The Online Library of Liberty

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• Errata
• Preface
• Suggestions to Readers
• Part I.: Introduction
• Part I, Chapter I: Income and Capital 2
• Part I, Chapter II: Money Interest and Real Interest
• Part I, Chapter III: Some Common Pitfalls
• Part II.: The Theory In Words
• Part Ii, Chapter IV: Time Preference (human Impatience)
• Part Ii, Chapter V: First Approximation to the Theory of Interest Assuming Each Person's Income Stream Foreknown and Unchangeable Except By Loans
• Part Ii, Chapter VI: Second Approximation to the Theory of Interest Assuming Income Modifiable (1) By Loans and (2) By Other Means
• Part Ii, Chapter VII: The Investment Opportunity Principles
• Part Ii, Chapter VIII: Discussion of the Second Approximation
• Part Ii, Chapter IX: Third Approximation to the Theory of Interest Assuming Income Uncertain
• Part III.: The Theory In Mathematics
• Part Iii, Chapter X: First Approximation In Geometric Terms
• Part Iii, Chapter XI: Second Approximation In Geometric Terms
• Part Iii, Chapter XII: First Approximation In Terms of Formulas
• Part Iii, Chapter XIII: Second Approximation In Terms of Formulas
• Part Iii, Chapter XIV: The Third Approximation Unadapted to Mathematical Formulation
• Part IV.: Further Discussion
• Part Iv, Chapter XV: The Place of Interest In Economics
• Part Iv, Chapter XVI: Relation of Discovery and Invention to Interest Rates
• Part Iv, Chapter XVII: Personal and Business Loans
• Part Iv, Chapter XVIII: Some Illustrative Facts
• Part Iv, Chapter XIX: The Relation of Interest to Money and Prices
• Part Iv, Chapter XX: Objections Considered 73
• Part Iv, Chapter XXI: Summary
• Appendix to Chapter I
• Appendix to Chapter X
• Appendix to Chapter Xii
• Appendix to Chapter Xiii
• Appendix to Chapter Xix
• Appendix to Chapter Xx
• Bibliography

PART IV, CHAPTER XXI

SUMMARY

§1. Interest and Purchasing Power of Money

We have seen that, theoretically, the rate of interest should be subject to both a nominal and a real variation, the nominal variation being that connected with changes in the standard of value, and the real variation being that connected with the other and deeper economic causes.

As to the nominal variation in the rate of interest, we found that, theoretically, an appreciation of 1 per cent of the standard of value in which the rate of interest is expressed, compared with some other standard, will reduce the rate of interest in the former standard, compared with the latter, by about 1 per cent, and that, contrariwise, a depreciation of 1 per cent will raise the rate by that amount. Such a change in the rate of interest would merely be a change in the number expressing it, and not fundamentally a real change. Yet, in actual practice, for the very lack of this perfect theoretical adjustment, the appreciation or depreciation of the monetary standard does produce a real effect on the rate of interest, and that a most vicious one. This effect, in times of great changes in the purchasing power of money, is by far the greatest of all effects on the real rate of interest. This effect is due to the fact that the money rate of interest, while it does change somewhat according to the theory as described in Chapters II and XIX, does not usually change enough to fully compensate for the appreciation or depreciation. The inadequacy in the adjustment of the rate of interest results in an unforeseen loss to the debtor, and an unforeseen gain to the creditor, or vice versa as the case may be. When the price level falls, the interest rate nominally falls slightly, but really rises greatly and when the price level rises, the rate of interest nominally rises slightly, but really falls greatly. It is consequently of the utmost importance, in interpreting the rate of interest statistically, to ascertain in each case in which direction the monetary standard is moving and to remember that the direction in which the interest rate apparently moves is generally precisely the opposite of that in which it really moves.

It should also be noted that in so far as there exists any adjustment of the money rate of interest to the changes in the purchasing power of money, it is for the most part (1) lagged and (2) indirect. The lag, distributed, has been shown to extend over several years. The indirectness of the effect of changed purchasing power of money comes largely through the intermediate steps which affect business profits and volume of trade, which in turn affect the demand for loans and the rate of interest. There is very little direct and conscious adjustment through foresight. Where such foresight is conspicuous, as in the final period of German inflation, there is less lag in the effects.

§2. The Six Principles

But the more fundamental theory of interest presupposes a stable purchasing power of money so that the real and nominal rates coincide. In that case the rate is theoretically determined by six sets of equations or conditions: the two Opportunity Principles; the two Impatience Principles; and the two Market Principles. The last pair may be said to cover prima facie supply and demand.

(A) The market must be cleared—and cleared with respect to every interval of time. (B) The debts must be paid.

The other two pairs represent the two sets of forces, one objective and the other subjective, behind supply and demand. The subjective pair expresses the influence of human impatience or time preference.

(A) The rate of time preference depends on the character of the various individuals concerned and on each individual's prospective income, its size, time-shape and risk.(B) Each individual's rate of time preference tends, at the margin of choice, to harmonize with the market rate of interest. Human impatience to spend and enjoy income is crystallized into the market rate.

The objective pair expresses the influence of investment opportunities.

(A) Each individual is encompassed about by opportunities to change the character of his prospective income stream. (B) At the margin of choice, any additions to an individual's future income at the cost of more immediate income constitutes a return over that cost, the rate of which return over said cost is also crystallized into the market rate of interest.

So the rate of interest is the mouthpiece at once of impatience to spend income without delay and of opportunity to increase income by delay.

Thus both from the subjective and the objective field appear prototypes, one of each for every individual, of the market rate of interest.

That rate, i, is equal to every individual's degree of impatience or rate of time preference, f, and also to his investment opportunity rate or rate of return over cost, r.

Yet these equations are not enough to make the problem determinate without those of the other four sets of determining conditions (clearing the market, repaying debts and empirical dependence of impatience and investment opportunity).

Much less is it possible to determine the rate of interest from the subjective side alone, through time preference, or from the objective side alone, through investment opportunity, or "productivity", or "technique of production".

The full explanation requires both (as well as the market principles) in order that there may be as many independent equations as unknown variables in the problem. Moreover there is not merely one rate of interest; there are many, one for each interval of time. And even so the explanation is full only under the theoretical conditions presupposed. If we pass beyond the presuppositions in order to approximate closer to the actual world, we find that, to be determinate, the problem requires more and more equations of a more and more empirical nature. This is especially true as (1) we introduce risk with its innumerable and omnipresent ramifications, involving in particular a multiplicity of rates of interest even for the same period of time; and as (2) we extend our view to admit variations in all other prices besides the rates of interest, involving thereby the whole economic equilibrium, not only of the loan market but of all markets, each interacting on every other; and as (3) we extend our view from one theoretical market to the actual markets of the whole world, involving thereby all the relations of international trade; and as (4) we take account of any other factors which may not be included in the foregoing specifications so as to take account, in particular, of all "institutional" influences, laws, politics, banking practices, government finance and so on to the end.

In the economic universe, as in astronomy, every star reacts on every other. From a practical point of view we cannot ignore the many perturbations. But from the theoretical point of view we gain clearness, simplicity and beauty, if we allow ourselves to assume certain other things equal, and confine our laws to a little part of the whole, such as the solar system.

From such a point of view, the second approximation is the most instructive, rather than the first which rules out the important element of investment opportunity, or than the third which becomes too complicated and vague for any complete theoretical treatment.

