The Theory of Interest
PART II, CHAPTER VI

TABLE 2 The Three Optional Income Streams 




A For farming 
B For forestry 
C For mining 

1st yr.  $ 450  $ 000  $2000 
2nd yr.  450  000  1800 
3rd yr.  450  300  1600 
4th yr.  450  400  1400 
5th yr.  450  500  1200 
6th yr.  450  500  1000 
7th yr.  450  500  800 
8th yr.  450  500  600 
9th yr.  450  500  400 
10th yr.  450  500  200 
11th yr.  450  500  000 
etc. 
etc. 
etc. 

Present Value  $9000  $8820  $9110 
The particular income stream selected will tend to leave its impress on the time shape of the total income stream of the individual who owns it. For, as was seen in Chapter I, the total net, or final, income stream of any individual during any interval of time is simply the sum total of the items of income flowing during that interval from all the articles of property belonging to him. Hence, if one selects the mining use for his land, whereby the income stream gradually decreases, its tendency will be to produce a similarly decreasing trend in the total income stream enjoyed by the individual. This tendency may be counteracted, of course, by some opposing tendency, but will have full sway if the income from all other capital than the land remains the same in value and time shape. It is true that the direct income from the mine is not itself real income, but consists of services which, relatively to some other capital source, are disservices, thus constituting intermediate income or interactions. But those items are readily transformed, through a chain of credits and debits, into real, and then into enjoyment income. Thus the ore of the mine is exchanged for money, and the money spent for enjoyable services or for commodities which soon yield enjoyable services, so that the real income closely copies in time shape *58 the original intermediate income from the mine.
The possessor of the mine, however, is not compelled thus to copy in his real income the mine's fluctuations of physical or natural income. He may counteract any fluctuations in his whole net income which may be caused, in the first instance, by the choice of income C rather than B, or A. Or, if he prefers, he may further exaggerate those fluctuations. In fact he may make the time shape of his income follow any model he likes. He may do this as described under the first approximation, by either borrowing or lending in suitable amounts and at suitable times along his income stream; or, more generally, by buying and selling income streams or parts of income streams so as to fashion the time shape of his own final net enjoyable income to suit himself.
He may, for instance, so far as time shape is concerned, achieve an even flow of income such as he could get from the farm use of his land. But he will not on that account choose this farm use in preference to the mining use; for the mining use has the larger present value, and the undesirable time shape of its income stream, under our present hypothesis, can be very easily remedied. For instance, he may lend some of the proceeds of its earlier output and in later years be paid back with interest.
Of course, his loan at five per cent does not alter in the least the figure $9110, the discounted value at five per cent of all the ten items of income ($2000, $1800, $1600, $1400, $1200, $1000, $800, $600, $400, $200); it simply adds to the later of these ten figures and subtracts from the earlier ones. The present value of the additions is necessarily equal to the present value of the subtractions; for the additions are the repayments, while the subtractions are the loans, and the present value of any loan equals that of its repayment.
We may totally separate, therefore, in thought the two choices made by the land owner, namely, (1) the choice of C (mining) in preference to A and B on the ground of greater present value, and, (2) the choice of time shape. If, as just supposed for illustration, the second sort of choice is that of an even income stream, it will be at the rate of $455.50 a year perpetually. That is to say, the mine owner will lend at interest $1544.50 the first year (all but $455.50 out of his original mining income of $2000); in the second year he will lend $1344.50 (all but $455.50 out of his original $1800); and so on. When the ninth year is reached, he ceases to lend further, for the mine then yields only $400. Instead, he then ekes this out by $55.50 returned from the previous loans. Likewise, in the tenth year he ekes out the $200 from mining by $255.50 returned from loans. Thereafter he will get nothing further from mining; but his loans will have accumulated a sinking fund (of $9110) to take the place of the mine and from this fund he can annually derive a 5 per cent revenue of $455.50. Consequently, the net result of the double choice (mining use and even time shape) is to increase the perpetual income of $450 offered by farming to a perpetual income of $455.50. This new perpetual annuity has exactly the same time shape as that derived from the farming use, but is larger by $5.50 per annum.
Incidentally it may be observed that this mining income, thus evened out by financing into a uniform $455.50 per year, exceeds the uniform farming income of $450 in exactly the same ratio as the present value ($9110) of the mining income exceeds that ($9000) of the farming income.
The following table exhibits the operations in detail:
TABLE 3
Mining and Farming Use Compared  

Owner Receives from Mine 
Of which He Lends 
Leaving for Real Income 
As Against Which the Farming Use Would Have Yielded 

