The Theory of Interest

Fisher, Irving
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New York: The Macmillan Co.
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1st edition.
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Assuming Each Person's Income Stream Foreknown and Unchangeable Except by Loans

§1. Hypotheses of First Approximation


IN the last chapter we reached three conclusions:

(1) that the rate of time preference, or impatience for present over future goods, is, in the last analysis, a preference for present over future enjoyment income, or, let us say, real income;
(2) that the degree of impatience depends, for any given individual, upon the character of his real income-stream—in particular, on its size, time shape, and probability;
(3) that the nature of this dependence differs with different individuals.


The question at once arises: will not the actual degrees of impatience of different individuals necessarily be very different, and if so, what relation do these different rates have to the market rate of interest? Is the market rate of interest a sort of average of these individual degrees of impatience, or does it equalize them?


It is doubtless true that the different rates of impatience of different individuals who are not connected through a common loan market do vary widely. In a nation of hermits, in which there existed no mutual lending and borrowing, individuals would be independent of each other. But, among ourselves who have access to a common loan market, borrowing and lending do, at least, tend to bring into equality the marginal rates of impatience in different minds. Absolute equality is not reached even among those making use of such a market; but this is because of the limitations of the market and, in particular, because of the risk element. This element will be considered in the third approximation, but for simplicity of exposition is omitted in the first two approximations.


Here we shall assume a perfectly competitive market, one in which each individual is so small a factor as to have, singly, no perceptible influence on the rate of interest, and in which there is no limitation on the amount of lending and borrowing other than that caused by the rate of interest itself. The would-be borrower is thus supposed to be able to obtain as large or small a loan as he wishes at the market price—the rate of interest. He is not cut down to $5,000 when he is willing to borrow $100,000, merely because he cannot furnish enough collateral security or a satisfactory endorser. He can buy a loan as he can buy sugar, as much or as little as he pleases, if he will pay the price.


In the actual world, of course, no such perfect market exists. While many people in New York City can obtain as large loans as they wish, there are thousands who are unable to obtain any at all. The price of a loan is paid not in the present, as the price of sugar is paid, but in the future. What the lender gets when he makes the loan is not payment but a promise of payment, and the future being always uncertain he needs some sort of assurance that this promise will be kept. We are assuming in the first and second approximations that there will never be a lack of such assurance. This amounts almost to assuming that there is no risk in the world. The element of risk is assumed to be entirely lacking, both with respect to the certainty of the expected income streams belonging to the different individuals, and with respect to the certainty of repayment for loans. In other words, we assume that each individual in the market is free to give up any part of his income during one period of time to some other person in consideration of receiving back an addition to his own income during another period of time.


We assume further that thus to buy and sell rights to various parts of his income stream is the only method open to any individual to alter his income stream. Such trading between present and future dollars may be in the form of loans, since a loan is the sale of future money for present money, or it may be in the guise of buying and selling bonds or other securities conveying title to fixed sums of money. In any case the trading reduces itself to buying and selling titles to future income. Prior to such exchange, the income stream of each individual is assumed to be fixed in size and shape. Each capital instrument which he possesses, including himself, is assumed to be capable of only a single definite series of services contributing to his income stream. Each individual is a stipendiary with a definite income which he receives and spends according to a foreknown schedule—so much next year, so much the year after, and so forth.


Thus the assumptions of our first approximation are: (1) that each man's income stream is initially certain and fixed; (2) that he is a negligible element in a vast and perfect competitive loan market; (3) that he has free access to this market, whether as borrower or lender, to any desired extent, at the market rate; (4) that his sole method of modifying his future income stream is through such borrowing or lending (or, more exactly and generally, through trading income).

