Appendices

That the ratio of the net income from the machines to their capital-value is equal to the rate of interest used in calculat ing the value of each individual machine is a necessary truth and may be shown mathematically as follows:For simplicity, let us assunle that each machine yields its income in a single item at the end of each year. If a machine when new is to last m years and yields a certain annuity of a dollars each year, the value (VI) of this machine is found by discounting the terminable annuity of a dollars for m years at a rate of interest i. This value will be a a a a VI =1 +i +(1 +l)2 +... +(1 +~).~l +(1 +i)'" (1) The gross annual income of a plant consisting of m machines will be ma. The net income of the plant, assuming that one. machine wears out and is replaced annually, will be found by deducting from this gross income the cost, 'VI, of a new machine'. This annual net income = ma - VI.

The value of the plant of m machines can now be found by discounting. the future mcome which the plant will yield. Let us assume that the plant is "kept up" for n years, after which it is allowed to run down until exhausted. The period of running down will be m years, the life of the newest machine. We assume, of course, that whether kept up or run Iling down, the plant yields for each machine a dollars annu'ally. Under these conditions the value of the .plant is the discounted value of two series of income: (1) n years of income of ma - 'VI per year, while the plant is kept up, and (2) m year$. of income which gradually shrinks from ma the first year 347 348 APPENDIX TO CHAPTER II (2) when all the luachines are in use, to (m--l)a the second year, after one machine has dropped out, (m - 2)a the third, etc., to a in the mth year, after which time the plant will cease to exist. We have, then, the present value of ma - VI for each of n years, and the present value for m more years of ma, (m -1) a, (m - 2) a, · · ·a. The present value of these succes sive sums is evidently [ ma-V l ma-Vl ma-vlJ 1 + i + (1 + i)2 +...+ (1 + t)n + [ ma (m-l)a a ] (1 +it+! + (1 +i)n+2 +...+(1 +i)n+m ' which may also be written (ma-v 1{1 ~ i+ (l;i?+ ... + (11itJ+ (l:ti)[1:i+ (7;i~2+ '" + (11tiJ· Of the two terms of which this expression consists, the first is the more important if the rate of interest, i, has a finite positive value, but the second is the more important if that rate is zero. In the former case, rthe longer the plant is kept up (i.e. the larger n is) the smaller will the second term be come; for the divisor of this second term, (1 +i)n, will increase indefinitely and the other factors, a and the square bracket, remain constant. Hence, as n increases indefinitely, this second term becomes more and more negligible and approaches zero as a limit. That is, the value of a plant whose up-keep is indefi nitely maintained is equal to the first~ of the two terms. This first term becomes, when n is indefinitely great,!

(3) which expresses the value of the plant. In other words, the value of the plant is the capitalization of its annual net in come, ma - VI. Or, again, the annual income ma - VI divided by the value of the plant (ma - VI) ~ will equal the rate of interest i. t 1 For proof, see The Nature of Oapital and Income, Appendix to Ch. XIII, § 3.

PRODUCTIVITY THEORIES 349 The same result applies to a plant which contaips more or less machines than m, since the size of the plant will affect both income and capital alike. § 2 (TO CR. II, § 7) Discussion of the Case of Zero Interest as Applied to the Valuation of Reconstituted Capital. The case of a zero rate of interest offers a peculiarity not presented under ordinary circumstances. In all other instances of perpetual up-keep, ,the net income capitalized gives the entire capital-value. This was shown in§ 1 of this Appelldix. But in the case of zero interest the proposition is not true, as may best be shown by mathematics. In § 1 of this Appendix the expression for the value of a plant of m machines to be kept up for n years and then allowed to run down during m years was found to be [ ma - VI ma - VI ma, - VIJ+ 1 +i +(1 + i)2 + ... + (1 + i)" [ ma (m-l)a a J (1 +i)7&+l+(1 +;110+2+...+(1 +i)"+m • In the previous section it was assumed that i was finite and positive, from which it followed that when n was indefinitely great the second square bracket became negligible. But under our present assumption that ·i is zero, the term is not negligible; on the contrary, it is the first square bracket which now van ishes. To show this we observe that formula (1) of§ 1, giving the value of each machine, reduces, when i = 0, to a a a VI = - +- + ... + - ,1 1 2 1m =ma, whence ma-v1=O.

Hence the first term in equation (2), being the product of ma - VI (zero) by a finite number, is zero. The second term of (2) reduces, when i = 0, to !!'..[~+m-1+ ... +1.J ora[m(m+l)J.1n 1 1 2 1m 2 350 APPENDIX TO CHAPTER II which, since the first term is zero, represents the entire value of the m machines. The result is now independent of n. If m = 10, this expression becomes 55a. The value of such a plant is then fifty-five times the annual yield of each machine. If this yield is $100, its value is $5500, which agrees with the calculation in the text.

APPENDIX TO CHAPTER IV BOHM-BAWERX'S THEORY § 1 (TO Cn. IV, § 2) Nature of Various Means-Arithmetical, Geometrical, Harmonical,etc. In general, a mean, a, of a number of magnitudes, aH a·2, as, etc., is defined by an equation connecting these magnitudes and a in such a manner that if all of the magnitudes, au a2' a3, etc. are equal to each other, the value of a given by the equation will be equal to each of them. That this concept applies to the arithmetical, geometrical, and harmonical means is evident. These means nlay be defined by the following formulre, where, for convenience, the number of elements, aH a2, etc., averaged is restricted to three. This restriction, which may be very readily removed, is adopted solely for brevity. (1) A °th t· 1 - - - - at+ a2+ a S. rl me lca , a + a + a =a1+ a2 + ag or a= 3 (2) Geometrical, ii ii Ii = at a2 as or a= {Iat a2 as 0111111 _ 3(3) HarmonIcal, -=+:+-==-+-+- or a=l 1 1a a a a,. U2 ag _ + _+ _ al a2 as The weighted arithmetical mean is given by the formula - + - + - - W1al + w2a2 + wsUa"wla w2a wga =W1llt+ w2a2+ waag or a = ------Wl+W 2 +WS where the "weights" are the coefficients Wu W2, Ws• This is the mean employed by Bohm-Bawerk in the example given, the elements averaged, ai' a2, Us, etc., being the different ages of the labor, 10 years, 9 years, 8 years, 7 years, etc., and the' weights being the amount of labor, $20, $20, $5, $5, etc.

The formulm for both the geometrical and the harmonical averages may also be modified by introducing "weights." 351 352 APPENDIX TO CHAPTER IV By varying the formula we may evidently invent an infinite number of new kinds of means. Thus the formula _ a U,2 a+ 1 + -va=a1 +1 +-va;defines a as a sort of mean, though a complicated (and unsymmetrical) one, of aI' a2, a3• § 2 (TO CR. IV, § 2) Case Illustrating Futility of Measuring Average Production Period. Bohm-Bawerk's chosen concept, which was doubtless adopted purely for convenience, that a given application of labor will yield its return in a single sum all at once, is far too simple to cover the facts as actually found. On the contrary, both the labor of forming instruments and their return are spread over a considerable period of time. This distribution in time may take any form, and some of its forms would render use less the simple arrangement of Bohm-Bawerk of production periods into a series of varying duration.

Suppose, to take an extreme case, that a particular applica tion of labor issues in two items of income, namely: $5 ten years after date, and $100 one hundred years after date; while another application of labor issues in only a single item worth $15 in twenty-five years. In this case it becomes impossible to call one of the production periods longer than the other; for whereas the second is definitely 25 years long, the first may be measured as any period between 10 and 100 years, according to the nlethod employed for averaging 10 and 100. Moreover, it is not true that one of the alternatives will be chosen if the rate of interest is high, and the other if the rate of interest is low, as would be the case if they were subject to Bohm Bawerk's series. The application of labor which issued in the $5 and $100 would, oddly enough,be the most economical if the rate of interest were either very high or very low, whereas the other alternative would be chosen in case the interest were at a more moderate rate. Thus, if the rate of interest were 5 %, the present value of the $15 due in 25 years would be $4.43, and that of the two items, $5 in 10 years and $100 in 100 years, would be $3.83. On the other hand, if the rate of interest were 1 %, the value of the $5 and $100 alternative would be $11.70 and of the $15 alternative $41.28.

\_-~ ~ BOHM-BAWE{K,g THEORY 353 § 3 (TO CR. IV, § 3) Showing how Periods· of Production which are Relatively Long but· Unproductive are Eliminated. That long processes (assuming their length to be measur able) are more productive than short processes is, as BohIn Bawerk says, a general fact, not a necessary truth. The reason lies in selection. It is not true that, of all possible produc tive processes, the longest are the most productive; but it is true that, of all productive processes actually employed, the longest are also the most productive. Noone will select a long way unless it is at the same time a better way. All the long but unproductive processes are weeded out. The follow ing illustration will make the process clear: Suppose that by means of 100 days' labor invested to-day we can obtain a product of 100 units one year hence, or of 250 two years hence, of 50 three years hence, of 300 four year~ hence, of 250 five years hence, of 320 six years hence, of 100 seven years hence, of 300 eight years hence, etc.,- a series which we take quite at random. Out of this series of choices there will be eliminated those of 3, 5, 7, and 8 years, for each of these is outclassed by preceding choices. Thus, the 5-year period yielding 250 will be overshadowed by the 4-year period yielding 300; for this prospective return, being not only larger but earlier, will have a higher present value.

Eliminating, then, these ineligible cases, .we have left, .. to choose from, the 1, 2, 4, and 6 year periods. Of these,. that one will be chosen of which the return will have the highest present value; and the present value will depend on the rate of interest. If interest is at 5 %, it will be profitable to in vest the 100 days' labor so as to mature in four years. AB is the discounted value of 300 at5 %for four years, it being found by the discount-curve BO drawn at 5 %from O. Since this curve paSRes above the tops, 0', 0", 0''', of all the other vertical lines, this present value (at 5 %) of 300 in four years will be the maximum of the present values of all the returns, 2.A.

354 APPENDIX TO CHAPTER IV 100, 250, 50, etc. But if the rate of interest sinks to 2 %, as indicated by the discount-curve B'G', the point of maximum return is shifted forward to six years; for the discount-curve B' 0' at 2 %drawn through 0' now passes above the tops (0, 0", 0''', etc.) of all other lines, hence a six-year period will be chosen. If, on the other hand, the rate of interest were 10 %, a similar construction would show that the two-year period would be selected, as the highest discount-curve would then pass through 0". But in no case will the highest discount curve touch the top of one of the short lines, 100, 50, 250, 100. B-l· -0' 9' B 300 ~5q 320 800250, -Q~'" :LO:O 100 ))0 A 1 '2- ;3' '4i B '6 .rr ,s.years FIG. 25. § 4 (TO CB. IV, § 4) Mathematical Refutation of Bohm-Bawerk's Claim as to Ground of Preference for Present over Future Investment of Labor. Let the products obtainable by processes of 1, 2, 3, etc.

years be PH P2, P8' etc., and the" marginal utilities reduced in perspective" beginning in 1888 be UI, U2' Us, etc. Then, A MONTH'S LABOR AVAILABLE IN 1888 YIELDS IN 1889 YIELDS For the Marginal Amount of economic Units of utility value of Units Marg. Value period product reduced entire utility in persp. product 1888 PI Ul PIUl - Ul 1889 P2 U2 P2U2 Pl U2 PIU2 1890 ps Us P3U3 P2 US P2US 1891 P4 U4 P4U4 Ps U4 PSU4 etc. etc. etc. etc. etc. etc. etc.