§3. The Nature of Investment Opportunity

In the second approximation—which, as we have just noted, contains all that is most typical in the theory of the rate of interest—the distinctive factor is the rate of return over cost or the investment opportunity rate. This is also the most difficult factor to picture, isolate, and disentangle from the rate of interest which it helps determine. Therefore, it is a matter of great importance pedagogically to make that distinction clear. The investment opportunity rate is distinct from the market or loan rate of interest because an investment opportunity is distinct from a loan. Investment opportunity, as here used, does not include a mere loan at the market rate of interest nor any other purchase-and-sale transaction made merely on the basis of the market rate. The definition of investment opportunity is specially framed to exclude mere loans. It is any opportunity of an individual to modify his prospective income other than by merely lending or borrowing (or the equivalent, buying or selling) at the market rate of interest.

Under this definition and the assumptions employed in the theory there can never be any doubt as to whether a given proposed transaction is an investment opportunity or a market loan or purchase. In the case of a market loan or purchase the individual cannot vary the rate of interest by any act of his, such as varying the size of his transactions. Under our assumptions of a perfect market his influence on the market rate is not only unconscious but infinitesimal and therefore entirely negligible in our analysis in which his motivity is of the essence. In the case of an investment opportunity, on the other hand, he can vary the rate of return by varying the size of his operations.

This contrast between the theoretical constancy of the one and the variability of the other, in relation to individual action, is due to the fact that in the public market the individual is a negligible element, while an investment opportunity is more private and personal to him or his group. The former is typified by the purchase, say, of a Liberty bond, or other standard securities. The latter is typified by building a factory, improving a sales organization, deepening the shaft of a mine—cases where the marginal rate of return is under the control of the individual since he sets the margin.

Of course it is true that, in almost every such operation, there are elements of purchase and sale in which the market rate of interest is an implicit ingredient, but as long as the operation is not exclusively a mere market interest affair and contains other ingredients, the rate is subject to variation with the extent of the operation and so is to be called a rate of return over cost. We are here interested in those other ingredients which produce the variability and thus differentiate such a rate from the market rate of interest. They are the non-commercial or non-trading ingredients; they concern production and technique rather than trade. They deal not with the market place, but with nature, environment, and the refractory conditions which surround and hamper us in our efforts to secure income. They exist even when no market exists, when a Robinson Crusoe, a hermit, or an isolated ranchman battles with soil and the elements for his daily bread.

The rate of return over cost, under the law of diminishing returns, is thus far more elementary and primeval than the rate of interest, and however incrusted that rate of return may become with other elements which grow out of modern market conditions it is still the basic objective condition underlying our problem.

Thus the rate of return over cost is distinguished from the rate of interest (1) by varying with the extent of the individual's investment; (2) by being consciously recognized, as thus variable115 and controllable, by the individual; (3) by being, therefore, a personal and individual matter and not altogether a public market matter; (4) by being directly related to producing as contrasted with trading.

§4. Investment Opportunity for Society as a Whole

In modern society hermits and self-supporting ranches are so rare that we cannot find any important cases where investment opportunities exist in pure primitive form and apart from the alloy of trading. In fact, the most typical investment opportunities are not only full of such alloys but are tied up with market financing operations. Almost every big investment opportunity is married to a productive loan.

The best picture on a big scale of investment opportunity, divested so far as may be of all ancillary market features, is to be found by considering society as a whole instead of the individual.

Society as a whole cannot borrow or lend as an individual can. This world can, for instance, add nothing to this year's income by a loan from elsewhere and subtract this amount with interest from future income. Yet it can and does vary and control the total income stream according to investment opportunity.

This picture, in the large, of society arranging, modifying, adjusting its total income stream as between this year and later years is the most important picture we can draw of investment opportunity not only because it automatically leaves out borrowing and lending, or buying and selling, but also because it automatically reduces the picture of income to its fundamental terms of real or, as I prefer to call it, enjoyment income and its obverse, labor pain. We do not have to think so vividly, as we do in the case of an individual, of money items and intermediate processes. We can without difficulty fix our attention on the final consumption. Society is like Robinson Crusoe picking and eating his berries, however complicated may be the apparatus which intervenes between the labor of picking and the enjoyment of eating.

Society may add to or subtract from its income stream at will at any period, present or future. But beyond a certain point every addition at one period must be at the cost of a subtraction at some other period. If future income is added, the increment so added is a return on and at the cost of a decrement in less remote income. The rate of return over cost is thus a social phenomenon of great significance. There are two and only two ways in which society may effect the present cost and the future return. It may effect the present cost by exerting more present labor or by abstaining more from present consumption; and it may realize the future return over that cost either in the form of more future consumption or of less future toil.

Both the present and the future adjustments are effected by changing the use made of capital instruments including land and human beings. That is, the labor, land, and other capital of society may be used in many optional ways and in particular may be invested for the early or remote future.

If the capital instruments of the community are of such a nature as to offer a wide range of choice, we have seen that the rate of interest will tend to be steady. If the range of choice is narrow, the rate of interest will tend to be variable. If the range of choice is relatively rich in future income as compared with the more immediate income, the rate of interest will tend to be high. If the range of choice tends to favor immediate income as compared with more remote future income, the rate of interest will tend to be low.

Thus, for the United States during the last century, its resources were of such a character as to favor future income. This is true, for a time at least, in every undeveloped country, and, as we have seen, gives the chief explanation of the fact that the rate of interest in such localities is usually high. The same is true of countries recovering from war. Today, for instance, Germany resembles a pioneer country. Her present income is necessarily low, but her prospect of a higher and increasing income in a few years is very great. The range of choice is dominated by "low today and high tomorrow."

The range of choice in any community is subject to many changes as time goes on, due chiefly to one or more of three causes: first, a progressive increase or decrease in resources; second, the discovery of new resources or means of developing old ones; and third, change in political conditions.

Under the first head may be noted the impending exhaustion of the coal supply in England, as noted by Jevons and other writers. This will tend to make the income stream from that island decrease, at least in the remote future, and this in turn will tend to keep the rate of interest there low. Under the second head, the constant stream of new inventions, by making the available income streams rich in the future, at the sacrifice of immediate income, tends to make the rate of interest high. This effect, however, is confined to the period of exploitation of the new invention, and is succeeded later by an opposite tendency. During the last half century the exploitation of Stephenson's invention of the locomotive, by presenting the possibility of a relatively large future income at the cost of comparatively little sacrifice in the present, tended to keep the rate of interest high. As the period of extensive railroad building is drawing to a close, this effect is becoming exhausted, and the tendency of the rate of interest, so far as this particular influence is concerned, is to fall.

On the other hand, the invention of the automobile, and the inventions and discoveries in electricity and chemistry have succeeded the railroads as a field for investment and have required new sacrifices of immediate income for the sake of future income. Thus, as fast as the first effect of any one invention, tending to raise interest, wears off and is succeeded by its secondary effect in lowering interest, this secondary effect is likely to be offset by the oncoming of new inventions.

As to the third head, political conditions which affect the rate of interest, such as the insecurity of property rights which occurs during political upheaval, as in Russia recently, tend to make the pure or riskless rate of interest low. At the same time it adds an element of risk to most loans, thereby diminishing the number of safe and increasing the number of unsafe loans. Hence the commercial rate of interest in ordinary loans during periods of lawlessness is likely to be high. Reversely, during times of peace and security, the riskless rate of interest is comparatively high, while the commercial rate tends to be low.

