The Theory of Interest
1.  This term, investment opportunity, seems to be the nearest expression in popular language to suggest or denote the technical magnitude r employed in this book. The full expression for r is the rate of return over cost, and both cost and return are differences between two optional income streams. So far as I know, no other writer on interest has made use of income streams and their differences, or rates of return over cost per annum. The nearest approximation to this usage seems to be in the writings of Professor Herbert J. Davenport, particularly his Economics of Enterprise, (Macmillan, New York, 1913) pages 368, 379, 381, 394, 395, 396, 410, 411. [Original text reads Economics of Entreprise—Econlib Ed.]
Part I, Chapter 1
2.  The Nature of Capital and Income (first published in 1906) was primarily intended to serve as a foundation for The Rate of Interest which immediately followed it. It was my expectation that the student would read the former before reading the latter.
But now, for the convenience of those who do not wish to take the time to read The Nature of Capital and Income, I have written this first chapter summarizing it. I have availed myself of this opportunity to redistribute the emphasis and to make those amendments in statement which further study has indicated to be desirable.
A friendly critic, Professor John B. Canning, suggests that The Nature of Capital and Income should have been called "The Nature of Income and Capital" and that the subject matter should have been presented in reverse order, inasmuch as income is the basis of the concept of capital value and is, in fact, the most fundamental concept in economic science.
While it might not be practicable to employ the reverse order in such a complete presentation as I aimed to make in The Nature of Capital and Income, I have, in this chapter, where brevity may justify some dogmatism, adopted Professor Canning's suggestions. This radical change in mode of presentation may induce some who have already read that book to review it now in the reverse order employed in this chapter. I hope also that some who have not read it may be moved, after reading this chapter, to read The Nature of Capital and Income in full. I have tried, in this chapter, to confine myself merely to those conclusions most essential as a preliminary for proceeding to the consideration of the origins, nature and determinants of the rate of interest.
3.  The first writer to employ the concept of events as fundamental in interest theory appears to have been John Rae, whose book, originally published in 1834, is commented on elsewhere.
4.  Later in this chapter we shall see that these three sorts of income are all of a piece, parts of the entire economic fabric of services and disservices. Which of the three comes out of our accounting depends merely on which groups of these services and disservices are included in our summation.
5.  Even this is not homogeneous as a measure of subjective enjoyment; for a dollar to the poor and a dollar to the rich are not subjectively equal. See my A Statistical Method for Measuring "Marginal Utility" and Testing the Justice of a Progressive Income Tax. Economic Essays contributed in honor of John Bates Clark, pp. 157-193.
6.  Including, of course, all benefits or services whatever from the possession of the wealth such as the option to subscribe to stock, now often attached to bonds, or the privilege attaching to certain bonds which permits National Banks to use the bonds for the security of National Bank notes. Some of these benefits may be very indirect and related to whole groups. A man seeking voting control as a benefit who already possesses 49 per cent may pay a specially high price for a few more shares of stock for the benefit of raising his holdings to 51 per cent. Or, a man may include in the benefits of his wealth the fun of running the business, or the social standing he thinks it gives him, or political or other power and influence, or the mere miserly sense of possession or the satisfaction in the mere process of further accumulation. However indirect, unusual, or bizarre the benefit, the principle still holds that the value of any capital good or goods is derived solely from the prospect of future benefits.
7.  Possibly it would help to adjust our mental attitudes to this changed point of view if we could change the name of income to outcome, or output. Income suggests coming toward us while outcome suggests coming from the source. Thus the outcome from a farm is the net value of its crops; the outcome from a railway company is its dividends, etc.
Under this new procedure, we credit each item of income as outcome from its source and debit every negative item. Negative items of income are outgo. If we could change this name also, we would call it ingo, or input.
It is a mere clerical matter of bookkeeping thus to credit to its source every service rendered as so much outcome (or income) and debit it with every disservice rendered, as so much ingo (or outgo).
Having suggested these new terms, however, so that the student may mentally, or literally by lead pencil, substitute them for the old, I shall hereafter, for simplicity, adhere uniformly to the original terminology, using the term income even when we are thinking merely of its coming from its capital source while the recipient is forgotten.
8.  They may be so regarded in cases where labor is paid by the piece and the laborer is free to stop work at any point.
9.  See The Nature of Capital and Income, Chapters VII-X.
10.  Except, as already stated in a previous footnote, he derives in addition to this obvious income other less tangible and more subtle income from the sense of possession, prestige, power, etc., which go with great wealth.
11.  For fuller treatment of this subject the reader is referred to: The Nature of Capital and Income; Are Savings Income, Journal of American Economic Association, Third Series, Vol. IX, No. 1, pp. 1-27; The Income Concept in the Light of Experience, privately printed as English translation of article in Vol. III, of the Wieser Festschrift, Die Wirtschaftstheorie der Gegenwart, Vienna, 1927, 29 pp.
12.  In all three examples, each month's income is represented by a rectangular column or bar. In the last two cases, the resultant row of bars makes a series of flat tops or steps. But by taking days instead of months, we come nearer to a sloping curve which is a better and simpler ideal picture. Hereafter we shall use such continuous curves. But they may always be thought of as made up approximately of a series of columns or bars. For fuller discussion of such charts, see The Nature of Capital and Income, p. 204 ff.
13.  Cf. Fetter, Frank A., Interest Theories Old and New, American Economic Review, March, 1914, pp. 76 and 77; Fetter, Principles of Economics, pp. 122-127; Davenport, H.J., Interest Theory and Theories, American Economic Review, Dec., 1927, pp. 636, 639.