BOHM-BAWERK'S THEORY 355 We shall show that the labor available in 1888 is more valu able than that in 1889, provided only U1 > U2 > Us > U., etc.; that is, that the maximum of the first series of pu's, relating to 1888, is greater than the maxim um of the second series, relating to 1889 (assuming of course that maxima exist). To prove this, select the maximum of the second series. Sup pose it to be PaU.. This is necessarily less than PaUs in the first series; for since u. < Us by hypothesis, it follows that PSU'4 < PaUs. That is, there necessarily exists in the first series a term greater than the greatest term in the second series. .A. fortiori must the greatest term in the first series exceed the greatest in the second series. In other words, the value for 1888 exceeds that for 1889, provided only the marginal utili ties descend, whether or not the productivities ascend.

APPENDIX TO CHAPTER V ApPRECIATION AND INTEREST § 1 (TO CR. V, § 2) History of Theory of Appreciation and Interest Investigation shows that the present writer was by no means the first to' conceive the relation between appreciation and interest. Apparently the earliest was the anonymous author l of a remarkable pamphlet entitled: "A Discourse Concern ing the Currencies of the British Plantations in America," Boston, 1740 (reprinted in the Economic Studies of Amer ican Economic Association, 1897). He writes:" The Arguments current amongst the Populace in favour of Paper Money, are, "I. In most of the Paper Money Colonies one of the principal Reasons alleged for their first Emissions; was, to prevent Usurers imposing high, Interest upon Borrowers,from the Scarcityof Silver Money. It is true, that in all Countries the increased Quantity of Silver, falls the Interest or Use of Money; but large Emissions of Paper Money does naturally rise the Interest to make good the sinking Principal: for Instance, in the Autumn, A. 1737, Silver was at 26s. to 27 s. per Ounce, but by a large Rhode Island Emission, it became in Autumn 1739, 29s. per Oz. this is 7 per Cent. Loss of Principal, therefore the Lender, to save his Principal from sinking, requires 13 per Cent. natural Interest (our legal Interest being 6 per Cent.) for that Year. In Autumn A. 1733, Silver was 228.

per Oz. by large Emissions it became 27 s. in the Autumn, A. 1734; is 22 per Cent. loss of Principal; and the Lender to save his Principal; requires 28 per Cent. natural Interest for that Year. Thus the larger tht Emissions, natural Interest becomes the higher; therefore the Advocates for Paper Money (who are generally indigent Men, and Borrowers) ought not to complain, when they hire Money at a dear nominal Rate. " If Rills were to depreciate after a certain Rate, Justice might be done to both contracting Parties, by imposing the Loss which the Principal may sustain in any certain Space of Time (the Period of Payment), upon the Interest of a Bond or Price of Goods: but as Depreciations are uncer tain, great Confusions in Dealings happen. " 1 Now identified as the physician, William Douglass. 356 APPRECIATION AND INTEREST 357 John Stuart Mill expressed the same view,l as have also Robert Goodbody,2 Jacob de Haas,3 and Professor John B.

Clark.· A principle which apparently has been independently discovered by each of these economic students and quite pos sibly by others,' is likely to be of some importance. The present writer published in 1896 a monograph 6 in which he worked out the relation between interest and appreciation in quantitative form, its application to special cases, its statis tical verification, as well as its significance in the theory of interest and in the practical problem of regulating the stand ard of deferred payments. The major part of the material contained in this monograph is reproduced in Chapter V, Chapter XIV, and this Appendix. That the appreciation or depreciation of money does actually 1 Principles of 1'0Utical Economy, Book 3, ·Ch. 23, § 4. [A single paragraph. ] ~Mr. Robert Goodbody, Broker, New York, has for years in his trade..;letters maintained the doctrine that the rate of interest is high when money is depreciating, and low when money is appreciating. This he discovered about 1876, when the decline in silver was attracting attention.

He was then much interested in the higher mathematics, and as he ex pressed it, "accident or something caused me to differentiate the equation of imports and exports of any country, not with respect to time, but with respect to the variation of the standard of value. .The result was that I found that the fraction formed by the ratio of call money as nu merator and time money as denominator was smaller when the money standard was falling and larger when it was rising." 8 "A Third Element in the Rate of Interest," Journal of the Royal Statistical Society, -March, 1889. [An extended discussion, with statistics. ] 4 ~"rhe Gold Standard in the Light of Recent Theory," Political Science Quarterly, September, 1896. [Applied to the bimetallic con troversy.] 6 Mr. Byron W. Holt has cited other cases in which the relation be tween appreciation and interest has· been recognized. In his paper entitled" Interest and Appreciation" (Sound Currency, Vol. V, 'No. 22, 1898) he mentions Senator Jones of Nevada, Professor T. N. Carver, now ,of Harvard, David I. Greene of Hartford, and Professor H. H. Powers, formerly of Leland Stanford.

6 "Appreciation and Interest: a study of the influence of monetary appreciation and depreciation on the rate of interest, with applications to the bimetallic controversy·· and the theory of interest." Publications of the American Economic Association, 1896, Vol. XI, No.4, pp. 331 442.

358 APPENDIX TO CHAPTER V influece the rate of interest is now well recognized by those who have given attention to the subject.! § 2 (TO CR. V, § 3) Formula Connecting the I~ates of Interest in two Diverging Standards. In order to state the general relation between the rates of interest and appreciation or depreciation, let wheat fall in gold price (or gold rise in wheat price) so that the quantity of gold which would buy one bushel of wheat at the beginning of the year will buy 1 +a bushels at the end, a being therefore the rate of appreciation of gold in terms of wheat. Let the rate of interest in gold be i, and in wheat be j, and let the principal of the loan be D dollars or its equivalent B bushels. Our alternative contracts are then: For D dollars borrowed, D + Di or D (1 + i) dollars are due in 1 yr. For B bushels borrowed, B +Bj or B (1 +j) bushels are due in 1 yr. and our problem is to find the relation between i and j, which will make the D (1 + i) dollars =c= the B (1 +j) bushels. 2 At first, D dollars =c= B bu.

At the end of the year, D dollars =*B (1 +a) bu. Hence at theend of the year D (1 +i)dollars =* B(l +a) (1+'i) bu. Since D (1 + i) is the number of dollars necessary to liqui date the debt, its equivalent B (1 +a) (1 +i) is the number of bushels necessary to liquidate it. But we have already desig nated this number of bushels by B(l +j). 1 See Professor Marshall's testimony, Indian Cu.rrency Rep01"t, 1899, Pt. II, p. 169; Graziani, Studi 8ulla teoria dell' interresse, Turin, 1898, pp. 120-29; and Joseph F. Johnson, Money and Currency,Boston (Ginn), 1905, p. 158. But the subject has as yet attracted little attention in the business journals. See The Bond Record, April, 1896; a.lso the first number of Moody's Magazine, 1905, in which the "Symposium" and editorial on the effects of increasing the supply of gold are partly de voted to the relation between monetary depreciation and the rate of interest. The same material together with much else of importance is assembled in The Gold Supply and Prosperity, by Byron W. Holt, New York (Moody Publishing Co.), 1907. See also J. P. Norton, "The Depre ciation of Gold," Yale Review, 1906-7, pp. 29;3-306.

2 The symbol * signifies" equivalent to." .

APPRECIATION AND INTEREST 359 Our result, therefore, is : Dollars Bushels Bushels at the end of one year D (1+ i)::o=B(1 +j) = B(1+ a) (1 +'i), (1) which, after B is canceled, discloses the formula: l+i=(l+a)(l+i), (2) or j = i +a +ia. (8) or in words: ·The rate of interest in the relatively depreciating 8tandard is equal to the sum of three terms, viz. the rate of inter est in the apprec,iating standard, the rate of appreciation itself, and the product of these two elements. Thus, to offset the appreciation, the rate of interest must be lowered by slightly more than the rate of appreciation. l We may introduce depreciation in a similar manner. Instead of saying gold appreciates at the rate a, relatively to wheat, we may say, wheat depreciates at the rate d, relatively to gold. 2 This means that wheat has sunk in terms 'of gold in the ratio 1 to 1 - d, and reasoning similar to the foregoing shows that 1 +·i =(1 - d) (1 +j). (4) Equations (2) and (4) may be conveniently combined, thus: 1+· l+a 1 l+~=-l-=l-d· (5) Since 1 i a is the ratio of the value of gold at the end of the year to its value at the beginning (all in terms of wheat), that is, the ratio of divergence of the two standards expressed in wheat, while _1_ is the same ratio of divergence expressedl-d in gold, and since 1 +i .is the" amount" of $1 put at interest for one year, while 1 +j is the "amount" of one bushel; we may state equation (5) as follows: - .

1 Professor Clark (Political Science Quarterly, September, 1895) im plies that 1 ok appreciation is offset by less than 1%reduction of interest. But in making his calculation he has failed to "compound." The numeri cal illustrations of the eighteenth century pamphle~eer (supra) are also erroneous. E.g. instead of 28 °k the figure should be 29.32 °k. Professor Marshall (P'rinciples of Economics, Vol. I, 3d ed., p. 674) gives a correct example, designed to show the losses from a fluctuating currency. 2 The relation between d and a is (1 + a) (1 - d) = 1, which is evi dent from equation (5), or may be easily shown independently.

360 APPENDIX TO CHAPTER V The ratio of divergence between the standards equals the ratio between their "amounts." This is, perhaps, the simplest mode of conceiving the rela tion, and stress is laid upon it, because it brings into promi nence the "amount," or l'atio of future payment to present loan, a magnitude which in most questions of interest plays a more important role than the rate of interest itself. Equation (5) gives the relation between i and j in terms of a or d. From it follows the value of j in terms either of i and a or of i and d, and also the value of i in terms either of j and a or ofj and d, thus:(7) (6) (5) whence l+j=l+a 1 1 +i 1 =l-d' . . +. i+dJ = 1, +a ~a = -_.l-d · · d ·d j-at=J- -J =--.l+a It follows that j exceeds i by more than the rate of appre· ciation, which in turn is more than the rate of depreciation (i.e., j - i> a>d). or § 3 (TO CR. V, § 4) Formulre, when Ra.tes of Interest and of Appreciation are Reckoned oftener than Yearly.

In case we take the half-year instead of the year as the inter val for compounding the rates of interest and of appreciation, it may readily be shown that the formula 1 +j = (1 + i) (1 +a) gives place to 1~~=(1+~)(1+ ~), whence it also follows that instead of j = i +a + ia, we have the relation .. ia J=~+a+2 In case the interest and appreciation are compounded quar terly, the formula becomes .. iaJ=t+a+4"' APPRECIATION AND INTEREST 361 and so on. At the limit, when the rates of interest and appre ciation are reckoned continuously, the last term vanishes and the formula becomes simply j = i + a. § 4 (TO CH. V, § 5) Case of Partial Payments. First, consider the case in which lio interest is paid until the end of the term of years. Let .us suppose, for instance, a savings bank which receives $100, gold standard, and repays the depositor in :five years at 5 % compound interest. Let there be an alternative standard, say wheat, worth, at the beginning of the loan, $1 per bushel; but suppose that, in terms of wheat, gold is known to appreciate constantly by 1 % per annum. What would be the rate of interest in terms of wheat? If the repayment were to be made in one year, the equi valent of the 5 % would be a rate of interest in terms of wheat of 6h %, since the "amount" of a dollar of gold put at interest one year would be $1.05, and this would be worth, in bushels of wheat, 1.05 multiplied by 1.01, or 1.06;1)" bushels.