§5. Time Preference

We turn now to the remaining factor, namely, the dependence of time preference of each individual on his selected income stream. We have seen that the rate of preference for immediate as compared with remote income will depend upon the character of the income stream selected, but the manner of this dependence is subject to great variation and change. The manner in which a spendthrift will react to an income stream is very different from the manner in which the shrewd accumulator of capital will react to the same income stream. We have seen that the time preference of an individual will vary with six different factors: (1) his foresight; (2) his self-control; (3) habit; (4) the prospective length and certainty of his life; (5) his love of offspring and regard for posterity; (6) fashion. It is evident that each of these circumstances may change. The causes most likely to effect such changes are: (1) training to foster a realization of the need to provide against the proverbial "rainy day"; (2) education in self-control; (3) formation of habits of frugality, avoiding parsimony on the one hand and extravagance on the other; (4) better hygiene and care of personal health, leading to longer and more healthful life; (5) incentives to provide more generously for offspring and for the future generations; (6) modification of fashion toward less wasteful and harmful expenditures for the purpose of ostentatious display.

These various factors may act and react upon each other, and may affect profoundly the rate of preference for present over future income, and thereby influence greatly the rate of interest. Where, as in Scotland, there are educational tendencies which instill the habit of thrift from childhood, the rate of interest tends to be low. Where, as in ancient Rome, at the time of its decline, there is a tendency toward reckless luxury, competition in ostentation, and a degeneration in the bonds of family life, there is a consequent absence of any desire to prolong income beyond one's own term of life, and the rate of interest tends to be high. Where, as in Russia, under the Czars, wealth tended to be concentrated and social stratification to be rigid, the great majority of the community, on the one hand, through poverty and the recklessness which poverty begets, tends to have a high rate of preference for present over future income, whereas, at the opposite end of the ladder, the inherited habit of luxurious living tends, though in a different way, in the same direction. In such a community, the rate of interest is likely to be unduly high.

§6. Conclusion

From the foregoing enumeration, it is clear that the rate of interest is dependent upon very unstable influences many of which have their origin deep down in the social fabric and involve considerations not strictly economic. Any causes tending to affect intelligence, foresight, self-control, habits, the longevity of man, family affection, and fashion will have their influence upon the rate of interest.

§7. The Future

From what has been said it is clear that, in order to estimate the possible variation in the rate of interest, we may, broadly speaking, take account of the following three groups of causes: (1) the thrift, foresight, self-control, and love of offspring which exist in a community; (2) the progress of inventions; (3) the changes in the purchasing power of money. The first cause tends to lower the rate of interest; the second, to raise it at first and later to lower it; and the third to affect the nominal rate of interest, in one direction and the real rate of interest in the opposite direction.

Were it possible to estimate the strength of the various forces thus summarized, we might base upon them a prediction as to the rate of interest in the future. Such a prediction, however, to be of value, would require more painstaking study than has ever been given to this subject.

Without such a careful investigation, any prediction is hazardous. We can say, however, that the immediate prospects for a change in the monetary standard seem to be toward its stabilization; that this will tend toward a general prosperity, the main effects of which should be in the direction of lowering the rate of interest; that changes in thrift, foresight, self-control, and benevolence are for the most part likely to intensify these factors and thus to lower the rate of interest; and that the progress of discovery and invention shows now a tendency to increase in speed, the immediate result of which should be to raise the rate of interest but finally to lower it.

APPENDIX TO CHAPTER I

§1 (to Ch. I, § 1)
Quotations from Professor Canning's book

THE importance to the accountant of a clear and consistent concept of income and of capital is emphasized by Professor John B. Canning in his book, The Economics of Accountancy; A Critical Analysis of Accounting Theory.

APPENDIX TO CHAPTER X

§1 (to Ch. X, § 2)
[Geometric representation of incomes for three years]

IF we proceed from the consideration of two years to that of three, we may still represent our problem geometrically by using a model in three dimensions. Let us imagine three mutually perpendicular axes from an origin O called respectively OX', OX'', OX''', and represent the income combination or income stream for the particular individual by the point P, whose coördinates c', c'', and c''' are the three years' income installments with which the individual is initially endowed. Then through the point P draw, instead of the straight line in the previous representation, a plane ABC cutting the three axes in A,B, and C. This plane has a slope with reference to the two axes OX' and OX'' of equal to 1 + i' (unity or 100 per cent plus the rate of interest connecting the first and second years), and has a slope with reference to the axes OX'' and OX''' represented by equal to 1 + i'' (unity plus the rate of interest connecting the second and third years). Now suppose the space between the axes to be filled with willingness surfaces laminated like the coats of an onion, such that for all points on the same surface, the total desirability or wantability of the triple income combination or income position represented by each of those points will be the same. These surfaces will be such as to approach the three axes and the planes between them, and also such that the attached numbers representing their respective total wantabilities shall increase as they recede from the origin. The plane ABC drawn through P at the slope fixed by the rates of interest just indicated will now be tangent to some one of the willingness surfaces at a point Q, which is the point at which the individual will, under these conditions, fix his income situation, for every point on the plane ABC will have the same present value, and every point on this plane is available to him by borrowing and lending (or buying and selling) at the rates i' and i'', but not all of them will have the same desirability, or wantability. He will select that one which has the maximum wantability, and this will evidently be the point Q, at which the plane is tangent to one of the family of willingness surfaces. This point will be such that the rates of time preference will be equal to the rate of interest.

So much for the individual. The market problem determining the rate of interest is here solved by finding such an orientation for the various planes through the given points called P's as will bring the center of gravity of the tangential points, the Q's, into coincidence with the fixed center of gravity of the P's.

To proceed beyond three years would take us into the fourth dimension and beyond. Such a representation cannot be fully visualized, and therefore has little meaning except to mathematicians.

APPENDIX TO CHAPTER XII

§1 (to Ch. XII, § 1)
Algebraic expression of rate of time preference

IF W signifies wantability, or utility, or desirability and this year's income is signified by X' and next year's by X'', then DW' may be taken to signify the present wantability of, or want for, a small increment, DX' of money this year and will be the want-for-one-more unit of money this year. Also DW'' is the present want for a small increment, DX'', of money available next year, and will be the present want-for-one-more unit of money available next year.

Exact mathematical theory requires that the marginal wants per unit of money are the limits of those ratios when the increments approach zero as their limits or lim and lim ordinarily written in the differential calculus: and which, for short, may be called w' and w''.

The rate of preference, f, for a unit of present money over a unit of next year's may be defined as

§2 (to Ch. XII,§3)
Equality of marginal rate of time preference and rate of interest implies that desirability of income stream is made a maximum

ASSUME at first that only two years are considered. The fact that total desirability or wantability of the individual, as reckoned at the beginning, depends on the amount of income this year and next year may be represented by the equation

W'' = F(c' + x', c'' + x''),

where W'' represents his total wantability, and the equation represents this W'' as a function of his income stream consisting of c' + x' this year and c'' + x'' next year. This W'' is represented in Chapters X and XI by the numbers attached to the several Willingness or Wantability lines, each representing a certain level of wantability of Individual 1. But as we shall here consider only one individual, we omit the subscript numbers, 1, 2,..., n. The individual under consideration will attempt to adjust x' and x'' so as to maximize W. We are to prove algebraically that the condition that W shall be a maximum implies also that the rate of interest i shall be equal to the individual's rate of preference f. The condition2 that W shall be a maximum is that the total differential of W or of its equal F(c' + x',c'' + x''), called below F() shall be zero; thus

where the ∂'s represent the partial differentials with respect to x' and x''.

From this equation it follows that

The left-hand number of this equation is 1 + i, as may be seen by differentiating the equation of the loan as originally stated, viz.:

This differentiation yields= 1 + i.