§2. Income Prescribed


Such assumptions are, of course, highly theoretical. They imagine a world in which incomes are produced spontaneously, as mineral water gushes from the spring. They picture these income-bearing agents as pouring forth their income streams at rates which follow a foreknown, rigid, and unchangeable schedule. There being no flexibility in the flow from any one article, there is no flexibility in the scheme of combined flow from the whole group possessed by an individual. His total real income is scheduled in advance with no possibility of modification except by borrowing or lending, buying or selling.*49


The abstract nature of this hypothesis need not greatly trouble us for two reasons: the first, and in itself quite sufficient reason, is that most of the elements of this hypothesis will be abandoned when we reach the second approximation. It is adopted temporarily merely for simplicity of exposition. Secondly, the hypothesis might easily be made more realistic without changing its essential features. We might even alter our hypothesis of a rigidly prescribed income stream to the hypothesis of an income stream which, while it may be decreased at will, cannot be increased beyond a fixed amount at each period of time. While free to decrease his income in any period, its possessor would not do so unless thereby he could secure an increase in some other period.


Not only is such an hypothesis quite thinkable, it is probably actually approximated in primitive communities. In our own day most men have opportunities (quite apart from lending to others at interest) to secure much real income in future years by temporarily sacrificing a little immediate real income as, for instance, by investing labor in building a house or machine. But we can readily suppose a situation such that this year's production and next year's production would be almost independent of each other. This situation is true of most animals and even of man in the hunting and fishing stage, and before that stage even more markedly when his only implements were his hands. And even in our own civilization, many are mere stipendiaries, virtually without any opportunity to add to future income except by lending at interest.


The essence of the hypothesis therefore on which the first approximation rests is that we are not to be bothered by the possibility of a man's thus increasing his income in one period through decreasing it in another except through the process of trading some of one year's income with another person for some of another year's income.


We may in fact for practical purposes picture the income stream of each person as thus fixed for only a few years, and assume that expectation of income beyond those years so indefinite as to have no effect on the present rate of interest. Such a community has been approximated in former years by the typical American army camp isolated in a western community in which each inhabitant, or family, had a prescribed income. A series of such hypotheses will lead us through successive approximations to an eventual picture of actuality.

§3. Equalization of Impatience


Under the hypothetical conditions which have been stated for the first approximation, the rates of time preference for different individuals will, by the process of borrowing and lending, become perfectly reconciled to the market rate of interest and to each other, for if, for any particular individual, the rate of preference differs from the market rate, he will, if he can, adjust the time shape of his income stream so as to bring his marginal preference rate into harmony with the interest rate. A man who, for a given income stream, has a rate of preference above the market rate will sell some of his surplus future income in return for an addition to his meager present income, i.e., he will borrow. This will have the effect of enhancing his want for one more dollar of future income and decreasing his want for one more dollar of present income. The process will continue until the rate of preference of this individual, at the margin, is equal to the rate of interest. In other words, under our hypothesis, a person whose preference rate exceeds the current rate of interest will borrow up to the point at which the two rates will become equal.


On the other hand, the man, whose temperament or whose income stream or both give him a preference rate below the market rate, will buy future income with some of his abundant present income, i.e., he will lend. The effect will be to increase his preference rate until, at the margin, it harmonizes with the rate of interest.


To put the matter in figures, let us suppose the rate of interest is 5 per cent, whereas the rate of preference of a particular individual is, to start with, 10 per cent. Then, by hypothesis, the individual is willing to sacrifice $1.10 of next year's income in exchange for $1 of this year's. But, in the market, he finds he is able to obtain $1 for this year by foregoing only $1.05 of next year's. To him this latter ratio is a cheap price. He therefore borrows, say, $100 for a year, agreeing to return $105; that is, he contracts a loan at 5 per cent, when he is willing to pay 10 per cent. This operation partly satisfies his hunger for present income by drawing on his future income, and thus reduces his time preference from 10 per cent to, say, 8 per cent. Under these circumstances he will borrow another $100, being willing to pay 8 per cent, but required to pay only 5. This operation will still further reduce his time preference, and so on through successive stages, until it is finally brought down to 5 per cent. Then, for the last or marginal $100, his rate of time preference will agree with the market rate of interest.*50


In like manner, if another individual, entering the loan market from the opposite side, has a rate of preference of 2 per cent, he will become a lender instead of a borrower. He will be willing to lend $100 of this year's income for $102 of next year's. As he can lend at 5 per cent when he would do so at 2, he "jumps at the chance," and invests, not $100 only, but another and another. But his present income, being drawn upon by the process, is now more highly esteemed by him than before, and his future income, being supplemented, is less highly esteemed; and under the influence of successive additions to the sums lent, his rate of preference for the present will keep rising until, at the margin, it will equal the market rate of interest.