This resul t, 6.J1f %, is as true for a series of years as for one year. This maybe seen by separating the contract into several contracts of one year each. If we imagine deposited to-day in separate savings banks $100 in gold, and its equiva lent, 100 bushels of wheat, they will amount in one year respectively to $1.05 at 5 %, and its equivalent, 106.05 bushels at 6ilf %. We may now regard these equivalent amounts as withdrawn, but immediately redeposited for one year. Then, with the same rate of interest in gold and the same relative appreciation, we shall obtain the same rate of interest in wheat, so that $105 and its equivalent, 106.05 bushels, will amount in one year respectively to $110.25 at 5 %, and its equivalent, 112.41 bushels at 6-h-%. In this way each successive pair of "amounts," including the last, will be equivalent. For simplicity we have considered only the case in which the debt is allowed to accumulate to the end. The most general case, however, is one in which the repayments are in install ments.

Suppose, as before, that the interest in gold is 5 %and that gold is known to appreciate 1%per annum relatively to wheat. A farmer mortgageshis land for.$1000, or its then equivalent, 362 APPENDIX TO CHAPTER V 1000 bushels of wheat, and agrees to pay annually the interest and such parts of the principal as he can save, making the repayment complete in seven years. Our problem is to find that rate of interest in wheat which will make the contracts in gold and wheat equivalent in every respect. The solution is precisely the same as before, viz. 6;0%. For, at the end of one year, the farmer's debt amounts to $1050 or its then equivalent 1060.50 bushels. Let us suppose that he finds himself able to pay, not only the interest, $50, but also $50 of the "principal," that is, $100 all together. The equiva lent of this in wheat is 101 bushels. Hence he can either pay $100 on $1050.00 leaving $950.00 or 101 bu. on 1060.50 bu. leaving 959.50 bu.

and, since the U amounts" $1050 and 1060.50 bu. are equiva lent and the deductions $100 and 101 bu. are equivalent, the remainders $950 and 959.50 bu. must also be equivalent; in fact, this may be seen directly, since, with gold appreciating 1 %, $950, originally worth 950 bu., beCOlnes worth 1 %more or 959.50 bu. Thus the farmer's remaining debt at the end of the first year is the same whether measured in ,vheat or gold, and since the same reasoning applies to the second year, third year, etc., the equivalence remains to the end of the contract. It is worth noting here that the $100 payment in gold would be regarded as consisting of half "interest" and half "princi pal," whereas the equivalent payment in wheat, 101 bu., will be regarded as 60.50 bu." interest," and 40.50 bu. "principal." The liquidation of the contract during the seven years may thus be supposed to take place in either of the following equiva lent ways:GOLD STANDARD (dollars) I~TEREST AMOUNT PA.YMENT PRINCIPAL REMAINING At beginning 1000.00 In 1 year 50.00 1050.00 100.00 950.00 In 2 years 47.50 997.50 97.50 900.00 In 3 years 45.00 945.00 145.00 800.00 In 4 years 40.00 840.00 150.00 690.00 In 5 years 34.50 724.50 174.50 650.00 In 6 years 27.50 577.50 277.50 300.00 In 7 years 15.00 315.00 315.00 0.00 APPRECIATION AND INTEREST WHEAT STANDARD (bushels) 363 INTEREST I AMOUNT PAYMENT PRINCIPA.L REMA.INING At beginning 1000.00 In 1 year 60.50 1060.50 101.00 959.50 In 2 years 58.05 1017.55 99.46 918.09 In 3 years 55.54 973.63 149.39 824.24 In 4 years 49.87 874.11 156.09 718.02 In 5 years 43.44 761.46 183.40 578.06 In 6 years 34.97 613.03 294.57 318.46 In 7 years 19.27 337.73 337.73 0.00 In these two tables, every entry in one is equivalent to the correspondingentry in the other except those in the interest columns.

We thus see that the farmer who contracts a mortgage in gold is, if the interest is ptroperly adjusted, no worse and no better off than if his contract were made in a "wheat" standard. This principle, that debts in different standards are equiva lent if the rates of interest in the two standards are properly adjusted, holds true, of course, no matter whether the "partial payments" are large, small, or none at all; no matter whether the interest payments are made in full, in part, or not at all. The principals in the two standards are not equivalent, except at the beginning, nor are the annual interest sums equivalent; but the excess of the burden of interest in one standard is accompanied by.a deficiency in the burden of the principal, and vice versa. § 5 (TO CH. V, § 5) Formulm for Cases of Compound Interest and Partial Payments. The general case is precisely similar. If a debt in either of two alternative standards isto accumulate at compound interest, the rates of interest in the two standards must, in order that the contracts in each shall be equivalent, conform to the for mula, 1 +j = (1 +a) (1 +i), which we found in the simpler case of a one-year debt.

To show this, resolve the contract into a series of one-year contracts. For the first year we have, by formula (1) of § 2 above, 364 APPENDIX TO CHAPTER V Dollars due Bushels due Bushels due D(l+i)¢B(l+j)=B(l+a) (l+i) In the second year the same formula applies except that in place of D, the principal is now D (1 + i), and in place of B, B (1 +j) or B (1 +a) (1 +-i). Making these substitutions in the formula, we obtain D (1 + i)2¢B (1 +j)2 == B (1 +a)2 (1 +i)2. And similarly the third year, D (1 + i)3¢B (1 +j)3 = B (1 +a)3 (1 +i)3, and so on. Each of the results evidently yields the formula 1+i=(1+a) (l+i). If a debt in either of two alternative standards is to be liquidated in "partial payments," the rates of interest in the two standards must, in order that the contracts in each may be equivalent, conform to the same formula. The reason is simply that equivalent payments sub tracted from equi valent "amounts" will leave equivalent remainders. The payment in any year forms the same frac tional part of the "amount" in the two standards. We may designate this fraction at the end of the first year by f, the second year by /" etc., and we have the following results:END OF FIRST YE.!.R Dollars Bushels Bushels AIDount, D (l+i)* B (l+j) = B (l+a) (1+£) Payment, ID (l+i):e= IE (l+j) = IE (l+a) (l+i) Remainder, (I-f) D (l+i):c=(l-f) B (l+j)= (I-f) B (l+a) (l+i) In like manner the unpaid remainder at the end of the second year can be shown to be Dollars Bushels (1-/') (i-f) D (1+i)2=c=(1-/') (1-f) B (l+j)~ Bushels =(l-f') (i-f) B (1+a)2 (1 +i)2, and so on for any number of years. Each result again yields the formula (1 +j) = (1 +a) (1 + i). Similar reasoning applied to each succeeding year yields the same formula.

The case in which there are no partial payments is met by putting f, I', equal to zero.

APPRECIATION AND INTEREST § 6 (TO CR. V. § 5) 365 Case of Separate Payments of Interest and Principal in one of the Two Standards and Equivalent Payments in the Other. Suppose alternative contracts in gold at 5 % and wheat. at 6llr %, and suppose that the interest in the gold contract is annually paid and the principal redeemed in ten years. The following tables. will show what are the equivalent operations in the wheat standard. LIQUIDA.TION IN GOLD STANDARD, CONSISTING OF ANNUAL IKTEREST ($50) AND FINAL PRINCIPAL ($1000). INTBRBST AMOUNT PAYMBNT PRINOIPAL DUE REM.AINING At beginning (Dollars) - - - 1000.00 In 1 year 60.00 1050.00 50.00 1000.00 In 2 years 60.00 1050.00 50.00 1000.00 In 3 years 50.00 1050.00 50.00 1000.00 In 4 years 50.00 1050.00 50.00 1000.00 In 5 years 50.00 1050.00 60.00 1000.00 In 6 years 50.00 1050.00 60.00 1000.00 In 7 ;years 50.00 1050.00 50.00 1000.00 In 8 years 50.00 1050.00 50.00 1000.00 In 9 years 50.00 1050.00 50.00 1000.00 In 10 years 50·00 1050.00 1050.00 0.00 EQUIVALENT LIQUIDATION IN WHEAT STANDARD; ANNUAL PA.YMENTS ARE LESS THAN INTEREST (60.50 Bu.) A.ND FINAL PAYMENT MORE THAN PRINCIPAL (1000 Bu.).

INTEREST AMOUNT PAYMENT PRINOIPAL DUE REMAINING At beginning (Bushels) - - - 1000.00 In 1 year 60.50 1060.50 60.60 1010.00 In 2 years 61.10 1071.10 51.00 1020.10 In 3 years 61.72 1081.82 51.52 1030.30 In 4 years 62.32 1092.62 52.03 1040.69 In 5 years 62.96 1103.55 52.55 1051.00 In 6 years 63.69 1114.69 53.08 1061.51 In 7 years 64.22 1125.73 63.61 1072.12 In 8. years 64.86 1136.98 54.14 1082.84 In 9 years 65.51 1148.35 54.68 1093.67 In 10 years 66.17 1159.84 1159.84 0.00 366 .APPENDIX TO CHAPTER V If we suppose, conversely, that interest in the wheat stand ard is annually met and the principal redeemed in ten years, the equivalent operations in the gold standard will be as shown below. LIQUIDA.TION IN WHEAT STANDARD, CONSISTING OF ANNUAL INTEREST (60.50 Bu.) .AND FINAL PRINCIPAL (1000 Bu.). INTEREST AMOUNT PAYMENT IPRINCIPAL DUE REMAINING At beginning (Bushels) - - - 1000.00 In 1 year 60.60 1060.60 60.50 1000.00 In 2 years 60.50 1060.50 60.50 1000.00 In 3 years 60.50 1060.50 60.50 1000.00 In 4 years 60.50 1060.50 60.50 1000.00 In 6 years 60.50 1060.60 60.50 1000.00 In 6 years 60.50 1060.50 60.60 1000.00 In 7 years 60.50 1060.50 60.50 1000.00 In 8 years 60.50 1060.50 60.50 1000.00 In 9 years 60.50 1060.50 60.50 1000.00 In 10 years 60.50 1060.50 1060.50 0.00 EQUIVALENT LIQUIDATION IN GOLD STANDARD; ANNUAL PAYMENTS .ARE GREATER THAN INTEREST ($50) AND FINAL PAYMENT LESS THAN PRINCIPAL ($1000).

INTEREST AMOUNT PAYMENT PRINCIPAL DUE REMAINING At beginning (Dollars) - - - 1000.00 In 1 year 50.00 1050.00 59.90 990.10 In 2 years 49.50 1039.60 59.31 980.29 In 3 years 49.01 1029.30 58.72 970.58 In 4 years 48.53 1019.11 58.14 960.97 i In 5 years 48.05 1009.02 57.56 951.46 In 6 years 47.67 999.03 66.99 942.04 In 7 years 47.15 989.19 56.43 932.76 In 8 years 46.63 979.39 55.87 823.52 In 9 years 46.17 969.69 55.32 914.37 In 10 years 45.71 960.08 960.08 0.00 § 7 (TO CH. V, § 5) Case of Separate Payments of Interest and Principal in both Standards. Let us next compare the liquidations in the two standards by the simple annual payment of interest in each (i.e. $50 in APPRECIATION AND INTEREST 367 the gold standard and 60.50 bu. in the wheat standard, not interequivalent) and in ten years, final payment of principal ($1000 and 1000 bu. not interequivalent). In this case the individual payments in the two cases do not correspond, but the present values of the debts, reckoned at any date whatever, are always identical. Thus, the present value, at the date of contract, of the interest and principal, separately computed, at 6 %and 6",0 %in the two standards respectively, will be :1_ Present value of all interest payments, Present value of principal due in 10 years, Present value of total, Dollars Bushels 386.09 < 444.24 613.91 > 555. 76 1000.00 =* 1000.00 If the present values were computed five years after the date of the contract, and the "amounts" of past interest were com putedfor the same point of time, the items would be:Interest (present value and amounts), Principal (present value), Total, Dollars Bushels 492.75< 595.88 783.53 > 745.50 1276.28 * 1341.38 The two sums here, though not equal numbers, are equiva lent magnitudes; for whereas at the outset $1 of gold and 1 bushel of wheat were equivalent, now, after five years of an nual appreciation of gold relatively to wheat at the rate of 1 %, we shall find $1 worth (1.01)'s bushels, or 1.051 bu., whence $1276.28 will be worth 1341.38 bushels.

