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Front Page Titles (by Subject) PART III.: THE THEORY IN MATHEMATICS - 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.: THE THEORY IN MATHEMATICS - 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

WE found in Chapter V that, if the degree of any given individual's impatience depended solely on his income stream, and if that stream could not be modified through the loan market or otherwise, his impatience could not be modified either. In such a world of hermits, each person would have his own individual rate of time preference, the various individual rates ranging from several thousand per cent per annum down to zero, or below. In such a world, since there would be no loans, there would be no market rate of interest.

But we also perceived that, as soon as our hermits are allowed to swap income streams, one man exchanging some of this year's income for some future income of another, then these myriad rates of time preference or impatience tend to come together toward a common rate, and, on the assumption that no risk attends these transactions, a uniform market rate of interest would actually be reached. In such a perfect loan market the degree of impatience of each person would become equal to that of every other person and to the rate of interest.

§2. The Map of This Year's and Next Year's Income

Expressing the problem with the aid of the graphic method, the determination of the rate of interest may be reduced to a simple problem of geometry, just as the problem of price may be shown by supply and demand curves.

To depict adequately the elements of the interest problem, however, a new kind of chart is required. Our first task is to see the relation of this new kind of chart to those hitherto used in this book. First, then, let us recur to Charts 1, 2, and 3, in Chapter I, which picture a person's income stream over a period of years. This sort of chart consists of a row of vertical bars representing the real income, as measured by the cost of living in successive periods—days, months, or years. To prevent confusion, let us, for our present graphic purpose, shrink these vertical bars into mere vertical lines, without breadth. Then, each year's income may be pictured as if it were all concentrated at a point of time, say in the middle of the month or year concerned. Since the rate of interest is usually expressed in per annum terms, it will simplify the discussion if these lines are drawn, as in Chart 23, disregarding all time units other than years. To make our picture still more concrete, specific figures may be attached to specific years, the person's income for 1930 being set at \$1000, that for 1931 at \$1200, and so on.

We are now ready to pass to the new and radically different method of representing the real income stream. In the chart just described, horizontal distance measures time, while vertical distance measures amount or size. The reader is now asked to shake off these conceptions. Moreover, throughout Chapters X and XI, he must be on his guard against their unconscious return. In the new charts there is no time scale; time is not measured at all; both axes measure amount of income. The horizontal axis represents the first year's income, the vertical, the second year's income. Thus the point P1 in Chart 24, through its latitude and longitude, stands in a sense for both years' incomes combined. It represents what may be called a given individual's income combination, income stream, income position, or income situation for the given pair of years. On Chart 24 may be shown a complete map of all possible income combinations, or income positions, so far as two years, or periods of time, are concerned. To represent the third year, so easily shown in Chart 23 under the old method, we should need in this new method a third dimension. The chart would then cease to be a chart and become a three dimensional model.1

If on the map for two years, we were to draw a straight line from the origin toward the "northeast", midway between the two axes, every point on it would have its longitude and latitude equal, that is, would represent different income situations in which the incomes of two years were equal. A poor man—poor in both years—would be situated near the origin, and a rich man far from the origin. A man who has less income this year than he expects to get next year would be situated above this midway line, his latitude (meaning next year's income) being more than his longitude (meaning this year's). If we move his position sufficiently to the left, so as to reduce his longitude (this year's income), he will be like a man stranded on a polar expedition—with rations run short, though he might be assured by radio of plentiful supplies next year. On the other hand, a man who is more abundantly provided with income this year than he expects to be next year would be situated below the midway line, his longitude being greater than his latitude.

In this way, within the northeast quadrant (the only one shown in the charts) we can, by fixing the point P at all possible positions, represent all possible combinations of this year's and next year's income.

§3. The Market Line

Assuming that the individual's incomes for all other years remain unchanged, we shall now study the effects, for this one man, of changing the income amounts of the two years pictured. These changes are assumed to be caused wholly by trading some of his income of one year for some of another man's income for the other year. Except for such trading, his income situation is supposed to be fixed. He has, let us suppose, a rigid allowance of \$1000 for this year and \$1200 for next year with no opportunity to change these figures except by swapping some of one year's income for some of another's.

Suppose, for instance, that at a rate of interest of 10 per cent the individual borrows \$100 in 1930 in return for \$110 which he is to pay back in 1931. In Chart 23, such changes would be represented by lengthening the 1930 vertical line from \$1000 to \$1100 and by shrinking the 1931 vertical line from \$1200, not to \$1100 but to \$1090. In Chart 25 these changes are represented by shifting the income position from P1, the originally fixed income position, to M1, whose longitude is \$100 more and latitude \$110 less.

If, as shown in Chart 26, the individual borrows a second \$100, promising to repay \$110, his income position shifts again, this time from M1' to M1'', that is, from this-year-1100-and-next-year-1090 to this-year-1200-and-next-year-980. Borrowing a third \$100 would bring him to M1''', (this-year-1300-and-next-year-870). Every additional borrowing of \$100 adds \$100 to this year's income and subtracts \$110 from next year's income. Chart 26 pictures these successive changes as a "staircase" of which each "tread" is \$100 and each "riser" is \$110. The stairs are steep. So long as there exists a rate of interest their descent is necessarily always faster than 45 degrees—that is, future income decreases faster than present income increases; the riser is more than 100 per cent of the tread—more by the rate of interest.2

If the rate of interest were zero, each \$100 borrowed would require only \$100 to be returned next year. The riser would equal the tread; the Market line would make an angle of 45° with the axis, and its slope would equal unity, or, in other words, 100 per cent. Thus in this new method of charting, a given rate of interest is represented by the algebraic difference between the slope of the given Market line, and the 100 per cent slope of the 45° zero interest line.

Evidently the line P1M1 is a straight line. It may be called the Market line, Loan line, or Rate of Interest line, and, if prolonged, will contain all the positions to which this particular individual (who may be called Individual 1) can shift his income position by borrowing or lending.

If, starting at P1, he shifts down this straight staircase (southeast) he is a borrower or a seller of next year's income, because he is adding to this year's income and subtracting from next year's. If, instead, he shifts up the staircase (northwest) he is a lender or buyer of next year's income, subtracting from this year's income and adding to next.

§4. The Willingness Line

So far we have used the new type of chart only to show how the individual can move—change his income situation. The next question is: in which one of the two directions on the Market line will he actually move? The answer depends on his degree of impatience compared with the market rate of interest. We have seen how the market rate of interest is represented graphically by the slope, relatively to 45°, of the Market line in Chart 26. We are now ready to make a similar graphic representation of the individual's rate of time preference or impatience. This is expressed by a series of curved lines showing on what terms, at any income position (such as P1), the individual would be willing to lend or borrow, say \$100. What one is willing to do and what he can do are two quite different things. The Market line of Chart 26 shows what he can do, while the Willingness lines, now to be described, will show what he would be willing to do.

Individual 1's impatience is such that he would be willing, if necessary, as shown in Chart 27, to borrow his first \$100 at 30 per cent, considerably above the market rate, 10 per cent. That is, he would be willing to sacrifice \$130 out of next year's income in order to add \$100 to this year's income. To get a second \$100 this year, he would not be willing to pay quite so much, but only, say, \$120. A third \$100 would be worth to him only \$110; a fourth still less, and so on. These willingness points make a curved staircase, the steps being from P1 to W1', from W1' to W1'', and so on, in which each tread is taken to be \$100, but the risers are decreasing as the borrower steps down southeasterly, being successively 130, 120, 110. The Willingness line, P1W1IV, is not a straight line like P1M1 in Chart 26, but a curved line. The steepness of each step, or ratio of risers to tread, or slope of the curves shows the degree of impatience at that particular step.

The Willingness line through P1 extends, of course, in both directions. It shows not only on what terms Individual 1 would be willing to borrow, but also on what terms he would be willing to lend. At point P1, he is willing to lend the first \$100 at 40 per cent, a second, at 50 per cent, and so on. Everyone theoretically is ready either to borrow or to lend according to the terms. Individual 1 is here represented as barely willing at P1 either to borrow \$100 at 30 per cent, or to lend \$100 at 40 per cent. In short, he would be willing to substitute the combination represented by any given point on his Willingness line for the combination represented by any other point on the same line. All points on that line are, by hypothesis, equally desirable to Individual 1. Each segment of the Willingness line, by the divergence of its slope from 100 per cent, shows the degree of impatience at the particular income position there represented.

Consequently, P1W1 might be called an Impatience line quite as well as a Willingness line3 for Individual 1. There would be another Willingness line, or W line, for any other individual. The W lines thus differ from the Market, or M, lines which are common to everybody in the market.

