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Front Page Titles (by Subject) PART III, CHAPTER XI: SECOND APPROXIMATION IN GEOMETRIC TERMS - The Theory of Interest, as determined by Impatience to Spend Income and Opportunity to Invest it

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Subject Area: Economics
Topic: Money and Banking

## PART III, CHAPTER XI: SECOND APPROXIMATION IN GEOMETRIC TERMS - Irving Fisher, The Theory of Interest, as determined by Impatience to Spend Income and Opportunity to Invest it [1930]

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The Theory of Interest, as determined by Impatience to Spend Income and Opportunity to Invest it (New York: Macmillan, 1930).

Liberty Fund, Inc. is a private, educational foundation established to encourage the study of the ideal of a society of free and responsible individuals.

### §1. Introduction

Graphic illustrations of the solutions of two economic problems incident to the attainment of economic equilibrium, assuming incomes fixed, have been given in Chapter X. One was an individual problem, the other a market problem. We found their solutions respectively to be:

(1) The income situation Q1 which Individual 1 will reach from his original income position P1 by borrowing or lending will be found where his borrower-lender motive is balanced, i.e., where one of his Willingness lines is tangent to the Market line M; and

(2) The rate of interest, or divergency slope of the Market line from 45° will be such that the center of gravity of all Q's, as above found, will coincide with that of all the P's.

In this chapter the point P, which was assumed to be arbitrarily imposed upon the individual, is replaced by a series of optional points among which he may choose. If this group of points is shrunk into a single point, the analysis of this chapter becomes identical with that of Chapter X. In other words, Chapter X represents a special case, while this chapter represents the general problem.

In Chart 35 are represented various possible points supposed to indicate the various income situations available to Individual 1 aside from any further shifts through borrowing or lending. Instead of having no choice but a fixed position as in Chapter X, he now has the opportunity to choose any one of many income positions, but will actually confine his choice to those positions represented upon the boundary line O1' O1IV. This may be called the Investment Opportunity line or briefly the O line for Individual 1. Every individual, of course, has his own O line.

### §2. The Investment Opportunity Line

The reason why we may exclude all points inside of this boundary line is evident. The inside points would never be chosen under any circumstances, since each inside point is excelled by some points on the boundary in respect to both years' incomes. Thus the point A in Chart 35 will certainly not be chosen if the individual has the opportunity to substitute any other point to the north or east of it, or between north and east.

But in no case can income be increased indefinitely. There are limits in whatever direction we try—whether this year, next year, or both. These limits make up the boundary line O1' O1IV. Chart 35 represents Individual 1 as having the opportunity to shift his income position on this map in an eastward direction only up to the position O1'. In other words, he can increase his income in the present year without changing his income in the next year only up to that limit O1'. Technical limitations, including personal limitations, are assumed to forbid his pushing to the right beyond O1'.

In the same way, starting again at A, he has the opportunity to move northward on this map—that is to increase next year's income without changing this year's—but only up to a certain limit O1IV. Or he can move in a somewhat northeasterly direction and better himself for both years at once, but again he can do this only up to a certain limit, O1'' or O1'''.

The boundary line O1' O1IV, made up of these limiting points may, of course, take various forms, but, for the present it will be assumed to be a curve concave toward the origin. It is simply a geometric picture of the technical limitations of an individual's income in the two years considered, assuming, as always, all other years' income to remain the same. It is the locus, or line, of options and may be called the Option line or the Opportunity line,9 and is designated as O1 for Individual 1.

### §3. The Individual's Adjustment Without Loans

What we have seen so far is that Individual 1, having discarded all income positions inside the Investment Opportunity line, has left as still eligible only the points on that curve.

As before, we assume that each individual is unconscious of having any influence on the market rate of interest. To fix our ideas suppose, as before, this rate to be 10 per cent. The only adjustments the individual can make are: (1) adjusting his position on the O line; (2) further adjusting on the M line. Problem (2) is analogous to that of Chapter X, so that Problem (1) is the only new one. The solution of Problem (1) will be found to point the way to the solution of the knottiest part of the interest problem, purposely omitted from the first approximation. This is the problem of investment opportunity, productivity, or technique of production in relation to the rate of interest.