14.  For instance, Dr. Dublin computes the total value of the "human capital" of the United States to be 1,500 billion dollars, or about five times the value of all other capital.
Part I, Chapter 2
15.  A brief outline of the history and theory of appreciation and interest is given in the Appendix to Chapter V of The Rate of Interest.
16.  Since, because of ignorance and indifference, appreciations and depreciations are, as a matter of fact, never fully foreknown and their relation to interest and other business phenomena only dimly perceived, they are only partially provided against in the rate of interest itself. As I have tried to show in Stabilizing the Dollar, in The Money Illusion, and in other writings, the best remedy is to standardize, or stabilize, the dollar as we have standardized every other important unit of measure employed in business.
17.  Appreciation and Interest, Publications of the American Economic Association, Third Series, Vol. XI, No. 4, Aug., 1896, pp. 331-442.
18.  The Rate of Interest, Appendix to Chapter V.
19.  This is strictly true only when the rate of interest in each standard is reckoned momently, or continuously. If, as in practice, the rate is reckoned quarterly, semi-annually, or annually, this equation is slightly altered. For the mathematical demonstration of this proposition, see Appendix to Sec. 3, Chapter V, of The Rate of Interest. For the significance of "continuous reckoning," see The Nature of Capital and Income, Chapter XII; also Appendix to Sec. 12 of Chapter XIII.
20.  Böhm-Bawerk, The Positive Theory of Capital, pp. 252 and 297; Landry, L'Intérêt du Capital, p. 49.
21.  That the appreciation or depreciation of money does actually influence the rate of interest, even though feebly, is now well recognised by those who have given attention to the subject. See Professor Marshall's testimony, Indian Currency Report, p. 169; and Johnson, Money and Currency. Boston, Ginn & Company, 1905.
22.  The need to show income and capital accounts in terms of value instead of money units is now coming to be recognized by progressive accountants. For a clear and complete statement of this new principle in accounting, the reader is referred to Ernest F. DeBrul. Unintentional Falsification of Accounts. National Association of Cost Accountants Bulletin, Vol. IX. Pp. 1035-1058. New York, National Association of Cost Accountants, 1928. The same theory is expounded by Henry W. Sweeney in his doctor's dissertation, Stabilized Accounting, accepted by Columbia University but not yet published. See also his article, German Inflation Accounting, in the Journal of Accountancy, February, 1928, pp. 104-116; Dr. Walter Mahlberg. Bilanztechnik and Bewertung bei Schwankender Währung. Leipzig, G. A. Gloeckner, 1923; E. Schmalenbach. Grundlagen Dynamischer Bilanzlehre. Leipzig, G. A. Gloeckner, 1925; Lucien Thomas. La Tenue des Comptabilités en Période d'Instabilité Monétaire. Paris, éditions d'experta, 1927; Lucien Thomas. La Révision des Bilans à L'Issue de la Période d'Instabilité Monétaire. Paris, éditions d'experta, 1928. Also compare Dr. F. Schmidt, Die Industriekonjunktur—ein Rechenfehler! Zeitschrift für Betriebswirtschaft, Jahrg. 1927, and Ist Wertänderung am ruhenden Vermögen Gewinn oder Verlust?, Zeitschrift für Betriebswirtschaft, Jahrg. 1928.
Part I, Chapter 3
23.  In Chapters XI and XII supply and demand curves will be derived from the principles of impatience and opportunity. It will also be shown that impatience is not to be associated with demand to the exclusion of supply, nor opportunity with supply rather than demand, nor vice versa.
24.  While this assertion is made dogmatically at this point, the proof of its soundness is contained in Chapter XX, §7, and in the Appendix to Chapter XX.
25.  Böhm-Bawerk, Capital and Interest, p. 342.
26.  The Nature of Capital and Income, Chapter XI.
27.  It is true, however, as will become more apparent later, that if the increase in productivity is foreseen, the rate of interest will be temporarily raised. But after the transition period is over and the supposed doubled productivity is thereafter going on at a steady rate, the rate of interest will fall back; in fact, other things equal, it will fall below what it was before productivity was increased.
28.  For fuller discussion see The Rate of Interest, pp. 38-51, and below Chapter XX, §7 and the Appendix to Chapter XX of this volume.
Part II, Chapter 4
29.  Or ophelimity, utility, wantability, or the want for one more unit. See The Nature of Capital and Income, Chapter III; also my articles, Is 'Utility' the Most Suitable Term for the Concept It Is Used to Denote? American Economic Review, June, 1918, pp. 335-337; and A Statistical Method for Measuring "Marginal Utility" and Testing the Justice of a Progressive Income Tax. Economic Essays Contributed in Honor of John Bates Clark, pp. 157-193.
30.  To be more specific, we obtain the rate of time preference for a present dollar over a dollar one year hence by the following process:
(b) the present want for one more dollar due one year hence; and then
(c) subtract (b) from (a); and finally
(d) measure the result (c) as a percentage of (b).
In terms of the usual illustrative figures, if a present dollar is worth 105 wantabs (want units) and a next year's dollar is now worth 100 wantabs, then the difference, 5, is 5 per cent of the latter.
For a more strictly mathematical formulation, see Appendix to Chapter XII, §1.
31.  See The Nature of Capital and Income, Chapter X.
32.  It is true, of course, that, in determining economic equilibrium, every variable theoretically affects every other, and the rate of interest, as one variable, must therefore be assumed to affect indirectly the price of everything else by affecting its supply and demand.