§5. The Two Lines Compared

The M line of Chart 26 and the W line of Chart 27 are brought together in Chart 28. From P1 we have a pair of lines, one (M1) showing by its divergence from the 45° slope the rate which Individual 1 can get, as a borrower, and the other (W1) showing by its slope the rate at which he is willing to borrow. Evidently if he is willing to pay 30 per cent for a \$100 loan when he can get it at 10 per cent, he will borrow. Evidently also he will continue to borrow as long as his willingness rate, i.e., his rate of time preference or impatience, is greater than the rate of interest, in other words as long as, at each stage, the Willingness line is steeper than the Market line.

All that has been done so far is to describe the income map, and to place on it a point P1 to represent the assumed fixed income situation of Individual 1, and to draw through this point P1 two lines, one a straight line, the Market line, showing the direction in which Individual 1 can move away from P1, and the other a curved line, the Willingness line, showing the direction in which he is willing to move. He is willing to borrow a first \$100 at 30 per cent, or lend a first \$100 at 40 per cent, but can do either at 10 per cent.

§6. The Whole Family of Market Lines

But before we make use of the contrast between the M and W lines to follow the individual as he moves his income position from P1 either northwest as a lender or southeast as a borrower, we must first complete our pictures of the two kinds of lines. Thus far only one line of each of the two kinds has been drawn. These are the only lines applicable to the given income position P1. But for every other income position in which we may place Individual 1, the Market line would encounter a different Willingness line.

In the same way, we could picture a series of pairs of Market and Willingness lines for any other individual, such an Individual 2, depicted in Chart 29. In this case, Individual 2 is represented at one position to be willing to lend at 2 per cent but to be able to do so at 10 per cent, the market rate; we have then the Market line P2M2 and the Willingness line P2W2. In the same way, we could draw P3M3 and P3M3, for a third individual and so on.

Under our assumption of a perfect market, all the Market lines P1M1, P2M2, P3M3, etc., are parallel to each other, their common divergence from a 100 per cent slope representing the one universal market rate of interest, here supposed to be 10 per cent.4 Since the Market lines are thus parallel, they are really impersonal or independent of particular individuals, and we may consequently picture the whole map (Chart 30), as covered with straight and parallel Market lines, like the conventional picture of a rainstorm, each line being slightly steeper than 45°, and its divergence therefrom indicating a rate of interest.

§7. Many Families of Willingness Lines

The Willingness lines, on the other hand, are personal, or individual. The particular slope in each case is, of course, dependent on the particular income position P1. A given Individual 1 is concerned with only one Willingness line at a time, out of his whole "family" of lines—the one line which passes through his actual income position at P1. By comparing his Willingness line at that particular point with the Market line, we have a graphic picture of the motives which decide whether he will borrow or lend.

There always exists for him, potentially, other Willingness lines, passing not through his actual income position P1, but through any other income position at which he could be imagined to be. These curves represent the various rates at which Individual 1 would be willing to borrow or lend if his income position were varied. Thus the number of Willingness lines for any one Individual is infinite, and every other individual will have his own family of lines.

We conceive, then, of a map for Individual 1 alone, covered with a family of such Willingness lines infinite in number, arranged so as to vary gradually from each to the next, like the lines of elevation on a geographic contour map of a mountain.5

While Individual 1 finds all points on any one Willingness line of equal desirability or "wantability," he would rather have his income position on lines more to the "northeast," or farther from the origin. Each Willingness line might be labelled with a number representing specifically the total desirability or "wantability" pertaining to each and all of the income positions on it. It is the locus or assemblage of these points (combinations of income for the two years) equally desirable in the estimation of Individual 1. A greater aggregate income in the two years may be offset in respect to the resulting total desirability by a less convenient distribution between the two years and vice versa.6

But we are not yet interested in such differences of level or total desirability between the Willingness lines. We are here interested only in the directions of the Willingness lines at different points, representing the different degrees of impatience pertaining to different income situations. These directions, or divergences from the 45° line, picture to us how the individual would be willing to borrow or lend under all possible income circumstances.

§8. A Typical Family of Willingness Lines

While we cannot, of course, tell exactly how any human being would act if far from the income situation in which he is actually placed, yet we know what in a way would be characteristic of him. Chart 31 is believed to represent roughly a family of Willingness lines typical of most human beings. It is best analyzed by following the dotted straight line SS' running northeast, midway between the axes and therefore comprising points which represent, in each case, equal incomes in both years.

Four characteristic properties may be noted:

(1) At S, taken to represent a point near the minimum of subsistence—the income situation, say, of a polar explorer who is marooned and hopeless—the Willingness line is nearly vertical. In this income position of extreme need a person would scarcely be willing to lend at all, for to give up one iota of this year's income in exchange for any amount promised for next year would mean too great a privation in the present. On the contrary, in order to get an added dollar today to help keep body and soul together this year, he would be willing to give not only 10 per cent or 25 per cent but perhaps as much as 100 per cent, or 1000 per cent, or every dollar of next year's income even though next year is no more promising than this year.

(2) At S', taken to represent an exceedingly large income—say \$1,000,000 for each of the two years—the Willingness line is nearly at right angles to the SS' line; that is, nearly at 45° to the two axes, for with such a large income the two years would seem almost on even terms. Presumably one with this income would not be willing, in order to get \$1000 more income this year, to give up much over \$1000 out of next year's income.

(3) As one passes between the two income situations above mentioned, from S to S', his rate of time preference will gradually decrease—from nearly infinity at S down nearly to zero at S', that is, the larger the income, other things remaining the same, the smaller the degree of impatience.

(4) Any one Willingness line grows steeper as it proceeds upward and leftward, changing from a nearly horizontal direction at its lower right end to a nearly vertical direction at the opposite end.

§9. Time Preference May be Negative

If these specifications are correct, some, at least, of this family of Willingness curves, especially those far distant from the origin, and low down—that is, representing small future and large present income—will have an inclination less steep than 45° to the horizontal axis and a slope less than 100 per cent. At such income positions the rate of time preference would be negative. It is sometimes said that it is a fundamental attribute of human nature to prefer a dinner or a dollar this year to a dinner or a dollar next year, but this statement is evidently too narrow. Unconsciously it confines our view to regions of the income map where present income is relatively small or future income relatively large. For a starving man it is notably true, that is, the Willingness lines that lie in the left part of the map are far steeper than 45°. Of a man expecting large future income it is also true; that is, the Willingness lines toward the top of the map are also very steep.

But if we turn our attention in the opposite direction—to the right, or downward, or both—we find regions on the map in which, if the foregoing description is correct, the curves flatten out and incline less than 45°; the man's income situation is such that he might even be willing to lend for nothing, or even less than nothing, simply because he would, in such a case, be so surfeited with this year's income and so short, prospectively, of next year's income that he would be thankful to get rid of some of this year's superfluity, for the sake of adding even a trifle to next year's meager real income. His situation would be like a Robinson Crusoe on a barren island, well supplied, but with foods that could serve him only this year. Such situations are rare in practice, but they are certainly imaginable and sometimes even occur. In such situations a man would be willing to save for the future without any incentive in the form of interest. But this takes us beyond our present point; for here we are concerned only with the Willingness lines, not with the Market lines.

§10. The Personal and Impersonal Influences on Impatience

By the aid of this map we can see, anew, and more clearly, that a man's actual degree of impatience depends on two circumstances:

(1) It depends on his "personal equation," the whole contour of his whole family of Willingness curves, representing what he would be willing to do under all sorts of income situations. The Willingness lines of a spendthrift are steeper and those of a miser less steep than the typical, or normal, man's family of curves.

(2) It depends on his particular income situation on the map which is represented by the letter P. A poor man is more impatient than a rich man of the same personal characteristics. A man with great expectations for the future but with little available for the present is more impatient than the man oppositely situated with respect to the future.

Both the family of Willingness lines and the position on the map are, of course, changing every minute. Only at one particular time does the map, with its set of curves for Individual 1, and a particular location P1, picture his individual circumstances. What we are doing here is to take a flash-light, as it were, of his income situation and his Willingness lines, and to analyze his behavior at the instant.

§11. Deciding Whether to Borrow or to Lend

Individual character and income together with the market rate of interest determine what the individual will do in any given situation. Chart 32 pictures graphically one step in the process by which an individual adjusts his degree of impatience to the market rate of interest, by borrowing or lending.

To summarize and repeat, with the interest rate at 10 per cent, Individual 1 may, or may not, be willing to take a \$100 loan, depending upon whether or not, while his position is at P1, his impatience is or is not greater than this 10 per cent. If, as portrayed by Chart 32, his impatience is greater—say, 30 per cent—and if he can obtain the loan at 10 per cent, he will be glad to do so. Let W1' be the position to which he is willing to shift in order to get \$100 added to his present income. This position would require a sacrifice of \$130 out of next year's income, which, however, he need not make for he can get his accommodation for \$110 of next year's income and find himself at M1'.

Individual 2, on the other hand, whether because of his temperament, or because of a different time distribution of his income, may behave quite differently. His position to start with, let us say, is P2 in Chart 33. If his impatience, under these circumstances, is only 2 percent, when the market rate is 10 per cent, he will not borrow, but will lend. He will continue to lend until his degree of impatience is as great as the market rate of interest, that is as long as his Willingness line, at each stage, is less steep than the Market line.