1st year  $2000  $1544.50  $455.50  $450 
2nd year  1800  1344.50  455.50  450 
3rd year  1600  1144.50  455.50  450 
4th year  1400  944.50  455.50  450 
5th year  1200  744.50  455.50  450 
6th year  1000  544.50  455.50  450 
7th year  800  344.50  455.50  450 
8th year  600  144.50  455.50  450 
9th year  400  55.50  455.50  450 
10th year  200  255.50  455.50  450 
11th year  000  455.50  455.50  450 
etc.  etc.  etc.  etc.  etc. 
Or, instead of wanting a perpetual even flowing income, the land owner may prefer as his model the time shape of the forestry income. He will not, however, on that account, choose this forestry use in preference to the mining use. He will simply lend at interest from the items of mining income all of his $2000 the first year, leaving no income for that year; likewise, all of his $1800 the second; all but $310 the third; all but $413 the fourth and all but $516 the fifth, and every succeeding year until the ninth year. He will then turn around and use $116 from his loans just described to eke out his $400 and bring up his income in that year to $516. The tenth mining item, $200, will likewise be brought up to $516 after which he will depend entirely on his outside loans at five per cent, deriving therefrom exactly $516 every year.
The result will then be a series of income items exactly similar to the B, or forestry, series but each item magnified in the ratio of $9110 to $8820, the present values respectively of C and B.
The following table exhibits these operations:
TABLE 4
Mining and Forestry Use Compared  

Owner Receives from Mine 
Of Which He Lends 
Leaving for Real Income 
As Against Which the Forestry Use Would Have Yielded 

1st year  $2000  $2000  $000  $000 
2nd year  1800  1800  000  000 
3rd year  1600  1290  310  300 
4th year  1400  987  413  400 
5th year  1200  684  516  500 
6th year  1000  484  516  500 
7th year  800  284  516  500 
8th year  600  84  516  500 
9th year  400  116  516  500 
10th year  200  316  516  500 
11th year  000  516  516  500 
etc.  etc.  etc.  etc.  etc. 
Since, therefore, any time shape may be transformed into any other time shape, nobody need be deterred from selecting an income because of its time shape, but everyone may choose an income exclusively on the basis of maximum present value. It will then happen that his income, as finally transformed, will be larger than it would have been if he had chosen some other use which afforded that same time shape.
All this is true under the assumption used throughout this chapter, namely, that after the most valuable option has been chosen, you can borrow and lend or buy and sell ad libitum and without risk. If this assumption is not true, if a person were cut off from a free loan market, the choice among optional income streams might or might not fall upon that one having the maximum present value, depending on the other circumstances involved, particularly his preferences as regards time shape.
Of course our assumption is a violent one, made in this second approximation, as in the first, in order to simplify the theory of interest. But already it must be evident that the principle involved has important practical applications. To a very considerable extent a modern business man, with access to loan markets, can choose from among the various options open to him on the basis of present value, and trust to loans or other financing to rectify any inconvenience in time shape.
The lines AB and A'B' in Chart 15 picture alternative income streams, of which the descending one, AB, has the larger present value. The choice will fall on AB, and if the individual prefers the time shape of A'B', he will then lend some of the early receipts from the income stream AB and receive back some of the latter, converting his income AB of undesirable shape into the income stream A''B'' which has the desired shape. This final income A''B'' combines the virtues of both the original alternative incomes AB and of A'B'; it possesses the superior shape of A'B' and the superior present value of AB. As compared with A'B' it has the same shape but a greater size.
In practice, of course, the two steps are usually made simultaneously, not successively. In fact, usually the borrowing or financing often precedes the choice of option, thus reversing the order of presentation here adopted for convenience of exposition. So it would be quite as true to say that the loan, with the choice of option it makes possible, is made to secure an increased income as it is to say that the loan is made to even up the distorted income given by the option chosen.
But were it not for the possibility here assumed of modifying the time shape of his income stream by borrowing and lending, or buying and selling, the land owner would not feel free to choose the one from among the optional income streams which possesed the highest present value. He might find it advantageous, or even necessary, to take one of the others, being scarcely able to live if his property offered only distant income. If his capital were all in the form of growing young forests, and he could not mortgage the future in some way, he would have to starve or give up some of his holdings. In actual life we find such people—people who are said to be "land poor." In fact, we are all somewhat hampered in the choice of options by difficulties and risks both in the choice of options and in the financing it requires.
But we see that, in such a fluid world of options as we are here assuming, the capitalist reaches his final income through the cooperation of two kinds of choice of incomes which, under our assumptions, may be considered and treated as entirely separate. To repeat, these two kinds of choice are: first, the choice from among many possible income streams of that particular income stream which has the highest present value, and, secondly, the choice among different possible modifications of this income stream by borrowing and lending or buying and selling. The first is a selection from among income streams of differing market values, and the second, a selection from among income streams of the same market value.
Since this double choice results, when made, in a perfectly definite income stream, it might seem that the situation does not materially differ from the case of the rigid income stream discussed in the first approximation. But the two cases do differ materially, for under the present hypothesis (of optional income streams) the particular choice made by the individual depends upon what the rate of interest is. A change in that rate may shift the maximum present value to some other option, or alternative income stream, and that shift reacts on the rate of interest.
In the example cited, if the rate of interest should be 4½ per cent instead of 5 per cent, the order of choice would be changed. The present value of the land for A (farming) would be $10,000, for B (forestry), $9920, and for C (mining), $9280. The farming use, or A, would now be the best choice. Again, if the rate of interest should be 4 per cent instead of 4½ per cent, the present value of the use of the land for A, farming purposes, would be $11,250; for B, forestry purposes, $11,300; and for C, mining purposes, $9450. In this case, B, the forestry use, would be chosen.
Thus, it would pay best to employ the land for mining if the rate of interest were 5 per cent, for farming if it were 4½ per cent, and for forestry if it were 4 per cent.
The three options open to the owner of the land at these three different rates of interest may be summarized as follows:
TABLE 5
Present Values of the Three Options at Three Different Rates of Interest  