In such an ideal loan market, therefore, where every individual could freely borrow or lend, the rates of preference or impatience for present over future income for all the different individuals would become, at the margin, exactly equal to each other and to the rate of interest.

§4. Altering Income by Loans


To illustrate this reasoning by a chart, let us suppose the income stream to be represented as in Chart 5, and that the possessor wishes to obtain, by borrowing, a small item X' of immediately ensuing income in return for a somewhat larger item X'' later on, X'' being the amount of X' at interest. By such a loan he modifies his income stream from ABCD to EBD. But this change will evidently produce a change in his time preference. If the rate of time preference corresponding to the income stream represented by the unbroken line is 10 per cent, the rate of preference corresponding to the broken line will be somewhat less, say, 8 per cent. If the market rate of interest is 5 per cent, it is evident that the person will proceed to still further borrowing. By repeating the operation several times he can evidently produce almost any required conformation of his income stream.

Figure. Click to enlarge in new window.


If, instead of borrowing, he wishes to lend (Chart 6) he surrenders from his present income stream the amount X' for the sake of the larger amount X'' at a later time. After the operations are completed and the final conformations of the income streams are determined, the rates of time preference are all brought into conformity with the market rate of interest.

Figure. Click to enlarge in new window.


In practice, of course, the adjustments are never perfect and, in particular, the income stream is never a smooth curve, such as it is here for convenience represented.


In practice, also, loans are effected under the guise of money. We do not confessedly borrow and lend real incomes, but money and credit. Yet money—that universal medium in practice and universal stumbling-block in theory—merely represents real income, or capitalized real income. A hundred dollars mean the power to secure income,—any income the present value of which is $100. When, therefore, a person borrows $100 today and returns $105 next year, in actual fact he secures the title to $100 worth of income—immediately future, perhaps—and parts with the title to $105 worth of income a year later. Every loan contract, or any other contract implying interest, involves, at bottom, a modification of income streams, the usual and chief modification being as to time shape.


One reason why we often forget that a money loan represents real income is that it represents so many possible varieties of real income. A fund of money is usually the capitalization not simply of one particular future program, or lay-out, of income but of a large number of optional income streams, and is not restricted, as in the first approximation, here considered, to a simple income stream and its modification by loans or their equivalent.


We may distinguish six principal types of individuals in a loan market—three borrowing types and three lending types. The first type of borrower (Chart 7) is supposed to be possessed of an increasing, or ascending, income stream AB, a fact which, in his mind, results in a rate of preference above the market rate. This leads him to borrow, and relatively to level up his ascending income stream toward such a position as A'B'. The second type of individual already possesses a uniform income stream AB (Chart 8), but having, a strong propensity to spend, he too experiences a rate of preference above the market rate, and will therefore modify his income stream toward the curve A'B'. The third type is shown in Chart 9 and represents even more of a spendthrift. This individual has also a rate of preference in excess of the market rate, in spite of his having a declining income stream pictured by the descending curve AB. By his borrowing, he obtains a curve A'B' of still steeper descent.

Figure. Click to enlarge in new window.


In a similar way, the three types of lenders may be graphically represented. Chart 10 represents a descending income AB, which the owner, by lending present income in return for future income, converts into a relatively uniform income A'B'; Chart 11 represents a uniform income converted, by lending, into an ascending income; and Chart 12 an ascending income converted into a still more steeply ascending income.