Thus, the angle at P between the M line and W line of Individual 1, or of any other individual, shows whether that individual is potentially a borrower or a lender. If the individual's W line is steeper than the M line, he will borrow; if it is less steep, he will lend.

Therefore, if this second individual lends \$100, while the first individual borrows \$100, both at 10 per cent, both will reap advantage. One will shift down a 10 per cent Market line from P1 to M'1, and the other will shift up the 10 per cent Market line from P2 to M'2.

If we had only these two persons trading this year's income for next year's income between themselves, the rate of interest agreed upon for a loan of \$100 this year would not, of course, necessarily be 10 per cent, but would be determined by the respective bargaining power of Individual 1 and Individual 2. The rate might fall anywhere between the 2 per cent, the lowest rate at which Individual 2 is willing to lend \$100, and the 30 per cent, the highest rate, at which Individual 1 is willing to borrow \$100. But we are less interested in such a special trade, or haggle than in the general market.

§12. Interest Fixed for an Individual

It is a fact long recognized by price theorists,7 that the theoretical determination of any price in a special trade or haggle between two persons, each of whom is conscious of his influence on that one price, is more complicated than in a full-fledged competitive market in which each individual is so small a factor as to be unconscious of his influence on the market price. We here assume such a general market, in which a single buyer, or lender, is so small a factor that he is not actuated by any consciousness of influencing the market rate of interest. Each person finds the rate fixed for him. We assume, for the purpose of the present illustration, that the rate so fixed for him by the market is 10 per cent.

§13. How an Individual Adjusts his Income Position to the Market

Under these assumptions it is clear that Individual 1 will borrow \$100 at precisely 10 per cent, and that Individual 2 will lend \$100 at precisely 10 per cent, the market rate. Their individuality will find conscious play only in determining how far they will make use of the market. Let us now see how far beyond \$100 each will borrow or lend at this fixed 10 per cent rate.

To do this we need merely compare Willingness lines with the Market line. Will Individual 1, who has shifted from P1 to M'1 along the 10 per cent Market line with a \$100 loan, again shift from M'1 for an additional \$100 loan, along this same 10 per cent Market line? He will do so only if he is still willing to pay more than the market rate.

Whether this is true can be tested by precisely the same process as before, namely by drawing from this new income position on the Market line a new Willingness line and ascertaining whether or not it is steeper than the Market line. And so, step by step, \$100 by \$100, as he shifts along the Market line, we can always test whether he will shift still further. Any individual has but one Willingness line, cutting through any one income position, but when he sets out from any P, he is going to shift along the Market line and, at each shift, he encounters a new Willingness line.

On Chart 34 are represented a number of W lines which are assumed to depict Individual 1's varying degree of impatience at different income positions. A study of the chart will show how Individual 1, in income position P1, will borrow, because his degree of impatience exceeds 10 per cent, the market rate of interest. His borrowing will continue, of course, until his degree of impatience is reduced to 10 per cent, at which point he no longer secures a net gain from borrowing. This point is located at Q1 where the M line is tangent to Individual 1's W line through that point. There at last he reaches a position where the next \$100 shift on the Market line is no longer less steep than his Willingness line but exactly as steep.

The same principles apply to Individual 2. The only differences are: first, that he shifts upward from P to his Q instead of downward—that is, he adds to next year's income at the expense of this year's income; and, secondly, that he acts in reference to a different family of Willingness lines entirely his own. Such a picture implies the utmost sensitiveness or fluidity of inducements and responses. There would be a continual readjustment of loans and borrowings, back and forth; practically every person would be either a borrower or a lender; the extent of his borrowings or loans would be very finely graduated, and constantly changing.

We have not yet pictured geometrically the whole problem of the rate of interest; but we have pictured the solution of the problem of how any one individual will adjust, under the ideal conditions assumed, his lending or borrowing to the market rate of interest. This simplified solution consists, we have seen, in finding Q at the point where the M line at a given rate of interest is tangent to one of the given family of W lines.

Having solved this individual problem, we now proceed to the market problem.

§14. Market Equilibrium

It may seem that little progress has yet been made toward the ultimate end of determining the rate of interest because of our initial assumption that the rate of interest is already fixed for the individual by the market formed by others. How did they fix it? Have we begged the whole question?

Only a few steps are now required to finish the whole market picture. It is true that we assumed a fixed 10 per cent rate in order to see how an individual would shift his income position to harmonize his individual degree of impatience with that, to him, fixed rate. Nevertheless, each individual, even if unconsciously, helps to make the market rate by the very act of shifting his income situation from a P position to a Q position.

This statement will be clear if we ask ourselves what would happen were we to suppose the "fixed" market rate to have been fixed too high, or too low. If we imagine the market rate to be very high, say 25 per cent, then, the bulk of individuals would try to lend and few would want to borrow. The aggregate of loans thus offered would exceed the demand and the interest rate would fall. Conversely, if the rate were too low, demand would exceed supply and the rate would rise. Since the total sums actually lent must equal, in the aggregate, those borrowed, the horizontal displacements of all the Q's in one direction must equal that of all the other Q's in the other direction. Some Q's, those of borrowers, are to the right of the corresponding P's. Others are at the left. As a group, they are neither. The average Q has the same longitude as the average P. The same is true as to latitude. In short, the geometric "center of gravity" of all the Q's must coincide with that of all the P's, in order that the loan market may be cleared.

In other words, while the market rate, as represented by the divergence of slope of the Market line, always seems fixed to the individual adjusting his income situation to that rate, nevertheless that rate is not really completely fixed independently of his own borrowing or lending. What the market does is to keep the Market line for different individuals parallel. There cannot be two rates in the same market at the same time—at least not in the perfect market here assumed.

But these parallel lines are always swinging a little back and forth to "clear the market." Each person's Market line may turn slightly about his P as a pivot. All Market lines turning together, that is keeping parallel, tend to reach the right inclination—that which clears the market and brings the center of gravity of the Q's into coincidence with that of the P's.

Thus, the economic problem of determining the rate of interest becomes the geometric problem of experimentally oscillating all the M lines until their common inclination brings the center of gravity of the contacts (Q's) into coincidence with the center of gravity of the P's.

We now have the complete geometric representation of the whole problem of the rate of interest under the assumptions of the first approximation—complete except that, to put the picture on a two-dimensional chart, we have had to add the restriction that "other things are equal" as to all years beyond the first and second.8

Thus the economic problem of determining the rate of interest is translated into the geometric problem of drawing a series of parallel straight lines through given points, P's, at such a slope as will make the center of gravity of the Q's coincide with the center of gravity of the P's. There is a one-to-one correspondence between the economic and the geometric problem, so that if the "map" is correct we have reduced the whole problem to one of geometry.

Incidentally, it may be remarked here that there is evidently nothing inherent in the geometrical construction as presented which necessitates a positive rather than a negative market rate of interest. A negative rate can theoretically emerge whenever the P's and their center of gravity are of sufficiently low latitude or great longitude, or both, so that the common slope of the Market lines tangent at the Q's with the Willingness lines, will be less than 100 per cent. That this is theoretically possible is evident from inspection, provided the Willingness lines do, as assumed, have inclinations at certain income positions less than 45°. The reasons why the rate of interest is seldom or never negative have chiefly to do with the conditions introduced under the second approximation and will be more apparent in the next chapter.

§15. The Four Principles as Charted

In this geometric picture we see that the four principles formerly stated in words (in Chapter V) are now interpreted geometrically on the "map" as follows:

(1) Impatience principle A (that each man's impatience or rate of time preference depends on his income stream) is represented by a family of Willingness lines for each individual.

(2) Impatience principle B (that each rate of time preference is assimilated to the market rate of interest) is represented by the tangency at each individual's point Q, thus making the slope of his W line at that point equal to the slope of the M line.

(3) Market principle A (that the market will be cleared) is represented by the fact that the aggregate horizontal shift from P's to Q's to the right (by all borrowers combined) must equal the aggregate horizontal shift from P's to Q's to the left (by all lenders combined); and also that the two aggregate vertical shifts representing next year's repayments of loans must likewise be equal, so that the centers of gravity of the P's and the Q's coincide.

(4) Market principle B (that all loans are repaid and at one rate of interest) is represented by the fact that the Market lines are straight and parallel.

§16. The Geometric Method

These charts do, for the ideas they illustrate, what supply and demand curves do for the ideas illustrated by them.

Like all graphic methods, the one here applied is intended to segregate basic tendencies from the rough-and-tumble of real life, and set these tendencies going as they cannot go in real life. It condenses a year's income into an infinitesimal time; it confines our variations to two years only; it disregards the element of risk; it pictures next year's income as a certainty; it disregards the lack of security that limits the ease with which an individual can slide his series of transactions along the M line; it assumes that the market is perfect.