We thus see that it would be just as much of a hardship to pay the higher interest in wheat during the whole period as to pay the more onerous principal in gold at last. § 8 (TO CR. V, § 5) Case of Perpetual Annuity. The case of a perpetual annuity may be given special consid eration. As is well known, the present value of a perpetual annuity is its "capitalized" value. Thus, if the rate of inter est is taken at 5 %, the present value of a perpetual annuity of $50 per annum is $1000. Applying the same principle to the 1The 8ymbol < is here used for " is less than the equivalent of~" and > for " is more than the equivalent of."

368 APPENDIX TO CHAPTER V wheat annuity of 60.50 bushels and extending the previous reasoning, we find that the two annuities are equivalent. At first sight this seems impossible, since 6"210 %is a higher rate of interest than 5 %. This is true, numerically, and it is also true that the early payments of 60.50 bushels are actually more valuable than $50. But after a certain time (in this particular case 19 years) the reverse is true. The 19th pay ment of $50 in gold is worth 60.40 bushels, while the 20th is worth 61.01 bushels. That is, the recipient of the wheat an nuity has at .first a slight advantage over the recipient of the gold annuity, which ceases and becomes a slight disadvantage after 19 years. To derive the formula for the time at which the relative values of the two annuities become reversed, let the rate of interest in gold be i, in wheat j; let the two annuities be Di and Bj, their capitalized values being D and B, (D ~ B at the beginning), and let x be the number of years in which Bj is as valuable as or more valuable than Di. Then At the end of ~ years, Bushels Dollars Bj~Di At the end of ~ +1 years, Bj < Di and since we know that in x years D:c=B(l+a)Z, and hence Di ~ Bi (1 +a)X; and likewise in re + 1 years, Di ~ Bi (1 + a)Z+1, we see that the previous inequalities become:BU8he]8 Bu~he15 or At the end of x years, Rj ~ Bi(l +a)Z At the end of re + 1 years, Bj < Bi(l + a)~+l which may be combined in the formu1a:i(l + aYl:~ < i(l + a)z+I, x <logj -logi < ~ + 1.

=log (1 + a) That is, ai is the integral part of the number logj -logi. log (1 +a) Thus, ifi = .05, a = .01, and hence also j = .0605, then logj - log i = 2.7818 - 2.6990 _ .0828 = 19.3. log (1 +a) .0043 - .0043 Hence x=19. (8) APPRECIATION AND INTEREST § 9 (TO CR. V, § 5) 369 Case in which the Rate of Appreciation changes each Year. In this case the rate of interest in one or both of the two standard.s will change also. Beginning with a numerical illustration, let us suppose that a syndicate offers the United States government an alternative loan in gold or silver. Let it be known that 100 gold dollars will remain at par throughout the first year, but in two years will be worth 150 silver dollars, that is, gold will" appre ciate," in the second year, 50 %relatively to silver; also that in the third and fourth years it will appreciate 10 %and 5 % respectively. We shall suppose that the rate of interest, if the contract be in gold, is 3 %for each year of the contract.

Our problem is to discoverwhat will be the rate of interest in silver. It is perhaps already evident that there will be a dif ferent rate for each year. If the contract were made for one year only, the rate of interest in silver would also be 3 %, since silver remains this year at par with gold. If the contract (or any unpaid part of it) were then renewed for a second year the rate of interest would be, by formula (3) : ~ j=i+a+ai = .03 + .50 + .015 = .545 = 54t%· In like manner, we may deduce the rate of interest in each year, with the following results:GOLD STANDARD ApPRECIATION SILVER STANDARD 1st year " 30/0 0% gOA> 2d year 3°,'0 50% 54lok 3d year 3% 10°,'0 13ySo-% 4th year 30/0 50/0 8lo~% etc. The question arises, can a single "average" rate of interest be substituted for the above irregular series of rates in the silver standard? We answer that such an average is not possible if the debtor has the option of arbitrary partial payments. If, for instance, the average were 20 %, and the government could payoff at any 2B 370 APPENDIX TO CHAPTER V time, it would evidently be tempted to refund the debt at the end of the second year, to which the lender would not agree.

If, however, the conditions as to repayment are stipulated for in advance, an average can easily be computed on the prin ciple of present values. Suppose the borrower agreed to extinguish the debt in four years by paying at the end of successive years 20, 40, 30, and 10 millions (these to include "interest"). The present value of these sums is 66.321 millions, which is therefore the amount of the loan received from the syndicate. This sum is obtained by adding the present values of several payments. The present value of 20 millions, due one year hence, is 1:g3=19.418 millions, and of 40 millions, due two years hence, is (1.03t~.545) =25.136 millions; for evidently if this be put at interest for one year at 3 %, and for the next at 54! %, it will amount to 40 millions. Like wise the third and fourth payments have present values of (1.03) (1.:~5) (1.133) = 16.639 millions; 10 = 5.128 millions.

(1.03) (1.545) (1.133) (1.0815) The sum of these four present values is 66.321 millions. Now if we compute the present values of the four payments on the basis of a uniform "average" rate of 20.26 % interest, we obtain the same SUill, thus:(1.:g26) = 16.631 millions (1.~026)2 =27.659 millions (1.:0~6)3 17.250 millions (1.2~~6)4 = 4.181 millions Total = 66.321 millions APPRECIATION AND INTEREST 371 The separate present values are llere fictitious, that is, no one of them is the actual present selling price of the future payment to which it refers, but the deviations so offset each other that their Bum is the actual present selling price of the whole set of future payments. It follows from principles already stated that the debt, 66.321 millions, can be liquidated by precisely the same payments (20, 40, 30, and 10 millions) whether the interest is reckoned separately at 3, 54!, 13/0, and 8TYo%' or uniformly at 20.26 %. In fact the details of the bookkeeping in the two cases are:At 8, Mi, 18i\, SN1J% At 20.26% uniformly (In Millions) (In Millions) nate Interest Amount Payment Principal Interest Amount Payment Principal -------- ----- ----- - - 66.82 - - - 66.32 In 1 year 1.99 68.31 20.00 48.31 13.44 79.76 20.00 69.76 In2 years 26.33 74.64 4:0.00 34.64 12.11 71.87 40.00 31.87 In 8 years 4.61 89.25 30.00 9.25 6.45 88.3~ 30.00 8.32 In4 years .75 10.00 10.00 0.00 1.68 10.00 10.00 0.00 We thus see that 20.26 % is the "average" of 3, 54t, 131\, and 8T\r\ %, in the sense that, reckoning interest by this "aver age," the same payments will cancel the same debt as if the separate rates were used. It is not identical with the arith metical average, which is 19.74 %.

To express the law of such an average symbolically, let us suppose that the rate of appreciation of one standard in terms of the other is foreknown to be ltt the first year, a2 the second year, as the third year, and so on; also, to be as general as possible, that the rates of interest in both stand ards are variable, being in the appreciating standard it the first year, i2 the second, etc., and in the depreciating standard, ju jl' etc. Let the final settlelnent occur in n years. Then, as above, we may regard the contract as equivalent to a series of one..year contracts successively renewed in .whole or in part, the only difference being that the terms are all made in advance. As equation (2) of § 2 applies to each of these contracts, we have 1 +it = (1 +at) (1 +i1) 1 +}2= (1 +a2) (1 + i 2) (9) 372 APPENDIX TO CHAPTER V To obtain an expression for the average rate of interest in either standard, i.e., ia (or ja), we require a given series of pay ments, D1 , D2 , ••• Dn , in the one standard (or t"heir equiva lent Eu B 2, ••• B n , in the other standard). The aggregate present value of these payments, reckoned by the separate rates of interest, i1 , i2, •••in (or ju j2 •..in) is D1 D 2 Dn 1+i1 +(1+1· 1 ) (1+i 2 )+ ... +(1+i 1 ) (1+i 2 ) ••• (l+i n ) (or the corresponding expression in terms of B's and j's).

Now the" average" rate ia must be such that if applied to the same set of payments it will produce the same sum of present values; that is, ia, is determined by D 1 D2 Dn 1 + ia + (1 +ia)2 + ... + (1 + ia )'\ = (10) D1 D2 Dn 1 +i 1 + (1 +i1) (1 +i 2) + ... + (1 +i I ) (1 +i 2 ) •••(1 +?:n)' and ja is determined by the corresponding formula in B's andj's. This equation has only one real and positive root or value of ia • It can readily be obtained by Horner's method.! We may call i a and ja the" present-worth-average" of iu i 2 , .••• in and ju j2' 000 in respecti velyo2 We may define the average rate of appreciation of one of the two standards in terms of the other as that rate which would connect the two average interest rates if the latter were actual (instead of averages of actual) rateso3 That is, if the 1 For, by substituting for -11 . the single letter x, and for -11 . '-11 . ,+ta + 'It! +1,2 etc., the letters Xl, X2, etc., the equation becomes: DIX + D 2x 2 + ... + Dnxn = DIXI + D2XIX2 + ·o. +D nXIX2 .00 Xno In the example given previously the equation becomes: 20 x + 40 x2 +30 x8 + 10 x4 =66.321, the required root of which is x =.83155, which, applied to x = 1~' gives ja = .2026.+Ja 21 + i a becomes the" geometrical average" of 1 +il, 1 + i 2, etc., when DI =D2 = 0.. =Dn-l =o.

8 It may be proved that this definition of aa satisfies the general condition of an average, viz. that aa reduces to aI, a2, etc., when the latter are all equal, whether it, i2, etc. (and jI, j2, etc.), be all equal or not. The proof is left to the reader.

APPRECIATION AND INTEREST 373 average rates of interest are i" andja, the average.appreciation, an, is given by the equation 1+ja=(1+i a ) (l+a a); ja - 'l~a or, aU. = 1 +ia • Thus, as in the example given above, suppose the average silver interest is 20.26 %and gold interest 3 %, so that ·2026 - · 03 1676aa= 1+ .03 =. , or 16.76 %. This average is not identical with the arithlnetical average of 0, 50, 10, and 5 %, which would be 16.~5 %, nor is it identical with that rate which, if uniform, would result in four years in the same divergence between silver and gold as was produced by the four successive rates 0, 50, 10, and 5 %; this would be 14.70%.

APPENDIX TO CHAPTER VII FIRST ApPROXIMATION (This Appendix should be read as a whole after Chapter VII.) § 1 A.s the preliminary statement of the theory of interest enunciated in Chapter VII contains the kernel of our theory, it will be worth our while, before proceeding to introduce the various complications necessary to complete it, to give the first approximation a full mathematical expression. This mathematical statement will serve to make the preceding results clearer and more sharply defined. It will also serve to demonstrate the important fact that the number of deter mining conditions is exactly equal to the number of unknown quantities, and therefore is adequate for fully determining those unknown quantities. Inasmuch as the equations are necessarily numerous and complicated, it will aid the reader in following them if we break the argument up into two steps, first considering an arti ficially simple case where there are only two years of income to be considered, and then passing to the general case where there are any given number of years.