The borrower changes his income curve by tipping it down in the future and up in the present. The lender tips his income curve in the opposite direction. Of the three types of borrowers and of lenders, the first in each group of three (see Charts 7 and 10), is the usual and normal case. In both these cases the effort is to transform the given income into a more uniform one, the rising curve (Chart 7) being lowered and the falling curve (Chart 10) being raised toward a common horizontal position. Chart 9 and Chart 12, on the other hand, represent the extreme and unusual cases of the spendthrift and the miser.


But whatever the personal equation, it remains true that, for each individual, other things being equal, the more ascending his income curve, the higher his rate of preference; and the more descending the curve, the lower the rate of preference. If the descent of the income stream is sufficiently rapid, the rate of preference could be made zero or even negative.*51

Figure. Click to enlarge in new window.


These foregoing types of income streams are, of course, not the only ones which could be considered, but they are some of the more important. To them we may add the type of fluctuating income, as represented in Chart 13, which may result in alternate borrowing and lending so as to produce a more nearly uniform income stream. Such financing over the lean parts of a year is often practiced when the income is lumped at one or two spots, such as dividend dates.


It must not be imagined that the classes of borrowers and lenders correspond respectively to the classes of poor and rich. The factors, environmental and personal, discussed in Chapter IV, will determine whether a man's rate of preference is high or low, and therefore whether he will become a spender or a saver.


When we come to the second and third approximations and have to study so-called productive loans, especially of risk-takers, or enterprisers as Professor Fetter calls them, we shall find still other influences determining whether a person shall be a borrower or lender or both. At present we are only at the first approximation where it is assumed there is no risk and no such series of opportunities to vary the income stream as lie at the basis of so-called productive loans.

§5. Altering Income by Sale


But borrowing and lending are not the only ways in which one's income stream may be modified. Exactly the same result may, theoretically, be accomplished simply by buying and selling property; for, since property rights are merely rights to income streams, their exchange replaces one such stream by another of equal present value but differing in time shape, composition, or uncertainty. This method of modifying one's income stream, which we shall call the method of sale, really includes the former, or method of loans, for a loan contract is, as Böhm Bawerk has so well said, at bottom a sale, that is, it is the exchange of the right to present or immediately ensuing income for the right to future or more remote income. A borrower is simply a seller of a note of which the lender is the buyer. A man who buys a bond, for example, may be regarded indifferently as a lender or as a buyer of property.


The concept of a loan may therefore now be dispensed with by being merged in the concept of a sale. Every sale transfers property rights; that is, it transfers the title to income of some kind. By selling some property rights and buying others it is possible to transform one's income stream at will into any desired time shape. Thus, if a man buys an orchard, he is providing himself with future income in the form of apples. If, instead, he buys apples, he is providing himself with similar but more immediate income. If he buys securities, he is providing himself with future money, convertible, when received, into apples or other real income. Inasmuch as the productive life of a mine is shorter generally than that of a railway, if his security is a share in a mine, his income stream is less lasting than if the security is stock in a railway, though at first it should be larger, relatively to the sum paid for it.


Purchasing the right to remote enjoyable income, as was explained in Chapter I, is called investing; while purchasing more immediate enjoyable income is spending. These, however, are purely relative concepts; for remote and immediate are relative terms. Buying an automobile is investing as contrasted with spending the money for food and drink, but may be called spending as contrasted with investing in real estate. And yet the antithesis between spending money and investing is important; it is the antithesis between immediate and remote income. The adjustment between the two determines the time shape of one's income stream. Spending increases immediate real income but robs the future, whereas investing provides for the future to the detriment of the present. There is often misconception in reasoning about spending and investing. For example, Henry Ford's remark has been widely reported: "No successful boy ever saved any money. They spent it as fast as they got it for things to improve themselves." In this remark Mr. Ford drew no hard and fast line between spending for personal enjoyment and investment for improvement. And there is no hard and fast line. Spending merely means expending money primarily for more or less immediate enjoyment. Saving or investing is expending money for more or less deferred enjoyment. Consequently, much of what is called spending might legitimately be called investment. Even the money we spend for food, clothing and shelter is in a sense really partly invested, since our lives and capacity to work can be preserved only by means of these necessaries. Just so with a set of books, or any other durable good which increases the efficiency and hence the earning power of the purchaser; it is an example, not of spending for consumption merely, but of saving through investment. Mr. Ford cites the example of Thomas Edison as spending his early earnings as fast as he made them. But Edison did this, not for food and display, but for experimentation that resulted in time and labor saving inventions which have benefited everybody. His outlays were in a way investments.