The right-hand member, being the ratio of this year's marginal wantability to next year's marginal wantability, is by definition equal to 1 + f. Substituting the new value for the right- and left-hand members, we have

1 + i = 1 + f,

whence it follows that i = f, which was to have been proved.

The same reasoning may now be applied to three or more years. The total wantability for any individual is a function of the total future income stream. In other words,

W = F(c' + x', c'' + x'',..., c(m) + x(m)).

The individual tries to make this magnitude a maximum. In terms of the calculus, this is equivalent to making the first total differential equal to zero namely,

This total differential equation is equivalent, according to well-known principles of the calculus to a number of subsidiary equations obtained by making particular suppositions as to the different variations. Let us, for instance, suppose that only x' and x'' vary in relation to each other and that x''', xiv,..., x(m) do not vary. Then in the above equation all terms after the second disappear and the equation reduces, as before, to

So that, again, 1 + i' = 1 + f', and therefore i' = f'.

This expresses the relation between the first and second years. If we wish, in like manner, to express the corresponding connection between the second and third years, let us assume that x' as well as xiv,..., x(m) are constant but that x'' and x''' vary. Then the first term of the equation and all after the third disappear, and the equation reduces to

In other words, 1 + i'' = 1 + f'', or i'' = f''. Similarly, i = f for every other pair of successive years.

We have here, in mathematical language, the reason that the point of maximum total wantability is also the point at which the marginal rate of time preference for a unit of each year's income over that of next year's income is equal to the rate of interest connecting these two years.3

APPENDIX TO CHAPTER XIII

§1 (to Ch. XIII, § 9)
Rate of return over cost expressed in the notation of the calculus

IN the notation of the calculus, the rate of return over cost, called in the text r₁', is defined in terms of the partial differential quotient with the opposite sign of next year's income with respect to this year's income of Individual 1. That is, by definition

and

.

Analogous formulas express the remaining r's for Individuals 1, 2... n.

§2 (to Ch. XIII, § 9)
Rate of return over cost derived by differential equations

The magnitudes of 1 + r₁', 1 + r₁'',..., 1 + r₁(m), or

,

may be expressed in terms of y₁', y₁'',..., y₁(m) by differentiating the equation for the effective range of choice, f₁ (y₁', y₁'',..., y₁(m)) = 0.

§3 (to Ch. XIII. § 7, also § 9)
Mathematical proof that the principle of maximum present value is identical with the principle that the marginal rate of return over cost is equal to the rate of interest.

THE mathematical proof that the principle of maximum present value of optional income streams is identical with the Investment Opportunity Principle B or that the rate of marginal return over cost is equal to the rate of interest is as follows:

The present value V₁ of any income stream y₁', y₁'',...,y₁(m), of Individual 1 or their combined discounted value is

The condition that this expression shall be a maximum is that the first differential quotient shall be zero. That is,

This last equation expresses the relations which must exist between dy₁', dy₁'',..., dy₁(m), in order that the income stream, y₁', y₁'',... y₁(m), may have the maximum present value.

This condition contains within itself a number of subsidiary conditions. To derive these, let us consider a slight variation in the income stream, affecting only the income items pertaining to the first two years, y₁', and y₁'' (the remaining items, y₁''',..., y₁(m), being regarded for the time being as constant) and let us denote the magnitudes of dy₁' and dy₁'', under this assumption of restricted variations, by dy₁' and dy₁''. Then, under the condition assumed of constancy of y₁''', y₁iv,..., y₁(m), dy₁''', dy₁iv,..., dy₁(m), are equal to zero, and the equation becomes

From this, it follows directly that

But the left-hand member of this equation is by definition one plus the marginal rate of return over cost. Since we have designated the rate of return over cost by r₁' we may substitute 1 + r₁' for the expression , and write the above equation thus:

1 + r₁' = 1 + i',

or thus:

r₁' = i'.

In other words, the condition that the marginal rate of return over cost is equal to the rate of interest follows as a consequence of the general condition that the present value of the income stream must be a maximum. This proposition and its proof are analogous to those in regard to desirability or wantability, which have already been discussed in the Appendix to Chapter XII, that the condition of maximum wantability is equivalent in the condition that the marginal rate of preference is equal to the rate of interest.

The same reasoning may be applied to any pair of successive years. Thus, if we assume variations in y'' and y''', without any variations in the other elements of the income stream, y', yiv,..., y₁m, the original differential equation becomes

or = 1 + i'',

or 1 + r₁'' = 1 + i'',

or r₁'' = i''.

All this reasoning implies, in using the differentiation process, that there is continuous variation, and that, at the margin, it is possible to make slight variations in any two successive years' incomes without disturbing the incomes of the other years.

§4 (to Ch. XIII, § 9)
Geometrical explanation of the proposition expounded in § 3 of this Appendix

BUT the foregoing proof by algebra may not appeal to many students as much as the proof by geometry.

We know (see Chapter X, § 3 and Chapter XI, §§ 4 and 5) that the present value of any income position on the Market line is the same as that of any other income position on that line.

It follows that the present value of any point on a given Market line is measured by the intercept of that line on the horizontal axis, for that intercept evidently measures the present market value of one particular point on the Market line (namely its lower end) and, as just stated, this must have the same present value as every other point on the Market line.

It follows that as the Market line is moved further away from the origin (keeping its direction unchanged) the intercept becomes greater, and thus the present value of every one of the points on the Market line becomes greater correspondingly.

When, therefore, the line is thus moved as far as possible, so that it thereby assumes the position of tangent to the Opportunity line, the present value of every point on it and, therefore, of that point of tangency must be greater than that of any other point on the Opportunity line, since any other such point will necessarily lie on a Market line nearer the origin.

The same proof applies in three dimensions, substituting Opportunity surface for Opportunity line and Market plane for Market line. By analogy the proof may be extended to n dimensions.

§5 (to Ch. XIII, § 9)
Maximum total desirability is found when rate of time preference is equal to the rate of interest

IN the last section was outlined a geometric proof that the income stream possessing the maximum present value is such that the rate of interest (connecting each pair of successive years) is equal to the rate of return over cost (for the same pair of successive years).

The geometric method also supplies a simple proof that the maximum total desirability, or wantability, is to be found in the case of that income stream which satisfies the above mentioned condition and, at the same time, has a rate of time preference equal to the rate of interest. In geometric terms for two dimensions this means that this most desirable income position or point is where the Market line, which is tangent to the Opportunity line, is tangent to a Willingness line.

Consider two parallel Market lines, one tangent to the Opportunity line and the other somewhat nearer the origin; and consider the two points Q and S where these two are respectively tangent to a Willingness line. We are to prove that the total desirability or wantability of Q is greater than that of S. Draw a straight line from the origin through S and produce it until it cuts the first Market line at, say, T.

It is evident, of course, that of all the points on any given Market line the point of tangency with a Willingness line is the most desirable income position. Therefore, Q is more desirable than T. We assume that the Willingness lines are such that the farther we recede along a straight line from the origin the more desirable the income situation. Therefore, T is more desirable than S.

Therefore, Q being more desirable than T, and T than S,Q is more desirable than S, which was to have been proved.

§6 (to Ch. XIII, § 9)
Walras and Pareto

WALRAS and Pareto probably deserve more attention in interest theory, as in general economic theory, than they have received.

Walras' interest theory forms an integral part of his theory of general economic equilibrium.4 His solution consists of a demonstration that the problem comprises a number of independent equations exactly equal to the number of unknowns, and that the mathematical solution of these simultaneous equations is a counterpart of the economic process by which the unknowns are determined in the market. There is thus no reasoning in a circle in the Walras system. The number of equations is exactly equal to the number of unknowns.

