# (p.159) Appendix: Optimum Saving

# (p.159) Appendix: Optimum Saving

Is it possible to say anything useful, on purely theoretical grounds, about the proportion of income which an economy should ideally save? In particular, does it depend upon the rate of return which is to be expected from those savings? These are tempting questions, but I think I can show that there is not much to be said about them in the way they have usually been discussed.

The problem is one of optimum, not actual, behaviour. It has therefore appeared to be appropriate to discuss it under the assumptions (1) that what will happen, as the result of one course of behaviour or another, can be accurately foreseen, (2) that the rate of return can be sufficiently represented by a single rate of interest, and (3) that the behaviour is that of a representative ‘consumer’, so that questions of distribution do not arise. One may have doubts about each of these assumptions, but they will not here be questioned. Suppose we grant them, what can we say?

It is natural to take it that our ‘consumer’ has an indifference map of the usual type, between consumption now and consumption in future periods. Since the dating of the consumption is all that concerns us, the consumption of each period can be represented by a single index, so much ‘value’. Let ξ_{t} be the consumption, so reckoned, that is planned for period *t*. Then at time 0, he chooses a *plan*

If *r* is the rate of interest, (1 + *r*)^{−1}, or λ, is the discount factor. The discounted value of the stream (ξ) is

*K*(ξ, λ).

Since (ξ) is chosen at discount factor λ, any alternative plan which has a discounted value less than *K*(ξ, λ) is rejected in favour of (ξ); so it is ‘shown to be’ on a lower ‘utility’ level. If (ξ + δ ξ) is on the same utility level as (ξ), it cannot be among those which are so rejected. Thus, as between two indifferent positions, we must have the Samuelson inequality

If (ξ + δ ξ) were that actually chosen at λ + δ λ, it would similarly follow that

It then follows from this pair of Samuelson inequalities (dividing one by the other, as is permissible since all the sums are positive) that the effect of a rise
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in the discount factor (or a fall in the rate of interest) is in the direction of *diminishing* the ratio

*P*is an index of the

*trend growth rate*

^{1}of the stream (ξ). So a fall in the rate of interest (a rise in λ) diminishes this growth rate. As between two indifferent positions, a rise in the rate of interest tends to increase the trend growth rate of the stream of consumptions.

That is all that follows from the Samuelson inequalities, as so far stated; it does not tell us very much. For consider how it applies to the comparison of steady states. In a steady state, the growth rates of all elements are equal to *g*; so the growth rates of all the ξs are equal to *g*, and that of *K* must also be equal to *g*. The increase in *K*, from period to period, equals *gK*. Income by definition equals *rK*; so if *s* is the proportion of income saved, *s* = *gr*. All that we have learned, from the proposition so far established, is that *g* will be larger the larger is *r*; but we cannot conclude from that *s* increases with *r. s* may rise or fall, or remain constant; the only thing that is ruled out is that *s* should fall so much (when *r* increases) that *sr* is diminished. And even this is only ruled out so long as we confine our attention to the substitution effect. There does not seem to be any reason why we should not have, on occasion, to include an ‘income’ effect, which would work an exception, even to our very weak proposition.

It must have been for some such reason as this that several writers on the theory of interest and saving (from Fisher and Ramsey^{2} to Tinbergen and
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Frisch^{3}) have sought to specify the utility function further. Are there any particular characteristics which we may assume for the utility function plausible enough to be reasonable in most (if not all) intertemporal utility problems, and strong enough to give more definite rules about behaviour? There is a particular form of function, which has emerged from their work, and which is thought to satisfy these requirements. Certainly it does give a much stronger theory, if it is acceptable. It is tempting (and eminent economists, from Ramsey onwards, have succumbed to the temptation) to use it for making prescriptions about saving—for laying down rules of behaviour about saving, which (it is held) a rational community ought to follow. The belief that there are such rules (and that they have, at least, some degree of authority) has spread widely among economists; it is not without influence even on those who are unacquainted with the details of the Ramsey theory. The assumptions upon which the theory is based should therefore be given a close examination. I believe that I can show that they are in fact very fragile—much too fragile to stand the weight that has been put upon them.