Popular usage has devised many other terms and phrases in this field, most of which, like spending and investing, while containing meanings of importance, include also the alloy of misconception. Thus, the phrase "capital seeking investment" means that capitalists have property for which they desire, by exchange, to substitute other property, the income from which is more remote. It does not mean that there is any hard and fast line between invested and uninvested capital, much less does it mean that the inanimate capital has of itself any power to seek investment. Again, the phrase "saving capital out of income" means not spending—reserving money which would otherwise be spent for immediate enjoyable income in order to exchange it or invest it for remoter income; it does not mean the creation of new capital, though it may lead to that. Many needless controversies have centered about the phenomenon of saving chiefly because neither saving nor income was clearly defined.*52


From what has been said it is clear that by buying and selling property an individual may change the conformation of his income stream precisely as though he were specifically lending or borrowing. Thus, suppose a man's original income stream is $1000 this year and $1500 next year, and suppose that he sells the title to this income stream, and, with the proceeds buys the title to another income stream yielding $1100 this year and $1395 next year. Although this man has not, nominally, borrowed $100 and repaid $105, he has done what amounts to the same thing; he has increased his income stream of this year by $100 and decreased that of next year by $105. The very same diagrams which were used before may equally well represent these operations. A man sells the income stream ABCD (Chart 5) and with the proceeds buys the stream EBD. The X' and X'' are, as before, $100 and $105, but now appear explicitly as differences in the value of two income streams, instead of appearing as direct loans and payments.

§6. Interest Ineradicable


Thus interest taking cannot be prevented by prohibiting loan contracts. To forbid the particular form of sale called a loan contract would leave possible other forms of sale, and, as was shown in Chapter I, the mere act of valuation of every property right involves an implicit rate of interest. If the prohibition left individuals free to deal in bonds, it is clear that they would still virtually be borrowing and lending, but under the names of selling and purchasing; and if bonds were tabooed, they could change to preferred stock. Indeed, as long as buying and selling of any kind were permitted, the virtual effect of lending and borrowing would be retained. The possessor of a forest of young trees, not being able to mortgage their future return and being in need of an income stream of a less deferred type than that receivable from the forest itself, could simply sell his forest and with the proceeds buy, say, a farm, with a uniform flow of income, or a mine with a decreasing one. On the other hand, the possessor of a capital which is depreciating, that is, which represents an income stream great now but steadily declining, and who is eager to have an increasing income, could sell his depreciating wealth and invest the proceeds in such instruments as the forest already mentioned.


It was in such a way, as for instance by rent purchase, that the medieval prohibitions of usury were rendered nugatory. Practically, the effect of such restrictive laws is little more than to hamper and make difficult the finer adjustments of the income stream, compelling would-be borrowers to sell wealth yielding distant returns instead of mortgaging them, and would-be lenders to buy such wealth instead of lending to the present owners. It is conceivable that explicit interest might disappear under such restrictions, but implicit interest would certainly remain. The young forest sold for $10,000 would bear this price, as now, because it is the discounted value of the estimated future income; and the price of the farm, $10,000, would be determined in like manner. The rate of discount in the two cases, the $10,000 forest and the $10,000 farm, must tend to be the same, because, by buying and selling, the various parties in the community would adjust their rates of preference to a common level—an implicit rate of interest thus lurking in every contract, though never specifically mentioned therein. Interest is too omnipresent a phenomenon to be eradicated by attacking any particular form of it; nor would any one undertake to eradicate it who perceived its substance as well as its form.