Walras' treatment of the problem of the determination of the rate of interest is very detailed and highly mathematical. For readers who are not familiar with his treatment I venture to attempt a brief summary. Walras assumes a market for capital goods as well as for services. He assumes that the prices of capital goods depend on the prices of their services. Since some capital goods last longer than others and all are subject to risk, he makes allowance for depreciation, amortization, and insurance.

His treatment combines the subjective and objective elements in a simple and direct manner. His cost of production equations correspond, in a general way, to my opportunity principles. His equation for the demand for savings corresponds, likewise, to my impatience principles.

Pareto's analysis of the problem of the rate of interest5 is along the lines laid down by Walras, although he was evidently not fully satisfied with Walras' treatment. Neither he nor Walras has developed a systematic theory of income but he shows, in effect, that the substitution of one income stream for another or, as he says, the "transformation in time" is only a particular case of a more general transformation which is dealt with in the theory of production. His indifference equations for consumers correspond in a general way to my impatience principles and the analogous equations for the obstacles in his treatment of production correspond likewise to my opportunity principles.

The fundamental differences between the approach of Walras and Pareto on the one hand and mine on the other seem to be four:

(1) Walras and Pareto determine the rate of interest simultaneously with all the other unknowns of the problem—the quantities of the commodities exchanged and the services used in their production and the prices of the commodities and the services, while I try to isolate the interest problem by assuming that most of such unknowns have already been determined and confine my discussion to the special factors directly affecting the rate of interest.

(2) They both treat what I call interactions or intermediate services along with the ultimate factors—our desires or tastes (les gôuts) and the obstacles which must be overcome to satisfy them—while I try at the outset to get the interactions canceled out, leaving only the income stream and (labor) sacrifice.

(3) Neither Walras nor Pareto has elaborated the concept and principles of an income stream.

(4) Neither has elaborated the concept or principles of opportunity as a choice from among a series of income streams, although it is, in part, implied in Pareto's treatment.6

APPENDIX TO CHAPTER XIX

§1 (to Ch. XIX, § 4)
Tables giving basic data

In Chapter XIV of The Rate of Interest "virtual," or "real," rates of interest were computed from "nominal," or "money," rates of interest by making adjustments for appreciation in the value of money calculated from index numbers of prices. In this book, the money rates of interest are adjusted directly to the rates of change in the general price level. These two methods, of course, yield identical results, since the one is the obverse of the other.

The average annual percentage changes in the general price level, given in the Tables VII to XI inclusive, are computed from the wholesale price indexes of the several countries. The index numbers for two dates, as 1825 and 1834, give us a measure of the price level at those two dates, and from these it is easy to calculate the average annual percentage change. The method is the same as that employed for finding the rate of interest by which $1, by compounding, will amount to a given sum in a given time. Theoretically, since the loans here included run usually perhaps thirty to ninety days, the quotations of rates of interest averaged should begin at the first of the two dates, and cease, say, sixty days before the second. But the index numbers are not always for definite points of time, nor can the interest quotations be subjected to such minute corrections without an immense expenditure of labor. Hence, the method adopted has been to average the rates for all the years of a period, e.g., for the ten years, 1824-1834. The annual percentage change in the price level is reckoned between those dates. If the index numbers present the price levels at the middle of 1825 and 1834, then the average interest rates ought in theory to include only the last six months of 1825 and the first four months of 1834. But it seems better to include too much at both ends than to omit the averages for 1825 and 1834 altogether, for the reason that an average is the more valuable the greater the number of terms included.

The real interest rates are obtained by subtracting from the money rate for any period the rate of annual change in the price level for the same period.

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§2 (to Ch. XIX, § 6)
Tables of interest rates

APPENDIX TO CHAPTER XX

§1 (to Ch. XX, § 17)
Waiting as a Cost

IF waiting were a cost like other costs, it would be subject to the law of discount, according to which the capital-value of any article of wealth is equal to the discounted value of its expected income less the discounted value of its expected outgo. The value of the tree which has been mentioned, taken, say, at the end of 14 years, will actually be about $2, and this is the discounted value of the $3 of income which the tree will yield at the end of eleven more years. According to what I believe to be the correct theory, this $3 is the only future item involved in this example. But according to the theory here criticised, this is not the case. Besides this positive item of income, $3 due in eleven years, we have to deal with a series of eleven negative items called "waiting" distributed through these eleven years, and amounting to the interest—about 10 cents for the first year and gradually increasing to 15 cents for the last year. If the waiting-items were bona fide annual costs—like, for instance, actual labor-costs of pruning the trees—the process of discount would properly be applied to them. That is, if these waiting costs really exist, they ought to be discounted and their discounted value ought to be deducted from the discounted value of the $3 of expected income. But we should then have to assign as the value of the tree not the correct figure of $2 but an incorrect figure of much less. The fact that we cannot thus discount so-called "waiting" costs as we discount all true costs is a proof that the "cost of waiting" even if we insist on calling it such differs radically from true costs.7

If we are to have any logical, usable self-consistent theory of income and capital, all items of income, positive or negative—the negative ones being "costs"—must be discountable.

But, as an answer to this objection, it might be argued by the abstinence theorists (if I may ascribe to them the best argument I can think of) that while waiting-cost is certainly not a discountable cost, nevertheless its inclusion in the list of costs obviates the necessity of discounting the other items of cost or of income. If all income and all cost items, including waiting, are counted at full value—not discounted at all—the capital may be valued simply by taking their net sum. Thus, to count "waiting" as a cost appears as an alternative and plausible method of keeping accounts. By this system we could apparently get rid of discounting and merely add and subtract items regardless of their situation in time. While this procedure obviates the objection to the abstinence theory of cost, so far as its application to capital value is concerned, it leaves objections equally great to its application to income. If waiting is a genuine economic cost, it must certainly be included on the outgo side of the income account. To show how this would apply to the cost of the tree, the following table is presented.

According to this method of accounting, we see that, during the year in which the sapling is planted, its cost consists of labor to the extent of $1, expended, let us say, at the beginning of the year, and 5 cents' worth of waiting suffered during the course of that first year. During the second year a waiting cost of about the same amount is incurred, and so on for each succeeding year, the cost of waiting gradually increasing, as the tables of compound interest would indicate, until in the fourteenth year it amounts to 10 cents, and in the twenty-fifth year to 15 cents. The total cost for the 25 years will then be $3, and the return to the planter at the end, from the sale of the tree, will also be $3. Consequently, if we take the whole period from the first application of labor to the final sale of the tree, the net income will be zero. This result is, to say the least, somewhat surprising, but not so much so as some other results of the same species of bookkeeping, as the following additional examples will show.

Suppose a person owns an annuity amounting to $100 a year for 10 years. According to the ordinary method of keeping accounts, his income consists of this $100 a year each year. But if we count the waiting as a cost, we shall find that the income for each year is less than $100. The owner of such an annuity will, during the first year, have to suffer "waiting" to the extent of $39, supposing interest is at 5 per cent; for this is the increase in value of his annuity during that year, due to his waiting for the future installments of income of which his annuity consists.8 His net income during that year, therefore, according to such accounting, is not $100, but $100 - $39, or $61. During the second year his income in this second year is somewhat greater, for the cost of "waiting" is only $35. His net income is, therefore, $100 - $35, or $65. Similar computations carried out for succeeding years are shown in the table on the following page.