For the simplest form of the theory we need *three* special assumptions; it will be convenient to begin by setting it out on that basis. Not all of these assumptions, we shall subsequently find, are essential, but there is one that is essential—the whole theory collapses without it. And there is a quite simple reason why that assumption is very hard to accept. It is therefore not surprising (as we shall also find) that the theory itself has some odd features; not all of the consequences that must be drawn from it make as good sense as they should. But it will be best to let all these points appear as we proceed on the basis of the three assumptions.

(1) The first of the three assumptions is *stationariness* (as Koopmans^{4} has called it). This ‘stationariness’, it must be emphasized, is a pure characteristic of the utility function; it has nothing (necessarily) to do with Stationary Equilibrium. What it means is that the (intertemporal) want‐system remains unchanged over time; as time moves forward, the whole want‐system moves forward with it. More precisely, the marginal rate of substitution, between consumption *t* _{1} periods *hence* and consumption *t* _{2} periods *hence*, remains the same at whatever date the choice between the two consumptions is made, provided that all the planned consumptions (dated so many periods *hence*) remain the same. Consumptions are valued according to their distance in time from the planning date, whatever the planning date may be. The stationariness assumption is that the system of wants is of this character, and that the plan chosen is one that *continues* to maximize utility in terms of that want‐system.

It is clearly by no means necessary, in a practical problem of planning, that this stationariness assumption should be valid. A simple case in which it (p.162) would not be valid would occur with a plan which incorporated an extension of education, which will modify (and is expected to modify) the wants of the community as time goes on. But the existence of exceptions does not prevent the stationariness assumption from being an interesting assumption. It is not this assumption which I want to criticize. What is the effect of assuming stationariness is one of the questions in this field to which we should certainly like an answer.

If the stationariness assumption (and nothing else) is incorporated into our analysis by Samuelson inequalities, we shall have to add to the inequalities (written above) which compare values discounted back to time 0, similar inequalities with values discounted back to each future date (only the consumptions subsequent to that date being included in each case). The plans will in fact have to be tested for indifference *all along the line*. Manipulating these in the same manner as before, we shall find that the substitution effect (of a rise in the rate of interest) must not only increase the trend growth rate of the whole stream

(2) The second assumption to be introduced is *homogeneity*. What is meant by this is the same as what is meant by homogeneity in the case of the homogeneous production function—an increase in consumptions (inputs) in the same proportion will increase utility (output) in a proportion that depends upon the utility (output) level, but is independent of the proportions in which the consumptions (inputs) are combined. Homogeneity of the utility function does not imply cardinality; it is a pure property of the indifference curves (or surfaces). Here, if (ξ) and (ξ′) are two streams that are on the same indifference level, we are to get another pair of indifferent streams when we multiply every item in (ξ) and every item in (ξ′) by any identical multiplier. It follows from this property that the marginal rate of substitution, between consumption at time *t* _{1} and consumption at time *t* _{2}, is entirely determined by the *ratios* between consumptions at these and at other dates. Consequently, if the rate of interest (which plays the part of a price system) is given, the ratios
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between planned consumptions (at different dates) will be determined, irrespective of the general level of consumption that is attainable over the whole sequence. So long as the rate of interest remains unchanged, an increase in total wealth will increase planned consumption, in all periods, in the same proportion.

In itself, this homogeneity assumption looks harmless; it amounts to no more than a bracketing together of ‘income’ and substitution effect; the ‘income effect’ is made manageable by being reduced to the simplest form that it can possibly take. But when this assumption is combined with the stationariness assumption, the result is drastic.

If the want‐system is stationary, and is also homogeneous, the only type of consumption plan that can be optimal, at a constant rate of interest, is a plan with a constant growth rate. For if

There is certainly no question that this is a convenient property, and to reach it in this way may perhaps persuade us to adopt it with a lighter heart. But even if we do accept it, it does not, in itself, give us any help in dealing with the problem with which we began. For it was precisely in relation to constant growth rate paths that our original difficulty came up most sharply. All that can be said, even when we have both of these first two assumptions, is that a rise in the rate of interest will tend to increase the growth rate of the optimum path—our former proposition, only made a little more precise, since we need no longer talk about ‘trends’ and have absorbed the old reservation about income effects. But the effect on the proportion of income saved remains as obscure as ever.