§7. "Marginal" Principle Is "Maximum" Principle


The fact that, through the loan market, the marginal rate of time preference for each individual is, by borrowing or lending, made equal to the rate of interest may be stated in another way, namely, that the total present desirability of, or want for, the individual's income stream is made a maximum. For, consider again the individual who modifies his original fixed income stream by borrowing until his rate of preference is brought into unison with the market rate of interest. His degree of impatience was at first, say, 10 per cent; that is, he was willing, in order to secure an addition of $100 to his present income, to sacrifice $110 of next year's income. But he needed to sacrifice only $105; that is, he was enabled to get his loan for less than he would have been willing to pay. He was therefore a gainer to the extent of the present desirability of, or present want for, $5 of next year's income. The second $100 borrowed was equivalent, in his present estimation, to $108 of next year's income, and the same reasoning shows that, as he pays only $105, he gains to the extent of the present desirability of $3 next year; that is, he adds this present desirability to the entire present total desirability of his income stream. In like manner, each successive increment of loans adds to the present total desirability of his income, so long as he is willing to pay more than $105 of next year's income for $100 of this year's income. But, as he proceeds, his gains and his eagerness diminish until they cease altogether. At, let us say, the fifth instalment of $100, he finds himself barely willing to pay $105; the present total desirability of his income is then a maximum, and any further loan would decrease it. A sixth $100, for instance, is worth in his present estimation less than $105 due next year, say $104, and since in the loan market he would have to sacrifice $105 next year to secure it, this would mean a loss of desirability to the extent of the desirability today of $1 due in one year. Thus, by borrowing up to the point where the rate of preference for present over future income is equal to the rate of interest, five per cent, he secures the greatest total desirability, or, so-called consumer's rent.*53


Similar reasoning applies to the individual on the other side of the market, whose rate of preference is initially less than the market rate of interest. He also will bring his present net total desirability to a maximum by lending up to the point where his rate of preference corresponds to the rate of interest. At the beginning, $100 this year has to him the same present desirability as, say, $102 due one year hence, whereas in the market he may secure not $102 but $105. It is then clear that by lending $100 he gains the present desirability of $3 due one year hence. By lending each successive $100 he will add something to the total present desirability of his income, until his rate of preference for present over future income is raised to a level equal to that of the rate of interest, five per cent. Beyond that point he would lose by further lending.

§8. Market Equilibrium


We are now in a position to give a preliminary answer to the question, What determines the rate of interest? Thus far we have regarded the individual only, and have seen that he conforms his rate of income-impatience to the rate of interest. For him the rate of interest is a relatively fixed fact, since his own impatience and resulting action can affect it only infinitesimally. To him it is his degree of impatience which is the variable. In short, for him individually, the rate of interest is cause, and his lending and borrowing is the effect. For society as a whole, however, the order of cause and effect is reversed. This change is like the corresponding inversion of cause and effect in the theory of prices. Each individual regards the market price, say, of sugar, as fixed, and adjusts his marginal utility, or desirability, to it; whereas, for the entire group of persons forming the market, the adjustment is the other way around, the price of sugar conforming to its marginal desirability to the consumer.*54 In the same way, while for the individual the rate of interest determines the degree of impatience, for society the degrees of impatience of the aggregate of individuals determine, or help to determine, the rate of interest. The rate of interest is equal to the degree of impatience upon which the whole community may concur in order that the market of loans may be exactly cleared.


To put the matter in figures: Suppose that at the outset the rate of interest is arbitrarily set very high, say, 20 per cent. There will be relatively few borrowers and many would-be lenders, so that the total extent to which would-be lenders are willing to reduce their income streams for the present year for the sake of a much larger future income will be, say, 100 million dollars; whereas, the extent to which would-be borrowers are willing to increase their income streams in the present at the high price of 20 per cent will be only, say, one million. Under such conditions the demand for loans is far short of the supply and the rate of interest will therefore go down. At an interest rate of 10 per cent the lenders may offer 50 millions, and the borrowers bid for 20 millions. There is still an excess of supply over demand, and interest must needs fall further. At 5 per cent we may suppose the market cleared, borrowers and lenders being willing to take or give respectively 30 millions. In like manner it can be shown that the rate would not fall below this, as in that case it would result in an excess of demand over supply and cause the rate to rise again.


