THE RATE OF INTEREST NATURE, DETERMINATION AND RELATION TO ECONOMIC PHENOMENA IRVING FISHER, PH.D. ITS PROFESSOR OF POLITICAL ECONOMY, YALE UNIVERSITY

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1 THE RATE OF INTEREST ITS NATURE, DETERMINATION AND RELATION TO ECONOMIC PHENOMENA BY IRVING FISHER, PH.D. PROFESSOR OF POLITICAL ECONOMY, YALE UNIVERSITY fgorfe THE MACMILLAN COMPANY 1907 All rights reserved

2 PREFACE THE problem of interest has engaged the attention of writers for two thousand years, and of economists since economics began. And yet, with the exception of what has been accomplished by Rae, Bohm-Bawerk, Landry, and some others, very little progress has been made toward a satisfactory solution. Even these writers can scarcely claim to have established a definitive theory of interest. While the value of their work is great, it is chiefly negative. They have cleared the way to a true theory by removing the confusions and fallacies which have beset the subject, and have pointed out that the rate of interest is not a phenomenon restricted to money markets, but is omnipresent in economic relations. The theory of interest here presented is largely based upon the theories of the three writers above mentioned, and may therefore be called, in deference to Bohm-Bawerk, an "agio theory." But it differs from former versions of that theory by the introduction explicitly of an income concept. This concept, which I have developed at length in The Nature of Capital and Income, is found to play a central role in the theory of interest. The difficult problem is not whether the rate of interest is an agio, or premium, for of this there can be no question, but upon what does that agio depend and in what manner? Does it depend, for instance, on the volume of money, the amount of capital, the productivity of capital, the "superior productivity of roundabout processes," the labor of the capitalist, the helplessness of the laborer, or upon some other condition? vii

3 v iii PKEFACE The solution here offered is that the rate of interest depends on the character of the income-stream, its size, composition, probability, and above all, in time. its distribution It might be called a theory of prospective provision of income. As in The Nature of Capital and Income, mathematics have here been relegated to appendices. These appendices are not, however, mere translations into mathematical language of the theory verbally expressed in the text. Mathematics can properly claim no place in economic discussions except as they add something not expressible, or at any rate only imperfectly expressible, in ordinary language. Parts of Chapters V and XIV with their appendices have appeared in somewhat different forms in Appreciation and Interest. My thanks are due to the American Economic Association for permission to use portions of this monograph unaltered. Since it appeared a decade ago, the view expressed in it, to the effect that appreciation of money should, and to some extent does, lower the rate of interest expressed in money, has gained considerable currency, though it is still unfamiliar to most persons. It has been thought wise to present again the statistical evidence in its favor, and to bring the statistics down to date. In the preparation of this book I have received important aid from many persons. For general criticism I am indebted to my wife, to my colleagues, Professors H. C. Emery and J. P. Norton, and to my friend Richard M. Kurd, President of the Mortgage-Bond Company of New York City. My thanks are also due to Finance Minister Bohm-Bawerk for his kindness in reading and criticising the chapter devoted to his theory of interest ; to Professor Clive Day for facts and references on the history of interest rates; to Dr. Lester W. Zartman for a large part o-

4 FIRST APPROXIMATION 387 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' is constant and #% etc., constant. Then the first term of the equation disappears and all after the third term, and the equation reduces to _dx'" BF( ) /SF( ) dx" = 8 as" / 8 x"' In other words, 1 + *'" = 1 +/", or t" =/" ; and so on for each succeeding year. We therefore see in mathematical language that the point of maximum desirability is also the point at which the marginal rate of preference for each year's income over the next year's income is equal to the rate of interest connecting these two years. We turn now to the geometrical interpretation. 7 ' In Figure 26, let the point P be found such that its coordinates are q f and Cj", the values of this year's income and next year's income respectively, c/ being laid off along the horizontal axis