Is it good bookkeeping to introduce a new and anomalous element of cost which results in making the net income of the annuitant not the $100 which he actually receives and which common sense recognizes as the income from the annuity but the queer sums given in the table, namely, $61, $65, $68, and so forth?9

To push this criticism to the limit, let us finally consider a perpetual annuity of $100 a year. In this case we shall find that the "cost of waiting" each year is the full $100, for the value of such an annuity, reckoned at 5 per cent, is $2000 reckoned at the beginning of each year, and $2100 reckoned at the end. If this annual $100 cost of waiting is to be regarded as a negative item of income and, like other costs, is to be subtracted from the positive income, we are forced to conclude that the owner of such a perpetual annuity receives each year no income whatever! For, if we deduct from the $100 of positive income the $100 cost of waiting, the remainder each year is zero! Yet a perpetual annuity is the simplest, purest case of income.

It should now be obvious that the theory which calls "waiting" a cost has worked out its own absurdity. If taken seriously and introduced into an accounting system it either interferes with the discount or capitalization principle or else distorts and even obliterates the income reckoning in its simplest, or most typical form, that of a perpetual annuity. It falsely simplifies the formula for valuing capital.

The idea that the value or price of an article should equal its cost seems to possess a certain fascination for many students of economics. That it is false has been sufficiently shown by Böhm-Bawerk through reasoning somewhat similar to the foregoing. That it is absurd when carried to its logical conclusion will be evident if we consider what happens if the same method of bookkeeping is carried out with respect to the future as well as the past. It is a poor rule which will not work both ways. This rule, applied to future expected income and outgo, yields the strange result that the capital value of any article instead of being less than its expected income is equal to it. Thus, to revert to the case of the tree, let us take its value at the end of 14 years. It is then worth $2, which, in the parlance of the abstinence theorists, is equal to its previous costs of production, consisting of $1 worth of labor plus $1 worth of waiting during the 14 years. It is also, in like manner, equal to the future income to be derived from it, which consists of $3 worth of actual receipts from the sale of the tree, due at the end of eleven more years, less the cost of waiting for those $3, which amounts to $1.

In the same way, the ten-year annuitant just considered has, at the beginning, property worth $772. This, according to proper bookkeeping, is the discounted value of the future income of $100 a year for 10 years, the total amount of which income is $1000. But, according to the abstinence theory, logically carried out, the income which the annuitant receives for the whole period is, as has been shown, not this $1000, but $772, which is just equal to the value of the property.10 Pursuing the method of limits, we find that, for the owner of a perpetual annuity, the same proposition would hold good. According to the true and ordinary method of reckoning, the total income from such an annuity is infinity, although its present capital value is only $2000. But according to the abstinence theorists the income itself is not infinite, but only $2000.

Those who are enamored of the alluring simplicity and neatness of the formula of the abstinence theorists, by which the capital value is not greater than past cost of production, but exactly equal to it, can scarcely be attracted by the exaggerated simplicity of the inverse theorem which is also involved, namely, that the capital value of any future expected income is not less than that income, but exactly equal to it also.

The fallacy of the abstinence theorists lies in the simple fact that waiting has no independent existence as a "cost." We can never locate it in time, nor estimate its amount, without first knowing some other more real and tangible costs. Waiting means nothing unless there is something to be waited for, and the cost of waiting can only be estimated in proportion to the magnitude of that which is so waited for. What is waited for is some payment or other event constituting income or outgo. But waiting for income or outgo is not itself income or outgo.

The mere accrual of value as we draw nearer the items constituting true income is neither income nor outgo but capital gain. The typical picture we should carry in our mind is of a saw-tooth curve consisting alternately of a gradual ascent along a discount curve, and a sudden drop as an income coupon is detached. The only income in this picture is the series of sudden drops, on which all the rest hangs. The gradual ascent in each saw tooth is not income; otherwise it would (largely) duplicate the true income. Nor is it outgo; otherwise it would (largely) negative the true income.

In the case of a bond selling at par these alternate ascents and drops are equal, and we carelessly speak of both as interest or as income. But the instant the bond sells above or below par we recognize the difference. If we follow this out we can scarcely go astray.

Even to those who do not formally accept any cost theory of interest, the interest itself will seem in some sense to be a cost, and in most books on economics, interest, however explained, is regarded as one of the costs of production. It is true that for a debtor who pays interest, the interest is, to him, a real cost, and is debited on his books. But we need only to be reminded of the debit and credit bookkeeping of the first chapter to see that this item is counterbalanced on the books of the creditor, to whom this interest is by no means a cost, but, on the contrary, an item of income. For society as a whole, therefore, even in the case of interest which is explicitly paid, it cannot be said that it constitutes a cost of production. In the case of a person who works with his own capital, the truth of this statement is even more evident. Economists who state that the independent capitalist must charge off interest as one of his costs of production seem to forget that such self-paid interest must be charged back again as income also. Labor sacrifice is quite different. It is a real cost and in no time bookkeeping can it be cancelled out. The fallacy of assuming that interest is a cost is doubtless due to the habit of regarding production from the point of view of the "enter priser." Since he usually pays interest, he comes to think of it purely as a cost.

I have devoted considerable space to the refutation of the abstinence theory so far as it is more than verbal, and collides with any workable theory of income, because its errors are so subtle and insidious as to beguile many of the best and most wary of economists.

BIBLIOGRAPHY

I. WORKS ON INTEREST THEORY.

1. Books:

BÖHM-BAWERK, EUGEN VON. Capital and Interest. London, Macmillan and Co., 1890. xlv, 431 pp.

———. The Positive Theory of Capital. Translated by William Smart, London, Macmillan and Co., 1891. xl, 428 pp.

———. Recent Literature on Interest (1884-1889). New York, The Macmillan Co., 1903. xlii, 151 pp.

———. Positive Theorie des Kapitales. Dritte Auflage, Innsbrück, Wagner'schen Universitäts-Buchhandlung, 1912. xxiii, 652 pp. Also Exkurse, 477 pp.

———. Kleinere Abhanalungen über Kapital und Zins. Wien und Leipzig, Hölder-Pichler-Tempsky A. G., 1926. viii, 585 pp.

BROWN, HARRY GUNNISON. Economic Science and the Common Welfare. Columbia, Missouri. Lucas Brothers, 1926. xiii, 273 pp. Especially Part II, Chapters III and IV, pp. 76-170.

CARVER, THOMAS NIXON. The Distribution of Wealth. New York, The Macmillan Co., 1904. xvi, 290 pp.

CASSEL, GUSTAV. The Nature and Necessity of Interest. London, Macmillan and Co., 1903. xii, 188 pp.

———. The Theory of Social Economy. New York, Harcourt, Brace and Co., 1924. xiv, 654 pp.

CLARK, JOHN BATES. Distribution of Wealth. New York, The Macmillan Co., 1899. xxviii, 445 pp.

DAVENPORT, H. J. Value and Distribution. University of Chicago Press, 1908. xi, 582 pp.

FETTER, FRANK A. Economic Principles. New York, The Century Company, 1915. x, 523 pp.

FISHER, IRVING. The Rate of Interest. New York, The Macmillan Co., 1907. xxii, 442 pp.

GONNER, E. C. K. Interest and Savings. London, Macmillan and Co., 1906. xv, 172 pp.

HEINZE GERHARD. Statische oder Dynamische Zinstheorie? Leipzig, Dr. Werner Scholl, 1928. viii, 165 pp.

HOAG, CLARENCE GILBERT. A Theory of Interest. New York, The Macmillan Co., 1914. x, 228 pp.

JEVONS, W. STANLEY. Theory of Political Economy. 3rd edition, London, Macmillan and Co., 1888. lvi, 296 pp.

