Part III. Second and Thir Approximations

till assumin that · come-streams' are certain that is can be definitely foreseeI4-.w~ now intro lice the hypothesis ~they are not fixed, but flexible; that is, that the owner fl.f any capital-wealth Qr ca.pit.a.1-prQpertyis not restricted to a single use to which he may put it, but has open to his 'IiIt:. .choice several different uses, each of which constitutes a separate optional income-stream. For instance, the owner of land may use it in more than one way. He ~ay use it to grow crops, graze animals, plant forests, extract minerals, support buildings, or for other purposes. Again, the owner of a building may use it for office purposes, for apartments, or for stores. Most raw materials can be used for anyone of a number of pur poses. Iron may be wrought into steel rails or into ma chinery, implements, tools, armor for ships, or girders for buildings. A derrick may be used for quarrying stone, building a house, or unloading a boat. A ship may be used to carry any sort of cargo, and over anyone of numerous different routes. Hammers, saws, nails, and other tools may be used in almost numberless ways.

Perhaps the most adaptable of all instruments of wealth is man himself. He may be simply a passive enjoyer or 137 138 THE RATE OF INTEREST [CHAP. VIII "transformer" ~of the services of other wealth, and as such derive his satisfactions in sensual, esthetic, intellectual, or spiritual ways; or, he may also be an active producer, and as such perform physical or mental work. If his work is physical, it may consist in anything from wielding a pick and shovel to the deft manipulation of the instruments employed in the jeweler's art. If his work is mental, he may be a bookkeeper, clerk, superintendent, director, law yer, physician, editor, teacher, or scientist. In consequence of such a range of choice, the ~me s~of productive instruments may result in very different income streams. Their energies may be directed at will to pro duce cheap frame houses or durable stone ones; to equip a city with horse cars, trolleys, or underground rapid transit; to secure an income-stream which shall consist largely of the pleasures of the table, or of the amusements of the theater, or of the gratification of social vanities, - in short, to select one particular income-stream out of a thousand possible income-streams differing in size, compo sition, and time-shape, as well as in probability, though in this chapter the element of uncertainty is supposed absent.

Owing to this great range of choice, the owner of capital may modify the income-stream he derives from it, not simply by the devices of borrowing and lending or of sell.. ing and buying, but also by changing the use or employ ment to which his capital is put. It should be noted, however, that this third method of modifying an income stream really includes the other two. Just as buying and selling virtually include borrowing and lending, so the change from one use of capital to another may be said to include buying and selling, and therefore also to include borrowing and lending. This is evident if we consider that one method of employing capital is to sell it. In fact, a merchant regards himself as "using" his stock in the ex clusive sense of selling it. This method of modifying the income-stream is therefore 1 See The Nature 0/ Capital and Income, Chap. X.

SEC. 2] SECOND APPROXIMATION 139 a general one. But, while it includes thei other methods, it includes much else so different from the methods of borrowing and lending or buying and selling that we shall need to distinguish the new method from the old. There are two principal reasons for keeping the new method separate. First, the former and narrower methods of modifying income-streams cannot be applied to society as a whole. Society as a whole cannot borrow and lend, nor buy and sell; and yet it can radically change the character of its income~stream by changing the employment of its capital. Secondly, when borrowing and lending or ordi nary buying and selling are employed to modify an income-stream, the present values of the original in come-stream and of the modified income-stream are the same. But when an income-stream is modified by a change in the use of the capital yielding it, its present value may not remain the same.

§ 2 The choice among the various optional income-streams will fallon the one which has the maximum desirability. As among income-streams of different sizes but alike in com position and time-shape, the most desirable will of course be the largest; as among income~streams of different com position but alike in other respects, the most desirable will be that in which the marginal desirabilities of the different con stituen ts are proportional to their several prices,inaccordance with a fundamental principle in the theory of prices; finally, as among income-streams differing in time-shape, the most desirable is found in accordance with the principles which govern the rate of interest. It is therefore with the differ ences in time-shape that we are here chiefly concerned. To illustrate these differences,let us suppose an individual possessedof a piece of land almost equally good for lumber ing, farming, or mining. He thus has the option of using 140 THE RATE OF INTEREST [CHAP. VIII it in anyone of three different ways: (1) in farming, which, let us say, will give him a regular and indefinite succession of crops with an income-stream of the type A in Figure 15; (2) for forest purposes, with very slight returns for the first few decades, and larger returns in the future, as indicated by the curve B; (3) for mining purposes, in which case we shall suppose that, as the mining plant is already set up and the richest ore lies close to the surface, the income is greatest for the early years and thereafter gradually decreases until the mine is exhausted. This is shown by curve C. What are the principles upon which the owner of the land chooses among these three income streams?

o A B· FIG. 15. We shall suppose, as heretofore, that there is a uniform rate of interest, and that any individual is free either to borrow or lend at that rate to any required amount. Under this hypothesis the choice among the options will simply depend on the one which gives the maximum present value, reckoned at the market rate of interest. Thus, if the use of the land for forestry purposes yields the following sums: zero for the first two years, $300 for the third, $400 for the fourth, $500 for the fifth, and $500 thereafter forever,-then the value of the land, if the rate of interest is 5 per cent., will be $8820. If the land is used for mining purposes, it will yield an income-stream of quite a different character, let us say, as follows: $2000 SEC. 3] SECOND APPROXIMATION 141 the first year, $1800 the second, $1600 the third, and so on diminishing annually by $200 to the point of 'exhaustion.

The present value of these sums is $9110. If, finally, the land is used for farming purposes and yields a net income of $450 a year perpetually, the present value will be $9000. Under these conditions the choice will evidently fallon the mining use, because, for mining purposes, the land is worth $9110, which is greater than $8820, its value for forestry purposes, and than $9000, its value for farming purposes. The particular income-stream selected will leave its im press on the time-shape of the total income~stream of the individual who owns it. For, as was seen in The Nature of Capital and Income,t the total final income-stream of any individual' is simply the sum of the incomes flowing from all the articles of property belonging to him. Hence, if one selects the mining use for his land, whereby the income'"9streamgradually decreases~ its tendency will be to produce a similar decrease in the total income-stream possessedby the individual. This tendency may, of course, be counteracted 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 income from the mine is not final enjoyable income, but consists of U interactions." But these interactions are readily transformed, through a chain of credits and debits, into final enjoyable income. 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 "enjoyable" income follows closely behind the "intermediate" income from the mine, and almost exactly copies it in time-shape.2 § 3 Yet the possessor of the mine is not compelled to copy in his-finalenjoyable income the mine's fluctuations of natural 1 Chaps. VII--X, XVII. j See The Nature 0/ Capital and Income, Chaps. VIII, IX, XVII.