33.  For a statement of the theory of valuation in general, see Walras, Éléments d'Économie Politique Pure; Pareto, Cours d'Économie Politique; also, Manuel d'Économie Politique.
34.  Cf. Landry, L'Intérêt du Capital, Chapter X, § 149, pp. 311-315; §150, pp. 315-317.
35.  Böhm-Bawerk, The Positive Theory of Capital, p. 249.
36.  For a theoretical discussion of marginal want as a function of various factors, see my Mathematical Investigations in the Theory of Value and Prices. For a mathematical formulation of impatience as a function of successive installments of income, see Appendix to this chapter, §7. See also Pareto, Manuel d'Économie Politique, p. 546 et seq.
37. Cf. Rae, Sociological Theory of Capital, p. 54; Böhm-Bawerk, The Positive Theory of Capital, Book V, Chapter III.
38.  To be exact, however, we should observe that lack of foresight may either increase or decrease time preference. Although most persons who lack foresight err by failing to give due weight to the importance of future needs, or, what amounts to the same thing, by estimating over-confidently the provision existing for such future needs, cases are to be found in which the opposite error is committed; that is, the individual exaggerates the needs of the future or underestimates the provision likely to be available for them. Such people stint themselves needlessly, even impairing health by insufficient food in their efforts to save for the dreaded future. Their lack of foresight in this case errs in under-estimating instead of overestimating future income and so makes them too patient instead of too impatient. But in order not to complicate the text, only the former and more common error will be hereafter referred to when lack of foresight is mentioned. The reader may, in each such case, readily add the possibility of the contrary error.
39.  Rae, Sociological Theory of Capital, p. 54.
40.  Rae, The Sociological Theory of Capital, pp. 53-54.
41.  See Rae, The Sociological Theory of Capital, p. 97.
42.  Tarde, G. Social Laws. English translation, New York, Macmillan and Co., 1899. Also Les Louis de l'Imitation. Paris, Germer Baillière et C., 1895.
43.  The Psychology of Socialism. English translation. London, T. Fisher Unwin, 1899. Also The Crowd.
44.  Social and Ethical Interpretations in Mental Development. New York, Macmillan and Co., 1906.
45.  Bagehot, Walter. Physics and Politics. New York, D. Appleton and Co., 1873, Chapter III.
46.  See The Sociological Theory of Capital, Appendix, Article 1, pp. 245-276.
47.  See The Nature of Capital and Income, Chapter XIV.
48.  One of the few defects in Rae's analysis of interest, or at any rate of his statement of it, is his emphasis on the accumulation of capital. Since this accumulation is merely in anticipation of future income, the emphasis belongs on the latter.
Part II, Chapter 5
49.  One consequence of this assumption (to secure which the assumption was really made) is that the capitalized value of each person's income at a given rate of interest will be unchangeable. He is, so to speak, on a definite allowance, and any trading, as by borrowing, or mortgaging the future, cannot make him richer or poorer. It can only shift his income in time (with interest). The theoretical significance of this constancy will appear in the second approximation.
50.  The above-mentioned 10 per cent and 8 per cent rates of time preference are not rates actually experienced by him; they merely mean the rates of preference which he would have experienced had his income not been transformed to the time shape corresponding to 5 per cent. As in the general theory of prices, this marginal rate, 5 per cent, being once established, applies indifferently to all his valuations of present and future income. Every comparative estimate of present and future which he actually makes may be said to be "on the margin" of his income stream as actually determined.
51.  This is the case mentioned by Carver (The Distribution of Wealth, pp. 232-236), when he remarks that a man with $100 in his pocket would not think of spending it all on a dinner today, but would save at least some of it for tomorrow. Whether these conformations of the income stream resulting in zero or negative preference may ever actually be reached so that the market rate of interest itself may be zero or negative is another question.
52.  Thus, by saving, some writers understand that capital necessarily increases, and hence the income stream is made to ascend; others, like Carver (loc. cit., p. 232), apply the term broadly enough to include the case where a descending income is simply rendered less descending. The latter view harmonizes with that here presented. Saving is simply postponing enjoyable income.
53.  We do not here need to argue as to the zero or starting point from which we measure net total desirability. The crest of a hill is the highest point whether the height is measured above sea level or from the center of the earth.
54.  See my Mathematical Investigations in the Theory of Value and Prices.
Part II, Chapter 6
55.  See Chapter I, or The Nature of Capital and Income, Chapter X.
56.  See my Mathematical Investigations in the Theory of Value and Prices.
57.  It is, of course, realized that the principles of price determination involve interest just as the principles determining interest involve prices. A complete picture of economic equilibrium includes every possible variable, each acting and reacting on the others. Theoretically we cannot determine the price of bread by itself and then go on to determine, each separately, other prices, including the rate of interest. Theoretically any analysis of one part of the economic organism must include an analysis of the whole, so that a complete interest theory would have to include also price theory, wage theory and, in fact, all other economic theory.
But it is convenient to isolate a particular element by assuming the other elements to have been determined. So this book is a monograph, restricted, so far as may be, to the theory of interest, and excluding price-theory, wage theory and all other economic theory. Afterward it will be easy to dovetail together this interest theory, which assumes prices predetermined, with price theory which assumes interest predetermined, thus reaching a synthesis in which the previously assumed constants become variables. But all the principles remain valid.