The principle by which the individual may shift his position along the Investment Opportunity line is very similar to the principle already set forth in Chapter X by which he shifts along the Market line. It will be recalled that the individual shifted along the Market or M line according to its slope when that slope is compared, at any point, with the slope of the Willingness or W lines. We saw that, if we suppose him situated at a point on the M line at which the Willingness line drawn through that point is steeper than the Market line, he will move away from that point downward, along the M line, that is, he will borrow; while, if situated where his W line is less steep than his M line he will move upward along the M line, that is, he will lend.

Similar comparisons apply to our present problem merely by substituting Opportunity line for Market line. Suppose Individual 1 to be situated, to start with, at O1' on the Opportunity line, as shown on Chart 36. He then has the opportunity to shift to any other point on that line as formerly he could shift along the Market line. Let us, as before, proceed by small steps of \$100 each. The first step is from O1' to O1''. The chart indicates that, by sacrificing \$100 of this year's income, he can add \$150 to next year's income, while he is willing to receive only \$115 as indicated by his Willingness line drawn through O1'. The \$50 net return he will receive is a 50 per cent rate of return over cost. This is his investment opportunity rate. He is willing to lend \$100 for a net return of \$15 or 15 per cent over cost. This measures his degree of impatience or rate of time preference. Evidently, as just hinted, he will seize the opportunity to invest for a 50 per cent return when he would be willing to take 15 per cent. This choice is represented on Chart 36 by following the Opportunity line from O1' to O1''.

If, as a second step, another \$100 can bring him \$140 while he would be willing to take \$120, he will seize that opportunity, too, and so move on to O1'''. That is, he will choose a 40 per cent investment opportunity when his degree of impatience is only 20 per cent. Thus, he may be pictured ascending a staircase on the Opportunity line. The successive steps, in this case, grow less steep as he proceeds. At each point he decides whether to take the next step or not by comparing its steepness with that of the W line at that point. The successive W lines will be more and more steep as he goes on investing successive \$100's, while the Opportunity line will become less and less steep.

When the point is reached where the Opportunity line is no longer steeper than the Willingness line, he will stop investing. The Willingness line through that point will have the same steepness as the Opportunity line, say 30 per cent. That is, the two curves will there be tangent. This point of tangency, R, is shown in Chart 37.

The reasoning which has just been used is evidently exactly like that used in Chapter X, the only important difference being that then we had a straight line to deal with to express what the individual can do while here we have instead a curved line, or at any rate, a line which need not be straight.

And the result of this reasoning so far is also similar. The stopping point is where the can line (in Chapter X, the Market line; here, the Opportunity line) is tangent to the Willingness line.

### §4. Individual Adjustment with Loans

Up to this point in the second approximation we have reasoned as though the Individual did not have freedom to borrow or lend in the loan market. We purposely excluded that possibility for the moment and went ahead as if the man were shut off from the loan market completely, so that any investment must be out of his own income and not be made with borrowed money. Were this the case (as in practice it often is) Chart 37 would correctly represent the result of the individual's shift. It would be a one-way shift, entirely along the Opportunity line.

But if now we return to the hypothesis of a perfect loan market, accessible to all concerned and to any extent desired, then Chart 37 does not fully picture our problem because it fails to take account of the fact that the individual not only can shift along the Opportunity line, but can also shift along a Market line by borrowing and lending. That is, he now has two "can" lines, both the Market line of Chapter X and the Opportunity line of this chapter.

Chart 38 pictures the double movement of Individual 1. Starting at O1', he moves along the Opportunity line to P1 where the Opportunity line becomes tangent to the M line, then along the M line to Q1 where the M line becomes tangent to a W1 line. That is, the fixed rate of interest will cause the individual so to shift that the marginal rate of return over cost (investment opportunity rate and the marginal rate of time preference (degree of impatience) will, each of them, be equal to the market rate of interest. Chart 38 depicts Individual 1's adjustment of his rate of investment opportunity and his degree of impatience to the market rate of interest. The rate of interest is, as always, represented in the slope of the M line, and the rate of return over cost is represented in the slope at P1 of the O1 line. These two slopes are the same, since the two lines are there tangent.

The slopes of the M line and the O1 line at P1 are identical since the two lines are tangent at that point. The degree of impatience is represented by the identical slopes of the M line and of the W1 line at their point of tangency Q1. Since the M line is a straight line, the slopes of the O1 line and the M line at P1 and the slopes of the M line and the W1 line at Q1 are all identical and the identity of the opportunity rate, the impatience rate, and the market rate is shown.

