formation lines or constant - opportunity - costs) the answer to the first question is very simple. While the process of saving still continues, the rate of Th I II I ... e re a ve mpo,,interest will be determined solely by the ance ot productlvllJ productivity of investment (the slope of and time preterence the transformation curve), and the psychical attitude will merely determine how much will have to be saved at every moment in order that the marginal rate of time preference may adapt itself to the given and constant productivity rate. The only role "time preference" plays in this particular case is that it determines how long it will take until a stationary position is reached. 1 To the second question there are two alternative answers. It is of course possible for saving to go on until there is no possibility of further investment. In this case the final equilibrium position will The llnal stationary be represented in the diagram by the point equlUbth;m where the end of the highest transformation line lies on the line Ow. Here investment comes to a standstill simply because, although possibly the person would still like to invest more, there are no further outlets for investment.

1 The situation here is of course exactly the same as in the more general case where the relative costs of two commodities are constant (i.e. independent of the quantities produced) and where, in conse. quence, their relative values are uniquely determined by their relative costs and cannot be affected (except in the very short run) by changes in their relative utilities.

228 Investment in a Simple Economy rT. II Since this case is quite irrelevant to the problems which will arise later under more realistic assumptions, we need not consider it further. The second possibility is that the process may come to an end before the investment opportunities are exhausted. It may be that at some point on the line Ow the trans formation line will coincide with the tangent to the in difference curve, and that in consequence there will be no inducement to move along that line to the right, i.e. to invest. In the present case this is possible only if the slopes of the indifference curves at the points where they cut the line Ow are not the same for all the curves, but gradually increase. So far we have had no occasion to make any explicit assumption in this regard. If, however, we now introduce the assumption just mentioned (i.e. that the slopes of the successive indifference curves at the points where they cut the line Ow become continually larger), it is evidently possible that in some such position where present and future incomes are equal, the time preference may be equal to the technical productivity of capital. In this case the ultimate stationary equilibrium will be reached with a positive rate of interest equal to the constant productivity of investment. The psychical attitude will merely determine at what income this point will be reached. The rate of interest in this state of final equilibrium will be determined solely by the productivity of investment.