58.  See The Nature of Capital and Income, Chapters VIII, IX, XVII.
59.  That this is the case under our present hypothesis is shown fully in Chapter XIII.
60.  The details of such a multiform equilibrium are given in mathematical terms in Chapter XIII.
Part II, Chapter 7
61.  Positive Theory of Capital, p. 277, footnote.
62.  In case the advantages (returns) precede the disadvantages (costs), as is the case when the merits of the mining use are compared with those of the farming use, the proposition must be reversed, as follows: The earlier advantage will be chosen only in case the rate of future costs over present returns is less than the rate of interest. In such a case it would be more convenient, in comparing the two options, to regard them in the reverse order, that is, to consider the advantages of the farming use over the mining use, so that the disadvantages may come first, i.e., the investment precede the returns. As long as the costs always precede the returns, we need only to consider whether or not the rate of return over cost exceeds the rate of interest.
63.  Inasmuch as we assume that the income from the forest is all to accrue at one time—the time of cutting—instead of being distributed over a long period, the income stream becomes a single jet and might here better be called income item.
64.  The Nature of Capital and Income, pp. 221-222.
65.  The Nature of Capital and Income, Chapter XIII.
66.  Del Mar, Alexander. Science of Money. New York, Macmillan & Co., 1896; George, Henry. Progress and Poverty. New York, Sterling Publishing Co., 1879. For a general criticism of this theory see Lowry, The Basis of Interest, American Academy of Political and Social Science, March, 1893, pp. 53-76
67.  See Chapter I or The Nature of Capital and Income, Chapters VII, VIII, IX, and X.
68.  Ibid., p. 317.
Part II, Chapter 8
69.  A somewhat similar treatment is that of Professor Harry G. Brown, Economic Science and the Common Welfare. Mathematical treatments substantially in harmony with mine are those of Walras and Pareto referred to in Appendix to Chapter XIII.
70.  H. G. Brown, Economic Science and the Common Welfare, pp. 137-145.
71.  p. 232. See also Carver, T. N. The Place of Abstinence in the Theory of Interest, Quarterly Journal of Economics, Oct., 1893, pp. 40-61.
72.  Brown, Economic Science and the Common Welfare, pp. 137-147.
73.  See The Nature of Capital and Income, Chapter XIV, No. 4.
74.  Rae, The Sociological Theory of Capital, p. 47.
75.  The Nature and Necessity of Interest, p. 122.
Part II, Chapter 9
76.  For a more complete treatment of the relation of risk to the interest yield of securities, see The Nature of Capital and Income, Chapter XVI.
77.  Even in the first and second approximations the rate of interest for different periods would not necessarily be the same.
78.  See Carver, The Distribution of Wealth, p. 256, and Cassel, A Theory of Social Economy, pp. 246-247.
79.  See The Nature of Capital and Income, Chapter XVI, §6.
80.  Ibid., Appendix to Chapter XVI, p. 409.
81.  Smith, Edgar L. Common Stocks as Long Term Investments. New York, The Macmillan Company, 1924.
82.  Van Strum, Kenneth, Investing In Purchasing Power. Boston, Barron's, 1925.
83.  See The Money Illusion. Also When Are Gilt-Edged Bonds Safe? Magazine of Wall Street, Apr. 25, 1925.
84.  See The Nature of Capital and Income, Chapter XVI.
85.  See the writer's Economics as a Science, Proceedings of the American Association for Advancement of Science, Vol. LVI, 1907.
86.  See Appendix to Chapter IX, No. 1.
Part III, Chapter 10
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.  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.  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.  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.  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.  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.  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.  A more complete expression, in mathematical terms, applying to any number of years, is given in Chapter XII.
Part III, Chapter 11
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.  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.  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.  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.  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.  See Chapter VIII, §4, for discussion of these examples.
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.  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.)
Part III, Chapter 12
17.  In Chapter V, §9, and in Chapter IX, §9, first of the three lines, in each case.
18.  In Chapter X, §15.
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.  The term "year" is used for simplicity, but "month" or "day" would be equally admissible and would do less violence to facts.
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.  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.
Part III, Chapter 13
23.  These have already been expressed in words in Chapters VI and IX and geometrically in Chapter XI.
24.  For the mathematical statement on this equivalence see Appendix to this chapter (Chapter XIII), §3.
25.  See Appendix to this chapter (Chapter XIII), §1.
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,
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
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.  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.  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.  For brief comparison of Chapters XII and XIII with other mathematical formulations see Appendix to this chapter (Chapter XIII).
Part III, Chapter 14
30.  See The Nature of Capital and Income, Appendix to Chapter XVI.
31.  See my article, Economics as a Science, in Proceedings of the American Association for the Advancement of Science, Vol. LVI, 1907.
32.  See Mitchell, Wesley C., Quantitative Analysis in Economic Theory. American Economic Review, Vol. XV, No. 1, March 1925, pp. 1-12.
Part IV, Chapter 15
1.  See Chapter I; also The Nature of Capital and Income, Chapter IX.
2.  See my Mathematical Investigations in the Theory of Value and Prices. New Haven, Yale University Press, 1926.
3.  This is, of course, highly theoretical; it assumes a competitive market free from legal and other restrictions.
4.  See Cannan, Edwin, Theories of Production and Distribution. London, P. S. King & Son, 1903.
5.  Pareto, Cours d'Êconomie Politique, Vol. II, Book III.
6.  King, W. I., The Wealth and Income of the People of the United State, New York, The Macmillan Co., 1915.
7.  Mitchell, W. C., King, W. I., Macaulay, F. R., Knauth, O. W., Income in the United States. New York, National Bureau of Economic Research, Inc., 1922.
Knauth, O. W., Distribution of Income by States in 1919. New York, Harcourt, Brace & Co., 1922.