### §5. The Double Adjustment Discussed

In such a double adjustment, P1, the point of tangency on the Investment Opportunity line, has to be found first and Q1, the point of tangency on a W1 line last, for there is only one Opportunity line and only one point on it at which the slope corresponds to the rate of interest; while there are an infinite number of W lines with a point on each having that slope or direction.10

It is worth noting that the point P1 thus located on the Opportunity line will be quite different from the Point R on that line shown in Chart 37 when the individual was assumed to be cut off from loans.11 The two may differ in either direction.

It is also to be noted that the W1 lines always say the last word, that is, fix the final income position at Q1, the point of tangency of the M line to a W1 line. All other income positions represent points reviewed in Individual 1's mind but rejected in favor of Q1. The point P on any individual's Opportunity line is merely a point in transit toward Q, which is the final point of equilibrium.

If we wish to be even more realistic, our individual need not be pictured as traveling along the Opportunity line at all, even on a non-stop flight to Q1. He may, more properly, be pictured as making a more direct jump, across lots, directly from O1' his income position on the Opportunity line to Q1.

The reader may, starting at O1', trace the individual by small steps of combined \$100 investments and loans. Thus the first \$100 step would carry him from O1' to B. Successive investments and borrowings of equal amounts would increase the individual's next year's income while leaving his present year's income the same as before. On the chart his income position would move first from O1' to B and then step by step in a vertical line above B. But he will not necessarily confine his borrowings to the amount of his investments. The chart represents a man whose impatience leads him to borrow for this year's consumption the amount represented by CF. His borrowings represented by the horizontal difference between O1' and P1 (that is the distance CE) is what is often called a productive loan, while the horizontal difference between O1' and Q1 (that is the distance CF) is what is called a consumption or convenience or personal loan.12

Properly speaking, however, no part of the loan is itself productive. It is the investment which is properly to be called productive. To shift along the M line adds nothing to the total present worth of the individual, for it merely substitutes \$110 next year for \$100 this year, or a series of such sums, and each \$110 next year has the same present worth as \$100 this year. A shift along the Opportunity line, however, does add to a man's present worth. Up to the last \$100 invested, each \$100 yields more than \$110 next year and so possesses a greater present worth, reckoned at 10 per cent, than \$100.

The sole advantage of any shift along the M line alone is to gain not more market worth, but to gain in convenience—to reach a greater total desirability. This is true in both the first approximation and the second. Every loan, merely as such, is a shift on the M line alone, and is in itself always a convenience loan. Strictly speaking, no loan, as such, is "productive."

It is only in so far as the loan makes a difference in the other shift, that along the O line, that it can claim to be called a productive loan, and it is quite true, in the case pictured, that the loan does make such a difference. That is, we call the loan productive because, without it, the investment would not be made, or would not be so great—because it would be inconvenient (or even impossible) to invest so much out of this year's income.

The essential effect of a so-called productive loan is to enable the individual (under our hypothesis of perfect fluidity and no risk) to disregard entirely what has been called the time shape of the income stream P, that is, the proportion of this to next year's income represented by P. It enables him to push P as far to the left as he wishes without threatening him with starvation, or causing him any inconvenience. He need practice no abstinence. For whatever P lacks in this year's income may be made up by loans, that is, by use of the Market line. In fact, P may be pushed even to the left of the vertical axis, a position of negative this year's income, which is physically impossible except as simultaneously offset by a loan so as to bring him back again to a position of real income this year.

In short the investment, or O shift, affects the size of income as measured in present worth of the entire income position while the loan, or M shift, affects its final shape.

The Chart 38 is evidently only one type among many and the reader who wishes to pursue the subject into special cases will find it easy to do so by varying the curves to suit himself.13

### §6. Market Equilibrium

Just as in the first approximation, so in this second approximation there are two successive problems:

• (1) How the individual reacts to a given rate of interest.
• (2) How market equilibrium determines that rate.

The first of these two problems having now been solved, we are ready for the second, the market problem—to show how market equilibrium is established. This is precisely as in Chapter X, except that the M line, instead of rotating about a fixed point P, now rolls around the O line.