Thus, the rate of interest registers in the market the common marginal rate of preference for present over future income, as determined by the supply and demand of present and future income. Those who, to start with, have a high degree of impatience, strive to acquire more present income at the cost of future income, and thus tend to raise the rate of interest. These are the borrowers, the spenders, the sellers of property yielding remote income, such as bonds and stocks. On the other hand, those who, to start with, have a low rate of preference, strive to acquire more future income at the cost of present income, and so tend to lower the rate of interest. Such are the lenders, the savers, the investors.

Not only will the mechanism just described result in a rate of interest which will clear the market for loans connecting the present with next year, but, applied to exchanges between the present and the more remote future, it will make similar clearings. While some individuals may wish to exchange this year's income for next year's, others wish to exchange this year's income for that of the year after next, or for a portion of several future years' incomes. The rates of interest for these various periods are so adjusted as to clear the market for each of the periods of time for which contracts are made.

§9. Four Principles


If we retain our original assumption that every man is initially endowed with a rigidly fixed or prescribed income stream which can be freely bought and sold and thereby redistributed in time, the foregoing discussion gives us a complete theory of the causes which determine the rate of interest, or rather, the rates of interest, there being, theoretically, a separate rate for each time period. These rates of interest would, under these circumstances, be fully determined by the following four principles, to which all the magnitudes in the problem of interest must conform:

A. Empirical Principle


The rate of time preference or degree of impatience of each individual depends upon his income stream.

B. Principle of Maximum Desirability


Through the alterations in the income streams produced by loans or sales, the marginal degrees of impatience for all individuals in the market are brought into equality with each other and with the market rate of interest.


This condition B is equivalent to another, namely, that each individual exchanges present against future income, or vice versa, at the market rate of interest up to the point of the maximum total desirability of the forms of income available to him.

A. Principle of Clearing the Market


The market rate of interest will be such as will just clear the market, that is, will make the loans and borrowings or, more generally expressed, purchases and sales of income equal for each period of time.

B. Principle of Repayment


All loans are repaid with interest, that is, the present value of the payments, reckoned at the time of contract, equals the present value of the repayments. More generally expressed, the plus and minus alterations or departures from a person's original income stream effected by buying and selling at two different points are such that the algebraic sum of their present values is zero.


Will these four sets of conditions determine the rate of interest? And why should there be so many conditions? Ought not one single condition to suffice?


These are really questions in mathematics. It is a fundamental principle that in order to solve an equation containing only one unknown quantity only one equation is necessary; and that to solve one containing two unknowns, two independent equations are needed; and so on, one additional equation for each additional unknown quantity introduced.


In the present problem we are trying to determine only one unknown, the rate of interest. But we can do so only by determining, at the same time, the other unknowns that are involved. To say that the rate of interest is equal to Smith's marginal rate of impatience is saying something, but not enough. It merely expresses one unknown, the rate of interest, in terms of another unknown, Smith's marginal rate; and two unknowns cannot be determined by one equation or condition. If we add that the rate of interest must also equal Jones' rate of impatience, while this statement gives us another equation it also adds another unknown and three unknowns cannot be determined by two equations; and so on. If we include Jones and everybody else in the market, we shall still be one equation short. This is equivalent to saying that the second set of conditions (Impatience Principle B) is not enough.


In a market comprising 1000 persons there will be, as our unknowns, not only the rate of interest, but 1000 rates of impatience, and the additions to or deductions from the income of these 1000 persons in each period of time. The rate of interest and these thousands of variables act and react on each other and the determination of each can be accomplished only with the determination of all the rest.


In Chapter XII this problem is stated in mathematical formulas such that the number of equations is exactly equal to the number of unknown quantities.

Notes for this chapter

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.
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.
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.
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.
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.
See my Mathematical Investigations in the Theory of Value and Prices.

Part II, Chapter 6

End of Notes

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