Leven, Maurice, and King, W. I., Income in the Various States; Its Sources and Distribution, 1919, 1920, and 1921. New York, National Bureau of Economic Research, Inc., 1925.
8.  Stamp, Sir Josiah, Wealth and Taxable Capacity. London, P. S. King & Son, Ltd., 1922. Also, British Incomes and Property. London, P. S. King & Son, 1916.
9.  Rae, The Sociological Theory of Capital, Chapter XIII.
10.  See Day, Clive. The Dutch in Java. New York, The Macmillan Co., 1904, Chapter X.
11.  See Gonner, Interest and Savings.
Part IV, Chapter 16
12.  Report on Recent Economic Changes, National Bureau of Economic Research, p. 87.
13.  Ibid., Chapter XII.
14.  The economic effects of invention, and particularly its effects upon the rate of interest, were well treated by John Rae, The Sociological Theory of Capital, Chapter IX, pp. 132-150; Chapter X, pp. 151-203.
Part IV, Chapter 18
15.  The Sociological Theory of Capital, pp. 128-129.
16.  My colleague, Professor Clive Day, finds that interest rates in Java have advanced rather than declined since the early years of the twentieth century. He cites as the best source for recent conditions the Great Investigation (Onderzoek naar der mindere Welvaart de inlandsche Bevolking op Java en Madoera. Batavia, 1912, IX bl Dl II. 1, page 66). The report states that on small loans under 5 florins unsecured, the annual rate runs to several hundred per cent. On secured loans, say of 25 florins more or less, the rate varies from 36% to 60%. These rates are notably higher than his estimate of 40% quoted in The Rate of Interest, p. 292.
17.  Professor J. S. Lawrence, of Princeton University, informs me that needy negroes often pay at the end of the week, $7.00 for $5.00 borrowed at the beginning of the week, or more than 40% per week.
18.  See Bloch, Ivan, The Future of War. Translated by R. C. Long. New York, Doubleday and McClure Co., 1899, p. 205. It appears that the peasant would sell a promise to labor a short time in the future at one third the current wages! See also Lanin (pseud.), Russian Finance, Fortnightly Review, Vol. LV, February, 1891, pp. 188, 190, 196, for typical and extreme cases. Inostranietz, L'Usure en Russie, Journal des Economistes, Vol. XVI, Ser. 5, 1893, pp. 233-243, states that the rates paid by poor peasants to well-to-do peasants are frequently 5 per cent per week.
19.  The Sociological Theory of Capital, pp. 71-72.
20.  The Sociological Theory of Capital, pp. 64, 95-99, 129. Rae's authority for Roman interest rates is Boucher, Histoire de l'Usure, p. 25.
21.  Seligman, Edwin R. A., Principles of Economics. London, Longmans, Green and Co., 1907, p. 404.
22.  Brown, The Development of Thrift.
23.  The Sociological Theory of Capital, pp. 88-89 and 92.
24.  For details as to thirteen typical loans of this character, see U. S. Bureau of Labor Bulletin, No. 64, May, 1906, pp. 622-633. Thus "loan 1," 143 per cent, "loan 3," 224 per cent, "loan 7," 156 per cent. For later facts see Ryan, Usury and Usury Laws; also the publications of the Russell Sage Foundation on Small Loans, especially Raby, The Regulation of Pawnbroking.
25.  Longfield, Mountifort. The Tenure of Land in Ireland in Probyn's Systems of Land Tenure in Various Countries. London, Cassell, Petter, Galpin & Co., 1881, p. 16.
26.  From a letter to the author from Professor E. W. Kemmerer; see also his article in The Business Monthly, Pittsburgh, April, 1907, p. 2.
27.  On the uncertainties of Indian life, see Rae, The Sociological Theory of Capital, pp. 69 and 70.
28.  Hurlbert, H. B. The Passing of Korea. New York, Doubleday, Page and Co., 1906, p. 283.
29.  See Appendix to Chapter XIX, § 1.
30.  Plehn, Carl C. Notes Concerning the Rates of Interest in California. Quarterly Publications of the American Statistical Association, September, 1899, pp. 351-352.
31.  Plehn, p. 353.
32.  See Message of the Governor of the State of Nevada, 1879.
33.  Mines and Quarries, 1902. Special Report U. S. Census, p. 255.
34.  See later Messages of the Governor of the State of Nevada.
35.  Zartman, Lester W. The Investments of Life Insurance Companies. New York, Henry Holt and Co., 1906, p. 103.
36.  Giffen, Growth of Capital, p. 44.
37.  Ibid., p. 35.
38.  Ibid., pp. 40-42.
39.  See U.S. Bureau of Labor Bulletin, No. 64, May, 1906, pp. 622 ff.
40.  The average rates for each month plotted on the chart were compiled from the daily rates published in The Financial Review.
41.  The Review of Economic Statistics, January, 1923, pp. 17-27.
42.  See Palgrave, Bank Rate and the Money Market, p. 97.
Part IV, Chapter 19
43.  Moses, Bernard, Legal Tender Notes in California, Quarterly Journal of Economics, October, 1892, p. 15.
44.  This table has been obtained by the aid of the usual brokers' bond tables. In the case of currency bonds, it was only necessary to deduct accrued interest (if any) from the quoted price and look in the table for the interest which corresponds to the price so found and the number of years to maturity. This gives the rate in terms of "currency." In the case of coin bonds, since the quotations are given in currency, it is necessary to divide the quoted price by the price of gold in order to obtain their price in gold (i.e., "coin"), and then proceed as above indicated. We thus get the rate in terms of "coin." The quotations of prices of bonds and gold are the "opening" prices for the months named, and are taken from the Financial Review and its annual summary, The Commercial and Financial Chronicle, 1895, the (New York) Bankers' Magazine and the Bankers' Almanac. After 1884, January quotations were not always available.