The problem, then, is simply to draw a set of straight M lines, one for each individual, each person's M line being tangent to his Opportunity line at a point P, all such M lines being parallel to each other, to find on each of them the point Q at which it is tangent to a W line of the person concerned, then to roll these straight lines around said Opportunity lines, while still keeping them all parallel, until they so slant that the center of gravity of the Q's shall coincide with the center of gravity of the P's. This slope, thus determined, signifies the rate of interest which will clear the market.

Let us recapitulate. We have given:

• (1) The Market lines, just as in the first approximation.
• (2) The families of Willingness lines, one family for each individual, just as in the first approximation.
• (3) The Opportunity lines, one only for each individual, that is, a series of points takes the place of the single point P1 in the first approximation.

We also have, correspondingly, three rates:

• (1) The market rate of interest represented by the slope (over and above that of 100 per cent) of each and every straight Market line.
• (2) The degree of impatience, or rate of time preference, one of each person, represented by the slope of the Willingness lines and depending on his income situation, as finally determined after all adjustments have been made.
• (3) The rate of return over cost, or the investment opportunity rate, one for each person, represented by the slope of the Opportunity line and depending on the position chosen on it.

The charts of this chapter interpret the second approximation exactly as the charts of Chapter X interpreted the first approximation, but with two new investment opportunity principles added to the four principles common to both Chapters X and XI and already geometrically interpreted in Chapter X. That is:

The Investment Opportunity Principle A is represented by the Opportunity line.

The Investment Opportunity Principle B is represented by the tangency of the Opportunity line with the Market line, so that the marginal rate of return over cost is equal to the rate of interest.

This last principle, combined with Impatience Principle B, means that each individual so adjusts his position (first along the Opportunity line to P and then along the Market line to Q) that the Market line PQ shall be tangent to the first at P and to the second at Q. This Q will be his income situation finally chosen. To clear the market the Q's must be so chosen that their center of gravity coincides with that of the P's.

### §7. The Nature of the Opportunity Line Discussed

This chapter differs from Chapter X chiefly in the introduction of the concept of investment opportunity which is depicted on the charts as the Opportunity line, or O line. Just what does this line represent in the real world? Is there any distinction between investing in the opportunities offered by man's environment and lending at the market rate of interest? Is not lending, or buying a bond, just as truly investing as digging an oil well, building a factory, or making shoes? Reserving the merely verbal part of the answer, let us first go to the main question as to the possibility of definitely distinguishing the two lines.

Under the assumptions explained in Chapters V, VI, VII, and VIII, there is a clear distinction between an O line and an M line. The O line, unlike the Market lines, is not straight, is not common to all individuals, and is not a family of lines but a single line. It may be defined as the limiting line of a group of points which represent all the optional income situations available to an individual who neither borrows nor lends. Every one has opportunities to shift along his O line at a rate above or below the market rate of interest, even if it be merely in the degree of care he gives his clothes, his house, his fences, or even his food. At a certain stage it is literally true that a "stitch in time saves nine". That is, mending one's clothes yields 900 per cent. But beyond a certain point mending one's clothes, or a roof, painting a house, or tilling the soil will not repay the cost. Each activity has its marginal point and enters into the construction of every person's O line. An individual's Opportunity line is a composite of his separate potential activities—what he might do if he chose.

Of course the O line cannot be drawn without the aid of valuations which involve the market principles and so involve the rate of interest. The farmer who encounters the law of diminishing returns in agriculture buys machinery and labor and sells grain. His O line is thus some-what dependent on the prices of machinery and, since the price of every good is a discounted valuation, it depends on the rate of interest. Only in a primitive or imaginary Robinson Crusoe land can we get a pure case of investing successive amounts of this year's income for the sake of getting a diminishing return in future years without the presence of some buying or selling as an ingredient in the make-up of the O line. It is largely because the element of the rate of interest is almost omnipresent in the valuations entering into the O line that the other and essential ingredient of technical limitations has been overlooked so generally. Even the farmer does some of that omnipresent trading, but besides this trading with other men, he is dealing with nature—the soil, the seasons, the weather, insect enemies, and all the rest. Every investment in his farm will have a variable decreasing return as contrasted with the (to him) constant return to be got in the loan market. Yet every investment in his farm will somewhat imply an interest element and will theoretically change as the interest rate changes. Thus, strictly speaking, his O line is not to be pictured as immovable like a rock but as subject to some slight change with every change in the slope of the M line. Nevertheless this fact evidently does not alter the principles by which the slope of the M line is determined. The M line still rolls around an O line, even if that curve changes a little as it rolls.