142 THE RATE OF INTEREST [CHAP. VIII income. He may, for instance, prefer as his model an even flow of income such as he could get from the farm-use of his land. He will not, however, on that account choose this farm-use in preference to the mining-use; for the min ing-use has the larger present value, and the undesirable time-shape of its income-stream can be remedied by the methods explained in the previous chapter, - by lending at interest the proceeds of its earlier output and postponing enjoyable income to later years; or, more generally, by buying with the early proceeds such property as will yield returns at such future times as are most desired, in short, by "investing" instead of "spending." 1 The difference is merely that if he "spends" the yield from his mine, he is exchanging it for property from which enjoy able income comes promptly, whereas if he "invests," he is exchanging it for property from which enjoyable income comes more tardily. If he "spends" the mine's income as fast as he receives it, for food, clothing, shelter, travel, amusements, his "enjoyable" income simply shadows the " intermediate" income from the mine; but if he "invests"

the mine's income in more durable forms, such as furniture, or still better, dwellings, or stocks and bonds, his enjoy able income lags further behind the income from the mine on which it depends, and by proper manipulation can be distributed in time in any desired manner, - for instance, evenly, as above supposed. Since the mining-use has the higher present value, there is an advantage in selecting it rather than the farm-use which has the more desirable time-shape; for after the mining income is converted into the same time-shape as the farming income, it will be greater in magnitude, in the ratio of their present values,2 9110: 9000. 1 See Chap. VII, § 4. 2 This is evident from the principles explained in The Nature of Cap ital and Income, Chap. XIII; for the mining income, after conversion by investment at 5 per cent., will still have the same present value, $9110, and the even income of which this is present value is at 5 per cent., $455.50, instead of the $450 which the farm-use yields. The SEC. 3] SECOND APPROXIMATION 143 Again, it may be that the mine owner prefers, not a steady, but an ascending income-stream, and as in the case just cOIlsidered,he may secure·such an income by modify ing the income by means of properly graduated investments of the early parts of the mine's income. He can secure, if he likes, exactly the same time-sllape as though he had chosen the forestry use, with the advantage that his income will be larger. Thus, he may invest all of his first two years' income of $2000 and $1800 respectively, $1290 in the third year, $987 in the fourth year, and so on, reducing his annual investments by the proper gradations; and, proceed ing at the proper time to "realize" on these investments, he may obtain, as the final result of these operations, an income of precisely the same time-shape as that which he would have obtained from the forestry use. But the size of the income will be larger in the ratio of the present values of the mining and forestry income-streams, 9110: 8820. The following table exhibits these operations:As AGAINST RECEIVES FROM WHICH SO THAT WHAT THE FROM MINES HE INVESTS HIS INCOME FOREST USE IS WOULD HAVE YIELDED 1st year 2000 2000 000 000 2d year 1800 1800 000 000 3d 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 mine owner needs simply to invest annually the excess of· h.is income above $455.50; namely, $1544.50 in the first year, $1344.50 in the second year, etc. When the ninth year is reached the investment ceases, for the mine then yields only $400. This is then eked out by $55.50 from the amounts previously invested, and the same methods are pursued thereafter.

144 THE RATE OF INTEREST [CHAP. VIII Since any time-shape may be transformed into any other no one need be deterred from selecting an income because of its time-shape, but may choose it exclusively on the basis of maximum present value. On the other hand, were it not for the possibility 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 three optional employments of his land which possessed the highest value, but might be forced to take one of the others. We assume in this chapter that, after the most valuable option has been chosen, it is possible to borrow and lend or to buy and sell ad libitum. It will then happen that his income as finally transformed will be larger than it could have been B/,..--FIG. 16. if he had chosen some other use which afforded that same time-shape.

To illustrate this by a diagram, let AB and A'B' in Figure 16 be alternative income-streams, of which the descending income-stream AB has a larger present value than the as cending income-stream A'B' . The choice will then fall on AB, even though the individual prefers the time-shape of SEC. 4] SECOND APPROXIMATION 145 the other income-stream A'B'. He will then lend some of the early receipts from the income-stream AB and receive back some of the later, converting his income AB of unde sirable shape into the income-stream A"B" which has the desired shape. Consequently 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 worth of AB. As compared with A'B' it has the same shape but a greater size. We see, then, that the capitalist reaches his final income through the cooperation of two separate kinds of choice of incomes,- first, the choice of the income-stream which has the highest present value, and second, the choice among different possible modifications of this income-stream by borrowing and lending or buying and selling. These two kinds of choice are distinguished. from each other by the fact that the first is a selection among optional incomes of different market values, and the secondis a selection among optional incomes of the same market value.

§4 Since this double choice, when it is made, results in a perfectly definite income-stream, it might seem that the situation does not materially differ from the case of a rigid income-stream discussed in the preceding chapter. But the two cases differ materially; for in the present case of optional income-streams, the particular choice depends upon the rate of interest. A change in that rate may shift the choice of maximum present value to some other alter native. Thus, in the example cited, if the rate of interest should be 4i per cent. instead of 5 per cent., the order of choice would be changed. The value of the land for fores~ try use would be $9920, for farming use, $10,000, and for mining use, $9280.. The farming use would now be the best choice. Again, if the rate of interest should be 4 per cent. instead of 4! per cent., the present value of the use of the L 146 THE RATE OF INTEREST [CHAP. VIII land for forest purposes would be $11,300, for farming purposes, $11,250, and for mining purposes, $9450. In.

this case the forestry use would be chosen. We see, then, that it pays best to employ the land for mining if the rate of interest is 5 per cent., for farming if it is 4t per cent., and for forestry if it is 4 per cent. The various options open to the owner of the land at different rates of interest are summarized in the following table:PRESENT VALUE AT OPTIONAL USES 5% 4!°t'o 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 choice of income-streams. A high rate of interest will encourage investment in the quickly returning in comes, whereas a low rate of interest will encourage invest ment in incomes which yield distant returns. As the busi ness 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 according as interest is at one rate or another. Conse quently the existence of optional uses of capital introduces a new variable into the problem of interestdetermination.

To the individual, the rate of interest will determine the choice among his optional income-streams; but for society, rthe order of cause and effect is reversed, - the rate of in terest. will be influenced by the existence of the options. To trace this influence is the purpose of the present chapter.

SEC. 5] SECOND APPROXIMATION § 5 147 At first sight it may appear that we are reasoning in a circle: the rate of interest depends on individual rates of preference; the rates of preference depend on the time shapes of individual income-streams; and the choice of 'f these time-shapes of income~streams depend, we have just ~, on the rate of interest itself. It· ~. perfectly true that the rate of interest depends on a seri., of factors which finally depend on the rate of interest. Ye.\\his series is not the vicious circle it seems, for the last step i&not the inverse of the first. To distinguish be~\Yeen a true and a seeming example of a circular dependence~ we may contrast the following two simple problems: 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 about the height of the child. If we are told that the child's height differs from his father's by twice itself, the problem is circular and insoluble, for the last step is reducible to the first, being merely a concealed inversion of it. The problem essen tially states that the father's height is three times the child's and the child's one third of the father's, ~an ob vious circle. But if the dependence of the father's height on the child's were essentially different from that of the child's on its father's, there would be no circle. Thus, suppose as before that the father is three times as tall as the child, but that the child's height differs from the father's by four times the child's, less four feet. This sounds as circular as the first problem, - the father's height is expressed in terms of the child's, and the child's in terms of the father's; but here the second expression is not 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 each of these magnitudes is specified in terms of the other does not constitute a vicious circle.

148 THE RATE OF INTEREST [CHAP. VIII The same is true in our present problem. Real examples of circular reasoning in the theory of interest are common enough, and many of them have, in fact, been noted in earlier chapters, but the dependence above stated, of interest on the range of options and of the choice among those options on interest, is not a case in point. The logical principle holds true that any problem is determinate if only there are as many determining conditions as there are unknown quan tities; it is only necessary that these conditions shall be " independent"; in other words, that no one shall be derivable from the others. That this is mathematically the case under our present hypothesis is shown fully in the Appendix to this chapter. For our present purpose we need only present the matter to the reader's imagination by a series of successive approximations. To find out the rate of interest on which the market will finally settle, let us try successively a number of different rates. First, suppose a rate of 5 per cent. This rate will determine the choice of income-streams for each individual.