Let us suppose, therefore, that the rate of preference for this year's income over next, in the case of each individual, can be expressed as dependent alone on the amount of this year's and next year's income, the incomes of all the future years being regarded, for the sake of argument, as fixed. Sup pose also that the income of each year is concentrated at one point, say the middle of the year, making the two such points just a year apart, and that borrowing and lending are so re stricted as to affect only this year's and next year's income. Let 11 represent the rate of preference for this year's over next year's income, for individual No.1, and let his original endowment of income for the two years be respectively c/ and ct". This original income-stream, ~', c1 ", is modified by borrowing this year and repaying next year. The sum bor374 ~~~ftlt'S'!' ·:APPROXIMATION 375 rowed this year is ~', the value of which is yet to be deter mined. This sum is therefore to be added to the present income ~'. Next year the debt is paid, and consequently the income CI" for that year is reduced by the sum paid. For the sake of uniformity, however, we shall regard both modifica tions algebraically as additions, the addition Xl' to the first year's income being a positive quantity, and the addition, which we shall designate by Xl'" to the second year's income being a negative quantity. Thus, if $100 is borrowed this year and $105 repaid next year, ~l' is +100 and re/' is - 105. Thus the first year's income is changed from c/ to cl ' +Xl' and the second year's from CI" to cl " +Xl". By this notation we avoid the necessity of employing minus signs. Then the first con dition determining interest, namely, that the rate of preference for each individual depends upon his income-stream, is repre sented by the following equation:Ji = FI(CI' + ~', cl " + Xt"), which expresses Ji as a function of the income of the two years. In case the individual lends instead of borrows, .the same equation may be taken to represent the resulting relation between his rate of preference and his income-stream as modi fied by lending,. the only difference is that in this case the particular value of x' is negative and of x", positive.

In like manner, for individual No.2 we have the equation j;, =F2 (c2' +':£2" c2" +':£2")· For the third individual, fa = Fa (ci +xa', ca" +xa"), and so on up to the last or nth individual, for whom the equa tion will be < In = FA (cfa' + Xn', C,a" +cn"). These n equations therefore express the first of the four con ditions mentioned at the close of Chapter VII. The second condition, that the marginal rates of preference of the n different individuals for present over future income shall be equal to each other and equal also to the rate of inter est, is expressed by the continuous equation:Ji=J2=fs···= ... =fn=i.

376 APPENDIX TO CHAPTER VII These equations hold true, as we saw in Chapter VI, because if a particular f should be greater than its corresponding i, the individual would become a borrower, and if f should be less than i, he would become a lender. In the former case the effect of his borrowing would be to reduce his f until it became equal to i. In the latter case the effect of his lending would be to increase his f until it became equal to i likewise. We have, then, as a result of borrowing or lending, the equation f = i, and as the same applies to every individual, all the f's are equal to i and therefore to each other also. The third condition, namely, that the market must be cleared, or that the loans and borrowings must be equal, is expressed by the following two equations: Xl' + X2' + + + xn ' = 0, a;" +'X2" + + + xn " = o. That is, the total of this year's borrowings is zero (lendings being regarded as negative borrowings), and the total of next year's returns is likewise zero (payments being regarded as negative returns).

The fourth and last condition, that for each individual this year's loans and next year's returns discounted are equal, is fulfilled in the following equations, each corresponding to one individual: Xt"xl'+l~=O, +~ x"1'l'9'+_2_-0 ~.2 1 .-,+~ x"1'l'9'+_3__ 0 ~3 1 .-,+~ X', ~ '+-"-_0o.V" 1 .-.+11 §2 We now proceed to compare the number of equations with the number of unknowns. There are evidently n equations in FIRST APPROXIMATION 377 the first set, n in the second, 2 in the third, and n in the fourth, making in all 3n+ 2 equations. The unknown quantities are flJ j;, Is, ·.. , · .. , fn' ah', 'X2', 'Xs', •••, •••, 'Xn', Xl'" 'X2" ros", •••, •••, 'X,,,", and finally, i, · making in all3n + 1 unknowns. or n unknowns, or n unknowns, or n unknowns, 1 unknown, We have, then, one more equation than necessary. But ex amination of the equations will show that they are not all independent, since anyone equation in the third and fourth sets may be determined froIn the others of those sets. Thus, to determine the first equation of the third set, add together all.the equations of the fourth set. The.additiongives In this equation we may substitute zero for the numerator of the fraction, as is evident by consulting the second equation of the third set. Making this substitution, the above equation becomes which .is identical with the first equation of the third set.

Since·we have here derived the first equation of the third set from all the other equations of the third and fourth sets, the equations are not all independent. It follows, therefore, that of the 3 n + 2 equations, one may be dispensed with (namely, any one of the equations of the third or fourth sets), so that there are left only 3n +1 independent equations, which are there .. fore exactly equal in number to the unknown quantities to be determined. There are, therefore, just sufficient equations to determine those unknown quantities, namely, thef's, or rates of preference for different individuals, the re"s and 'X"'s, or the loans and their repayments, and finally, i, the rahe of interest'. §3 In order to obtain an explicit expression for i, we may" solve with respect to i" all of the preceding equations that can be 378 APPENDIX TO CHAPTER VII so treated. Thus, .to take one of .the equations-from the third "set, namely, ~' +t~ i = 0, it is evident that, solving for i, this equation may be written · -ah" 1t=--- .

~l' To interpret this, we recall that Xl' is a sum borrowed this year (say $100) and thus added to this year's income, and Xl'" being the sum "added" next year, and hence a negative quantity (say -105), it follows that - Xl" is the positive quantity (as 105) returned. Hence - ~" is the ratio (as 105) of the sum reXl' 100 turned to the sum borrowed, or the ratio of exchange x',between this year's and next year's goods, and ~ -1 Xt ( as 105 -1, or 5 %) is the premium above par of the rate of100 exchange. Since this premium is the definition of the rate "of interest, the equation i = - Xt -1 is merely an equation reI' of definition. The results of the proposed transformations may be summa rized as follows: i=ft =12 = =1"" = F I(cI ' +~l" <1."+xl")=F2(C~' +X2',C2" + x2")= = Fn(etc.) -~" -~" -x"=__1__ 1 = __2__ 1 =..,=__1'&_-1. ~ ~ ~' These three equations give the value of i, subject to the addi tional condition: ~'+x~/+ ,.. + ... +x,/=O, the other equation of the same type as the last being omitted as the one superfluous equation. The equation above written could be itself dispensed with by substituting in the previous equations the value of IJ;' derived from it, namely, -x2'- ••• -x,,'.

These equations state that the rate of interest, under the conditions of the problem, will be equal to the rates of time preference for all the individuals, as well as equal also to a FIRST APPROXIMATION 379 certain definite function of the income-streams as finally de termined by the loans and to the premium of exchange of this year's income in terms of next year's income. These con- . ditions, taken together with the condition that the "suDlB·lent must be equal to the sums borrowed-that is, that the rate of interest must be such as just to clear the market -will yield a complete determination of the rate of interest, which is the object of our search. It may be remarked in passing that the first of the three continuous equations given above is closely analogous to the third; in fact, the first may be called the subjective proto type of the third, which may, in like manner, be called the objective expl'ession of the first. The third equation states the definition of the rate of interest as the premium in the ratio of exchange between this year's and next year's income, and the first states that the rate of interest is equal to the premium in the relative desirability of the two. It is not diffi cult to express the analogy algebraically by putting the f's equal to the excess above unity of the ratio between the mar ginal desirability of this year's income!and the marginal desir ability of next year's income.

If we desire only to obtain the simplest expression for deter mining i, the above equations may be condensed still further. The first continuous equation may be omitted altogether. For, to omit it will evidently rid us of as many unknown quantities as equations, namely, n. Again, since the third continuous equation is evidently nothing more than a defini tion of the rate of interest, .we may, if we choose, omit the "letter i and employ the expression - ~ -1 in its place. We Xl' shall then have, instead of the three continuous· equations above expressed, simply the following continuous equation:4" ~". X"--_1= __2 -1= ... =-....!L-1 Wt' x2' w,.." = F1 (CI' +Xl" CI" +Xl") = F2(C2' +~:/, C~"+~2") This equation, together with the equation for clearing tIle market, ~' + x2' + ... +a;,/ == 0, will fully determine the rate 380 APPENDIX TO CHAPTER VII of interest. But, as was shown, the equation for clearing the market may be omitted, if we first obtain from it the value of one of the unknowns it contains, say, Xl" and substitute this value in the previous continuous equation. The continuous equation, so amended, will then of itself yield a complete solu tion of our problem.

§ 4 In the preceding solution, the loan transactions were sup posed to extend over two years only. This restriction was made in order that the mathematics might be as simple as possible in our first formulation. We shall now remove this restriction and proceed to the case in which more than two years (let us say m years) are involved. We shall assume, as before, that the x's, representing loans or borrowings, are to be considered of positive value when they represent additions to income, and of negative value when they represent deduc tions. The equations in the first set will now be in several groups, of which the first is : h' = FI , (cI ' +Xl" cl " +a;", cI '" +Xl"" •••Cl(m) + a; (m»), j;'= F2' (C2' + X2', C2"+ X2", C2'" + X2"', •••C2(m) + X2(m»), in' = Fn, (Cn' + x.,,,', Cn" + xn ", cn'" + X n"',·· • c,/m) + xn(m». These equations express the rates of preference of different individuals (Ji.' of individual No.1, h' of individual No.2, etc.) for the first year's income compared with the next. To ex press their preference for the second year's income compared with the next there will be another group of equations, namely: Ji" = F I" (</t" + 4.", Cl'" +Xl"" •••Cl(m) + X1(m»), 12" = F2" (C2" + X2",···· · • C2(m) + X2(m)), In" = Fn" (Cn" + Xn",····· ••••• ••Cn(m) +xn(m)).

For the third year there will be still another group, formed by inserting "",,, in place of """, and so on to the m -1st year; for the m - 1st year is the last one which has any ex change relations with the next, since the next is the mth or last year. There will therefore be m -1 groups of equations, and since each of these (m -1) groups contains n separate equa tions, there are all together n(m -1) equations in the entire set.

FIRST APPROXIMATION 381 iff =!I" =j;" =13"= ... =fn". Turning to the second set, we first observe that we are now compelled to assume a separate rate of interest for each year. The rate of interest connecting the first year with the. next will be called i', that connecting the second year with the next, iff, and so on up to i(m-l). Under these conditions we shall find, as before, that the rates of timepreference for each year will be reduced to a uniform level for all the different individuals in the com munity, - a level equal to the rate of interest. Algebraically expressed, this condition is contained in several continuous equations, of which the first is i' =Ji.'=}2' =fs' = ... =fn'· This expresses the fact that the rate of timepreference of the first year's income compared with next is the same for all the individuals, and is equal to the rate of interest between the first year and the next. A similar continuous equation may be written with reference to the timepreferences and interest as between the second year's income and the next, namely:Since the element of risk is supposed to be absent, it does not matter whether we consider these secondyear ratios as the ones which obtain in the minds of the community to-day, a year in advance, or those which will obtain next year; under our assumed conditions of no risk, these are necessarily identical.

A similar set of continuous equations applies to time-ex change between each succeeding year and the next, up to that connecting the m - 1st and the mth year. There will therefore be m - 1 continuous equations. Since each continuous equa tion is evidently made up of n constituent equations, there are in all n(m-l) equations in the second set of equations. The third set of equations, which expresses the "clearing of the market," will be as follows:Xl' +X').' + + xn' = 0, Xt" +x2" + + Xn" = 0, Xt(m) +X2(m) + ... + Xn(m) = o. There are here m equations ..