LANDRY, ADOLPHE. L'Intérêt du Capital. Paris, V. Biard and E. Brière, 1904. 367 pp.

PARETO, VILFREDO. Cours d'Économie Politique. Lausanne, F. Rouge, 1896 and 1897. Tome Premier, viii, 430 pp. Tome Second, 426 pp.

———. Manuel d'Économie Politique. Paris, V. Giard and E. Brière, 1909. 695 pp.

RAE, JOHN. The Sociological Theory of Capital. New York, The Macmillan Co., 1905. lii, 485 pp.

SAX, EMIL. Der Kapitalzins. Berlin, Julius Springer, 1916. viii, 249 pp.

WALRAS, LÉON. Éléments d'Économie Politique Pure. Lausanne, F. Rouge, 1900. xx, 491 pp.

2. Articles:

ANSIAUX, M. Le Phénomène de L'Intérêt et son Explication. Revue de L'Institut de Sociologie. Deuxième Année, Tome I, Bruxelles, 1921-1922, pp. 47-57.

BILGRIM, H. Analysis of the Nature of Capital and Interest. Journal of Political Economy. Vol. XVI, March, 1908, pp. 129-151.

BÖHM-BAWERK, EUGEN VON. Capital and Interest. Quarterly Journal of Economics. Vol. xxi, November, 1906, pp. 1-21; February, 1907, pp. 247-282.

BORTKIEWICZ, L. VON. Der Kardinalfehler der Boehm-Bawerkschen Zinstheorie. Jahrbuch fuer gesetzgebung, Band 30, 1906, pp. 61-90, Leipzig, Duncker und Humblot, 1906.

CARVER, T. N. The Place of Abstinence in the Theory of Interest. Quarterly Journal of Economics, October, 1893, pp. 40-61.

CHAPMAN, S. J. Must Inventions Reduce the Rate of Interest? Economic Journal, Vol. XX, September, 1910, pp. 465-469.

DAVENPORT, H. J. Interest Theory and Theories. American Economic Review, Vol. XVII, No. 4, December, 1927, pp. 636-656.

DAVIES, G. R. Factors Determining the Interest Rate. Quarterly Journal of Economics, Vol. XXXIV, May, 1920, pp. 445-461.

FETTER, FRANK A. Interest Theories Old and New. American Economic Review, Vol. IV, No. 1, March, 1914, pp. 68-92.

———. Clark's Reformulation of the Capital Concept, in Economics Essays Contributed in Honor of John Bates Clark, pp. 136-156, New York, The Macmillan Co., 1927.

FISHER, IRVING. Professor Fetter on Capital and Income. Journal of Political Economy, Vol. XV, July, 1907, pp. 421-434.

———. Are Savings Income? Journal of the American Economic Association, Vol. IX, No. 1, April, 1908, pp. 1-27.

———. A Reply to Critics. Quarterly Journal of Economics, Vol. XXIII, May, 1909, pp. 536-541.

———. Capital and Interest. Political Science Quarterly, Vol. XXIV, No. 3, 1909, pp. 504-516.

———. Capital and Interest: Reply to Professor Veblen. Political Science Quarterly, Vol. XXIV, September, 1909, pp. 504-516.

———. The Impatience Theory of Interest. Scientia, Vol. IX, April 1, 1911, pp. 380-401.

———. The Impatience Theory of Interest. American Economic Review, Vol. III, No. 3, September, 1913, pp. 610-615.

FLUX, A. W. Irving Fisher on Capital and Interest. Quarterly Journal of Economics, Vol. XXIII, February, 1909, pp. 307-323.

GONNER, E. C. K. Considerations about Interest. Economic Journal, Vol. XVIII, March, 1908, pp. 42-51.

GRAZIANI, AUGUSTO. Capitale e Interesse. Società Real di Napoli, 1925, pp. 33-92.

LANDRY, ADOLPHE. Irving Fisher: The Rate of Interest. Revue d'Economie Politique, 23 Année, 1909, Bulletin Bibliographique, pp. 156-159. Paris, L. Larose and L. Tenin, 1909.

LORIA, A. Irving Fisher's Rate of Interest. Journal of Political Economy, Vol. XVI, October, 1908, pp. 331-332. Reply by Irving Fisher, same issue, pp. 532-534.

LOWRY, DWIGHT M. The Basis of Interest. American Academy of Political and Social Science, March, 1892, pp. 53-76.

SCHUMPETER, JOSEPH. Eine "Dynamische" Theorie des Kapitalzinses. Zeitschrift für Volkswirtschaft, Sozialpolitik und Verwaltung, 1913, pp. 599-639. Vienna, Manzche, K. U. K. Haf-Verlags und Universitätsbuchhandlung, 1913.

SHAPOSCHNICOFF, N. von. Die Böhm-Bawerksche Kapitalzinstheorie. Jahrbüchern für Nationalökonomie und Statistik, Dritte Folge, Bd. XXXIII (LXXXVIII), Jena, Gustav Fischer, pp. 433-451.

TAUSSIG, F. W. Capital, Interest and Diminishing Returns. Quarterly Journal of Economics, Vol. XXII, May, 1908, pp. 333-363.

VEBLEN, T. Fisher's Rate of Interest. Political Science Quarterly, Vol. XXIV, June, 1909, pp. 296-303.

II. OTHER WORKS DEALING WITH INTEREST.

1. Books:

ADLER, KARL. Kapitalzins und Preisbewegung. Leipzig, Duncker und Humblot, 1913. 48 pp.

BECKHART, BENJAMIN H. The Discount Policy of the Federal Reserve System. New York, Henry Holt and Co., 1924. xii, 604 pp.

BOUCHER, PIERRE B. Histoire de L'Usure. Paris, Chaigneau, 1806. 215 pp.

BROWN, MARY W. The Development of Thrift. New York, The Macmillan Co., 1900. x, 222 pp.

CANNING, JOHN B. The Economics of Accountancy. New York, The Ronald Press Company, 1929. viii, 367 pp.

DICK, ERNST. The Relation Between the Rate of Interest and the Level of Prices. Distributed by H. R. Scott, Kodaikanal, S. India, March, 1928. 83 pp.

EDIE, LIONEL D. Economics: Principles and Problems. New York, Thomas Y. Crowell Co., 1926. xx, 799 pp.

———. Money, Bank Credit, and Prices. New York, Harper & Brothers, 1928. xiv, 500 pp.

FETTER, FRANK A. Modern Economic Problems. New York, The Century Company, 1917. xi, 498 pp.

FISHER, IRVING. The Income Concept in the Light of Experience. English translation of article in Die Wirtschaftstheorie der Gegenwart, Vol. III of the Wieser Festschrift, Vienna, 1927. 29 pp., in translation.

———. The Nature of Capital and Income. New York, The Macmillan Co., 1927. xxi, 427 pp.

GIFFEN, ROBERT. The Growth of Capital. London, George Bell and Sons, 1889. 169 pp.

GRIMES, JOHN ALDEN and CRAIGUE, WILLIAM HORACE. Principles of Valuation. New York, Prentice Hall Inc., 1928. xvii, 274 pp.

KOCK, KARIN. A Study of Interest Rates. London, P. S. King, 1929. 264 pp.

MONTAGNE, JEAN. Le Capital. Paris, Albin Michel, 1919. 253 pp.

NORTON, JOHN P. Statistical Studies in the New York Money-Market. New York, The Macmillan Co., 1902. vi, 180 pp.

PALGRAVE, R. H. INGLIS. Bank Rate and the Money Market. New York, E. P. Dutton and Co., 1903. xxiii, 237 pp.