The landowner formerly supposed will, as we have seen, choose the mining'-use. 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. The amount which the wouldbe lenders are will ing to lend at 5 per cent. out of this year's instalment of their chosen income-stream will be perfectly definite, and likewise the amount which 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 for the given rate of interest,5 per cent., will be definite quantities. Should it happen that the demand for loans is less than the supply, it follows that 5 per cent. cannot be the correct solution of the rate of interest, for it is too high to clear the market.

SEC. 6] SECOND APPROXIMATION 149 In that case, let us suppose a rate of 4 per cent. Follow ing the same reasoning as before, we find that the landowner will now select the forestry use for his land. Other capital ists will select likewise their definite income-streams, and on the basis of these income-streams there will be the con sequent desire to borrow and lend. Should it then happen that the demand and supply of loans, on the basis of 4 per cent., are not equal, but that this time the demand ex ceeds the supply, it is a proof that not 4 per cent. is the true solution, but some higher rate. By again changing our trial rate back toward 5 per cent. we may evidently reach some intermediatepoint, let us say4! per cent., at which rate not only will all individuals··choose definite income.. streams, but also, at the same time, the demand and supply of loans engendered by these income-streams will exactly clear the market.

The introduction, therefore, of flexibility into our in come-stream still leaves the problem of interest entirely determinate. Though the income-streams are now a matter of choice, there is one definite choice corresponding to each rate of interest. The particular rate of interest which will solve the problem is that which will both deter mine the choice among income-streams differing in present value, and also bring it about that individual departures from such income-streams shall mutually cancel each other, YJ - in other words, that the markets for loans and sales shall be cleared. § 6 For the determination of the rate of interest we have therefore to modify the various conditions as given in the previous chapter. The modifications which are introduced are, (1) that in place of. the single fixed income-stream formerly· assumed, there now exists a given range of choice between different income..,streams;·and (2) that whereas formerly the individual had no".choice of income-streams, 150 THE RATE OF INTEREST [CHAP. VIII he now chooses out of those available the one which pos sesses the maximum present value. We therefore have six conditions determining the rate of interest, as follows: (1) There exists for each individual a given series of possible income-streams among which he may choose; (2) Each individual's preference rate· depends upon his income-stream, - its size, shape, composition, and proba bility; (3) The rates of preference of different individuals must be equal to each other and to the rate of interest in the market; (4) Out of all availableincome-streams,that one is selected which has the maximum present value for the rate of interest finally determined; (5) The rate of in terest must be such as will equalize supply and demand, or exactly clear the market; (6) The additions to and deduc tions from each incomErstream, brought about by borrowing and lending or buying and selling, must be such that their net present value is zero.

As to the first condition, viz., the existence of a range of choice, it is worth noting that some of the optional income streams would never be chosen under any circumstances. These are the income-streams the present value of which could not be the maximum,' no matter what the rate of interest might be. We have seen that the land, in our example, would be most profitably employed for farming, for mining, or for forestry, according to the rate of interest. But it would not be employed, let us say, for a quarry, no matter what might be the rate of interest. The optional uses which are thus out of the question may be called ineligible. We need consider only the eligible options. § 7 The six conditions for determining interest just enu merated differ from those given in the preceding chapter chiefly by the introduction of number four, - that the use of capital which yields the maximum present value will SEC. 7] SECOND APPROXIMATION 151 be selected. This additional condition is of so much impor tance that it should be restated in two other forms.

To illustrate these, let us .recur to the example of the land, which could be used in anyone of three ways. We found that when the rate of interest was 4 per cent., the use chosen·would be forestry, as this possessed the greatest present value. If we now compare, year by year, the in come from the land when used for forestry purposes with the income which it might have yielded if used in one of the other ways, - as farming, - we shall see that in some years there is an excess in favor of the forest use, and in other years a deficiency, as shown in the following table: ANNUAL VALUE OF USES FOR DIFFERENCE IN FAVOR OF FORESTRY FARMING FOREST USE 1st year · · - 450 -450 2d year · · - 450 -450 3d year · · · 300 450 -150 4th year . 400 450 -50 5th year · 500 450 +50 6tliyear 500 450 +50 7th year · 500 450 +50 8th year · 500 450 +50 9th year · · 500 450 +50 10th year . 500 450 +50 11th year . · 500 450 +50 Each year after • · 500 450 +50 Here we see that for the first four years there is a com parative disadvantage or sacrifice (amounting to $450, $450, $150, $50 in successiveyears) from the use of the land for forest purposes as compared with farm uses, but that this disadvantage is made up later by an advantage or return of $50 per annum. If we now· take the total present value, at 4 per cent., of the deficienciesmarked with a minus sign, we shall obtain $1024, whereas the present value of the excesses (continuing in perpetuity), indicated by a positive 152 THE RATE OF INTEREST [CHAP. VIII sign, will be $1070. Thus the present value of the gains exceeds the present value of the sacrifices by the difference between $1070 and $1024. In other words, as reckoned in present estimation, the gains outweigh the sacrifices. We may say, therefore, that, the rate of interest being 4 per cent., forestry is preferable to farming because of a surplus of advantages over disadvantages reckoned in present value. But if the rate of interest were 4! per cent. we should find the present value of the sacrifices to be $1017, and the present value of the gains, $930, showing a preponderance of the sacrifices. That is, if the rate of interest is 4! per cent., the sacrifice in using the land for forestry rather than mining outweighs the gains. The land would, therefore, in that case, not be used for forestry purposes.

The general principle is, therefore, that out of the various income~streams at the disposal of the capitalist, he chooses the most advantageous, or, more fully expressed, the one which, compalGd with any other, offers advantages which, reckoned in"present estimation at the given rate of interest, outweigh the disadvantages; and this is evidently merely a new formulation of the original principle that the use chosen will be that which has the maximum present value at the given rate of interest. § 8 There is yet a third method of stating this principle. This method may also best be shown by an example. We have seen in the previous illustration that if the rate of interest is 4 per cent., the net advantage is in favor of the forest use; and if the rate of interest is 4! per cent., the advantage is in favor of the farming use. It is evident that at some intermediate rate of interest the comparative advantages of the two uses would be equal. This inter mediate rate is approximately 4.2 per cent. To show the nature and importance of such an equalizing rate, we ma~r SEC. 8] SECOND APPROXIMATION 153 vary the example given to the following simple illustra tion: ANNUAL VALUE OF USES FOR DIFFERENCE IN FAVOR OF FORESTRY FARMING FORESTRY 1st year 000 100 -100 2d year 210 100 +110 3d year 100 100 000 4th year 100 100 000 Each subsequent year 100 100 000 In this case the equalizingrate is 10 per cent. .If the two income-streams be both discounted at 9 per cent., the for estry use will have the greater present value, $1112, as against $1111 for the farming use. If 11 per cent. is used, the scales are turned and the farming use is the more(valu able, being worth $909, as against $908 for the forestry.