382 APPENDIX TO CHAPTER VII The fourth set of equations, expressing the equivalence of loans and repayments, or more generally, the fact that for each individual the total "additions" to his income· stream, alge braically considered, will have a present value equal to zero, as expressed in the following equation: , x" Zt'" Zt(m) ~ + l~i' + (l+i')(l+i") + ... + (l+i')(l+i") ... (l+i< ..-l» =0. Similar equations will hold for each of the n individuals:x"X'+_2_+... =O 2 l+i' , x"roa' + _3- +...= 0,l+i' x"ro '+-"-+ ... =0," l+i' making in all n equations. We therefore have as the total number of equations the following: n(m -1) equations of the first set n(m -1) equations of the second set m equations of the third set n equations of the fourth set 2 mn + m - n equations in all. We next proceed to count the unknown quantities. First as tof's: for individual No.1 these will beji',Ji", ... ,!i<m-l) the number of which is m -1, and as there are an equal number for each of the n individuals there will be in all n(m -1) un knownf's.

As to x's, there will be one for each of the m years for each of the n individuals, or mn. As to i's, there will be one for each year up to the next to the last, or m -1. In short there will be n(m - 1) unknown 1'8, mn unknown ~'s, m - 1 unknown i's, or 2 mn +m - n -1 unknown quantities in all. Comparing this with the number of equations, we see that there is one more equation than the number of unknown quantities. This is due to the fact that not all the equations are independent.

FIRST APPROXIMATION 383 This may be shown if we add together all the equations of the fourth set, and substitute in the numerators of the fractions thus obtained their value as obtained from the third set, namely, zero. We shall then evidently obtain the first equa tion of the third set. Consequently we may omit anyone of the equations of the third and fourth sets. There will then remain just as many equations as unknown quantities, and our problem is exactly determined. In the preceding analysis, we have throughout assumed a rate of interest between two points of time a year apart. A more minute analysis would involve a greater subdivision of the income-stream, and the employment of a rate of interest be tween every two successive elements. This will 'evidently occasion no difficulty except to increase the number of equa tions and unknowns. § 5 The elaborate system of equations which is involved when m years instead of two years are considered introduces very few features of the problem not already contained in the simpler set of equations first given. The new feature of chief impor tance is that, instead of only one rate of interest to be deter mined, there are now a large number of rates. It is too often assumed, in theories of interest, that the problem is to deter mine·" the" rate of interest, as though one rate would hold true for all time. But in the preceding equations we have m-1 sepa rate magnitudes, i', i", i''', ... i(m-l). Is there any tendency at work to make these rates of interest equal? Is the rate of in terest which expresses the ratio of exchange between this year's and next year's income normally equal to, or nearly equal to, the rate of interest which expresses the ratio of exchange between next year's income and the year after? Bohm..Bawerk 1 put this question, and answered it affirmatively, stating that a species of arbitrage transactions tended to produce this result.

If, however, we examine his reasoning closely, we shall see that the only proposition he has proved is that, if the rate of inter est expressing the premium on the goods of 1888 as compared with 1889 is equal to the premium of 1889 over 1890, then a contract for the exchangeof the goodsof 1888for those of 1890 will take place at this same rate. But what is really needed 1 Positive Theory of Oapital, p. ~80.

384 APPENDIX TO CHAPTER VII is to know whether (as Bohm-Bawerk assumes) the rate of interest connecting the years 1888 and 1889 is the same as the rate of interest connecting the years 1889 and 1890, and if so, why? Under the hypothesis of a rigid allotment of future income among different time intervals, there is nothing to prevent great differences in the rate of interest from year to year, even when all factors in the case are foreknown. This is clear from· the fact that, by a suitable distribution of the values of CH C2, etc., there may be produced any differences in the values of i', iff, iff', . etc. If the total enjoyable income of society should be fore known to be, in the ensuing year, 10 billion dollars, in the fol lowing year one billion, and in the third year 20 billion, and there were no way of avoiding these enormous disparities, it is very evident that the income of the middle year would have a very high valuation compared ,vith either of its neighbors, and therefore that the rate of interest connecting that middle year with the first year would be very low, whereas that con necting it with the third year would be very high. It might be that the members of such a community would be willing to exchange $100 of their plentiful 10 billions for the first year, for only $101 out of their scarce one billion of next year, but would be glad to give, out of the third year's still more plenti ful 20 billions, $150 for the sake of $100 in the middle and lean year. The reason that, in actual fact, such abrupt and large variations in the rate of interest as from 1% to 50% are not more frequently encountered is that the supposed sudden and abrupt changes in the income-streamseldom occur. The causes which prevent their occurrence are:(1) The fact that history is constantly repeating itself.

For instance, there is regularity in the population, so that at any point of time the outlook toward the next year is very similar to what it was at any other point of time. The indi vidual may grow old, but the population does not. .As indi viduals are hurried across the stage of life, their places are constantly taken by others, so that, whatever the tendency in the individual life to Inake the rates of preference go up or down, it will not be cumulative in society. Relatively speak ing, society stands still. Again, the processes of nature recur in allnost ceaseless regularity. Crops repeat themselves in a yearly cycle. Even FIRST APPROXIMATION 385 when there are large fluctuations in crops, they are seldom world-wide, and a shortage in the Mississippi valley may be compensated for by an unusually abundant crop in Russia or Asia. The resultant regularity of events is thus sufficient to maintain a fair uniformity in the income-stream for society as a whole.

(2) The fact that the income-stream is not fixed, but may be modified in other ways than by borrowing and lending. The nature of these modifications are considered ill Chapter VIII. § 6 Let us now return, for fuller discussion, to the second con dition, that the rates of preference of the different individuals are equal to each other and to the rate of interest. It was shown in Chapter VII that when the individual de termined his income stream so that his marginal rate of pref-' erence for present over future income was equal to the rate of interest, he thereby maximized his present" total desirability." The two statements, that his preference rate is equal to the interest rate, and that his" total desirability" is a maximum, are thus interequivalent, and either may be deduced from the other. Mathematically this may be shown either by geometry or by algebra. ,",'Ve shall begin with the algebraic method.

Assume at first that only two years are considered. The fact that total desirability depends on the amount of income this year and next year may be represented by the equation U=F(c' + ro',c" + x"), where U represents total desirability or utility of an individ ual, and the equation represents this U as a "function" of his income-stream consisting of d + x' this year and e" +x" next .year. As we shall here consider only one individual, we omit the subscript numbers, 1, 2, etc., previously used to dis tinguish different individuals. .The individual will attempt to adjust x' and x" so as to maximize U. By the theory of dif ferential equations, the condition' that U shall be a maximum, is that the" total differential" of U or of its equal F (c' + xf, e" + x") shall be zero, thus dU=8F( )dx,+BF( )dx"=O, Bat 8x" 20 3.86 APPENDIX TO CHAPTER VII where the 8's represent the" ,partial differentials" with respect to x' and x", and the blank parentheses stand for (c' + x", e" + x").

From this equation it follows that _ dxr, = '8F( ) joF( ). dx' OX' OX" The left-hand member is 1 + i, as may be seen by differen tiating the equation of the loan as originally stated, viz. : I ro" dx"x' + -- =O. This differentiation yields - -, =1 + i.l+i ~ The right-hand member, being the ratio of this year's marginai desirability to next year's marginal desirability, is by defini tion equal to 1 +f. Substituting the new value for the right and left members, we have l+i=l+}; whence it follows that i = f. The same reasoning, applied to three or more years, may now be expressed. The total desirability for any individual is a function of the total future income-strealu. In other words, U = F (e' +x', e" + x", e'" + x''', etc.). The individual tries to make this magnitude a maximum. In terms of the calculus, this is equivalent to making the first total differential equal to zero, namely, dU=oF( )dx,+oF( )dx,,+oF( )dx"'+etc.=O ox' '0 x" '0 x'" · This total differential equation is equivalent to a number of subsidiary equations obtained by making particular suppositions as to the different variations. Let us, for instance, suppose that only x' and x" vary in relation to each other and that x"', XiV, etc., do not vary. Then in the above equation all terms after the second disappear and the equation reduces to _dx" =~/oF( ).

dx' 0 x' 8 x" In other words, 1 +i' = 1 +f, and therefore i' = f.

FIRST APPROXIMATION 387 This expresses the relation between the first and second years. If we wish in like manner to express the corresponding connection between the second and third years, let· us assume . that re'is cOt;lstant and reiV, etc., constant. Then the first term of the' equation disappears and all after the third term, and the equation reduces to dx'" of() jBF(. ) -dx" = oz" ~. In other words, 1 +iff = 1 +f', or iff =/"; and so on for each succeeding year. We therefore see in mathematical language that the point of maximum desirability tis also the point at which the marginal rate of preference for each year's income over the next year's income is equal to the rate of interest connecting these two years. 10 Ao. Ct" ""'-----------"--- ........-- X'o cI ' § 1 We turn now to the geometrical interpretation. In Figure 26, let the point P be found such that its coordinates are ci' " . -'10X 1020804050-60. N B FIG. 26.

and Cl", the values of this year's income and next year's in come respectively, Ct' being laid off along the horizontal axis 388 APPENDIX TO CHAPTER VII OXl, and CI" being laid off vertically. In the same way any other income-stream may be represented by another point, the coordinates of which represent the values of this year's and next year's income respectively. By borrowing or lending, the income-stream P is changed to another point Q to be determined. We shall assume, as before, that the modifica tion of this income-stream, cI ', cl ", through borrowing and lending or buying and selling applies only to the first and second years; all subsequent years are therefore omitted from our calculations. The income-stream that is the com bination of the two magnitudes cI ', C2", - the fixed income with which the individual is supposed to be endowed, - is represented by the point P. But this income-stream, CI', Cl'" he modifies by adding algebraically ~', through borrowing (or lend ing, if Xl' has a negative value) this year, and adding next year, when the loan is paid, the sum Xl", which is equal to ro/ put at interest, but of opposite sign; in ot herwords, Xl" = - x' (1+ i) x"or Xl' + 1~ =O. It is first proposed to determine Xl' and ~", +~ assuming that the rate of interest is a fixed and given magni tude. The new income-stream, c' + m', e" + ~", will be repre sented by a new point, Q, different from P.

To find this point, Q, draw the line AB through the~ point P at a slope determined by the given rate of interest, namely, so that g~ =1 +i. Then the new point will lie somewhere upon this straight line. For (1) the present value of the modified income-stream Ct' + 0;.', Ct" + Xl" is the saIne as that of the original income-stream CI', cI"; and (2) the straight line AB is the" locus" or assemblage of all points the present values of the income-streams represented by which are the same as the present value of the income-stream represented by the point P through which AB is drawn. That (1) the present values of the original and the modified income-streams are the same is due to the fact that the loan Xl' and its repayment xi" are equivalent in present value. This may be seen by transforming the formula for the present value of the modified income-stream, as follows:FIRST APPROXIMATION 389 since the magnitude in parenthesis is zero by the original hypothesis as to xI' and Xl".

That (2) the line AB locates all points of present value "Cj,' + 1~i is evident from analytical geometry, and may be shown, among other ways, as follows:The present value of the original income-stream C-t', cI " is equal to the length OA. For OA=OO+OA c"_1'_'+ 1 -VI l+i' "for OO=Ct' by construction and O.A= lC+i by the similar triangles .AOPand .AOB, whose proportional sides give ~= OBOA OAc" .=1+i, or OA= __1 __ l+i Similar reasoning applied to' any other point on the line AB will show, in like manner, that the present value of the income represented by that point is also OA. Hence eve1·Y point on the straight line AB represents an income-combination or income-stream having the same present value, OA, as that of the income-stream c', e" represented by P. Similar reason ing shows that no point out of AB'represents an income-stream of this present value. Therefore the individual who is pos sessed of the income-stream represented by P, and who modifies this income-stream by. borrowing and lending, merely shifts the point representing his income-stream along the line AB, as from P to Q. Which of all the points on this line open to his choice will he select? Evidently he will select that one which will give, for him, the maximum·present de sirability. In order to determine this point, let us suppose that the desirability corresponding to every point upon the plane is indicated by a number attached, and that through the points which have equal desirability, lines of equal de sirability, like isothermal lines on a weather map, are drawn. l These may be called iso-desirability lines. If any two points 1 Of. the writer's" Mathematical Investigations in the Theory of Value and Prices," Part II, Transactions of aonn~ Academy, New Haven, 1902, p.68.