(3) The third assumption is that of *independence*. This is the point at which we go over to Cardinal Utility; but it is not the cardinality that is important—it is the independence which is taken to go with it. A general assumption of cardinality would itself impose no additional restriction; but there is an
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additional restriction when the cardinal utility function is assumed to take the particular form of a sum of *separated* utilities

*only*. A cardinal measure of that particular form does imply an ordinal property: the marginal rate of substitution between consumption at time

*t*

_{1}and consumption at time

*t*

_{2}(being the ratio between the marginal utilities of these consumptions) is made to depend upon these two consumptions only; consumptions at other dates do not affect it. If we have that ordinal property, it must be possible to put the utility function (if we choose to do so) into the form which has just been given;

^{5}if we can have the separated form, we must have the ordinal property. The two are strictly equivalent.

It is independence (in this sense) which, when added to the other assumptions, works the transformation. The consequences of the combination are very far‐reaching indeed.

Take homogeneity and independence together. Homogeneity says that all consumptions are to increase in the same proportion when there is a change in capital value (and no change in interest); independence says that the marginal utilities of the consumptions (which are to keep the same proportions to one another since there is no change in interest) are each of them dependent upon its own consumption only. These things can only happen together if all of the (separated) marginal utility curves have the same elasticity; and since we might have started from any combination of consumptions, they must have the same elasticities at all points of the curves, which can only happen if each curve is a curve of constant elasticity—the same constant elasticity for each separated curve. Thus the marginal utility of consumption at time *t* must be given by the formula

*q*

_{t}is a constant, that may vary from one curve to another.

At an optimum position, this is to be proportional to the discount factor, which we have been writing as λ^{t}. In this case, however, it is neater to work with continuous time. Let us accordingly write the discount factor as *e* ^{−rt}, where *r* is (now) an *instantaneous* rate of interest. It follows at once that ξ_{t} must be proportional to

If there is also to be stationariness, the path must be a constant growth path; and this formula will only give a constant growth path if *q* _{t} itself has a constant rate of growth (or decline) over time. Now the *qs* are the weights that are given to future utilities to make them comparable with present; it follows from the stationariness assumption that these weights can only differ because of delay (as *t* increases). It is commonly accepted that the delay will diminish the (present) utility of future consumptions; let us grant that (at least provisionally). We may therefore write *q* _{t} = *Ce* ^{− pt}, where *p* is to be constant (the rate of time‐preference). Finally, therefore, we have

^{6}In so far as this formula is acceptable, it is undoubtedly more informative than the bare rule that the bare rule that the growth rate rises with the rate of interest, with which we started.

But is it acceptable? Let us begin by noticing some of its implications.

The familiar *g* = *sr* is still valid (along a constant growth path) even though we are using instantaneous rates.^{7} Thus, for the proportion of income saved, we have (from the Champernowne formula)

*p*= 0) we have

*s*= η, so that the proportion of income saved is independent of the rate of interest; (2) if

*p*is positive, there will be zero saving when

*r*=

*p*, while if

*r*>

*p*, the saving‐proportion will be larger, the larger

*r*, up to a maximum at

*s*= η, as before. These are quite sharp conclusions; but they are odd conclusions. The more one reflects upon them, the odder they seem.

It must surely be supposed that consumption must always be positive; *s* < 1. But it has just been shown that if *p* = 0, *s* = η; so that the model will only make sense, when *p* = 0, if ν < 1; the marginal utility curves must be *inelastic*. If *p* > 0, we can have *s* < 1, with η < 1, if the rate of interest is not too high; but we shall find consumption going negative at high rates of interest. This is
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intolerable; it must therefore be concluded (and has been concluded in fact by Ramsey and his followers) that we must take η < 1. But there is no intuitive reason why the marginal utility curves should be inelastic; it is odd^{8} that we should have to make them inelastic in order to make sense of the theory.