At the intermediate rate of 10 per cent., the two uses are equivalent in present value, both being worth exactly $1000. Since 10 per cent. is the rate which equalizes the advantages and disadvantages of the two alternatives in present value, it is the rate at which the third column in the table will have a present value of zero. Again, it is the rate which the $110 yields on the -$100, or the rate "realized" to the investor who, by choosing the forestry use, relatively sacrifices $100 this year, but obtains a com pensating return of $110 next year. Such a rate is therefore called the rate of return on sacrifice. These terms are ap plied exclusively to the comparative merits of two alter native income-streams. By" sacrifice" is meant the com parative loss from one's income-stream at first, caused by substituting one use of capital for another; and by "return" is meant the comparative gain which later accrues by rea son of this same substitution.

To return to the original example and the table in §'7, the equalizing rate was 4.2 per cent. This was the 154 THE RATE OF INTEREST [CHAP. VIII rate of return on sacrifice of the forestry use when com pared with the farming use. It is the rate which makes the series of future returns, $50, $50, etc., indefinitely, equivalent in present value to the first sacrifices, $450, $450, $150, and $50. It follows 1 that if the latter series of sums were successively deposited at 4.2 per cent. in a sav ings bank, they would "earn" for the depositor the former series of sums. In short, 4.2 per cent. is the rate which an investor· "realizes" who in the first four years sacrifices suc cessively $450, $450, $150, and $50, and receives as return in succeeding years, $50, $50, etc. In general, the rate of return on sacrifice is a supposed rate of interest which will make equal the present values of the "sacrifices"

and "returns" involved in comparing one optional income-stream with another. It is not, of course, to be confused with the actual rate of interest. Now if the actual rate of interest is 4 per cent., while the rate of return on sacrifice which would be realized by choosing the forestry rather than the farming use is 4.2 per cent., it would evidently be profitable to choose forestry. As the investor might put it, he would be getting more than the market rate, - getting 4.2 per cent. instead of 4 per cent. If, however, the rate of interest in the market is 4.5 per cent., it would not pay to choose the forestry use, for to do so would, comparatively to the farming use, re turn only 4.2 per cent. In this case the prospective investor would evidently prefer to choose the farming use, and then lend his money at 4.5 per cent. Recurring to the former table, we see that had he chosen the forest use instead of the farming use he would have sacrificed during the first four years successively $450, $450, $150, and $50.

He may, if he likes, put these very sums at interest in a savings bank and make 4.5 per cent. upon them, whereas, had he chosen the forest use, he would have received only 4.2 per cent. In other words, when a man can invest at 1 See The Nature 01 Capital and Income, Chap. XIII.

SEc.8} SECOND APPROXIMATION 155 4.5 per cent. by lending, he will not invest at 4.2 per cent. by choosing forestry rather than farming. Out of all possible employments of his capital, the capitalist will choose that one which, compared with any other, has advantages worth the disadvantages, - returns worth the sacrifices. This means that the rate of return on sacrifice will exceed the rate of interest. In case the advantages precede the disadvantages, as when the merits of the mining use are compared with those of the farming use, the proposition must be reversed, as follows: The earlier advantage will be chosen only in case the rate of later sacrifice on present return is less than the rate of interest. In such a case it would be more conven ient, in comparing the two uses, to regard them in the opposite order, that is from the point of view of the advantages, not of the mining use over the farming use, but of the farming use over the mining use. This will m.ake the sacrifices precede the returns. As long as the sacrifices always precede the returns, we need only to con...

sider whether or not the rate of return on sacrifice exceeds the rate of interest. If it does,~ the optional income stream which, compared with another, yields such return on sacrifice will be chosen in preference; otherwise it will be rejected.! 1 Of course it· is possible to have two alternative uses so related that the sacrifices are not grouped together in one mass and the re turns in another, but are intermingled. Thus, the first few years may offer advantages, the following, disadvantages, those following still later, advantages, and so on in alternating succession. In such a case, if the market rate of interest is 4 per cent. and the rate which equalizes the gains and sacrifices is 4.2 per cent., in order to decide which of the optional income-streams ought to be chosen, it would be necessary to consider the effect of a slight variation from the 4.2 per cent. rate used in discounting the comparative advantages and disadvantages. Let the rate change from 4.2 per cent. to 4.1 per cent., i.e. toward the actual rate 4 per cent. If the .effect of such a change is to make the advantages outweigh the disadvantages, in present value, it is a proof that the income-stream possessing these advantages and disadvantages is preferable to the one being com pared with it. In such a case it is much more convenient not to 156 THE RATE OF INTEREST [CHAP. VIII The condition, therefore, determining the choice between options may be stated in anyone of three ways, namely: (1) Out of all options that one is selected which has the maXimUITl present value, reckoned at the market rate of interest; (2) Out of all options that one is selected of which the advantages over any other outweigh, in present value, its disadvantages, when both are discounted at the market rate of interest; (3) Out of all options that one is selected wl1ich, conlpared with any other option, yields a rate of return on sacrifice greater than the rate of interest.

§ 9 Let us now apply the third mode of statement to the case in which the range of choice is not confined to a few options, but extends to an infinite number. This case is really more like the facts of life than the imaginary case of a few options, such as the farming, mining, or forestry uses of land. As a matter of fact, each of these is not a single use, but a whole group of optional uses. Thus, the farmer may cultivate his farm with any degree of intensity; and for each particular degree of intensity he will have a different inconle-stream. He may, for instance, invest $100 worth of labor in the present, in order that in six months he may have a larger income than otherwise, by $200. If the rate of interest is 4 per cent. (reckoned semiannually), he would evidently prefer this option; for it diminishes his present income by $100 and increases his income six months later by $200, being 100 per cent. in six months, whereas the interest for that time is only 2 per cent. Another course would be to invest, not $100, bllt $200, in present cultiva tion. The extra $100 would add to his returns in a half year's time something less than the $200 yielded on his consider at all any equalizing rate, such as 4.2 per cent., but to recur to one of the preceding methods. In practice, however, such per plexities seldom or never arise.

SEC. 9] SECOND APPROXIMATION ;" 157 first $100, let us say $150. This also would be a good in vestment, yielding him 50 per cent. return when the rate of interest is 2 per cent. And so each successive choice, compared with its predecessor, shows a law of decreasing returns for additional sacrifice. Thus, if he invests, not $200, but $300, the third $100 thus sacrificed will add to his returns in six mont.hs, let us say $120. Here is a gain of 20 per cent., whereas the rate of interest is only 2 per cent. As another option, he may sacrifice a fourth $100 for the sake of a return of an additional $110; in like manner he may sacrifice a fifth $100 for the return of an additional $105; a sixth $100 for an additional $103; a seventh $100 for $102. Thus far, each successive option is preferred to its predecessor; for, as compared with its predecessor, each option yields more than 2 per cent., which is the rate of interest for six months. The next opti~n is to sacrifice an eighth $100 for an additional $101 in six months.

Evidently, it will not-be to the farmer's interest to take this last step; he will stop at the previous step, at which he gets a 2 per cent. return on the last sacrifice of $100. As we saw in the preceding section, each successive option is chosen as long as the rate of return on sacrifice of that option, compared with the previous option, is greater than the rate of interest, and that use is rejected at which the rate of return on sacrifice becomes less than the rate of in terest. The intensiveness of his farming is thus deter mined by the rate of interest. lie chooses that degree of intensiveness which gives his income-stream the maximum present value, - which is the same thing as choosing that degree at which the rate of return on sacrifice is equal to the rate of interest. The various possible income-streams are represented in Figure 17. Income-stream A is large for the first six months, and for the second six months very small. The next income-stream B is $100' smaller than A for the first six months, and $200 larger for the last six months.