390 APPENDIX TO CHAPTER VII on one and the same iso-desirability curve be compared, it will be seen that one of them represents an income richer this year and poorer next year than the income represented by the other; the superiority of the former income over the latter this year exactly compensates for its inferiority next year, so that, all considered, the two incomes are equally desirable. It is clear that these iso-desirability lines will constitute a "family" of curves, each approaching the axes OX' and OX", and that the numbers attached increase in magnitude as the curves recede from the origin O. The curve drawn nearest the axes is labeled "10" at each end, to signify that all points upon that curve have a desirability to the individual represented by 10. The point P evidently has a desirability between 10 and 20. As we proceed from P to'\vard B along the line AB, it is evident that the desirability first increases and then decreases, reaching the maximum at Q, where the line AB is tangent to one of the family of curves. This point Q is therefore the point sought, and represents the income-combina tion which has the maximum present desirability. Thus, for the individual, the solution of the problem of how much to borrow or lend is determined geometrically by drawing through the point P, representing his fixed original incolne-endowment, a straight line at the slope OB = 1 +i', and finding the pointOA Q upon this line at which it is tangent to some one of the num berless curves of equal desirability. Q differs from P in having one of its coordinates larger (by Xl") and the other smaller (by - Xl').

The fact that the iso-desirability curves at the right slope less than 45° interprets the fact, which should not be lost sight of, that if this year's income is sufficiently abundant or next year's income sufficiently scarce, or both, that this year's goods may exchange for ne4 t year's on less than even terms; that is, that the rate of timepreference may be negative. § 8 Not only is it true tha,t Q represents the point of maximum desirability, but also that at the point Q the rate of preference Ji. is equal to the rate of interest. First we observe that the rate of preference for present over future income at any point depends upon the slope of the iso-desirability curve which FIRST APPROXIMATION 391 passes through that point. To make this clear, let us consider, on any of the iso-desirability curves, such as 70-70, the point M and an adjacent point N. The substitution of the income combination represented by N for that represented by M in volves the sacrifice of an alnount of this, year's income repre sented by MS, which lnay be called - ~x', for the sake of an addition to next year's income represented by NS, or 4x".

If the points M and N are indefinitely near together, we may represent MS by - dx', and NS by dro". The loss in desirability by' surrendering MS is represented by the difference in number between the iso-desirability curves through M and S, namely, 70 - 60, or 10. Likewise, the gain in desirability by the addition of SN to next year's in come is represented by the difference between the numbers corresponding to iso-desirability curves through Sand N, also 70-60, or 10. In other words, the loss in desirability through the surrender of MS is equal to the gain in desira bility through the addition of SN, ,or, the desirability of the loss of MS of this year's income equals the desirability of the gain,of NS of next year's income. Since, therefore, 8M and SN, or - ~x' and ~x", are the amounts of income for the two respective years which possess equal desirability, it is evident that the degree of desirability per unit of income for this year and next year will be in the inverse proportion. Thus, if 8M is two hundred and SN is three hundred dollars' worth of in come, this means that, for the particular individual for whom the figure is drawn, having an income-stream represented by the point M, $200 taken from this year's income would be exactly compensated for in present estimation by the addition of $300 to next year's income. Hence the desirability per dollar of the present income is 1t tilnes the desirability per dollar of next year's incoDle.

Symbolically these relations are:=0, =0,or Desirability of ~ x' + desirability of ~ x" =0. d · A' d· "O eSIr. ~X A I eSIre ~ x J1 " r J1 x' ~ x + 4X" x ~u ~u -- J1x'+--~x" ~x' ~ x" a -u auor --,,: -,,- = ~ro": -~~, = NS:MS ~ x~x 392 APPENDIX TO CHAPTER VII That is, the desirability of an additional present dollar is to that of an additional dollar next year, as NS to MS, or the slope of the line joining .jlf and N. If the points M and N are indefinitely near together, the slope will be the slope of the iso-desirability curve through M. Thus the slope of any of the curves in the diagram at any point is the geometrical rep l"esentation of the relative valuation of present and future income which the individual feels when in possession of the income-stream represented by that point. We have already specified that the slope of the straight line AB represents the ratio of exchange of this and next year's income. Thus the slope of the curves represents the subjective, and the slope of the line the objective, ratio of equivalence for the two years.

The former slope, the ratio of the marginal desirability of this year's income to the marginal desirability of next year's income, is, by our previous definition, 1 +ii, just as the latter slope is l+i. Applying these ideas to the particular point Q, it is clear that the slope of the iso-desirability curve through Q is eqnal to the slope of the straight line BA. But the slope of the desirability curve is 1 +ii, and the slope of the straight line AB is 1 + i, therefore, for any individual, 1 +.h= 1 +i, or .Ii = i. Hence the individual who 1110difies his income from P by a loan at the rate i will shift it to a point Q, such that his sub jective rate of preference Ji which corresponds to that point will be equal to the objective rate of interest i,-or, speaking geo metrically, so that the "slope" of his curves will be made equal to the" slope" of the market. We have presented the geometrical method in considerable detail, in the belief that it is well worth mastering. It will be found especially helpful when extended so as to apply to the more complicated problem discussed in the Appendix to Chapter VIII.

§ 9 If we proceed from the consideration of two years to that of three, we may still represent our problem geometrically by using three dinlensions. Let us consider three mutually pel" pendicular axes OX', OX", OX"', and represent the incomeFIRST APPROXIMATION 393 combination or income-stream for the particular individual by the pointP, whose coordinates e', e", and e'" are the three years' inconle-installments with which the individual is initially endowed. Then through the point P draw, instead of the straight line in the previous representation, a plane ABO cut ting the three axes in .A., B, and O. This plane has a slope with reference to the two axes OX' and OX" of g~ equal to l+i', and has a slope with reference to the axes OX" and OX'" represented by 00 equal to l+i". The letters i' andOB iff here represent, as before, the rate of interest in the exehange of this and next year's goods, and of next year's and the year after. Now suppose the space between the axes to be filled with iso-desirabilitycurved surfaceslike the coats of an onion, such that for all points on the same surface the total desirability of the income,;,stream represented by those points will be the same. These surfaces will be such as to approach the three axes and the planes between theIn, and also such that the attached numbers representing their respective total desira bilities shall increase as they recede from the originO. The plane ABO drawn through P at the slope fixed by the rates of interest, as just indicated, will now be tangent to some one of . the iso-desirability surfaces at a point Q, which is the point at which the individual will fix his income. For every point on the plane ABO will have the same present value,and everypoint on this plane is available to him by borrowing and lending (or buying and selling) at the rates i' and i", but not all·of them will have the same desirability. He will select that one which has the maximum desirability, and this will evidently be the point Q, at which the plane is tangent to one of the family of iso-desirability surfaces. Reasoning similar to that given. for two dilnensions will show that this point will be such that ft.' = i' and.ft."= i' '.

To proceed beyond three years would take us beyond the limitations of space; for we should then need in our represen tation more than three dimensions. Such a representation is of little meaning except to mathematicians, since it cannot be fully visualized. For the practical purpose of visualization, the simple geometrical representation in two dimensions, though limited to two years, is the most helpful.

394 APPENDIX TO CHAPTER VII § 10 Having shown the geometrical representation as applied to a particular individual, we now proceed to show how the rate of preference is determined for a series of individuals. To recur to the geometrical representation in Fig. 26, where only two variables are considered, the problem is as follows:Given a number of different individuals, each with his own separate point P and his own separate set of iso-desirability curves, we are required to draw through these points straight lines parallel to each other at such a slope as to "clear the market," in other words, such that the sum of the x"s for the different individuals shall be zero, and, as implied thereby, that the sum of the ~'''s shall also be zero. It is evident that, according as the slope of the lines AB changes, the points of ~angency, the Q's, for the different indi viduals will vary, which means that the amount borrowed or lent, namely, x' and x", will change. We have then a swarm or group of fixed points, the P's, and another swarm of variable points Q's. By rotating the lines each about its pivot P, and so that all remain parallel to each other, we can evidently shift the position of the second swarm of points, the Q's. The solu tion is found by fixing upon such a slope of the lines that the center of gravity of the Q swarm is brought into coincidence with the fixed center of gravity of the P swarm. The slope of the lines AB which will accomplish this result is the rate of interest which will just clear the market; for the horizontal deviations, Xl" X~/, etc., between the P and Q for each different indi vidual will then be self-canceling, their algebraic sum being zero, and the same is true for the vertical deviations Xl'" x2", etc.

For three dimensions, we have precisely similar determina tions. The problem of the rate of interest is here solved by finding such an orientation for the various planes through the points called P's as ,viII bring the center of gravity of the tangential points, the Q's, into coincidence with the fixed center of gravity of the P's.

APPENDIX TO CHAPTER VIII SECOND ApPROXIMATION (This Appendix should be read as a whole after Chapter VIII.) § 1 In the Appendix to Chapter VII we found that the number of equationsavailablefor determiningthe rate of interestwas equal to the number of unknown quantities, and therefore that the rate of interest and the other associated variables were determinate under the assumption there made. This assump tion was that all. income-streams were unalterable, except as they.could be modified ·by borrowing and lending, or buying and selling. We now introduce, in place of such a fixed income stream, the hypothesis of a range of choice between different income-streams. This, however, does not destroy the deter minateness of the interest problem; for along with the ne·w variables introduced, we find an equal number of new equations. Let us, then, state and count the equations which, under our several hypotheses, determine the rate of interest. The in come-stream, we must remember, no longer consists of known and fixed elements, cI', CI", cl ''', etc., as assumed in Chapter VII, elements which can be modified only by exchange; it now con sists of unknown and variable elements which we shall designate by YI', y/', YIlt', etc. This elastic income-stream may be mod ified in two ways: by the variations in these y's, as well as by . the method which we found applicable for rigid income streams, namely, the method of exchange - borrowing and lending or buying .and selling. The alterations effected by the latter means we shall designate as before by Xl" Xl", Xl"" etc., for successive years. These are to be algebraically added to the original income-items (the y's), deductions being included in this addition by assigning negative values. .,The income stream as finally determined is therefore expressed by the installments, YI' +Xl" YI" +Xl'" Yl'" +Xl"" etc.

395 396 APPENDIX TO CHAPTER VIII One of the determining conditions stated in Chapter VII is that the individual rates of preference are functions of the in come-streams. Algebraically stated, this condition gives the equations: 11' = FI, (YI' +Xl" YI" + ~it", ... Yl (m) +Xt(m»), 12' = F9:(Y2'+x2', Y2" +ft2", •••Y2(m) + ft2(m), f - F. .(y , + x' y" + x" ... y (m) + X (m))J,,' - n n ",.. " , fI, ft· These equations express the individual rates of preference for the first year's income compared with the next. To express the preference for the secondyear's income compared with the next, there will be another set of equations, namely:f" _ 77T (y'f + ,.."." y'" + (¥'_,,, •..Y (til) +,..".(tn»)J 1" - 11 1" 1 &(.11' I ""'1' I ~l' f~1 = F 2" (Y2" +X2", •••••••••••••••••Y2(m) +x2('11&»), f~" = F",,(Y,,"+X,,", Y,,(m) + 'XfI,('11&»). For the third year, as compared with its successor, there would be another similar set, with" ",,, in place of " "", and so on to the next to the last or (m -1) year as compared with the last. Since each of these m -1 groups of equations con tains n separate equations, there are all together n (m -1) equa tions in the entire set.