It further follows that if the marginal utility curves are inelastic, the total utility function (for the individual dated consumption) must be such that there is a limit beyond which utility cannot rise however much consumption increases; utility cannot increase indefinitely. Ramsey's ‘Bliss’ is an essential character of the model; the whole construction depends upon it. But (again) it is not obvious intuitively why the utility function must have this property.

There is a final point, which was not noticed by Ramsey.^{9} If the utility function, *U* _{t}(ξ_{t}), for the individual dated consumption must have this form, the utility of the whole stream, which is the sum of *U* _{t}(ξ_{t}) from *t* = 0 to *t* = infinity, will not be finite unless *p* > 0. But it is the total utility, in the sense of this sum, which is being maximized. One cannot maximize something which is infinite. So that unless *p* > 0, the whole construction breaks down.

Thus it is not fair to proceed as Ramsey did, to treat zero time‐preference as a criterion for rationality, and to pour scorn upon the weakness of our telescopic vision which makes us unwilling to save at the high rates which (it is alleged) would be appropriate for *p* = 0. If there is no time‐preference, then (on the theory) it is necessary that the curves should be inelastic; but if the curves are inelastic, it is impossible that *p* = 0.

What has happened? We must go back to the assumptions and look at them again. One which is clearly weak is homogeneity. Can we drop that, and get something that is more acceptable?

If we drop homogeneity, but maintain stationariness, we shall drop the constant growth path. But perhaps it is that which ought to be abandoned.

If we drop homogeneity, but maintain independence, we are not tied down to a particular utility function; we can give the marginal utility curves another form. Quite general forms are hard to handle; but there is one (which was considered by Ramsey) which is quite simple, and which is surely an improvement. This would make the marginal utility of consumption become infinite, not at zero consumption, but at some positive ‘subsistence’ level. We can then keep something like the ‘constant elasticity’ form, but shift the whole curve to the right. That is to say, for the marginal utility of consumption at time *t*, we should have

It can be shown that if *g* is this growth rate, the proportion of income saved (at time *t*) is given by

What then of independence? It is more and more apparent that it is independence that is the key assumption. As long as we maintain independence we are bound to get something like the Ramsey results; but what is the case for the independence assumption? As soon as we face up to it, and consider (quite directly) what it implies, it becomes apparent that the case for making it is very weak indeed.

If the successive consumptions (ξ_{t}) have independent utilities, the amount of present consumption which the chooser will be willing to give up, in order to be able to increase consumption in year 5 from so much to so much, will be independent of the consumptions that are planned for years 4 and 6. It will be just the same, whether the increase in year 5 is to be a sudden spurt, out of line with its neighbours, or if it is needed to fill a gap, to make up a deficiency (that would otherwise have occurred in year 5), so raising the consumption of year 5 up to the common level. This is what is implied by the independence assumption; when it is stated in those terms, surely it must be said that it cannot be accepted. The sacrifice which one would be willing to make to fill a gap must normally be much greater than what it would be worth while to incur for a mere extra. There is indeed a sense in which there is a rapidly falling ‘marginal utility of consumption’ in the particular period. But it is not due to the inelasticity of an independent ‘marginal utility curve’; it is due to the complementarity between the consumption that is planned for the particular
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period and that which is planned for its neighbours. It is nonsense to assume that successive consumptions are independent; the normal condition is that there is a strong complementarity between them.

It is not to be denied that there are some kinds of saving which are directed towards particular future expenditures, so that the complementarity with neighbouring consumptions may for them be rather weak. But the clearest case of this is saving for a particular event (as for one's own old age, or for the marriage of one's children); and though there may be independence in these instances, it is abundantly clear that with them the stationariness assumption will not hold. Even with them we do not have independence and stationariness. Other sorts of saving‐up, as for the purchase of durable consumer‐goods (in the days before consumer credit) are not, I think, exceptions to the rule of complementarity.

Any saving which is not just saving‐up—saving which is a pure exchange of present for future satisfactions, satisfactions that are not inherently different save that the one is present and the other is future—must, because of the complementarity, take the form of the substitution, for the present consumption, of some sort of a *flow* of consumption in the future. This is, surely, how it appears to the ordinary man; this is why what is offered to him (nearly always) is interest, or dividend, or annuity, on his savings. The institutional arrangements are a practical recognition of the complementarity that is in question. Saving of this kind (and it is the only kind for which the stationariness assumption is appropriate) cannot be adequately analysed by considering the present and *one particular future date* in isolation.