Again, the Willingness lines should not be drawn as continuous curves. They are actually rough and jagged, so that, for this reason alone, the nicety of adjustment which would obtain under the assumption of continuity is lost. We know also that most individuals require a considerable stimulus even to start sliding along a Market line. Besides the height of the rate of interest, there is the trouble of negotiating a loan, establishing a line of credit, and practical considerations without end. One result is that in order to reverse one's direction on the Market line, a bigger rise or fall of the interest rate would be needed than the charts as here used would suggest. It takes a push to dislodge the individual from P in either direction. The same sort of considerations cause his position Q to be determined without the nicety of precision suggested by the continuous curves.

But all these and other practical considerations do not destroy the fact that each of our four determining conditions represents a reality—a real tendency even when in actual practice balked or neutralized.

The relationship of the rate of time preference to income is analogous to that of marginal utility or cost to consumption or production. In order to show how the marginal desirability of sugar in the case of Individual 1 is related to his consumption of sugar, we employ a curve, which, under certain assumptions, becomes the familiar demand curve for sugar. Such a curve has come into universal use.

Why has no similar curve been used to indicate the corresponding relationship between time preference (a marginal desirability derivative) and income? There are many reasons, but perhaps chief is the difficulty of finding a suitable graphic method for variables so diverse and related to each other in so complicated a manner. The map of the Willingness, or Impatience, lines partly solves this problem. So far as two periods of time are concerned, it "puts on the map" the whole problem of interest.

§17. Relation to Supply and Demand

Some students familiar with demand and supply curves as applied to the loan market may feel that they can get their bearings better if the exact relation is shown between these and the "map" here used. Therefore, it seems worth while here to bridge the gap between these two sorts of representations just as, at the outset, the gap was bridged between the map and pictures of the income stream earlier in this book. We may readily and completely derive the curves of demand and supply from the map and the constructions which have been drawn on it.

The individual demand curve of Individual 1 is found as follows: Rotate the straight line PQ about P as a pivot, that is, draw a series of PQ's from P at varying slopes. On each such PQ find Q, the point of tangency with a W line. The horizontal displacement of Q to the right of P is the loan which Individual 1 is willing to take at the rate of interest represented by the slope of PQ.

Thus we have both coordinates (namely, interest rate and amount of loans demanded at that rate) given by the map. Having these coordinates, we merely need to plot them on a separate sheet in the usual way.

In the same way we may construct every other individual's demand curve. The aggregate curve of all individuals (by adding all demands at a given interest rate) gives the total demand curve in the market.

The supply curves are constructed similarly; the only difference being that for supply we use the horizontal displacement of Q to the left of P, instead of to the right.

Of course, at any given slope near the slope of the market rate, some individuals will have a right, and others a left, displacement, and at the market rate itself the two displacements are equal in the aggregate. This is true where the supply and demand curves intersect.

Evidently the map gives us the same relationships as the ordinary supply and demand curves and much more. The supply and demand curves, for instance, give us only the displacements, or differences in income position, as between P and Q, while the map gives the whole income position of both points. And while we can reconstruct, as above, the demand and supply curve from our map, we cannot reconstruct our map from the supply and demand curves.

It may also be noted here that the supply curve is derived from the map in spite of the absence, in this first approximation, of any investment opportunity or productivity element. The significance of this fact will be more apparent in Chapter XI where under the second approximation this element is introduced.

§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.

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.

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.

§1. Case of Two Years and Three Individuals

IN this chapter, the four principles constituting the first approximation, previously expressed verbally17 and geometrically,18 will be expressed algebraically. Inasmuch as the equations, the solutions of which express the solution of the interest problem, are necessarily numerous and complicated, we shall first consider a simplified special case where there are to be considered only two years in which there is income and three individuals who borrow or lend. We shall then pass to the general case where there are any given number of years and any number of individuals.

In the simplified case, we assume, therefore, that each individual's degree of impatience for this year's over next year's income can be expressed as dependent solely on the amount of this year's and next year's income, the incomes of all other future years being disregarded. We also assume, for simplicity, that the income of each of the two years is concentrated at the middle of the year, making the two points just a year apart, and that borrowing and lending are so restricted as to affect only this year's and next year's income.

Let f1 represent the marginal rate of time preference19 for this year's over next year's income for Individual 1 (this is the slope at Q of a Willingness line relatively to the 45° line). Let his original endowment of income for the two years20 be respectively

c1' and c1''.

(These are the longitude and latitude of P in Chapter X.) This original income stream, consisting merely of the two jets, so to speak, c1', c1'', is modified by borrowing this year and repaying next year. The sum borrowed this year is called x1' (this is the horizontal shift from P to Q). To represent the final income of this year this sum x1' is therefore to be added to the present income c1'. Next year the debt is to be paid, and consequently the income finally arrived at for that year is c1'' reduced by the sum thus paid. For the sake of uniformity, however, we shall regard both additions to and subtractions from pre-existing incomes algebraically as additions. Thus, the addition x1' say \$100, to the first year's income is a positive quantity, and the addition, which we shall designate by x1'', to the second year's income, is a negative quantity—\$105. The first year's income is, therefore, changed from

c1' to c1' + x1',

and the second year's from

c1'' to c1'' + x1''.

(Just as c1' and c1'' are the longitude and latitude of P in Chapter X, so c1' + x1' and c1'' + x1'' are those of Q.) By the use of this notation we avoid negative signs and so the necessity of distinguishing between the expressions for loans and repayments or for lenders and borrowers.

§2. Impatience Principle A (Three Equations)

The first condition determining interest, namely, Impatience Principle A, that the rate of preference for each individual depends upon his income stream, is represented for Individual 1 by the following equation:

f1 = F1 (c1' + x1', c1'' + x1'')

which expresses f1 as dependent on, or, in mathematical language, as a function of the two income items of the two respective years, F1 being not a symbol of a quantity but an abbreviation for "function." In case the individual lends instead of borrows, the equation represents the resulting relation between his marginal rate of preference and his income stream as modified by lending; the only difference is that, in this case, the particular numerical value of x' is negative and that of x' positive. The equation is the algebraic expression for the dependence of the slope of a Willingness line on the income position of Individual 1.

In like manner, for Individual 2, we have the equation

f2 = F2 (c2' + x2', c2'' + x2''),

and, for the third individual,

f3 = F3 (c3' + x3', c3'' + x3'').

These three equations therefore express Impatience Principle A.

§3. Impatience Principle B (Three Equations)

Impatience Principle B requires that the marginal rates of time preference of the three different individuals for present over future income shall each be equal to the rate of interest, and is expressed by the following three equations:21

f1 = i
f2 = i
f3 = i

where i denotes the rate of interest. These three equations are best written as the continuous equation:

i = f1 = f2 = f3.

(These equations express the fact that at the Q's the slope of the Willingness line is the same as of the Market lines.)

§4. Market Principle A (Two Equations)

Market Principle A, which requires that the market be cleared, or that loans and borrowings be equal, is formulated by the following two equations:

x1' + x2' + x3' = 0,
x1'' + x2'' + x3'' = 0.

That is, the total of this year's borrowings is zero (lendings being regarded as negative borrowings), and the total of next year's repayments is likewise zero (payments from a person being regarded as negative payments to him).

§5. Market Principle B (Three Equations)

Market Principle B requires that the present value of this year's loans and the present value of next year's returns, for each individual, be equal. This condition is fulfilled in the following equations, each corresponding to one individual:

§6. Counting Equations and Unknowns

We now proceed to compare the number of the foregoing equations with the number of unknowns, for one of the most important advantages of an algebraic statement of any economic problem is the facility with which, by such a count, we may check up on whether the problem is solved and determinate. There are evidently 3 equations in the first set, 3 in the second, 2 in the third, and 3 in the fourth, making in all 11 equations. The unknown quantities are the marginal rates of time preference, the amounts borrowed, lent and returned, and the rate of interest as follows:

f1, f2, f3, or 3 unknowns,
x1', x2', x3', or 3 unknowns,
x1'', x2'', x3'', or 3 unknowns,

and finally,

i, or 1 unknown,

making in all 10 unknowns.

We have, then, one more equation than necessary. But examination of the equations will show that they are not all independent, since any one equation in the third and fourth sets may be determined from the others of those sets. Thus, if we add together all the equations of the fourth set, we get the first equation of the third set. (namely, x1' + x2' + x3' = 0). The addition gives

In this equation we may substitute zero for the numerator of the fraction (as is evident by consulting the second equation of the third set). Making this substitution, the above equation becomes

x1' + x2' + x3' = 0,

which was to have been proved. Since we have here derived one of the five equations of the last two sets from the other four, the equations are not all independent. Any one of these five may be omitted as it could be obtained from the others. We have left then only ten equations. Since no one of these ten equations can be derived from the other nine, the ten are independent and are just sufficient to determine the ten unknown quantities, namely, the f's, x''s, x'''s and i.