The other options are also indicated. Incomtrstream H 'is 158 THE RATE: OF INTEREST [CHAP. VIII the one chosen, because, as compared w.its predecessor, its disadvantage is $100 for the first six months. and its advantage $102 for the second six months - just en~h to "compensate for interest." We therefore reach the conclusion that where the options are indefinite in number, the option chosen, compared with a neighboring option with which it was in competition, yields a rate of return on sacrifice equal to the rate of interest. FIG. 17. This rate of return, computed on the basis of two alterna tive income-streams closely neighboring upon each other, we shall call the marginal rate of return on sacrifice. It follows that, when there is a continuous range of choice, we may substitute for the statement that the choice will fall upon the option of maximum present worth, the better statement that the choice will fall on the option whose t marginal rate of return on sacrifice, reckoned relatively to a 'neighboringoption, is equal to the rate 01 interest.

SEC. 10] SECOND APPROXIMATION § 10 159 We have introduced a new magnitude into our discus sion; namely, the rate of return on sacrifice, and especially the particular value of this rate of return called the mar ginal rate of return 011 sacrifice. This marginal rate of return on sacrifice comes close to being a "natural rate of interest." By means of it we are enabled to admit into our theory the elements of truth contained in some of the claims of the productivity theories, the cost theories,l and Bohm-Bawerk's theory of the technique of production. The example just given of the farmer who selects, out of a series of income..,streams,that for which the return on sac rifice is equal to the rate of interest, perfectly exemplifies the theory of John Rae.2 According to Rae, all instru ments may be arranged in an order depending on the rate of return on cost. Some instruments return double the cost of their formation in a year; in other words, the rate of return on sacrifice is 100 per cent. Others return 50 per cent., 20 percent., alld so on in descending order. In any community, Rae says, instruments will be "wrought up" ,to the point at which the rate of return on sacrifice corresponds t~ Rae's equivalent for what we have called the "rate of preference," and this, in turn, as we have seen, is equal to the rate of interest. It will be evi dent to every student of Rae that the preceding discussion accords with Rae's idea that those instruments which most promptly· yield returns are formed first, and that the less rapidly returning instruments are successively formed until the margin is reached which corresponds to the rate of interest. The statement of Rae that, for a certain' cost of formation, an instrument will yield a certain return, is merely a form of our statement that a certain decrease of present income will be accompanied by a certain increase 1 See Rae, The Sociological Theory of Capital, Chapters IV to VI.

'Cf. also Landry, L'Interet du Capital, 1904, Chapter III; Carver, Distribution of Wealth, New York, 1904, p. 230.

160 THE RATE OF INTEREST [CHAP. VIII in future income. The relation between the immediate decrease and the future increase will vary within a wide range, wherein the choice will fall at the point correspond ing to the ruling rate of interest. The subject is one which may be looked upon from many points of view, and it is important that these points of view should be thoroughly coordinated. In the example above given, the farmer was supposed to invest to-day and receive returns six months afterward. Consider now a case in which the returns are repeated regularly each year. Let us suppose that our farmer possesses some swamp land in a primitive state. He has a large range of choice as to the method of utilizing this land. He can allow it to remain a swamp or, by clearing and draining it, convert it into crop-yielding land, the yield being in proportion to the thoroughness with which the clearing and draining are accomplished. Under the first llse, let us suppose that he derives a perpetual net income of $50 a year, and let us suppose that, at an immediate cost of $100 for clearing and draining, he can secure a perpetual net income from crops of $75 a year. As between these two choices, the second involves a decrease of immediate in come of $100, and an increase in annual income thereafter, from $50 to $75, or of $25. In other words, the invest ment of $100 will yield him 25 per cent. per annurn. Evi dently, if the rate of interest in the market is 5 per cent., it will pay him to make such an investment. ' N ext sup pose that a second $100 invested in improving the swamp would cause the crop returns to be $90 instead of $75 a year, or $15 more than before. Evidently, the investment of the second $100 yields 15 per cent., and is also a lucrative one, when we consider that the rate of interest is only 5 per cent. A third $100 may increase the annual crop to $100 instead of $90, an excess of $10 as compared with 'the previous investment, or a yield of 10 per cent. A fourth $100 invested will cause the annual crop to be $105, giving an increase of $5 and a yield of 5 per cent. A ~th $100 will SEC. 11] SECOND APPROXIMATION 161 cause the crop to increase to $108, giving an increase 'of $3 and a return of 3 per cent. Evidently it will pay the farmer to invest in draining and improving his swanlp up to the fourth $100, but not to the fifth $100. Rather than invest this fifth $100 and receive-thereon an annl1al income of $3 a year, he would prefer to invest $100 in the savings bank and receive 5 per cent. a year. ~ }.~ r In other words, the intensitywith whichhe will improve1~-....j/l1 ~d cultivate his land is det~rmined?y the current rate Qt rI ~ , !nterest. Should the rate of Interest In the market fall from ,'"b the 5 per cent. just assumed to 2 per cent., it would then pay him to invest the fifth $100. For, evidently, if nee,d be, he could borrow $100 at 2 per cent. and receive from his land a return of 3 per cent. As Rae has so clearly pointed out, in communities where the rate of interest is low~ swamps will be more thoroughly improved, roads better made, dwellings more durably built, and all instruments "worked up" to a higher degree of efficiency and a lower marginal return than in a community where the rate of interest is high.

The same illustrations which have been given will serve to set the present theory in line with that of Adolphe Landry in his Theorie de l' Inter~t. He states that one of the conditions determining the rate of interest is the' "productivity of capital," in the peculiar sense which he gives to this phrase. The process described by LaJ;ldry by which the productivity is assimilated to whatever rate .Q! interest happens to rule the market.L..virtually corre sponds to tIle successive selection of income-streams as outlined in the preceding examples. § 11 Our next case will serve to show how the element of truth already pointed out in the productivity theory of Del Mar and George fits into the theory here propounded. This theory is that the rate of interest ·corresponds to the rate of growth of animals and plants. In Chapter III we 162 THE RATE OF INTEREST [CHAP. VIII saw that the time of cutting a forest will be that at which it is growing at a rate equal to the rate of interest. Thus, if nine years from the planting of the forest it contains 900 cords of wood, while in ten years it contains 1000 cords, in eleven years, 1050, and in twelve years, 1071, and if the rate of interest is 5 per cent., the cutting will occur between the tenth and eleventh years. This choice is determined by precisely the same principle that has already been enunciated; namely, that the particular income-stream selected will be that which has the maximum present value; Of, in other words, that which is such that the marginal rate of return on sacrifice will be equal to the rate of interest.

To show how this principle applies to the cutting of the forest, let us consider as the first option the cutting of the forest at the end of nine years, when the income-stream consists of the single item, -the production of 900 cords of wood.l The second option is cutting the forest at the end, of ten years, and receiving an income item of $1000. The two alternatives may be put in the tabular form previously em ployed for the case of forestry and farming, as follows: OPTIONAL INCOMES FROM FOREST lO-YEAR 9-YEAR DIFFERENCE PLAN PLAN IN FAVOR OF IO-YEAR PLAN 1st year 000 000 2d year 000 000 -- -- -- -- 9th year 000 900 - 900 10th year 1000 000 +1000 1 Inasmuch as we assume that the income from the forest is all to accrue at one time - the time of cutting - instead of being dis tributed over a long period, the phrase It income-stream" might here better be replaced by U income i tern."