What are being compared are present sacrifice, and *flow* of future satisfaction. If the complementary is perfect, so that the time‐shape of the future consumption‐flow is taken as fixed (and this, though an extreme assumption, is a better assumption than the assumption of independence), we can treat the planned future consumption as a single good, and represent the whole choice on a two‐dimensional diagram. But it is a choice between sharply different things: once‐for‐all consumption in the present period, and a stream of consumption exteding into the indefinite future. Nothing more can be laid down about such a choice than we are accustomed to say about the choice between any two commodities (or commodity bundles). The supply curve of saving agaisnt the rate of interest may be rising from left to right or may turn back on itself (but for nothing more than the usual Walrasian reason).

When we particularize, splitting up the stream of consumptions into particular dated items, we do not change the situation fundamentally, once we allow for the complementarity. But there can be as much complementarity as is possible, and the Samuelson inequalities will still hold. Thus we come back to the rule about a rise in the rate of interest increasing the trend growth rate—the ‘poor thing’ with which we started.

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I should like to emphasize, in conclusion, that we get no further than this by assuming ‘rationality’. If the question is simply one of the choice between two ‘commodities’, it is equally rational to give a high value, or a low value, to one in terms of the other. The only thing that is irrational is to act in such a way as must result in future damage, and to leave it out of calculation. Thus it is ordinarily irrational to ‘waste one's substance’—to indulge in a high rate of current consumption, which must be followed, at some point, by a fall to a much lower level. Violent contractions are exceedingly painful, and it is foolish to expose oneself to such pain by lack of foresight. But as for the amount which should be sacrificed now in order to bring about a rise in the future *stream*—that is a matter on which wisdom may have more than one opinion.
(p.170)

## Notes:

(1)
*P* is the elasticity of *K*(ξ, λ) with respect to λ. Written in full, it is

*Value and Capital*(1939), pp. 186–8, I called it the

*Average Period*of the stream, for I wanted to show that it is the proper generalization of the Average Period which appeared in the work of Böhm‐Bawerk and in the early work of Hayek. I did, however, go on to point out in a footnote that there is a simple relation between it and what I there called the

*crescendo*of the stream, what I would now prefer to call the trend growth rate. ‘The best numerical definition for the crescendo of a stream of values is the rate of expansion of a stream, continuously expanding by the same proportion in every period, which has the same average period as the original stream. This rate of expansion is related to the average period by a simple formula’ (p. 188).

It is easy to show that the Average Period of a stream with constant growth rate *g**, extending into an indefinite future, is 1/(*r* − *g**). Since *r*, the rate of interest, is here only used for purposes of computation, we should keep it unchanged when comparing one stream with another. Thus a rise in *P* will always correspond to a rise in *g**.

(2)
Irving Fisher, *Rate of Interest* (1907), *Theory of Interest* (1930); F. Ramsey, ‘Mathematical Theory of Saving’ (*Economic Journal*, 1928).

(3)
J. Tinbergen, ‘Optimum Savings’ (*Econometrica*, 1960); R. Frisch, ‘Dynamic Utility’ (*Econometrica*, 1964).

(4)
T. C. Koopmans, ‘Stationary Ordinal Utility and Impatience’ (*Econometrica*, 1960). As will be evident, I owe a great debt to this subtle paper.

(5)
That the correspondence works both ways is becoming familiar through the use that is being made of the independence assumption in other connexions, by such writers as R. H. Strotz, ‘The Utility Tree’ (*Econometrica*, 1957), and I. F. Pearce, *A Contribution to Demand Analysis* (1964).

(6)
According to D. H. Robertson, *Lectures on Economic Principles*, vol. ii, p. 79.

(7) It will be remembered that we are not distinguishing between profits and wages; capital value is the discounted value of the whole stream of consumptions; income is the interest on this capital value.

(8) As Tinbergen has noticed (op. cit.).

(9) I owe it to the paper by Koopmans, where it is established in much more general terms.