Options 
Present Value at 

5% 
4½% 
4%  
For forestry  $8,820  $ 9,920  $11,300 
For farming  9,000  10,000  11,250 
For mining  9,110  9,280  9,450 
Thus a change in the rate of interest results in a change in the relative attractiveness of different optional income stream opportunities. A high rate of interest will encourage investment in the quickly returning incomes, whereas a low rate of interest will encourage investment in incomes which yield distant returns. As the business man puts it, when interest is high, he can less afford to wait for a remote return because he will "lose so much interest." An investor will, therefore, make very different choices among the various options open to him, according as interest is at one rate or another.
Consequently, the existence of various options to use one's capital introduces a new variable into the problem of interest determination. For the individual, the rate of interest will determine the choice among his optional income streams, but, for society as a whole, the order of cause and effect is reversed—the rate of interest will be influenced by the range of options open to choice. If we live in a land covered with young forests or otherwise affording plenty of opportunities for distant income but affording few opportunities for immediate income (as was the case in the pioneer days in this country) the rate of interest will, other things being equal, be very much higher than in a land full of nearly worked out mines and oil fields or otherwise affording many opportunities for immediate but few opportunities for remote income.
We are thus coming in sight of a principle, applying to interest determination, new in our study, the principle of opportunity to invest, not simply by lending but by changing the use of one's capital. This new principle, largely physical or technical, is just as important as the psychical principle of human impatience. It is really old in the sense that, implicitly, it has been recognized in almost all theories of interest, and explicitly in those of Rae, Landry, Walras, and Pareto. To trace this new influence on interest is the special purpose of the second approximation.
At first sight it may appear to those not familiar with the mathematics of simultaneous equations and variables that the reasoning is circular; the rate of interest depends on individual rates of impatience; these rates of impatience depend on the time shapes of individual income streams; and the choice of these time shapes of income streams depends, as we have just seen, on the rate of interest itself.
It is perfectly true that, in this statement, the rate of interest depends in part on a chain of factors which finally depend in part on the rate of interest. Yet this chain is not the vicious circle it seems, for the last step in the circle is not the inverse of the first.
To distinguish between a true and a seeming example of a circular dependence we may cite simple problems in algebra or mental arithmetic. Suppose we wish to find the height of a father who is known to be three times as tall as his child. To solve this we need to know something more about these two heights. If we are told in addition that the child's height differs from his father's by twice itself, the problem is really circular and insoluble, for the additional condition is really reducible to the first, being merely a thinly veiled inversion of it. The problem essentially states (1) that the father's height is three times the child's and (2) that the child's is onethird of the father's—an obvious circle.
But if the dependence of the father's height on the child's is essentially different from—independent of—the dependence of the child's on its father's, there is no circle. Thus supposing, as before, that the father is three times as tall as the child, let us stipulate in addition that the child's height differs from the father's by four times as much as the child's less two feet. This may sound as circular as the first statement—the father's height is expressed in terms of the child's, and the child's is expressed in terms of the father's; but the second stipulation is not now reducible to the first. The heights are entirely determinate, that of the father being six feet and that of the child, two. The mere fact that both of these magnitudes, the father's height and the child's height, are specified each in terms of the other does not constitute a vicious circle. The general principle, as Cournot and other mathematical economists have often pointed out, is simply the well known algebraic principle of simultaneous equations. In order that the equations may determine the unknown quantities involved, there must be as many independent equations as there are unknown quantities, although any or all of these equations may contain all the unknowns. (The equations are independent if no one of them can be derived from another or the others.) Many an example of economic confusion and wrong reasoning could be avoided if this fundamental principle of mathematics were more generally applied.
This mathematical principle of determinateness applies in our present problem. Real examples of circular reasoning in the theory of interest are common enough, but the dependence, above stated, of interest on the range of options and the dependence of the choice among them on interest is not a case in point, for this last determining condition is not derivable from the others.*59