§7. Case of m Years and n Individuals

We may now proceed to the case in which more than three individuals (let us say n individuals) and more than two years (let us say m years) are involved. We shall assume, as before, that the x's, representing loans or borrowings, are to be considered of positive value when they represent additions to income, and of negative value when they represent deductions.

§8. Impatience Principle A (n(m - 1) Equations)

The equations expressing Impatience Principle A will now be in several groups, of which the first is:

f1' = F1' (c1' + x1', c1'' + x1'',..., c1(m) + x1(m)),
f2' = F2' (c2' + x2', c2'' + x2'',..., c2(m) + x2(m)),
...................................................
...................................................
fn' = Fn' (cn' + xn', cn'' + xn'',..., cn(m) + xn(m)).

These n equations express the rates of time preference of different individuals (f1' of Individual 1,f2' of Individual 2,... fn' of Individual n) for the first year's income compared with the next.

To express their preference for the second year's income compared with the next there will be another group of equations, namely:

f1'' = F1'' (c1'' + x1'', c1''' + x1''',..., c1(m) + x1(m)),
f2'' = F2'' (c2'' + x2'', c2''' + x2''',..., c2(m) + x2(m)),
...................................................
...................................................
fn'' = Fn'' (cn'' + xn'', cn''' + xn''',..., cn(m) + xn(m)).

For the third year there will be still another group, formed by inserting the superscript ''' for '', and so on up to the year (m - 1), for the year (m - 1) is the last one which has any exchange relations with the next, since that next is the last year, or year m. There will therefore be (m - 1) groups each of n equations, like the above group, making in all n (m - 1) equations in the entire set.

§9. Impatience Principle B (n(m - 1) Equations)

To express algebraically Impatience Principle B22 we are compelled to recognize for each year a separate rate of interest. The rate of interest connecting the first year with the second will be called i', that connecting the second year with the third, i'', and so on to i(m-1). Under this principle, the rates of time preference for all the different individuals in the community for each year will be reduced to a level equal to the rate of interest. This condition, algebraically expressed, is contained in several continuous equations, of which the first is:

i' = f1'= f2' =... = fn'.

This expresses the fact that the rate of time preference of the first year's income compared with next is the same for all the individuals, and is equal to the rate of interest between the first year and the next. A similar continuous equation may be written with reference to the time preferences and the rate of interest as between the second year's income and the next, namely:

i'' = f1'' = f2'' =... = fn'.

Since the element of risk is supposed to be absent, it does not matter whether we consider these second-year rates of interest and time preference as the ones which are expected, or those which will actually obtain, for, under our assumed conditions of no risk, there is no discrepancy between expectations and realizations.

A similar set of continuous equations applies to time-exchange between each succeeding year and the next, up to that connecting year (m - 1) with year m. There will therefore be m - 1 continuous equations of the above type. Since each such continuous equation is evidently made up of n constituent equations, there are in all n (m - 1) equations in the second set of equations.

§10. Market Principle A (m Equations)

The next set of equations, expressing Market Principle A, represents the clearing of the market. These equations are as follows:

There are here m equations.

§11. Market Principle B (n Equations)

The equations for Market Principle B express the equivalence of loans and repayments, or, more generally, the fact that for each individual the present value of the total additions (amount borrowed, or lent) to his income stream, algebraically considered, will equal zero. Thus, for Individual 1, the addition the first or present year is x1', the present value of which is also x1', the addition the second year is x1'', the present value of which is

The addition the third year is x1''', the present value of which is

This is obtained by two successive steps, namely, discounting x1''' one year by dividing it by 1 + i'', thereby obtaining its value not in the present or first year but in the second year, and then discounting this value so obtained by dividing it in turn by 1 + i1''. The next item xIV is converted into present value, through three such successive steps, and so on. Adding together all the present values we obtain as resulting equations for Individuals 1,2,... n:

Similar equations will hold for each of the other individuals, namely:

making in all n equations.

§12. Counting Equations and Unknowns

We therefore have as the total number of equations the following:

• n (m - 1) equations expressing Impatience Principle A,
• n (m - 1) equations expressing Impatience Principle B,
• m equations expressing Market Principle A, and
• n equations expressing Market Principle B.

The sum of these gives 2 mn + m - n equations in all.

We next proceed to count the unknown quantities (rates of time preference, loans, and rates of interest): First as to the f's:

For Individual 1 there are f1', f1'',..., f1(m-1), the number of which is m - 1, and, as there is an equal number for each of the n individuals, there will be in all n (m - 1) unknown f's.

As to the x's, there will be one for each of the m years for each of the n individuals, or mn.

As to the i's, there will be one for each year up to the last year, m - 1. In short there will be

n (m - 1) unknown f's,
mn unknown x's,
m - 1 unknown i's,

or 2 mn + m - n - 1 unknown quantities in all. Comparing this number with the number of equations, we see that there is one more equation than the number of unknown quantities.

This is accounted for, as in the simplified case, by the fact that not all the equations are independent. This may be shown if we add together all the equations of the fourth set, and substitute in the numerators of the fractions thus obtained their value as obtained from the third set, namely, zero. We shall then evidently obtain the first equation of the third set. Consequently we may omit any one of the equations in the last two sets. There will then remain just as many equations as unknown quantities, each independent (that is non-derivable from the rest), and our solution is determinate.

In the preceding analysis, we have throughout assumed a rate of interest between two points of time a year apart. A more minute analysis would involve a greater subdivision of the income stream, and the employment of a rate of interest between each two successive time elements. This will evidently occasion no complication except to increase enormously the number of equations and unknowns.

§13. Different Rates of Interest for Different Years

The system of equations thus involved when n persons instead of three and m years instead of two are considered introduces very few features of the problem not already contained in the simpler set of equations for two years and three persons. The new feature of chief importance is that, instead of only one rate of interest to be determined, there are now a large number of rates. It is usually assumed, in theories of interest, that the problem is to determine "the" rate of interest, as though one rate would hold true for all time. But in the preceding equations we have m - 1 separate rates of interest, viz., i', i'',..., i(m-1).

Under the hypothesis of a rigid allotment of future income among different time intervals, which is the hypothesis of the first approximation, there is nothing to prevent great differences in the rate of interest from year to year, even when all factors in the case are foreknown and there is every opportunity for arbitrage. By a suitable distribution of the values of c1, c2,..., cm, there may be produced any differences desired in the magnitudes of i', i'',..., i(m-1). Thus if the total enjoyable income of society should be foreknown to be 10 billion dollars in the ensuing year, 1 billion in the following year, and 20 billion in the third year, and if there were no way of avoiding these enormous disparities in the social income, it is very evident that the income of the middle year would have a very high valuation compared with either of its neighbors, and therefore that the rate of interest connecting that middle year with the first year would be very low, whereas that connecting it with the third year would be very high. It might be that a member of such a community would be willing to exchange \$100 of the plentiful 10 billions for the first year, for only \$101 out of the scarce 1 billion of next year, but would be glad to give, out of the third year's still more plentiful 20 billions, \$150 for the sake of \$100 in the middle and lean year.

In actual markets we find some influence of such differences between future years (as looked at today) in the differences between short term and long term interest rates.

The reason why, in actual fact, no abrupt or large variations in the rate of interest, such as from 1 per cent to 50 per cent, is ordinarily encountered is that the supposed sudden and abrupt changes in the income stream seldom occur. The causes which prevent their occurrence are:

(1) The fact that history is constantly repeating itself. For instance, there is regularity in the population, so that, at any point of time, the outlook toward the next year is similar to what it was at any other point of time. The individual may grow old, but the population does not. As individuals are hurried across the stage of life, their places are constantly taken by others, so that, whatever the tendency in the individual life for the rates of preference to go up or down with age, it will not be cumulative in society. Relatively speaking, society stands still.

Again, the processes of nature recur in almost ceaseless regularity. Crops repeat themselves in a yearly cycle. Even when there are large fluctuations in crops, the variations are seldom world-wide, and a shortage in the Mississippi Valley may be compensated for by an unusually abundant crop in Russia or Asia. The resultant regularity of events is thus sufficient to maintain a fair uniformity in the income stream for society as a whole.

(2) The tendency toward uniformity is also favored in real life by the fact that the income stream is not fixed, but may be modified in other ways than by borrowing and lending as in accordance with investment opportunities. The significance of these modifications is algebraically considered in the next chapter.

§1. Introduction

THE object of this chapter is to express in algebraic formulas the six principles comprising the second approximation.23 In Chapter XII we assumed that all income streams were unalterable, except as they could be modified by borrowing and lending, or buying and selling rights to specified portions of these income streams. In the second approximation now to be put into formulas, we substitute for this hypothesis of fixity of the income streams the hypothesis of a range of choice between different income streams.

The income stream of Individual 1 no longer consists of known and fixed elements, c1', c1'', c1''', etc., in successive periods but of unknown and variable elements which we shall designate by y1', y1'', y1''', etc. (y1' and y1'' are the coördinates of the Opportunity line).