SEC. 12] SECOND APPROXIMATION 163 The last column shows that the ten-year plan, compared with the nine-year plan, involves a sacrifice of $900 in the ninth year which might be secured by the nine-year plan, but involves a return of $1000 in the tenth year. The rate of return on sacrifice would thus be a little over 11 per cent. If the rate of interest in the market is 5 per cent., it would evidently pay to "wait," or to choose the cutting in the tenth year rather than the ninth year. The· next option would be to cut in the eleventh year, which, as compared with the second alternative, would involve a sacrifice of $1000 in the tenth year and a return of $1050 in the eleventh year - in other words, a rate of return on sacrifice of 5 per cent. Evidently, then, it would be a matter of indifference whether the forest was cut in the tenth or eleventh year, inasmuch as the rate of return on sacrifice in one alternative as compared with. the other would be exactly equal to the rate of interest.

Similar reasoning shows that the choice of t4e next option, that of cutting the forest in the twelfth year, would yield a return of 1%-10' or 2 per cent. Inasmuch as 2 per cent. is less than the rate of interest, this alternative would be rejected. Should, however, the rate of interest fall to 2 per cent., or below,it is clear that the time of cutting the forest would be postponed until the rate of increase in stumpage value was reduced to correspond to the rate of interest. § 12 The same example will serve to show the bearing of Bohm-Bawerk's discussion as to the influence of the H roundabout process" upon the rate of interest. Accord ing to him, it is at the option of society to invest to-day's labor in·any one of m~ny different processes.. bringing re turns in different lengths of time, let us say, nine years, ten' years, eleven years, etc.; and he premises that ·the returns in these successive years will increase, but at a diminishing 164 THE RATE OF INTEREST [CHAP. VIII rate, let us say, in the order of the numbers already given: $900 for the ninth, $1000 for the tenth, $1050 for the eleventh, $1071 for the twelfth, etc. That use will be selected, as B6hm-Bawerk has pointed out, which has the maximum present value; and also, as he points out, the lower the rate of interest, the remoter will be the" produc tion period" on which the choice will fall. If the rate of interest is 5 per cent., the choice will fallon the tenth or eleventh year; if the rate is 2 per cent., on the eleventh or twelfth year; and the lower the rate of interest the more "roundabout" will be the methods of production.

This is entirely valid under the hypothesis involved; namely, that there is a range of optional returns, each consisting of a definite return at a definite point of time, increasing as the production period increases, but at a decreasing rate. It is also true, as Bohm-Bawerk has pointed out, that not only does a lower rate of interest tend to the choice of remoter returns, but that, contra riwise, the choice of remoter returns tends to check the fall in the rate of interest; the reason, expressed in our own terminology, being that the choice of an income stream relatively large in the future and small in the present tends to increase the relative valuation of present as compared with future income. The existence of such a range of choice as Bohm-Bawerk assumes, therefore, tends to act as a buffer, checking the variations in the rate of interest. This effect of the operation of a range of choice will be again referred tb.

§ 13 Thus, the elements of truth which were found in the pro ductivity theory, in the cost-theory, and in B6hm-Bawerk's technique-of-production theory, all find a place under the head of the choice among optional uses of capital. In some cases, as in the example illustrating the theories of Henry George and Bohm-Baw~rk, the selection of one SEC. 13] SECOND APPROXIMATION 165 option rather than another involves, as its effect on the income-stream, the mere omission of one item of income and the substitution of another. In other cases, as in the examples illustrating the theory of John Rae and Adolphe Landry, the selection of one option rather than another .involves the application of labor, or the incurring of cost of some other sort, for the sake of a future return. But in all cases there is a choice among optional income-streams, - a decision how to adjust the income-stream at different periods, whether or not to decrease it at one time in order to increase it,at another. It matters not in what way or at what periods of time the flexing of the income-stream occurs. It may be, as in the case of the farmer contemplating the planting of a crop, that the income is flexed or varied at merely two points of time, as seed time and harvest; or, as in the case of clearing a swamp, there may be a decrease of present income for the sake of an increase of the income of all succeeding years; or there may be any other arrangement of sacrifices and returns. But in all cases we have to. deal simply with a range of choice among income-streams' of different conformations. If this range of choice were limited to a few options, the best state ment of the principle which governs the.selection would be that the income...,streamhaving the maximum present worth would be selected. But if there isa varied or continuous range of choice, the preferable method of stating the prin ciple is that the income-stream will be selected which, as compared with the neighboring streams, will yield a rate of return on sacrifice equal to the rate of interest.

To a person who has never tried to co~ect them, many of the theories of the authors just compared seem to have no vital relation. But they are seen to be connected as soon as we look at them in the light of the concept of an income-stream. The problems of choosing when to cut a forest, of what length to make a production period, tp what degree of intensiveness to cultivate land, or how far to improve a piece of land, are all problems of choosing the 166 THE RATE OF INTEREST [CHAP. VIII best out of innumerable possible income-streams. In each problem the rival income-streams present differences as to size, shape, composition, or probability, - especially shape. In respect to shape, they can best be compared by means .... _--_.....---FIG. 18. of diagrams. Figures 18 to 21 show typical ways in which the income-stream may conceivably be subjected to slight FIG. 19. variation. The unbroken line in each case indicates the income-stream chosen, and the dotted line a neighboring FIG. 20.

possible choice. Figure 18 may be taken as applying to the planting of a crop; Figure 19 to the draining of a swamp; -'FIG. 21. Figure 20 to the cutting of a forest; and Figure 21 to a case of alternating sacrifices and returns.

SEC. 14] SECOND APPROXIMATION 167 To students of physics, it will be interesting to obesrve that the identity of the principle of maximum present value with the principle that the marginal rate of return on sacrifice is analogous to the identity between the prin ciple of minimum energy and D'Alembert's principle. A suspension bridge assumes the form which will bring its center of gravity at the lowest possible point; this is in accordance with the principle of minimum energy. It is clear that the various parts of the structure, so to speak, compete with each other in the effort each to reach the lowest possible point. The result is a compromise; no part reaches the lowest point for itself but is held above it by the sagging of other parts. If from the position of equilibrium a slight displacement of any de scription is imagined, it requires that the depression of some parts is offset by the elevation of others, the work being done by the one set being equal to that done upon the other set; this is in accordance with the principle of D'Alembert (principle of virtual displacements). The income··curve is like the curve of the hanging bridge re versed. The effort is to raise it as high as possible so that its present value is a maximum. But its various parts compete with each other in the attempt each to reach the point highest in present value. The result is a compro mise; no part reaches the highest value possible for itself but is kept from so doing by the other parts. If from the position of equilibrium a slight displacenlent of any de scription is imagined, it requires that the elevation of some parts is offset by the depression of others, the 'present value of the gains being equal to the present value of the losses. This is equivalent to saying that the rate of re turn on sacrifice is equal to the rate of interest.