For our present purpose we need only present the matter to the reader's imagination by a process of trial and error. To find the rate of interest on which the market will finally settle, let us try successively a number of different rates. First, let us suppose a rate of 5 per cent. This rate will determine the choice between options for each individual. The land owner formerly supposed will, as we have seen, choose C, the mining use, because the present value of the income so obtained ($9110) exceeds the present values of the rival uses. Every other individual in the market, in like manner, will select that particular use for his capital which will give him the maximum present worth. With these choices made, the different individuals will then enter the market of loans or sales, desiring to modify the time shapes of their income streams to suit their particular desires.
As a result of all these choices, the total amount which all the wouldbe lenders are willing to lend at 5 per cent out of this year's instalment of their chosen income stream will be perfectly definite, and likewise the total amount which all the wouldbe borrowers are willing to take. This we saw in the preceding chapter. In other words, the demand and supply of loans for the present year, at the given rate of interest, 5 per cent, will both be definite quantities. Should it happen that the supply of loans exceeds the demand, it would follow that 5 per cent could not be the correct solution of the rate of interest, for it would be too high to clear the market.
In that case, let us try again; suppose a rate of 4 per cent. Following the same reasoning as before, we now find that the land owner will select the forestry opportunity for his land because the present value ($11,300) of the income from forestry—now reckoned at 4 per cent—will exceed that of the two rival income possibilities. Other capitalists will likewise select their best option from among those available to them and on the basis of these income streams—not the same as before under 5 per cent. In a word, there will now be a different supply and demand. The land owner, for instance, instead of lending, may now borrow (or sell securities) to even up his income stream. Should it then happen that the demand and supply of loans, on the basis of 4 per cent, are still not equal, but that, this time, the demand exceeds the supply, it would be a proof that not 4 per cent is the true solution, but some higher rate. By again changing our trial rate—part way back toward 5 per cent, we may evidently reach some intermediate point, let us say 4½ per cent, at which rate not only will each individual choose the best use of his capital—that having the highest present worth—but also, at the same time, the demand and supply of loans engendered by all such choices will exactly clear the market, i.e., bids and offers at the given rate will be equal. Likewise, the same clearing will be worked out for next year and for all years.*60
The introduction, therefore, of flexibility into our income stream still leaves the rate of interest entirely determinate, even though the income streams are now, in the second approximation, not fixed or rigid but subject to choice, and even though that choice will depend on the rate of interest itself.
For the determination of the rate of interest we must now, therefore, in the second approximation, add two new principles to the four principles already given in the first approximation described in the previous chapter.
There exists, for each individual, a given specific set or list of optional income streams to choose from, differing in size and time shape (but without any uncertainty as to what will happen if any particular one is chosen).
Out of this list of options each individual will choose that particular income stream possessing the greatest present worth when calculated by means of the rate of interest as finally determined by these six conditions.
The degree of impatience, or rate of time preference, of any given individual depends upon his income stream as chosen by him and as modified by exchange.
Each person, after or while first choosing the option of greatest present worth, will then modify it by exchange so as to convert it into that particular form most wanted by him.
This implies, as we have seen, that each person's degree of impatience, or rate of time preference, will at the margin, be brought to equality with the market rate of interest and, therefore, with the marginal preference rates of all the other persons.
The rate of interest must be such as will clear the market, that is, equalize supply and demand. That is, for every time interval, the additions to some individuals' incomes caused by borrowing or selling must balance the deductions from others caused by lending or buying.
The loans must be equivalent in present worth to repayments, or, more generally, the additions to any individual's income, brought about by borrowing or selling, in some time intervals must be equivalent in present worth to the deductions from his income in other time intervals brought about by lending or buying.
Thus we see that the rate of interest is determined by two principles of investment opportunity as well as by two principles of impatience and by the two selfevident market principles.
More briefly stated, the rate of interest is determined so as (1) to make the most of opportunities to invest, (2) to make the best adjustment for impatience and (3) to clear the market and repay debts.
In short, the theory is thus one of investment opportunity and human impatience, as well as exchange.
But while we have reached the two chief theoretical foundations of our subject, we are still, of course, far from the real world. The real world is vastly more complex than the imaginary world described in this chapter. In particular, we still need to take account of risk. This we shall do in the third approximation.
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 pricetheory, 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.
Part II, Chapter 7
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