The Theory of Interest
PART III, CHAPTER XIII

For  Impatience  Principle  A, n(m  1)  equations 
"  "  "  B, n(m  1)  " 
"  Market  "  A, m  " 
"  "  "  B, n  " 
For  Investment  Opportunity  Principle  A, n equations  
"  "  "  "  B, n(m  1)  " 
The sum total of these is 3n(m  1) + 2n + m, or 3mn + m  n.
To compare this number with the number of unknowns, we note that all the unknowns in the first approximation are repeated;
the  number  of  f's  being  n(m  1) 
"  "  "  x's  "  mn 
"  "  "  i's  "  m  1 
making a total of 2mn + m  n  1 carried forward from the first approximation.
In addition, the new unknowns, the y's and the r's, are introduced. There is one y for each individual for each year, the total array of y's being
The number of these y's is evidently mn.
There is one r for each individual for each pair of successive years, i.e., firstandsecond, secondandthird, etc., and nexttolastandlast years, the total array of r's being
The number of these r's is evidently n(m  1).
In all, then, the number of new unknowns, additional to the number of old unknowns carried forward from the first approximation, is mn + n(m  1), or 2mn  n.
Hence we have:
number of old unknowns, 2mn + m  n  1, 
+ number of new unknowns, 2mn  n, 
= total number of unknowns, 4mn + m  2n  1, 
as compared with 3mn + m  n equations.
The reconciliation of these two discordant results is effected by two considerations. One reduces the number of equations. Just as under the first approximation, we have one less independent equation in the two sets expressing the Market Principles than the apparent number, thus making the final net number of equations
The other consideration is quite different. It subtracts from the number of unknowns. This can be done because each r, of which there are n(m  1), is a derivative from the y's. By definition r is the excess above unity of the ratio between a small increment in the y of next year to the corresponding decrement in the y of this year. The same applies to any pair of successive years. This derivative is, more explicitly expressed, a differential quotient.*25
The reader not familiar with the notation of the differential calculus will get a clearer picture of the inherent derivability of the r's from the y's by recurring to the geometric method in Chapter XI. There y' and y'' are shown as the coördinates ("latitude" and "longitude") of the Opportunity line, while r is shown as the tangential slope of that line. It is evident that, given the Opportunity line, its tangential slope at any point is derived from it. It is not a new variable but is included in the variation of y' and y'' as the position on the curve changes.
If, now, we subtract n(m  1), the number of the r's, from 4mn + m  2n  1, we have, as the final net number of unknowns,
which is the same as the total net number of independent equations. *26 Thus the problem is fully determinate under the assumptions made.
We have already seen (Chapter XI, §9), that zero or negative rates of interest are theoretically possible. In terms of formulas all that is needed to make the rate of interest zero is that the forms of the F and f functions shall be such as to produce this result. This implies that these functions shall have solution values equal to zero.
Of course it would be possible that interest, impatience, and return over cost for one particular year might be zero or negative without this being true for other years. If they were zero for all the years, we should have the interesting result that the value of a finite perpetual annuity (greater than zero per year) would be infinity. No one could buy a piece of land for instance, expected to yield a net income forever, for less than an infinite sum. A perpetual government bond from which an income forever was assured would have an infinite value. Since this is quite impracticable, we thereby reduce to an absurdity the idea that it is possible to have at one and the same time:
1. A zero rate of interest for each year forever; and
2. a perpetual annuity greater than zero per year.
But the absurdity is lessened or disappears altogether if either:
1. The zero rate of interest is confined to one year; or
2. no perpetual annuity greater than zero per year is possible.
Unusual conditions may easily reduce the rate of interest for one year to zero. As to an unproductive or barren world, like the hardtack island, only a finite totality of income would be possible; a perpetual annuity even of one crumb of hardtack a year would be impossible.
While this and the previous chapter are largely restatements in terms of formulas of Chapters X and XI in terms of diagrams, which, in turn were largely restatements of Chapters V and VI in terms of words, nevertheless, these formula chapters have a value of their own, just as did the geometric chapters.
In particular, the formula method has value in showing definitely the equality between the number of equations and the number of unknowns, without which no problem of determining variables is ever completely solved.
It is for this reason that these restatements are included in this book. In fact, if I were writing primarily for mathematically trained readers, I would have reversed the order, giving the first place to the formulas, following these with the charts for visualization purposes, and ending with verbal discussion. Each method contributes its distinctive help toward a complete understanding of what is, at best, a difficult problem to encompass by any method at all. I have, therefore, included in these formula chapters, as in the geometric ones, several points not well adapted to the more purely verbal presentations of Chapters V and VI.
Two corollaries follow. One is that any attempt to solve the problem of the rate of interest exclusively as one of productivity or exclusively as one of psychology is necessarily futile. The fact that there are still two schools, the productivity school and the psychological school, constantly crossing swords on this subject is a scandal in economic science and a reflection on the inadequate methods employed by these wouldbe destroyers of each other. Each sees half of the truth and wrongly infers that it disproves the existence of the other half. The illusion of their apparent incompatibility is solely due to the failure to formulate the problem literally and to count the formulas thus formulated.
The other corollary is that such a formulation reveals the necessity of positing a theoretically separate rate of interest for each separate period of time, or to put the same thing in more practical terms, to recognize the divergence between the rate for short terms and long terms. This divergence is not merely due to an imperfect market and therefore theoretically subject to annihilation by arbitrage transactions, as BöhmBawerk, for instance, seemed to think. They are definitely and normally distinct and due to the endless variety in the conformations of income streams. No amount of mere price arbitrage*27 could erase these differences.
Thus, there should always be, theoretically, a separate market rate of interest for each successive year. Since, in practice, no loan contracts are made in advance so that there are no market quotations for a rate of interest connecting, for example, one year in the future with two years in the future, we never encounter such separate year to year rates. We do, however, have such rates implicitly in long term loans. The rate of interest on a long term loan is virtually an average*28 of the separate rates for the separate years constituting that long term. The proposition affirming the existence of separate rates for separate years amounts to this: that normally there should be a difference between the rates for short term and long term loans, sometimes one being the larger and sometimes the other, according to the whole income situation.
The contention often met with that the mathematical formulation of economic problems gives a picture of theoretical exactitude untrue to actual life is absolutely correct. But, to my mind, this is not an objection but a very definite advantage, for it brings out the principles in such sharp relief that it enables us to put our finger definitely on the points where the picture is untrue to life.
The object of any theory is not to reproduce concrete facts but to show the chief underlying principles as tendencies. There is, for instance, the very real tendency for all marginal rates of time preference and all marginal rates of return over cost to equal the market rates of interest. Yet this is only a tendency, an ideal never attained.*29
which function is empirical and derivable from the opportunity function f already given.
Analogously we may express the equations of definition for r_{2}', r_{3}' .. r_{n}' and likewise for the corresponding r'''s, r''''s, etc., up to r^{(m)}'s making n(m1) 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), §15.
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.
Part III, Chapter 14
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