This elastic income stream may now be modified in two ways: by the variations in these y's, as well as by the method which we found applicable for rigid income streams, namely, the method of exchange, borrowing and lending, or buying and selling. The alterations effected by the latter means we shall designate as before by the algebraic addition of x1', x1'', x1''',..., x1m, for successive years. These are to be applied to the original income items (the y's), deductions being included by assigning negative numerical values. The income stream, as finally determined, will therefore be expressed by the successive items,

y1' + x1', y1'' + x1'', y1''' + x1''',..., y1(m) + x1(m).

§2. Impatience Principle A. (n(m—1) Equations)

Impatience Principle A states that the individual rates of preference are functions of the income streams, and gives the following equations:

f1' = F1' (y1' + x1', y1'' + x1'',..., y1(m) + x1(m)),
f2' = F2' (y2' + x2', y2'' + x2'',..., y2(m) + x2(m)),
.............
.............
fn' = Fn' (yn' + xn', yn'' + xn'',..., yn(m) + xn(m)).

But these equations express the various individuals' rates of impatience only for the first year's income compared with the next. (They are the slopes of the Willingness lines.) To express their impatience for the second year's income compared with the third, there will be another set of equations, namely:

f1'' = F1'' (y1'' + x1'', y1''' + x1''',..., y1(m) + x1(m)),
f2'' = F2'' (y2'' + x2'', y2''' + x2''',..., y2(m) + x2 (m)),
.............
.............
fn'' = Fn'' (yn'' + xn'', yn''' + xn''',..., yn(m) + xn(m)).

For the third year, as compared with its successor, there would be another similar set, with ''' in place of '', and so on to the (m - 1) year as compared with the last, or m year. Since each of these (m - 1) groups of equations contains n separate equations, there are in all n (m - 1) equations in the entire set expressing Impatience Principle A.

§3. Impatience Principle B (n(m - 1) Equations)

Impatience Principle B requires that the rates of time preference and of interest shall be equal. This relationship is represented by the same equations as given in Chapter XII, namely:

i' = f1' = f2' =... = fn',
i'' = f1'' = f2'' =... = fn'',
.............
.............
i(m -1) = f1(m-1) = f2(m-1) =... = fn(m-1).

Here are n(m - 1) equations expressing Impatience Principle B.

§4. Market Principle A. (m Equations)

The sets of equations which express Market Principle A, the clearing of the market, are also the same as before, namely:

x1' + x2' +... + xn' = 0,
x1'' + x2'' +... + xn'' = 0,
.............
.............
x1(m) + x2(m) +... + xn(m) = 0.

Here are m equations expressing Market Principle A.

§5. Market Principle B. (n Equations)

Market Principle B, the equivalence of loans and discounted repayments, is also represented algebraically as before, namely:

These are n equations expressing Market Principle B.

§6. Investment Opportunity Principle A. (n Equations)

The equations in the four sets just reviewed differ from the equations of Chapter XII only in the first set, which contain y's in place of c's. The c's were supposed to be given or known, but the y's are new unknown quantities. Consequently, the number of unknowns is greater than the number in the first approximation, whereas the number of equations thus far expressed is the same.

The additional equations needed are supplied by the two Investment Opportunity Principles, namely, Investment Opportunity Principle A, that the range of choice is a specified list of optional income streams, and Investment Opportunity Principle B, that the choice among the optional income streams shall fall upon that one which possesses the maximum present value.

The range of choice, i.e., the complete list of optional income streams, will include many which are ineligible—those which would not be selected whatever might be the rate of interest—whether that rate be zero or one million per cent. Excluding all ineligibles the remaining options constitute the effective range of choice which in Chapter XI is pictured as the Opportunity line.

If this list of options be assumed for convenience of analysis to consist of an infinite number of options varying from one to another, not by sudden jumps, but continuously, the complete list can be expressed by those possible values of y1', y1'',... y1(m) which will satisfy an empirical equation. There will be one such equation for each individual, thus:

f1 (y1', y1''... y1(m)) = 0,
f2 (y2', y2'',... y2(m)) = 0,
.............
.............
fn (yn', yn'',... yn(m)) = 0.

Here are n equations, expressing the Investment Opportunity Principle A. The form of each of these equations depends on the particular technical conditions to which the capital of the Individual concerned is subjected. It corresponds to the O line of Chapter XI except that only two years were there represented whereas here all m years are represented. Any one equation sets the limitations to which the variation of the income stream of a particular individual must conform. Each set of values of y1', y1'',... y1(m) which will satisfy this equation represents an optional income stream.

§7. Opportunity Principle B. (n(m—1) Equations)

Out of this infinite number of options the individual has opportunity to choose any one rather than any other. That particular one will be chosen for which the present value is greater than for any other, in other words, is the maximum.

If the options differ by continuous gradations this principle that the maximum present market value is chosen is the same24 as the principle that r1, the marginal rate of return over cost, shall be equal to i, the market rate of interest.

This is true for each year-to-year relation, so that we have, for Individual 1, the following continuous equations:

i' = r1' = r2' =... = rn',
i'' = r1'' = r2'' =... = rn''
...
...
i(m-1) = r1(m-1) = r2(m-1) =... = rn(m-1).

Here are n(m - 1) equations expressing Investment Opportunity Principle B.

§8. Counting the Equations and Unknowns

Collecting our various counts of the numbers of equations, we have:

 For Impatience Principle A,n(m - 1) equations " " " B,n(m - 1) " " Market " A,m " " " " B,n "
 For Investment Opportunity Principle A,n        equations " " " " B,n(m - 1) "

The sum total of these is 3n(m - 1) + 2n + m, or 3mn + m - n.

To compare this number with the number of unknowns, we note that all the unknowns in the first approximation are repeated;

 the number of f's being n(m - 1) " " " x's " mn " " " i's " m - 1

making a total of 2mn + m - n - 1 carried forward from the first approximation.

In addition, the new unknowns, the y's and the r's, are introduced. There is one y for each individual for each year, the total array of y's being

y1', y1'',...,y1(m),
y2', y2'',...,y2(m),
.............
.............
yn', yn'',..., yn(m).

The number of these y's is evidently mn.

There is one r for each individual for each pair of successive years, i.e., first-and-second, second-and-third, etc., and next-to-last-and-last years, the total array of r's being

r1', r1'',..., r1(m-1),
r2', r2'',..., r2(m-1),
.............
.............
rn', rn'',..., rn(m-1).

The number of these r's is evidently n(m - 1).

In all, then, the number of new unknowns, additional to the number of old unknowns carried forward from the first approximation, is mn + n(m - 1), or 2mn - n.

Hence we have:

 number of old unknowns, 2mn +m -n - 1, + number of new unknowns, 2mn -n, = total number of unknowns, 4mn +m - 2n - 1,

as compared with 3mn + m - n equations.

§9. Reconciling the Numbers of Equations and Unknowns

The reconciliation of these two discordant results is effected by two considerations. One reduces the number of equations. Just as under the first approximation, we have one less independent equation in the two sets expressing the Market Principles than the apparent number, thus making the final net number of equations

3mn + m - n - 1.

The other consideration is quite different. It subtracts from the number of unknowns. This can be done because each r, of which there are n(m - 1), is a derivative from the y's. By definition r is the excess above unity of the ratio between a small increment in the y of next year to the corresponding decrement in the y of this year. The same applies to any pair of successive years. This derivative is, more explicitly expressed, a differential quotient.25

The reader not familiar with the notation of the differential calculus will get a clearer picture of the inherent derivability of the r's from the y's by recurring to the geometric method in Chapter XI. There y' and y'' are shown as the coördinates ("latitude" and "longitude") of the Opportunity line, while r is shown as the tangential slope of that line. It is evident that, given the Opportunity line, its tangential slope at any point is derived from it. It is not a new variable but is included in the variation of y' and y'' as the position on the curve changes.

If, now, we subtract n(m - 1), the number of the r's, from 4mn + m - 2n - 1, we have, as the final net number of unknowns,

3mn + m - n - 1

which is the same as the total net number of independent equations.26 Thus the problem is fully determinate under the assumptions made.

§10. Zero or Negative Rates of Interest

We have already seen (Chapter XI, §9), that zero or negative rates of interest are theoretically possible. In terms of formulas all that is needed to make the rate of interest zero is that the forms of the F and f functions shall be such as to produce this result. This implies that these functions shall have solution values equal to zero.

Of course it would be possible that interest, impatience, and return over cost for one particular year might be zero or negative without this being true for other years. If they were zero for all the years, we should have the interesting result that the value of a finite perpetual annuity (greater than zero per year) would be infinity. No one could buy a piece of land for instance, expected to yield a net income forever, for less than an infinite sum. A perpetual government bond from which an income forever was assured would have an infinite value. Since this is quite impracticable, we thereby reduce to an absurdity the idea that it is possible to have at one and the same time:

• 1. A zero rate of interest for each year forever; and
• 2. a perpetual annuity greater than zero per year.