RABY, R. C. The Regulation of Pawnbroking. New York, Russell Sage Foundation, 1924. 63 pp.

RYAN, FRANKLIN W. Usury and Usury Laws. Boston, Houghton Mifflin Company, 1924. xxix, 249 pp.

WICKSELL, KNUT. Über Wert, Kapital und Rente. Jena, Gustav Fischer, 1893. xvi, 143 pp.

———. Geldzins und Güterpreise. Jena, Gustav Fischer, 1898. xi, 189 pp.

2. Articles:

BIRCK, L. V. Moderne Scholastik. Eine Kritische Darstellung der Böhm-Bawerkschen Theorie. Weltwirtschaftliches Archiv., 24 Bd., October, 1926, Heft 2, pp. 198-227.

BONN, H. Geld und Kapitalmarkte im Jahre 1924. Wirtschaftsdienst, Vol. X, Feb. 6, 1925, pp. 247-248.

BURGESS, W. RANDOLPH. Factors Affecting Changes in Short Term Interest Rates. Journal of the American Statistical Association, Vol. XXII, New Series, No. 158, June, 1927, pp. 195-201.

CASSEL, GUSTAV. The Future of the Rate of Interest. Skandinaviska Kreditaktiebolaget, January, 1926. Stockholm, P. A. Norstedt & Söner, 1926, pp. 1-4.

———. The Rate of Interest, the Bank Rate, and the Stabilization of Prices. Quarterly Journal of Economics, Vol. XLII, August, 1928, pp. 511-529.

———. Discount Policy and Stock Exchange Speculation. Skandinaviska Kreditaktiebolaget, October, 1928. Stockholm, P. A. Norstedt & Söner, 1928, pp. 57-60.

CLEVELAND TRUST COMPANY. Business Bulletin, June 15, 1928, and August 15, 1928.

CONRAD, OTTO. Der Kapitalzins. Jena, Jahrbücher für Nationalökonomie und Statistik, 3 Folge, Band 35, 1908, pp. 325-359.

CRUM, W. L. Cycles of Rates on Commercial Paper. Review of Economic Statistics. Prel. Vol. V, No. 1, January, 1923, pp. 17-29.

FETTER, FRANK A. Recent Discussion of the Capital Concept. Quarterly Journal of Economics, Vol. XV, November, 1900, pp. 1-45.

FISHER, IRVING. Appreciation and Interest. Publications of the American Economic Association, Vol. IX, No. 4, August, 1896, pp. 331-442.

———. What Is Capital? Economic Journal, Vol. VI, December, 1896, pp. 509-534.

———. The Rôle of Capital in Economic Theory. Economic Journal, Vol. VII, December, 1897, pp. 511-537.

———. The Rate of Interest after the War. Annals of the American Academy of Political and Social Science, Vol. LXVIII, November, 1916, pp. 244-251.

———. Comment on President Plehn's Address. American Economic Review, Vol. XIV, No. 1, March, 1924, pp. 64-67.

FRIDAY, DAVID. Factors which Determine the Future of the Rate of Interest: Economic Principles of Supply and Demand. Trust Companies, Vol. XXIII, July, 1921, pp. 9-12.

GIBSON, A. H. The Future Course of High-Class Investment Values. Bankers', Insurance Managers', and Agents' Magazine, January, 1923, pp. 15-34.

GIFFEN, SIR ROBERT. Accumulations of Capital in the United Kingdom in 1875-85. The Journal of the Royal Statistical Association, Vol. LIII, 1890, pp. 1-35.

HARGER, C. M. Problems of Interest Rates. Financial World, Vol. XXXII, June 23, 1919, p. 19.

INOSTRANIETZ, M. L'Usure en Russie. Journal des Économistes, 1893, Ser. 5, Vol. XVI, pp. 233-243. Paris, Administration et Redaction, Librairie Guillaumin et C., 1893.

JAY, PIERRE. Call Money Market in New York City and the Interest Rates Charged Therein. Economic World, Vol. XIX, April 10, 1920, pp. 511-513.

KEMMERER, E. W. War and the Interest Rate. Economic World, Vol. XVI, November 2, 1918, pp. 616-619.

———. Rediscounting and the Federal Reserve Discount Rate. American Bankers' Association Journal, Vol. XII, April, 1920, pp. 582-584.

LEVY, R. G. Du Taux Actuel de L'Intérêt et de ses Rapports avec la Production des Métaux Précieux et les Autres Phénomènes Économiques. Journal des Économistes, March, 1899, p. 334; April, 1899, p. 28.

MAGEE, JAMES D. Call Rates and the Federal Reserve Board. American Economic Review, Vol. X, March, 1920, pp. 59-65.

MITCHELL, W. C. Interest Cost and the Business Cycle. American Economic Review, Vol. XVI, No. 2, June, 1926, pp. 209-221. [Original text reads "MITCHELL, W. F."—Econlib Ed.]

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[115.][115] It is true that in the hard-tack case and some other extreme and hypothetical cases considered, it was assumed that for a certain interval the O curve was assumed to be straight. To include such a theoretical case, the statements in the text need a slight modification. But such extreme cases are not typical even in the theory and are probably never exemplified in practice.

[1.][1] American Economic Review, Vol. XIV, p. 64.

[2.][2] See any text book on the calculus, e.g. Wilson, E. B. Advanced Calculus. Boston, Ginn & Co., 1912, pp. 118-125.

[3.][3]The mathematical reader will note that the function F here representing total wantability W is vitally related to the function F in Chapters XII and XIII, representing the marginal rate of time preference f, since 1 + f is the ratio of the differential quotient of W relatively to this year's income to the corresponding differential quotient for next year's income.

[4.][4] Walras, Leon, Élements d'Économie Politique Pure.

[5.][5] Pareto, Vilfredo, Cours d'Economie Politique.

[6.][6] Cours d'Économie Politique, Vol. I, p. 314.

[7.][7] See Böhm-Bawerk, Recent Literature on Interest (1884-1899), p. 35.

[8.][8] This is evident, since the value of his annuity, capitalized at 5 per cent, reckoned at the beginning, is $772, whereas, reckoned at the end of the first year, before his $100 is paid, it is $811.

[9.][9] It may be of interest to note that this error is the inverse of, or complementary to, the more common one by which the net income is the $100 less the "depreciation." In the first year this would be $772 less $711, or $61, so that the "income" is $39. This sort of accounting, when, instead of depreciation, there is appreciation or savings, would make savings appear as income instead of capital. This savings, or depreciation, fallacy is especially discussed in Are Savings Income? American Economic Association Journal, April, 1908, and The Income Concept in the Light of Experience. It has been the subject of much controversy. Some economists who fall into this savings-are-income, depreciation-is-outgo fallacy in some parts of their system fall into the waiting-is-cost fallacy in other parts. Both cannot be right. Each exhibits the evil consequences which ensue from playing fast and loose with the concepts of capital and income. If we wish to indulge in such a metaphor as "I got it at the 'cost' of waiting," we can do so but only at the "cost" of inaccuracy. Neither of these so-called "costs" is more than a metaphor.

[10.][10] Lest the non-mathematical reader should be puzzled by this result, which seems to contradict the fact already brought out, that, under the pseudo-reckoning of the abstinence theorists, the net income is zero every year, it must be remembered that this zero income is repeated an infinite number of times, and that when we deal with infinity we can get reliable results only by the method of limits. The mathematical reader will find no difficulty in showing, by the method of limits, that there is a "remainder term" which will, in the supposed accounting, make the total income distributed through all eternity simply equal to the capital value, $2000.