§ 14 Up to this point one complication in the problem of interest has been carefully kept in the background, not 168 THE RATE OF INTEREST [CHAP. VIII because it invalidates any of the principles which have been developed, but because it seemed advisable not to distract attention from the essential features of the theory by intro ducing prematurely a factor which, after all, is more intri cate than important. This complication consists in the fact that not only, as we have seen, does the choice between different optional income-streams depend upon the rate of interest, but also that even the rangeof choice depends upon that rate. If the rate of interest is changed, a change is produced not only in the present values of the income streams but in the income-streams themselves. To recur to the illustration of the land which may be devoted to one of three uses, not only is it true that a change in the rate of interest from 5 per cent. to 4 per cent. will change the relative present values of the income-streams which consist of the farming, mining, and forestry uses of the land, but this change from 5 per cent. to 4 per cent. may also materially affect the three income-streams themselves.

The net income from any instrument of wealth is the difference between the total gross income and the outgo. But many of the elements, both of income and outgo, are materially dependent upon the rate of interest. This is true, whether the items of income and outgo are "final" or merely "intermediate." 1 In the case of intermediate income, or "interactions," a change in the rate of interest affects the income-stream directly, because, as has been shown elsewhere,2the valuation of an interaction involves the discount-process and is therefore dependent ·upon the rate of interest. Thus, the service of planting apple trees will be valued in part by discounting the value of the fu ture apples. Given the value of the apples, it is evident that the value of the planting will be high or low according as the discounting is reckoned at a low or a high rate of in terest. But even "final" income - the income secured from the apples, for instance - may be indirectly affected 1 See The Nature 01 Capital and Income, Chaps. VII-X.

2 Ibid., p. 317.

SEC. 14] SECOND APPROXIMATION 169 by a change in the rate of interest, through a redistribu tion in the amounts, combinations, and values of the vari ous items constituting final income, and hence in their values. It would lead us aside from our topic to follow·these lines of reasoning to the limit. It will sufficeto indicate in brief their application in the case of labor. The labor cost is one of the commonest elements of outgo in the income account connected with any group of capital. For· in stance, whether the land is used for farming, mining, or forestry, it must be worked by human beings, and the cost of the work will materially affect the values of the three income-streams. Now the cost of that work is wages, and, to the employer, takes the form of and·normally represents the discounted value of the ultimate enjoyable services to which the labor leads. Consequently, if interest varies, wages will vary. Thus, if the land is used for farming, the wages paid for planting crops will be gauged in the estimation of the farmer by discounting the value of the expected crops, and· will vary somewhat according as the discounting is at 5 per cent. or 4 per cent. In like man ner, the workers engaged in bridge building are paid the discounted value of the ultimate benefits which will accrue after the bridge is built; the wages of those engaged in making locomotives normally represent the discounted value of the completed locomotives, and hence (as the value of a completed locomotive is in turn the discounted value of its expected services) their wages represent the discounted value of the ultimate benefits in the series. In all these' cases, the rate of wages is the discounted value of some future product, and therefore tends to decrease as interest increases. But the effects in the different lines will be very unequal. Workers whose product matures rapidly, as in the case of domestic servants and in the case of those en gaged in putting the finishing touches on enjoyable goods, will have their wages comparatively little affected by the rate of interest. On the other hand, for laborers who are 170 THE RATE OF INTEREST [CHAP. VIII engaged in work requiring much time, the element of dis count applied to their wages is a much more important factor. If a tree planter is paid $1 because this is the discounted value, at 5 per cent., of the $2 which the tree will be worth when matured in fifteen years, it is clear that a change in the rate of interest to 4 per cent. will tend materially to raise the value of such labor. Sup...

posing the value of the matured tree still remains at $2, the value of the services of planting it would be, not $1, hut $1.15. On the other hand, for laborers engaged in a bakery or other industry in which the final satisfactions mature early, the wages are almost equal to the value of these products. If they produce final services worth $1, due, let us say, in one year, their wages, being the dis counted value of this sum at 5 per cent. per annum, would be 95 cents. Evidently in such a case a change in the rate of interest from 5 per cent. to 4 per cent. would only increase the wages from 95 cents to 96 cents. But it is clear that such unequal effects coming from a reduction in the rate of interest, as an increase from $1 to $1.15 in one industry, and from 95 cents to 96 cents in another, could not remain permanently. For the laborers engaged in the occllpations in which their work matured in a relatively short time, such as the bakers just mentioned, finding that their neighbors engaged in lengthier processes were receiving higher wages, would tend to desert their work for this more remunerative employment. The consequence would be that the amount of labor, and con sequently the amount of final enjoyable income, accruing from the shorter processes would be reduced, and that from the longer processes increased. The consequence of this, in turn, would be to raise the value of the earlier enjoyable income and lower that of the later. Therefore, in the end, the change in the rate of interest from 5 per cent.

to 4 per cent. would effect a redistribution in the values, not only of intermediate items of income, but in the values of the final items themselves. For the various final elements SEC. 15] SECOND APPROXIMATION 171 of income are bound together, as it were, by means of the competition of the preparatory services, such as those of the laborers just mentioned, and the consequent neces sity of equalizing the remuneration for these preparatory serVIces. In short,a change in the rate of interest will affect all income-streams flowing from given instruments of capital whether these streams consist of interactions or of final services. It will affect (1) the value of interactions, like tree-planting or bread-baking, because the rate of interest enters directly into the valuation of all interactions; and (2) the value of final income,enjoyments,becausemany in teractions, as, for instance, the services of laborers, may be used interchangeably in several different directions.

The effect of a change in the rate of interest, on the value of the interactions, will naturally be the more pro nounced, and will be greater in a country where lengthy processes are usually employed than in one where the shorter ones are common. If, for instance, laborers in a given country are engaged largely in building elaborate works like the Panama Canal, in planting forests, and otherwise investing for the sake of remote returns, a fall in the rate of interest will produce a considerable rise in wages; whereas, in a country where such lengthy pro cesses are unknown and workmen are chiefly employed in tilling the ground and performing personal services, a change in the rate of interest will scarcely affect wages and the values of other preparatory services at all. § 15 The complete discussion of this subject would lead us toa statement of the general theory of the "price of labor"

and of prices in general. For present purposes, it is only necessary to emphasize the bare fact that the range of choice between different income-streams is somewhat dependent upon .the rate of._.interest. If the modification 172 THE RATE OF INTEREST [CHAP. VI!I due to this fact were introduced into the tables previously given for the three different uses of land, we should find that the income-streams from using the land. for farming, forestry, and mining would differ according to the rate of interest. Thus, let us suppose, as before, that for a rate of interest of 5 per cent. the three optional income-streams are: FORESTRY FARMING MINING 1st year 000 450 2000 2d year 000 450 1800 3d year 300 450 1600 4th year 400 450 1400 5th year 500 450 1200 6th year 500 450 1000 7th year 500 450 800 8th year 500 450 600 9th year 500 450 400 10th year . 500 450 200 Thereafter . 500 450 000 In our previous discussion, when we changed the rate of interest from 5 per cent. to 4 per cent., we supposed the figures in this table to remain unchanged. The only change we had then to deal with was the change in their present values. Now, however, we admit the possibility of a change in tIle table figures themselves. If the rate of interest falls to 4 per cent., the product of forest, farm, and mine will be more nearly equal to the value of the ultimate services to which they lead. The value of lumber will be more nearly equal to the value of the houses it makes, and these to the value of the shelter they give; the value of wheat from a farm will be nearer the value of the bread it will make; and the value of ore from a mine will be nearer the value of the steel it will become, and this, in turn, more nearly equal to the values of those innumerable satisfac tions which come through the use of steel. These shiftings forward of the values of the intermediate income of forest, SEC. 15] SECOND APPROXIMATION 173 farm, and mine toward the values of the ultimate satis factions to which they lead, combined with possible read justments in the values of these satisfactions themselves -the values of house shelter, bread consumption, etc......:..