But the absurdity is lessened or disappears altogether if either:

• 1. The zero rate of interest is confined to one year; or
• 2. no perpetual annuity greater than zero per year is possible.

Unusual conditions may easily reduce the rate of interest for one year to zero. As to an unproductive or barren world, like the hard-tack island, only a finite totality of income would be possible; a perpetual annuity even of one crumb of hard-tack a year would be impossible.

While this and the previous chapter are largely restatements in terms of formulas of Chapters X and XI in terms of diagrams, which, in turn were largely restatements of Chapters V and VI in terms of words, nevertheless, these formula chapters have a value of their own, just as did the geometric chapters.

In particular, the formula method has value in showing definitely the equality between the number of equations and the number of unknowns, without which no problem of determining variables is ever completely solved.

It is for this reason that these restatements are included in this book. In fact, if I were writing primarily for mathematically trained readers, I would have reversed the order, giving the first place to the formulas, following these with the charts for visualization purposes, and ending with verbal discussion. Each method contributes its distinctive help toward a complete understanding of what is, at best, a difficult problem to encompass by any method at all. I have, therefore, included in these formula chapters, as in the geometric ones, several points not well adapted to the more purely verbal presentations of Chapters V and VI.

Two corollaries follow. One is that any attempt to solve the problem of the rate of interest exclusively as one of productivity or exclusively as one of psychology is necessarily futile. The fact that there are still two schools, the productivity school and the psychological school, constantly crossing swords on this subject is a scandal in economic science and a reflection on the inadequate methods employed by these would-be destroyers of each other. Each sees half of the truth and wrongly infers that it disproves the existence of the other half. The illusion of their apparent incompatibility is solely due to the failure to formulate the problem literally and to count the formulas thus formulated.

The other corollary is that such a formulation reveals the necessity of positing a theoretically separate rate of interest for each separate period of time, or to put the same thing in more practical terms, to recognize the divergence between the rate for short terms and long terms. This divergence is not merely due to an imperfect market and therefore theoretically subject to annihilation by arbitrage transactions, as Böhm-Bawerk, for instance, seemed to think. They are definitely and normally distinct and due to the endless variety in the conformations of income streams. No amount of mere price arbitrage27 could erase these differences.

Thus, there should always be, theoretically, a separate market rate of interest for each successive year. Since, in practice, no loan contracts are made in advance so that there are no market quotations for a rate of interest connecting, for example, one year in the future with two years in the future, we never encounter such separate year to year rates. We do, however, have such rates implicitly in long term loans. The rate of interest on a long term loan is virtually an average28 of the separate rates for the separate years constituting that long term. The proposition affirming the existence of separate rates for separate years amounts to this: that normally there should be a difference between the rates for short term and long term loans, sometimes one being the larger and sometimes the other, according to the whole income situation.

The contention often met with that the mathematical formulation of economic problems gives a picture of theoretical exactitude untrue to actual life is absolutely correct. But, to my mind, this is not an objection but a very definite advantage, for it brings out the principles in such sharp relief that it enables us to put our finger definitely on the points where the picture is untrue to life.

The object of any theory is not to reproduce concrete facts but to show the chief underlying principles as tendencies. There is, for instance, the very real tendency for all marginal rates of time preference and all marginal rates of return over cost to equal the market rates of interest. Yet this is only a tendency, an ideal never attained.29

§1. Introduction

THE second approximation fails to conform to conditions of actual life chiefly with respect to risk. While it is possible to calculate mathematically risks of a certain type like those in games of chance or in property and life insurance where the chances are capable of accurate measurement, most economic risks are not so readily measured.30

To attempt to formulate mathematically in any useful, complete manner the laws determining the rate of interest under the sway of chance would be like attempting to express completely the laws which determine the path of a projectile when affected by random gusts of wind. Such formulas would need to be either too general or too empirical to be of much value.

In science, the most useful formulas are those which apply to the simplest cases. For instance, in the study of projectiles, the formula of most fundamental importance is that which applies to the path of a projectile in a vacuum. Next comes the formula which applies to the path of a projectile in still air. Even the mathematician declines to go beyond this and to take into account the effect of wind currents, still less to write the equations for the path of a boomerang or a feather. If he should do so, he would still fall short of actual conditions by assuming the wind to be constant in direction and velocity.

Scientific determination can never be perfectly exact. At best, science can only determine what would happen under assumed conditions. It can never state exactly what does or will happen under actual conditions.31

We have thus far stated verbally, geometrically, and algebraically the laws determining interest under the simpler conditions first, when it was assumed that the income streams of individuals were both certain and fixed in amount, but variable in time shape, and, secondly, when it was assumed that the income streams were certain, but variable in amount as well as in time shape. We have also considered verbally the interest problem under conditions of risk found in the real world.

§2. The Six Sets of Formulas Incomplete

All that I shall attempt here is to point out the shortcomings of the six sets of formulas in the second approximation. Impatience Principle A in the second approximation is expressed by formulas of the type:

f = F (y' + x', y'' + x'',..., y(m) + x(m))

indicating that a person's impatience is a function of his income stream as specifically scheduled, indefinitely in the future.

Of course, in the uncertainties of actual life no such specific scheduling is possible. The equation is true but incomplete, as f is properly not only a function of one expected income program but of many possible such programs each with its own series of probabilities, and those probabilities are too vague even to be specifically expressed or even pictured by the person concerned. That is, the average person is merely aware that he is willing, say, to pay five per cent for a \$1000 loan because he thinks his future prospects will justify it. He vaguely expects that his present \$10,000 income will probably rise to \$20,000 within a few years, possibly to \$30,000—and possibly not rise at all. He could think of innumerable possibilities and these would imply many other variables than those above cited. Much might depend on the income of others besides himself and on the future size of his family, the state of their health, and other conditions without end. His own future income is the important matter, but that itself is dependent on all sorts of variables on which he will reckon summarily in a rule of thumb fashion.

We could, formally, rewrite for the third approximation the above equation so as to read:

f = F ()

merely refusing to attempt any enumeration of the innumerable variables inside the parentheses—among them being, perhaps, all the variables included in all the equations in the second approximation, including rates of interest as well as numberless other variables such as probabilities not there included. In so far as the latter, or new, variables enter, each of them requires a new equation in order to make the problem determinate. Such new equations would be merely empirical. Among the equations which would be needed, would be those expressing the y's and x's in terms of real income—that is, as the sum of the enjoyable services, each multiplied by its price. This would lead us into the theory of prices and the general economic equilibrium.

Impatience Principle B was expressed in the second approximation by formulas of the type:

i = f.

But now we must face not only one i but a series of i's, according as the market is the call loan market, the 60 to 90 day commercial paper market, the gilt-edge bond market, the farm mortgage market, and innumerable others, for each of which there is its own separate f and i.

These many magnitudes, including the i's, require still other empirical equations impossible to formulate satisfactorily, albeit we know in a general way that the rate on gilt-edge bonds is lower than on risky bonds, the rate on first mortgages lower than that on second mortgages, and that the long term and short term markets do influence each other. But these relations are too indefinite to be put into any equations of real usefulness, theoretical or practical.

Investment Opportunity Principle A was expressed by formulas of the type

f (y', y'',..., y(m)) = 0

This becomes

f () = 0

where the blank parenthesis stands for a multitude of unknowns (and unknowables) which could be discussed ad infinitum and each of which, in so far as it was not already included among the variables entering into our system of equations, would require a new empirical equation of some sort in order that the problem shall be determinate. Moreover, the f equation representing a man's ensemble of income opportunities is a composite of separate opportunities, the full and detailed expression of which would take us again into the theory of prices and general economic equilibrium.

Investment Opportunity Principle B, expressed in the second approximation by equations of the type

i = r,

would have to be replaced by as profuse a variety as replaced the i = f above.

A full statement of the margins of investment opportunity would include the margins of numberless individual enterprises and adjustments in the use of every item of capital. It would again lead us into the theory of prices and general economic equilibrium. Walras and Pareto have formulated systems of equations which do this and in which the theory of interest is merely a part of a larger whole.

Market Principle A, expressed in the second approximation by equations of the type

x1 + x2 +.... + xm = 0,

will remain true only if or in so far as performance of contracts corresponds to promises and expectations. In so far as, because of defaults, the equations fail of being precisely true, no useful mathematical relation expressing that failure seems possible.

The same is true of Market Principle B, expressed under the second approximation by formulas of the type

to say nothing of the fact that this type of equation will take many different forms in view of the variety of i's and in view of the probability factors.

The only explicit practical inclusion of such probability factors, mathematically, is to be found in the formulas of life insurance actuaries. But these are of little more than suggestive value in our present effort to express the determination of the rate of interest.

§3. Conclusions

We must, therefore, give up as a bad job any attempt to formulate completely the influences which really determine the rate of interest. We can say that the system of equations which has been employed would fully determine the rate of interest were it not for disturbing factors; that it does do so in combination with those disturbing factors; and that this amounts to saying that it expresses the fundamental tendencies underlying those disturbing factors.