45.  Specie payments were resumed in 1879 and thenceforward the price of gold was, of course, 100.
46.  The silver bonds or "rupee paper" were issued to raise loans in India, but they were also enfaced for payment in England, and in 1893-1894 some Rx. 25,000,000 were on the London books. Burdett's Official Intelligencer (1894), p. 75.
47.  Thus, in 1880 the average price paid in London for "rupee paper" of face value Rx. 1000 yielding 4 per cent, or Rx. 40 per annum, was £79. In order to find the rate of interest realized by the investor, we must translate £79 into silver. The average rate of exchange in 1880 was 20d. per rupee. Hence £79 were equivalent to 948 rupees. That is, speaking in terms of silver (or, more exactly, in terms of exchange on India), the price of a 4 per cent bond was 94.8, which, if the bond be treated as a perpetual annuity, yields the investor 4.3 per cent. In the same year, an India gold bond yielded 3.6 per cent.
48.  This table is made from averages of (usually ten) quotations distributed through each year, taken from the Economist, the Investor's Monthly Manual, and the (London) Banker's Magazine. The fourth column is founded on the table in the Report of the Indian Currency Committee (1893), p.27, but it is corrected to apply to calendar instead of official years.
49.  The quotation from which the interest was computed for 1895 and succeeding years is for 3½ per cent rupee paper. All previous quotations are for 4 per cent's. The 4 per cent's were repayable on three months notice; this notice was given in 1894, and the bonds redeemed or converted into 3¼ per cent's before the close of the year. To obtain the rate of interest realized, the London quotations in pound sterlings are first converted into rupees at the current rate of exchange, and the bonds are treated as perpetual annuities. The results differ from those given in the Investor's Monthly Manual, because the rupee is there converted at a conventional value, not the market value.
50.  From 1865 to 1880 inclusive the figure refer to 4 per cent's, repayable October, 1888, or later; those of 1881 are for 3¼ per cent's maturing in 1931 and those for 1885 to 1906 are for 3 per cent's maturing in 1948.
51.  The preceding comparisons serve only to establish the influence of the expected divergence between the two standards on the rates of interest, but afford no exact measure of that influence. In order to measure the extent to which the fall of silver was allowed for by investors, it would be necessary to examine the rates realized during specific periods. Somewhat unsatisfactory attempts to do this (both for the case above of American gold and "currency" bonds and for the case below of Indian gold and "rupee bonds") were made in my Appreciation and Interest, but are not reproduced here.
52.  The basic tables for computing the rate of price changes and the method of computation are given in the Appendix to this chapter §1.
53.  See Journal of the American Statistical Association. June, 1925, p. 81, footnote 3.
54.  The British bond yields here used were taken from an article by A. H. Gibson, The Future Course of High Class Investment Values, The Bankers', Insurance Managers' and Agents' Magazine, January, 1923, p. 15. Reprinted in revised form in The Spectator, March 7, 1925.
55.  The United States bond yields here used were taken from the Statistical Bulletin, 1928-1929, of the Standard Statistics Company. Percentage price changes were computed from the Wholesale Commodity Price Indexes of the United States Bureau of Labor Statistics.
56.  The theory of distributed influence and lag was developed incidentally in the course of my studies of price-trade relations reported in several papers. See (a) The Business Cycle Largely a "Dance of the Dollar," Journal of the American Statistical Association, December, 1923, p. 5, top paragraph; (b) Fluctuations in Prive-Levels in The Problem of Business Forecasting by Warren M. Persons, William T. Foster, Albert Hettinger. Boston, Houghton Mifflin Company, 1924, pp. 50-52; and particularly, (c) Our Unstable Dollar and the So-Called Business Cycle, Journal of the American Statistical Association, June, 1925, pp. 179-202.
57.  A price change P'm pertaining to the month tm exerts an influence, F(tm + l), whose intensity is proportional to 8 during tm+3, to 7 during tm+4,..., to 1 during tm+10, to 0 during tm+11.
Conversely the aggregate influence on the affected variable during the month tm consists of the various P'm-l which enter with the following weights tapering off in arithmetical progression; P'm-3 with weight 8, P'm-4 with weight 7,...., P'm-10 with weight 1, P'm-11 with weight 0. The numerical measure of this composite influence is:
the divisor 36 being the sum of the weights,
58.  Interest rates were taken from The Statistical Bulletin, 1928-1929, of The Standard Statistics Co. The price indexes are computed from the United States Bureau of Labor Statistics. See Appendix to this chapter, §2, for these quarterly data.
59.  In the form of distribution of influence used in this study, the remote price changes have comparatively little weight in the composite . For example, in the case of a 20 term influence-range straight line distribution,
It is evident, and I have tested it thoroughly, that the composite would be only slightly changed if the tail half of the distribution were omitted. The distribution in the above example would then be,
For the United States yearly data 1890-1927, the correlation of i and with an influence range of 20 years was r = + 0.857. For this distribution truncated by chopping off the end quarter, r = +0.843; the end half, r =+0.798; and the end three quarters r =+0.439.