will result in a change, say in the figures in the table from those just given for 5 per cent. to the following for 4 per cent.:FORESTRY FARMING MINING 1st year 000 500 2100 2d year 000 500 1900 3d year 350 500 1700 4th year 450 500 1500 5th year 600 500 1300 6th year 600 500 1100 7th year 600 500 850 8th year 600 500 650 9th year 600 500 450 10th year 600 500 225 Thereafter. 600 500 000 If, then, the rate is 5 per cent., the landowner will choose that use among the three which, computing from the figures in the first table, has the greatest present value; while if the rate is 4 per cent., he will choose that which, computing from the figures in the second tabl~, has the greatest present value. If, then, the rate is 5 per cent., he will choose min ing, since, as we saw in § 4, the present values, when we compute at 5 per cent., are: forestry, $8820; farming, $9000; mining, $9110; ·but if the rate is 4 per cent., he will choose the highest from the present values at 4 per cent., computed from the second table. These present values now are: forestry, $13,520; farming, $12,500; mining, $10,100.

Thus the owner will choose forestry. It is true in-this case that the change in the range of choice does not affect the final result. In § 4 the choice also fell on the forestry use. The only difference is that the particular 174 THE RATE OF INTEREST [CHAP. VIII figures of present values in our revised 4 per cent. com putation are different from their values in our original 4 per cent. computation. The present values at 4 per cent. for forestry, farming, mining, respectively:Under our present hypothesis are Under our former hypothesis were 13,520 11,300 12,500 11,250 10,100 9,450 But, whatever, the final outcome of all the readjust ments, it is evident that the introduction of the in fluence of the rate of interest on the range of choice does not in any material way affect the reasoning already given in regard to the determination of interest. Since the rate of interest will itself fix the range of choice, it will still be true that, once the range of choice is fixed for a given rate of interest, the individual will choose, as before, that use which has the maximum present value. On the basis of this choice he is then led to borrow or lend in order to modify his income-stream so that his rate of timeprefer ence may harmonize with the rate of interest. If, upon an assumed rate of interest, the borrowing and lending for different individuals actually cancel-one another, - in other words, clear the market, - then the rate of interest assumed is clearly the one which solves the problem; other wise the borrowing and lending will not be in equilibrium, and some other rate of interest must be selected. By suc cessively postulating different rates of interest, and remem bering that each rate carries with it its own range of options and its own set of present values of those options, we finally obtain that one which will clear the market.

We therefore conclude by repeating, slightly modified, the formulation of the theory stated in § 6: (1) Each individual has given a specific list of eligible optional income-streams (some of which depend upon the rate of interest); (2) The rate of preference for each individual depends upon the character of his income-stream; (3) All the individual rates of preference are, through the loan or sale market, equalized with one another and with the rate SEc.16J SECOND APPROXIMATION 175 of .interest; (4) Each individual selects, out of the range of choice of income-streams available at a given rate of interest, that particular one which has the maximum present value, - in other words, that one whose advantages over any other outweigh (in present value) its disadvantages, or, in still other words, that one which compared with others makes the rate of return on sacrifice greater than the rate of interest, - or, finally (if the options are in finitely numerous), that one wllich compared with neigh boring options makes the marginal rate of return on sacrifice equal to the rate of interest; (5) The demand and supply of loans must balance for each period of time; and (6) The loans returned must equal the loans obtained, with interest.

§ 16 Having completed the formal statement of the effect of the existence of a range of choice upon the rate of interest, it remains to point out a practical effect of such a range of choice. This effect is to diminish the fluctuations in the rate of interest. In a country where there is a large range of choice between optional income-streams, the rate of interest is apt to be steadier than in one where the income streams are relatively rigid and unalterable. If any cause tends to lower the rate of interest, the immediate effect will be to put a premium on those income-streams the return from which is in the remote future, -for instance, to put a premium on forestry uses rather than mining·uses of land. But the decision to choose such in come-streams tends to prevent the very fall in the rate of interest which caused the choice. For, by relatively over supplying the future with income, and undersupplying the present, such uses as forestry will tend to raise the relative valuation of present over future income, and therefore also to raise the rate of interest. The fall, therefore, in the rate of interest which led to the choice of remoter in176 THE RATE OF INTEREST [CHAP. VIII comes, is checked, and is not so great as it would be if no such options were open.

Conversely, a rise in the rate of interest will favor those options for which the income-streams are relatively im mediate, and will bring its own check; for the choice of such an income-stream will relatively impoverish remote income and enrich immediate income, and consequently tend to diminish the premium on the latter. The existence of a large number of available income streams, then, acts as a balance wheel which tends to check any excessive changes in the rate of interest. In terest cannot fall or rise unduly; for any such fluctuation corrects itself through the choice of appropriate income streams. If interest is high, descending income-streams will be 'chosen which tend to make interest low; while, if it is low, the reverse will be true. We see here another reason, in addition to those given in Chapter VII, for the fact that interest does not suffer very violent fluctuations. It is not only true, as was then pointed out, that natural processes are regular enough to prevent sudden and great changes in the income-stream; but it is also true that man constantly aims to prevent such changes. Man is not the slave of Nature's income; to some extent he is her master. He has many options among which he may choose. He possesses, within limits, the power to flex his income-stream to suit himself. For society the flexibility is due to the adaptability and ver satility of capital, -especially human capital commonly called labor; for the individual the flexibility is greater still, since he possesses a double range of choice. He may not only choose among different employments of capital, but he may choose among different ways of exchanging with other people, - he may borrow or lend, buy or sell, invest or spend. This power is virtually the power to trade in income; for under whatever form an exchange takes place, at bottom it is income, and income only, which is exchanged. In making his choice among different emSEC. 16] SECOND APPROXIMATION 177 ployments of capital, he relies on his power to remedy any undesirable time-shape, etc., by recourse to exchange.

The result of the double range of choice-that between optional employments of capital and between optional modes of exchanging incOlne - is that his income is flexible and controllable in a high degree. Not only may he select the most valuable income, but also the income which is most desirable in respect to time-shape. He need not even commit himself to a given time-shape for any great length of time, for by changing his expenditures and illvestments he can alter that time-shape at will.1 1 For further discussion of the subject of this chapter by means of mathematics, see Appendix to Ohap. VIII. N CHAPTER IX CLASSES OF OPTIONS §1 IN order to present a full view of what is meant by tl optional employments of capital," it will be worth our while to pause a moment and examine the different classes of options open to the capitalist. Options are of t~~bielkinds: (1) options among em ployments of capital which differ in ~, as, for instance, the options previously cited of using land for mining, farming, or forestry; (2) options among employments of capital which d~ffer in the degr~f cerHlLnty,as, for in stance, the choice of sailIng a ship over several routes differing in the constancy of wind and current; and (3) options among employments of capital which differ in ~e and time-shape.