The O lines have been exemplified by the law of decreasing returns in agriculture. Such a curve is concave toward the origin and represents a law of decreasing returns in the sense that each succeeding dose of \$100 invested out of this year's income will return less and less next year.

But may there not be a law of increasing return? That is, may not the curve be convex in parts instead of concave?

We may imagine the O line, bounding or enclosing the group of points representing the possible options, to be convex or to have any conceivable shape. It may be reentrant, jagged, discontinuous, straight in parts. It is largely for convenience that we have hitherto pictured it as concave, curved and continuous. But if it were otherwise, almost the same result would follow. The line PQ would still roll around it. The result would evidently be that, wherever the curve was re-entrant (convex toward the origin), the straight Market line, in rolling around the group of points, would jump across this chasm at the slightest provocation due to a change in the interest rate. These re-entrant parts would be as inoperative as if they did not exist, and only the points on which the rolling took place would really count in establishing equilibrium. What is left, after dropping out such re-entrant parts as ineligible, is thus the "envelope" of the group of points representing an individual's opportunities to invest rationally and must therefore be concave toward the origin. We are justified then in assuming the curved concave Opportunity line as typical.

As to the applicability of the term investment to a shift on the M line, this is a matter of choice of words. Undoubtedly it is so applied in ordinary usage. In fact, such investments are more commonly so called than any other. I have not been able to think of a short phrase in common use which will apply exclusively to an investment the return on which varies with each successive amount invested. Perhaps "investment with diminishing returns" or "investment involving exploitation," as distinct from investment by mere sale and purchase, would come nearer to conforming both to usage and to the requirements of the case. Yet the full phrase which I have provisionally adopted, investment opportunity, seems fairly correct in its implications. We seldom speak of buying a bond as an investment opportunity, but investing in new industrial, mining or agricultural enterprises, such as radio production, or in oil wells, or orange groves, is spoken of as a real opportunity because the return is not a standardized market figure but subject to technical conditions as to productivity.

### §8. Investment Opportunity and Impatience

We see then how distinct is the O line from the M line. It is still more distinct from the W line. The Willingness lines represent subjective conditions; the Opportunity line represents objective conditions. The O line of an individual is simply one curve, while the W line is one of many. There is some rate of time preference represented by the angle or slope of a W line on the charts, be it positive, negative, or zero, corresponding to every possible income position of an individual wherever on the chart it may be. But this is not true of the O line. There is only a limited region of options on the map, bounded by a single curve.

As already explained, if the opportunity area enclosed by the O line shrinks to a single point, there is no determinate tangent and we automatically revert to the first approximation in which there is no opportunity to choose from among options.

Thus the investment opportunity influence may, theoretically at least, vanish entirely and lead us back to the first approximation, but the impatience influence can never vanish. Practically however, investment opportunity never quite vanishes. There is always at least some flexibility in everybody's income, but in primitive society, the range of opportunity is relatively small. While the Opportunity line never entirely collapses into a mathematical point, yet, for a person in primitive society it is an almost negligible spot or ring and could exert only a negligible influence on the rate of interest, even if it were to double in diameter or were to change in form. In such a society the only important influences on the rate of interest must come largely from a change in the map, that is, in the distribution of impatience relatively to income.

But when, as in modern society, the range of investment opportunity is great, the slopes of the Opportunity lines exert a great and more controlling influence on the slope of the Market lines.

If the investment opportunity area is large so as to cause the Opportunity line to curve slowly, its relative fixity of slope indicates a relatively stable rate of interest. If the slope is absolutely constant and the same for all individuals, as in the case of the hard-tack island,14 represented by a 45° straight line, or the example of Professor Harry G. Brown's imaginary fruit trees, represented by a straight line steeper than 45°, this fixed shape may, within limits, fix the rate of interest absolutely, forcing it to agree with that fixed slope whatever may be the Willingness lines representing impatience. The limits within which this would be true may readily be charted by the reader.