60.  See Pearson and Elderton, The Variate Difference Method, Biometrica, Vol. XIV, 1923, pp. 281-309; W. M. Persons, Statistics and Economic Theory, Review of Economic Statistics, Vol. VII, No. 13, July, 1925, pp. 179-197; Anderson, The Decomposition of Statistical Series into Components, Journal of the Royal Statistical Society, Vol. XC, pt. 3, 1927, pp. 548-570.
61.  See especially Our Unstable Dollar and the So-Called Business Cycle, Journal of the American Statistical Association, June, 1925, pp. 181-202; A Statistical Relation Between Unemployment and Price Changes, International Labour Review, Vol. XIII, No. 6, June, 1926, pp. 785-792.
62.  Snyder, Carl, The Influence of the Interest Rate on the Business Cycle, American Economic Review, Vol. XV, No. 4, Dec., 1925, pp. 684-699; Interest Rates and the Business Cycle, American Economic Review, Vol. XVI, No. 3, Sept., 1926, pp. 660-663.
63.  Ayres, Cleveland Trust Co., Business Bulletin, June 15, 1928, and Aug. 15, 1928.
64.  Mitchell, Interest Cost and the Business Cycle, American Economic Review, Vol. XVI, No. 2, June, 1926, pp. 209-221; Supplementary Note on Interest Costs, American Economic Review, Vol. XVI, No. 3, Sept., 1926, pp. 451-452.
65.  See the author's The Money Illusion. New York, Adelphi Company, 1928, pp. 41-42.
66.  Prof. Knut Wicksell was one of the first to recognize the influence of interest rates upon prices. See his book, Geldzins und Güterpreise; Prof. Alfred Marshall, Prof. Gustav Cassel, Rt. Hon. Reginald McKenna, Chairman of the Midland Bank of London, Mr. R. G. Hawtrey, of the Treasury of Great Britain, and many other well known economists, bankers, and business men have emphasized that business activity is influenced and may be largely controlled by manipulation of the discount rate.
67.  Mr. and Mrs. K. G. Karsten, for example, in their forecasts, make use of commercial paper rates in computing their forecasts of wholesale prices and business activity. They find an r of—0.98 between (1) the logarithms of P' and (2) the logarithms of the deviations of commercial interest rates from bond yields.
68.  For instance, Our Unstable Dollar and the So-Called Business Cycle, Journal of the American Statistical Association, June, 1925, pp. 179-202.
69.  Statistical Studies in the New York Money Market, Chapters VII and VIII.
70.  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.
71.  Cleveland Trust Company, Business Bulletin, June 15, 1928.
72.  Conditions of this type emphasize the importance of thorough study of the institutional factors influencing market interest rates such as that made by Mr. A. W. Marget in his doctoral dissertation The Loan Fund presented to the faculty of Harvard University in 1926-1927.
Part IV, Chapter 20
73.  For historical development and detailed criticisms of interest theories the reader is referred to Böhm-Bawerk, Capital and Interest. A brief historical résumé is given in Cassel, The Nature and Necessity of Interest, Chap. I, pp. 1-67.
74.  These articles are included in the general bibliography at the end of the book.
75.  References to the authors of the views which I am stating impersonally in the text will be made, where possible, in footnotes. It is hoped that this procedure will prevent the impression that I am attempting any exhaustive critique of the interest theory of others.
76.  Davenport, H. J., Interest Theory and Theories. American Economic Review, Vol. XVII, No. 4, December, 1927, pp. 636-656.
77.  Ibid., p. 650.
78.  Fetter, Frank A., Clark's Reformulation of the Capital Concept. Economic Essays Contributed in Honor of John Bates Clark, pp. 151-152, footnote on p. 153; cf. also Flux, Irving Fisher on Capital and Interest, Quarterly Journal of Economics, Vol. XXIII, Feb., 1909, pp. 307-323.
79.  See my articles: Are Savings Income? Journal of the American Economic Association, Vol. IX, No. 1, April, 1908, pp. 1-27; Professor Fetter on Capital and Income, Journal of Political Economy, Vol. XV, No. 7, July, 1907, pp. 421-434; Comment on Professor Plehn's Address, American Economic Review, Vol. XIV, No. 1, Mar., 1924, pp. 64-67; and The Concept of Income in the Light of Experience, English reprint from Wieser Festschrift, Vol. III.
80.  A Reply to Critics, Quarterly Journal of Economics, Vol. XXIII, May, 1909, pp. 536-541.
81.  Fetter, Interest Theories, Old and New, American Economic Review, Vol. IV, No. 1, March, 1914, pp. 68-92.
82.  What is Capital? The Economic Journal, Vol. VI, Dec., 1896, pp. 509-534.
83.  The Rôle of Capital in Economic Theory, The Economic Journal, Vol. VII, Dec., 1897, pp. 511-537.
84.  Fetter, Clark's Reformulation of the Capital Concept. Economic Essays Contributed in Honor of John Bates Clark, p. 152.
85.  Seager, The Impatience Theory of Interest, American Economic Review, Vol. II, No. 4, December, 1912, pp. 834-851; Brown, Economic Science and The Common Welfare.
86.  The Impatience Theory of Interest, American Economic Review, Vol. III, No. 3, September, 1912, pp. 610-615.
87.  Clark, Distribution of Wealth, pp. 338-344.
88.  For example, Seager and Brown, cited above.
89.  Seager, cited above, p. 844.
90.  Seager, Brown, and Flux cited above; also Loria, Irving Fisher's Rate of Interest, Journal of Political Economy, Vol. XVI, Oct., 1908, pp. 331-332.