In short, the theory of interest in this book merely covers the simple rational part of the causes actually in operation. The other or disturbing causes are those incapable of being so simply and rationally formulated. Some of them may be empirically studied and will be treated in Chapter XIX. They pertain to statistics rather than to pure economics. Rational and empirical laws in economics are thus analogous to rational and empirical laws of physics or astronomy. Just as we may consider the actual behavior of the tides as a composite result of the rational Newtonian law of attraction of the moon and the empirical disturbances of continents, islands, inlets, and so forth, so we may consider the actual behavior of the rates of interest in New York City as a composite of the rational laws of our second approximation and the empirical disturbances of Federal Reserve policy together with numberless other institutional, historical, legal, and practical factors. All of these are worthy of careful study but are not within the scope of the main problem of this book.

In some cases, as in the theory of the moon's motions, the perturbations may be worked out with a high approximation to reality by combining rationally a number of elementary influences. Such resolution of empirical problems represents the highest ideal of applied science. But until that stage is reached there remains a wide gap between rational and empirical science, and the two have to be pursued by somewhat different methods. That is the case with economic science in most of its problems today.32

In respect to our present problem, while there is a great field for research, the only perturbing influence of transcendent importance is that of an unstable monetary standard, and, as was seen in Chapter II, even that would make nothing more than a nominal difference in the results if it were not for the "money illusion."

But with respect to this disturbance, theory and practice are miles apart. The disturbances of unstable money often reverse the normal operation of those supposedly fundamental forces which determine the rate of interest and are the chief subject of our study in this book.

[1.][1] The reader who wishes, after finishing this chapter, to pursue the geometric analogy into more dimensions than the two here considered may do so by reading the Appendix to this Chapter.

[2.][2] The steps could be drawn just as well on the under side of the line as shown by dotted lines on the chart. If the steps were to consist, not of successive \$100 loans, but of successive \$1 loans the steps to P1, M1', M1'', M1''', etc., would be a hundred times as numerous and correspondingly smaller.

[3.][3] The latter name was chosen chiefly because the initial "W," for willingness, is more convenient to use in the chart than the letter "I," especially as "I" is the initial also of interest and income.

[4.][4] So large a rate as 10 per cent is used in the charts because a line with a divergence from the 100 per cent slope of less than 10 per cent cannot be clearly seen.

[5.][5] Those familiar with a contour map will find the analogy a good one, since each Willingness line represents a level of desirability different from the others, the level or height being here conceived as measured in the third dimension, that is, at right angles to the page of the map.

[6.][6] It would, of course, be possible to present the Willingness lines in terms of total desirability or wantability without supposing any hypothetical borrowing or lending; this was done in the Rate of Interest (Appendix to Chapter VII). The Willingness lines were there called iso-desirability lines. They might also be called lines of indifference.

[7.][7] See Auspitz und Lieben. Untersuchungen über die Theorie des Preises. Leipzig, Dunker und Humblot, 1889, p. 405; Marshall, Alfred. Principles of Economics. London, Macmillan and Co., 1907, p. 332; also my Mathematical Investigations in the Theory of Value and Prices, p. 25.

[8.][8] A more complete expression, in mathematical terms, applying to any number of years, is given in Chapter XII.

[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.)

[17.][17] In Chapter V, §9, and in Chapter IX, §9, first of the three lines, in each case.

[18.][18] In Chapter X, §15.

[19.][19] The relation of f, representing the rate of time preference of any individual to the marginal desirability, or "wantability," of this year's, and of next year's income, is given in the Appendix to this Chapter, §1.

[20.][20] The term "year" is used for simplicity, but "month" or "day" would be equally admissible and would do less violence to facts.

[21.][21] Strictly speaking these equalities are true only when the individual is so small a factor in the market as to have no appreciable influence on the market rate of interest. The equality of f and i implies that the total desirability, or wantability, of the individual is a maximum. (See Appendix to this chapter (Chapter XII), §2.

[22.][22] This principle here expressed in "marginal" terms has been alternatively stated in words in Chapter V and in geometric terms in Chapter X as the principle of maximum desirability. The equivalence of the principle whether stated with reference to a maximum or to a marginal equality is obvious, but the mathematical reader may care to see it put in formulas as a "maximum" proposition as in the Appendix to this chapter (Chapter XII), §2.

[23.][23] These have already been expressed in words in Chapters VI and IX and geometrically in Chapter XI.

[24.][24] For the mathematical statement on this equivalence see Appendix to this chapter (Chapter XIII), &sect3.

[25.][25] See Appendix to this chapter (Chapter XIII), §1.

[26.][26] Instead of thus banishing the r's, an alternative reconciliation is to retain them but to add for each an equation of definition of 1 + r. Thus 1 + r1' (corresponding to the slope of the Opportunity Curve) is a derivative from the y''s and y'''s of the two successive years called this year and the next (in other words, a partial derivative) making 1 + r1' dependent upon (in other words, a function of) the y's That is,

r1' = f1(y1', y1'',..., y1(m)),

which function is empirical and derivable from the opportunity function f already given.

Analogously we may express the equations of definition for r2', r3' .. rn' and likewise for the corresponding r'''s,r''''s, etc., up to r(m)'s making n(m-1) equations of definition. In this way, retaining the r's we have 4 mn + m - 2n - 1 independent equations and the same number of unknowns.

The complication mentioned in Chapter VII, §10, that the income stream itself depends upon the rate of interest, does not affect the determinateness of the problem. It leaves the number of equations and unknowns unchanged, but merely introduces the rate of interest into the set of equations expressing the Opportunity principles. These equations now become

f1 (y1', y1'',...,y1(m); i'', i'',...,i(m)) = 0

etc., and their derivatives, the y functions, are likewise altered in form but not in number.

The mathematical reader will have perceived that I have studiously avoided the notation of the Calculus, as, unfortunately, few economic students are, as yet, familiar with that notation, and as it has seemed possible here to express the same results fairly well without its use. See, however, the Appendix to this chapter (Chapter XIII), §1-5.

[27.][27] It is true, of course, that the attempt to make an individual's income stream more even by trading one time-portion of it for another tends to even up the various rates of interest pertaining to various time periods. But this is not price arbitrage and not properly to be called arbitrage at all, being more analogous to the partial geographic equalization of freight charges in the price of wheat by international trade than to the equalization by arbitrage of wheat prices in the same market at the same time.

[28.][28] The nature of this average has been expressed in Appreciation and Interest, pp. 26 to 29, and The Rate of Interest, pp. 369 to 373. It is impossible to give a concrete example of an average of a rate of interest for a long term loan as an average of the year to year rates, because as already noted, the year to year rates have only a hypothetical existence. The nearest approach to the concrete existence of separate year to year rates is to be found in the Allied debt settlements, by which the United States agreed that Italy, France, Belgium, and other countries should repay the United States through 62 years, with specific rates of interest changing from time to time. These are equivalent to a uniform rate for the whole period according to theory. For instance, the proposed French debt settlement provided for annual payments extending over 62 years, beginning with 1926 at interest rates varying from 0 per cent for the first 5 years, 1 per cent for the next 10 years, 2 per cent for another 10 years, 2½ per cent for 8 years, 3 per cent for 7 years, and 3½ per cent for the last 22 years. The problem is to find an average rate of interest for the whole period which when applied in discounting the various payments provided for in the debt settlement, will give a present worth, as of 1925, equal to the principal of the debt fixed in that year, namely, \$4,025,000,000. Clearly no form of arithmetic mean, weighted or unweighted, will give the desired rate. A rough computation indicates that the rate probably falls between 1½ per cent and 1¾ per cent. Discounting the annual payments by compound discount gives a total worth in 1925 at 1½ per cent of \$4,197,990,000; at 1¾ per cent of \$3,893,610,000. These results show that 1½ per cent is too low, since the present worth obtained by discounting at this rate is greater than the principal sum which was fixed at \$4,025,000,000. The rate 1¾ per cent is too high because the discounted present worth is less than the principal.

Discounting the annual payments at 1.6 per cent we obtain \$4,072,630,000. We can now locate three points on a curve showing the interest rates corresponding to different present values. By projecting a parabolic curve through the three determining points, we find the ordinate of the point on the curve which has the abscissa of \$4,025,000,000 is 1.64. Hence the average rate of interest for the whole period, within a very narrow margin of error, is 1.64 per cent.

[29.][29] For brief comparison of Chapters XII and XIII with other mathematical formulations see Appendix to this chapter (Chapter XIII).

[30.][30] See The Nature of Capital and Income, Appendix to Chapter XVI.

[31.][31] See my article, Economics as a Science, in Proceedings of the American Association for the Advancement of Science, Vol. LVI, 1907.

[32.][32] See Mitchell, Wesley C., Quantitative Analysis in Economic Theory. American Economic Review, Vol. XV, No. 1, March 1925, pp. 1-12.