The most important result here is that the Opportunity line cannot be dispensed with in the theory of the rate of interest. It is something distinct from and in addition to the Impatience lines as well as to the Market lines. If those theorists who still insist on the subjective principle as the only principle of interest will try to picture its determination on this map, they will find it impossible to get any determinate direction of the Market lines without invoking the Opportunity lines. To adapt a simile of Alfred Marshall's, both blades of a pair of scissors are needed to make the scissors work.

### §9. Can Interest Disappear?

One use of this graphic method is to help us form a more complete picture of the problem as to whether the rate of interest may ever be zero or negative.

Just as there is a prevalent idea among the economically illiterate that all interest should be zero—should be abolished—so among the economically literate there is a prevalent idea that the rate of interest could under no imaginable conditions ever be zero or below. Let us then see, under the assumption of the second approximation, what are the conditions, if any, which will permit of a zero or negative rate of interest.

A zero rate of interest means, in our chart, that PQ has an inclination of 45°, that is, a slope of 100 per cent. Our question, therefore, is: must PQ necessarily be steeper than 45°. The slope of PQ depends entirely on the conformation of the O curve and the W curves of each person in the loan market. The less steep these curves are, the less steep will be the Market lines. We have seen that toward the southeast parts of the map the W curves are flatter than 45°, that is, a man with a relatively large income this year and a relatively small one next year would be willing, if he had to, to trade more than \$100 today to get only \$100 next year. Probably this is potentially true of everyone. It is also true that seldom if ever are actual income situations (Q's) located in this southeast region.

We turn now to the O line. For the average man in a progressive country and age, like America today, this will be steeper than in a retrograde country or in a decadent age—a country or an age in which the natural resources are becoming exhausted. But if we go sufficiently to the northwest, it will always be flatter than 45°, that is, if any investment opportunity be exploited far enough it will yield less future return than its immediate cost. This is not only true of land cultivation and extractive industries generally but of all industries. Everywhere, in the end, any law of increasing returns will give place to a law of decreasing returns. And if we keep pursuing these decreasing returns far enough there will always come a point where additional investment would be worse than useless or where the rate of return over cost is less than nothing. Even in such cases of extraordinary returns as the example of the Bell Telephone Company, to have tried to push the development faster than new construction could be built or than the public, even with every device of the advertiser, could absorb, would have been sheer waste.15

Thus the charts depict regions in which the O curve of each individual is less steep than 45° and regions in which his W curves are likewise less steep than 45°. But that fact does not itself prove that the resultant market rate of interest may ever actually be zero. For the flatter parts of the W curves are to the southeast, as shown in Charts 31 and 34, while the flatter parts of the O curve are to the northwest, as shown in Chart 35. If this relative position of the flatter W and O lines were peculiar only to a few individuals, negative interest might well exist. The P of such an individual might be in the northwestern part of the map and the Q in the southeastern, the Market line PQ sloping less steeply than 45° and being tangent at P to the O line and at Q to a W line. He would thus be a borrower, and there would be plenty of lenders.

But if, as is the truth, practically everybody else has the same sort of map, that is, with the parts of the O and W curves which are flatter than 45° located northwest and southeast respectively; and if we should draw everybody else's PQ at the same slope as above, we would have only borrowers and no lenders at interest rates pictured by such slopes. Everyone would be glad to borrow at negative rates of interest. But a rate of interest at which there is no lending would necesarily rise. It could not clear the market. It could remain negative only if a sufficient number of people had maps on which the W lines were flatter than 45° even in the northwest and O lines flatter than 45° even in the southeast. Otherwise the center of gravity of the P's and Q's could not coincide. But there is nothing inconceivable in having such a layout overlapping the flatter-than-45° regions. In other words, if enough persons in the market were sufficiently miserly, or their income opportunities were sufficiently unpromising, or both, then the rate of interest could be zero or below.

To meet these conditions would require either a change in average human nature as to impatience under given income situations, or a change in the future prospects of production and investment opportunity, due, say, to impending exhaustion of natural resources or retrogression generally, instead of progress, in the industrial arts.

