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# The Theory of Interest

 Fisher, Irving (1867-1947) Display paragraphs in this book containing:
 Editor/Trans. First Pub. Date 1930 Publisher/Edition New York: The Macmillan Co. Pub. Date 1930 Comments 1st edition.
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### APPENDIX TO CHAPTER XX

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

App.6.1

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

App.6.2

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

App.6.3

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

Income Account of Tree if Waiting is Cost
TRUE INCOME
ALLEGED OUTGO
ALLEGED NET INCOME
TRUE CAPITAL VALUE AT END OF YEAR
1st year \$0.00 Labor \$1.00
Waiting .05   \$1.05
2d year 0.00 "       .05   1.10
3d year 0.00 "       .05   1.15
* * * * * * * * * * * * * *    * *   *   *
14th year 0.00 "      .10   2.00
* * * * * * * * * * * * * *    * *   *   *
25th year 3.00 "      .15   3.00

Total \$3.00 \$3.00 00.0

App.6.4

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

App.6.5

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

Income Account of Annuity if Waiting is Cost
TRUE INCOME
ALLEGED OUTGO
ALLEGED NET INCOME
TRUE CAPITAL VALUE AT BEGINNING OF YEAR
1st year \$100 Waiting \$39 \$61 \$772
2d year 100 "    35 65 711
3d year 100 "    32 68 646
4th year 100 "    29 71 578
5th year 100 "    25 75 507
6th year 100 "    22 78 432
7th year 100 "    18 82 354
8th year 100 "    14 86 272
9th year 100 "     9 91 186
10th year 100 "     5 95 95

\$1000 \$228 \$772

App.6.6

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

App.6.7

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

App.6.8

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

App.6.9

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

App.6.10

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

App.6.11

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

App.6.12

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

App.6.13

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

App.6.14

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

App.6.15

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

App.6.16

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

### Notes for this chapter

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

### End of Notes

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