91.  Seager, cited above, pp. 842-3, 847.
92.  The Impatience Theory of Interest, American Economic Review, Vol. III, No. 3, September, 1913, pp. 614-615.
93.  Davenport, Value and Distribution.
94.  Cf. Seager and Flux, cited above, and particularly Fetter, Interest Theories Old and New, American Economic Review, Vol. IV, No. 1, March, 1914, pp. 69-72.
95.  Fetter, Interest Theories Old and New, American Economic Review, March, 1914, pp. 74, 76, 77.
96.  Ibid., p. 77.
97.  As is always possible in solving simultaneous equations we can, if we wish, express certain of the variables, such as those relating to productivity, or opportunity, namely the rates of return over cost, in terms of the other variables and thus seem to eliminate them. In my first book I tried, for the most part, thus to present the rate of return over cost, or as I then called it, the rate of return on sacrifice as determined through the rate of preference. But we can just as well in like manner eliminate time preference and present it in terms of rate of return.
98.  Landry, L'Intérêt du Capital, pp. 66-95.
99.  Seager, The Impatience Theory of Interest, American Economic Review, Vol. II, No. 4, December, 1912, pp. 835-837.
100.  The Impatience Theory of Interest. American Economic Review, Vol. III, No. 3, September, 1913, p. 610.
101.  Positive Theorie des Kapitales. "Meine These schränkt vielmehr diese Wirkung ausdrücklich auf 'klug gewählte' Verlängerungen ein und lässt überdies, indem sie ihr Zutreffen nur 'in aller Regel' oder, wie ich in der ersten Auflage sagte, 'im grossen und ganzen' behauptet, das Vorkommen von Ausnahmen offen." Exkurs I, p. 3.
"Wenn Fisher hiemit nicht mehr in Abrede stellen wollte, als was ich oben auf S. 3 des Exkurse I selbst in Abrede gestellt habe, dass nämlich nicht alle längeren Produktionsumwege nur deshalb, weil sie länger sind, auch produktiver sein müssen, so wären wir in vollem Einklang." Exkurs IV, p. 105.
102.  The Positive Theory of Capital, p. 266.
103.  In The Rate of Interest there is presented an exhaustive analysis of the validity of the proof presented by these tables of the theory of technical superiority advanced by Böhm-Bawerk. While it does not seem advantageous to repeat here this analysis, since it is available in my previous book to any reader who may be interested, it is pertinent to present the conclusions drawn therefrom.
104.  The Positive Theory of Capital, p. 268.
105.  The Positive Theory of Capital, p. 268.
106.  Positive Theory of Capital, p. 269.
107.  Positive Theory of Capital, p. 269.
108.  This is true under the assumption, implied in Böhm-Bawerk's tables, that the product is the same except as to the time of its availability, namely, that the series of figures called "units of product" are identical as shown in his tables.
109.  Positive Theory of Capital, pp. 269 and 270.
110.  See von Bortkiewicz, Der Kardinalfehler der Böhm-Bawerkschen Zinstheorie. Jahrbuch für Gesetzgebung, Verwaltung und Volkswirtschaft, 1906, pp. 61-90.
See also von Schaposchnicoff, Die Böhm-Bauerksche Kapitalzinstheorie. Jahrbücher für Nationalökonomie und Statistik. Verlag von Gustav Fischer, Jena, Dritte Folge, Bd. XXXIII (LXXXVIII), pp. 433-451.
111.  Positive Theorie des Kapitales. Dritte Aufiage, and Exkurse zur Positive Theorie des Kapitales. See especially Exkurse IV and Exkurse XII.
112.  Böhm-Bawerk claims that merely to find out the factors operative in a given problem is not the same thing as to explain those factors. He thinks that my theory of interest would be adequate only if the mathematical solution of the problem by means of simultaneous equations, and what he calls the "causal" solution were the same, or at least somewhat similar.
Of course, these two types of solution are not the same. The causal solution cannot be so simply conceived as to make one factor solely cause and another solely effect. The advance of all science has required the abandonment of such simplified conceptions of causal relationship for the more realistic conception of equilibrium.
Here, all factors are considered as variables. Any disturbance in one factor reacts on all the others, and the variations in these other factors react upon the factor of original disturbance. The mathematical solution of the problem of interest by means of simultaneous equations recognizes the mutual interdependence of all the factors in the interest problem and, at the same time, yields a determinate solution for the problem.
113.  See Appendix to this Chapter, §1.
114.  Veblen, Fisher's Rate of Interest, Political Science Quarterly, Vol. XXIV, June, 1909, pp. 296-303. Marget, The Loan Fund, a doctoral dissertation at Harvard University, 1926; Schumpeter, Theorie der Wirtschaftlichen Entwicklung, p. 363. Criticisms of a similar character have been received in private correspondence from several of those who read this book in manuscript, particularly Professors L. D. Edie, B. H. Beckhart, and C. O. Hardy.
Part IV, Chapter 21
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.
Appendix to Chapter 1
1.  American Economic Review, Vol. XIV, p. 64.
Appendix to Chapter 12
2.  See any text book on the calculus, e.g. Wilson, E. B. Advanced Calculus. Boston, Ginn & Co., 1912, pp. 118-125.
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.
Appendix to Chapter 13
4.  Walras, Leon, Élements d'Économie Politique Pure.
5.  Pareto, Vilfredo, Cours d'Economie Politique.
6.  Cours d'Économie Politique, Vol. I, p. 314.
Appendix to Chapter 20
7.  See Böhm-Bawerk, Recent Literature on Interest (1884-1899), p. 35.
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.  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.  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.