Finally, the Opportunity line can never get very much flatter than the 45° inclination, if as flat as that, so long as among our opportunities there are even the present possibilities of preserving food and other goods, that is postponing their uses. We can scarcely expect a time to come when we cannot do at least as well for the future as the shipwrecked sailors with their hard-tack. That is, as long as such an alternative exists as being able to postpone much of our present income by preserving the goods which yield it, the real rate of interest can scarcely get below zero.16

Our conclusion is that negative interest is theoretically possible, though in practice the necessary conditions never occur.

### §10. Does Interest Stimulate Saving?

Just as the map helps visualize the theoretical possibility, yet practical improbability, of negative interest, so also it helps us to see clearly the answer to the much debated question whether saving is stimulated by raising the rate of interest.

If the reader will draw on the map any desired family of Willingness lines, place the individual at any desired income situation (or draw an Opportunity line to indicate all possible positions), and then incline a ruler at 45° and rotate it about that point (or roll it around that line) he will note that the points of tangency of the ruler with the several Willingness lines will themselves constitute a curve. The savings (or lendings) are evidently represented by the horizontal displacement of Q to the left of P. Opportunity and Willingness lines may easily be so constructed that, as the ruler turns clockwise interest rises and the amount saved and lent out of this year's income will first increase and then decrease.

### §11. Relation to Supply and Demand Curves

In §17 of Chapter X it was shown how supply and demand curves can be derived from the M line and the W lines depicted in Chart 34. Supply and demand curves can equally well be derived from the M line, the O line and the W lines shown in Chart 38. A series of positions of PQ, with different slopes, gives us all the material needed, each slope giving a rate of interest and each horizontal spread between P and Q being the demand for loans (if Q is east of P) or supply of loans (if Q is west of P). The only difference is that P is not now fixed as in the first approximation, but shifts as PQ has different slopes.

It will be noted that the Opportunity line which embodies the technical or production elements in the problem has no more relation to the supply than to the demand, although this runs counter to the common notions that productivity rules one side of the market and time preference the other.

It will also be noted that the map gives us vastly more light on the analysis of interest than do the mere supply and demand curves. But even the map fails to give a complete picture because, in particular, it shows only two years. The truth seems to be that no complete visualization of this difficult problem is possible. The only complete symbolization which seems to be possible is in terms of mathematical formulas as in the next two chapters.

[9.][9] An option is represented by any one point in Chart 35 not outside the area bounded by A O1' O1IV. Only those points on the curve O1'O1IV are really eligible. The opportunity to move from one of these points to another implies two points on this line. If these two points are close together, the direction of one from the other is the slope of the tangent to the curve. Thus, the term "option" suggests a point on the curve while the term "opportunity" suggests the direction of the curve.

[10.][10] The point P1 may also be described as that income position, or option, on the Opportunity line which has the greatest present value, as has been shown in Chapter VII and as may also be shown geometrically if desired.

[11.][11] It would not then be true that he would choose the option of highest present value. That point is not R but P1. Thus it is not self-evident, as it might seem, that a man will always choose the income stream of highest present value in the market. He will only do so provided he has, in addition, perfect freedom to move from that situation by means of a loan market. Otherwise he might not be willing to suffer the inconvenience, say, of a very small income this year even if his expected income next year is very large. The only maximum principle preserved in all cases is not that of maximum market value but that of maximum desirability.

[12.][12] With another individual, on the other hand, the most desirable point might be very different. He might borrow only part of what he invests, or even not borrow at all but lend as well as invest. All depends on the particular shape and position of his O and W lines.

[13.][13] Of course, the shift along the O line depends entirely on where we suppose the individual to be situated on that line in the first place. If we wish, we may suppose him to start on the opposite side of P from that hitherto pictured, in which case he does not enter into a contemplated investment but withdraws from one.

[14.][14] See Chapter VIII, §4, for discussion of these examples.

[15.][15] Explanations of the law of diminishing returns sometimes miss this point by ignoring the time element. This element is always essential and especially in interest theory. Enlarging a factory may in the future lower costs and so in time increase the rate of return obtained or attainable, but we are here concerned with a hypothetical series of doses of costs or investments all relating to the same period of time such as the present year and with the return over these costs, say next year.

[16.][16] It is true that unstable money sometimes drives the real rate below zero unintentionally. (See Chapters II and XIX.) But the money rate cannot get below zero so long as the standard is, like gold, capable of being stored substantially without cost. (Chapter II.)