5 388 APPENDIX TO CHAPTER VII OX 1 and, c/' being laid off vertically. In the same way any other income-stream may be represented by another point, the coordinates of which represent the values of this year's and next year's income respectively. By borrowing or lending, the income-stream P is changed to another point Q to be determined. We shall assume, as before, that the modification of this income-stream, c/, C/', through borrowing and lending or buying and selling applies only to the first and second years; all subsequent years are therefore omitted from our calculations. The income-stream that is the combination of the two magnitudes C/, <%", the fixed income with which the individual is supposed to be endowed, is represented by the point P. But this income-stream, c/, c^', he modifies by adding algebraically a^', through borrowing (or lending, if a^' has a negative value) this year, and adding next year, when the loan is paid, the sum a?/', which is equal to a;/ put at interest, but of opposite sign ; in ot herwords, a^" = x' (1-h i) 6r xj -f ^ L- =. 0. It is first proposed to determine xj and #/', assuming that the rate of interest is a fixed and given magnitude. The new income-stream, c' + x', c" + a;", will be represented by a new point, Q, different from P. To find this point, Q, draw the line AB through the" point P at a slope determined by the given rate of interest, namely, so that TT-: = 1 -f- i. Then the new point will lie somewhere upon this straight line. For (1) the present value of the modified income-stream c/4-a/, c^'-ha^" is the same as that of the original income-stream c/, c/' ; and (2) the straight line AB is the " locus " or assemblage of all points the present values of the income-streams represented by which are the same as the present value of the income-stream represented by the point P through which AB is drawn. That (1) the present values of the original and the modified income-streams are the same is due to the fact that the loan #/ and its repayment x\" are equivalent in present value. This may be seen by transforming the formula for the present value of the modified income-stream, as follows :

6 408 APPENDIX TO CHAPTER VIII This is, in analytical geometry, the equation of a plane which cuts the T1 axis at a distance K from the origin, and cuts the Y" axis at a distance -fit(l + i') from the origin, and the Y'" axis at a distance K(l -f i') (1 -f- i") from the origin. Similar considerations will show, just as in the case of the previous representation in two dimensions, that the farther the plane is from the origin 0, the larger the present value of the choices represented by points in this plane. Our problem, therefore, consists merely in finding that point on the bounding surface which is also on the plane farthest from among the parallel planes just drawn. This is evidently the point of tangency, and may be called t as before. It is also clear that a change in the rates of interest i 1 or i" will change the slope of the tangent plane, and therefore the point of tangency t. The algebraic interpretation of this case will be similar to the algebraic expression already given for two variables. When we proceed to consider four or more years instead of simply two or three years, the geometrical representation fails us, since the mind has difficulty in picturing spaces of 4, 5,... and m dimensions. In order to show how the new equations which have just been expressed enter into the determination of the problem of interest, we construct Fig. 29, applying to the case only two unknown quantities y' and y". The incomes for the third and 6 succeeding years are regarded for the moment as fixed. The diagram refers to a particular individual, and shows (1) the curve WPZ, giving the effective range of choice among different options open to him, and (2) a series of curves for total utility or desirability, as explained in the Appendix to the preceding chapter. The line AB is drawn at a slope equal to 1 + i and tangent to the curve WZ at the point P. This line will be tangent to one of the family of desirability curves at some point Q. P represents, out of all the options, the particular income-stream chosen by the individual. This incomestream P is, of course, as yet unmodified by borrowing and lending or buying and selling. The point Q represents the income-stream as finally thus modified. The coordinates of P are y' and y", and the coordinates of Q are y' + a;', y" -f- x", where x' and x" are the (algebraic) additions to the income y'.

7 SECOND APPROXIMATION 409 y" by borrowing and lending or buying and selling. P and Q thus represent graphically the double choice explained in Chapter VIII. We saw there that the individual first chooses, among the various eligible income-streams of different present values, that which is of maximum present value. WZ now represents the series of eligible income-streams, and P the income-stream of maximum present value. Also, we saw that after the individual had chosen this income-stream he modified it by selecting another income-stream of the same present value but of maximum desirability. Q now represents this final choice. It is worth our while in passing to emphasize that the individual would not follow out this program unless the final step ^ of modifying his income-stream by exchange were open to him. For, if he were shut off from exchange (i.e. compelled to accept P, unmodified to Q), the income-stream of maximum present value (P) would not necessarily be that of maximum desirability. In this case the maximum desirability would evidently be found at S, the point of tangency between WZ and a curve