The next condition, that the rates of preference and of inter est will be equal, is the same as in the first approximation,I and is represented by the same n(m-1) equations, namely:i' =ii' =};' = =1,/, iff =Ji" =12" = =/,.", i(m-l) = ji(m-l) •••=/;.("'-1) = = In(m -1). The two sets of equations which express the" clearing of the market" and equivalence of loans and repayments will also be the same as before, and represented by the same m equa tions :Xl' +x2' + + xn ' = 0, Xl" +x2" + + xn" = 0, XI("') +X 2(m) + ... + X n (m) =0; 1 See Appendix to Ch. VII, § 4.

SECOND APPROXIMATION 397 and the same n equations: re " re If' re (m) • Xl' + 1 ~ i' + (1+ i') (l+i") + ... + (l+i') (l+~") .•(l+i(") =0, re " re (m) re'+ 2 + + 2 2 l+i' ... (l+i')(l+·i") .. (l+i m ) =0, re,a' + ... =0. §2 These four sets of equations are the same in number as the corresponding sets given in the Appendix to Chapter VII, namely, 2mn + m - n, or, for l'easons there given, only 2mn + m - n -1 independent equations. These equations dif fer from the.equations of the preceding Appendix only in the first set, which contain y's in place of c's. The c's were con stants, but the y's ·are unknown quantities. Consequently, the number of unknowns is greater than the number in the first approxilnation, whereas the number of equations thus far expressed is the same. We therefore need to seek for new equations to supply the deficiency. These additional equa tions are found from the condition that the choice among the optional income-streams will fall upon that one which possesses the .maximum present value. .

The range of choice, i,e. the complete list of optional income streams, will include many which are ineligible.' By an ineli gible income-stream is meant one which would not be selected whatever might be the rate of interest,-whether zero or one million per cent.,-being smaller for every year than some other stream on the list. Excluding these ineligibles, the remaining options constitute the effective range of cholce. This effective range of choice is subject to the" technical" limita tions of productive conditions, and constitutes the technical conditions which influence the rate of interest. If this list of options be assumed, for convenience of analysis, to consist of an infinite number of options varying from one to another, not by sudden jumps, but continuously, the complete ;,list can be expressed by those possible values of Yl', Yl", Yl(m) which will satisfy an empirical equation <PI (YI',Yl" ..• YI(m») =0, the form of which depends on the particular technical condi tions to which the. capital of individual No.1 is subjected, 398 APPENDIX TO CHAPTER VIII whether dependent on his personal characteristics or on the physical and technical conditions of his business. Thus the form of the function CPl will be one thing if the capital of the individual, which yields the y's, consists largely of mines which are failing, and quite another if it consists of forests recently planted. In the former case, the equation will be satisfied only by values of the y's such that the earlier y's (as y'l or Yl") are comparatively large and the latter y's (as Yl(m-l)or y(m»arecomparatively small; whereas in the latter case the series of y's must conform to the opposite condition. The equation, therefore, while it admits of an infinite number of arrays of y's, does not admit of their variation ad libitum. It represents the limitations to which the variation of the income-stream must conform. Each set of values of Yl', Yl", ..• y/ ffl ) which will satisfy this equation represents an optional income-stream.

Out of this infinite number of options, that particular one will be selected of which the present value is a maximum. Now the present value ~ of any income-stream '!II', '!II", ••• Yl(m), of individual No.1, is evidently y " y'" ~=Y/ +1 ~i'+ (1 +i'/(l +i") + etc. The condition that this expression shall be a maximum is that the first differential quotient shall be zero. That is dy" dy'" d~ = dYl' +1-/'-i' +(1 +i') (1 +i") + etc. =O. This last equation expresses the relations which must exist between dyt', dYl", dYl"', etc., in order that the income-stream Yl', y!", YI''', etc., may have the maximum present value. This condition contains within itself a number of subsidiary con ditions. To derive these, let us consider a slight variation in the income-stream, affecting only the income-installments of the first two years, Yl' and Yl" (the remaining installments, Yl"', etc., being regarded as invariable), and let us denote the values of dYl', dYl", under this assumption of restricted varia tion, by BYl', BYl". Then, remembering that, under the sup posed condition, dYl''', dylV , etc., will be zero, the above equation becomes SECOND APPROXIMATION 399 from which it is evident that _ ~Yl" -1' +1)"0Yl' - .. fl.

But the left-hand member of this equation is evidently the marginal rate of return on sacrifice as between next year's income and this year's income, or the ratio of the increase which may be effected in next year's income by a given sacri fice in this year's income. If we call the premium in this ratio of return rl', we may express - B..,Yl;' as 1 +rI', ~nd write VYI the above equation thus:1 + rl' = 1 + i', or thus:- r/=i'. or or or In other words, the condition that the marginal rate of return on sacrifice is equal to the rate of interest follow3 as aconseq?tence of the general condition that the present value of the income.. stream must be a maximum. This proposition and its proof correspond to those in regard to desirability, which have already been discussed in Appendix to Chapter VII, § 6, that the condition of maximum desirability is equivalent to the condition that the marginal rate of preference is equal to the rate of interest.

The same reasoning may be applied to successive years. Thus, if we assume variations in y" and y"', without any vari ations in the other elements of the income-stream, y', yiV, etc., the original differential equation becomes 8Yl" 8Yl'" l+i'+ (l+i') (l+i") =0, _ By}'"_ (1 +i") ~Yl" - , 1+r1"=1+i", r/'=i". Corresponding analysis applied to each successive year will show that the annual successive marginal rates of return on sacrifice are equal to the annual successive rates of interest. All this reasoning implies that there is a possibility of con tinuous variation, and that at. the margin it is possible to make slight variations in any two successive years' incomes without 400 APPENDIX TO CHAPTER VIII distul'bing the incomes of other years. The values of 1 +'rl ', 0" 0'"1 +'r1", etc., or- ~Yt, , - ~Y\, (the rate at which the second year's OYI 0Yl income may be increased by decreasing the first year's income, and the rate at which the third year's income may be increased by decreasing the second year's income, etc.), may be found in terms of Yl', Yl" •••Yl(m) by differentiating the equation for the effective range of choice, ePI(Yl', Yl", ••• Yl(m) = O. This differ entiation gives oyt" ,,,- 8' =If!l''(Yl,Yl ,etc.),Yl oy'f'- ~=Vtl" (Yl', Yl", etc.),°Yl etc.

Writing together the equations of partial differentiation, we have, as our new set of equations:1 + i'= 1 + 'f1' = tPl" (Yl', y/', etc.) = 1 +r2" = tf!2" (Y2', Y2', etc.) = 1 +'fn ' = tP", (Yn', Yn", etc.). These equations, 2 n in number, relate to the rates of interest and returnon-sacrifice only as between the first and second years. The following similar 2n equations relate to the rates between the second and third years: 1 + i"= 1 + rl"= .pI" (Yl", Yl"', etc.) = 1 + r2" = tf!2" (Y2", Y2''', etc.), etc. Exactly similar equations apply to each year as related to its successor, until we reach the final set, which connects the next to the last year with the last, viz.:1 + i(m-l) = 1 +'r1(m-l) = lfl(m-l) (Yl", Yl"', etc.). As there are here (m-1) sets of equations and 2 n in each set, the total number of equations in these sets is 2 n (nt -1). These equations, together with the n equations of effective range of choice for the different individuals, viz.:cPl (Yl', Yl", etc.) = 0, </>2 (Y2', Y2", etc.) = 0, SECOND APPROXIMATION 401 give therefore 2n(m-1)+n or2mn-n, the total number of new equations in addition to those repeated or adapted from Appen dix to Chapter VII. The number of independent equations, thus repeated or adapted from the previous Appendix was' 2mn+m-n-l. Hence we have:numbeD of old equations, 2 mn + m - n -1, + number of new equations, 2 mn - n, =number of total equations, 4: mn +m - 2 n -1.

Exalnination will show that the number of unknowns will also be 4 mn + m - 2 n - 1. For all of the 2 ·mn +m-n -1 unknowns previously used (in Appendix to Chapter VII) are here repeated, and in addition, the new unknowns, the y'sand, the r's, are introduced. There is one y for each individual for each year, the total array being Yl', Yl", y/m), Y2', y,,", Y2(m), y,..',Y,/', ... y"(m). The number of these y's is evidently mn. There is one r for each individual for each pair of succes sive years, i.e. first and second, second and third, etc., and next to last and last, the total array being . r1', rl",. • •r1(m-I), r2', r2",. • • r 2(m-l), The number of these r's is evidently n(1(t-1). In all, then, the number of new unknowns, additional to those of the previous Appendix, is mn+ n (m - ~) or 2 mn -n. Hence we have: number of old unknowns, 2 mn +m - n - 1, number of new unknowns, 2 mn - n, total number of unknowns, 4 mn +m - 2n -1, 'which total is the same as the number of independent equa tions. Therefore the problem of the rate of interest and related magnitudes is determinate under the oonditi~ns pre scribed.

The complication mentioned in Chapter VIII, § 14, that the' 2» 402 APPENDIX TO CHAPTER VIII income-stream itself depends upon the rate of interest, does not affect the determinateness of the problem. It leaves the number of equations and unknowns unchanged, but merely introduces the rate of interest into the set of equations express ing the influence of the technique of production. These now become ,I" (' " t "'f, t) 0't'1 Yl' Yl , e c., 'it , t , e c. = , etc., and their derivatives, the .p functions, are likewise altered. § 3 The intricate system of equations just stated may be better understood by means of a geometrical representation. First we shall represent the range of choice. Let us suppose, for simplicity, that only two years need to be considered, so y" b·. ~ ~__--a._......,;~~~-......_--- rl ,.0 A. a FIG. 27. that the only unknown y's or income-installments are y' and y", all the y's for succeeding years being regarded as fixed. In other words, let us suppose the case of a farmer who is consid ering the choice between different methods of cultivating his farm for this and next year, but does not take into consideraSECOND APPROXIMATION 403 tion any possible variations in the income from his farm for succeeding years. He has the options· of allowing his land to lie fallow both the years; or to lie fallow the first year .and yield an income the second year; or to yield an income the first year and .lie fallow the second; or to cost him a net loss this year in order to add to the income next year; or to yield him something both years, - from nothing up to the maximum pos sible, though the maximum for one year would be incompatible with the maximum for the other. The farmer has here a choice among an indefinite number of income-streams.

Let us in Fig. 27 measure y' alo;ng the axis 0 Y', and y" along the axis 0 Y". Then the point p has for its coordinates y' and y", that is, for its abscissa Oc = y', and for its ordinate, cp = y". This point thus represents one of the optional income streams. In· like manner we may represent all· other options by a series of other points, as shown by the dots in the diagram. Out of this swarm of points, each representing a particular option, only one will of course be chosen. The point to be selected will, as we know, be that which corresponds to the maximum pres ent value. .To find it we do not need to consider all the points in the entire swarm, for some' of them are evidently out of the ·question. Thus, if through the point p we draw the line Op and prolong it to p', it is evident that the combination or option represented by the point p' will have a higher present value than that represented by p, no matter what the rate of interest may be; for p' evidently has both of its co ordinates y' and y" larger than the coordinates of p. In other words, p' represents an income-stream which is larger than p both this year and next year, and must consequently have a larger present value, whether the rate of interest be 1 %or 100 %. Consequently, of all the points along the line Op we may disregard all except the one point (p') remotest from the origin 0, or on the boundary of the entire mass of points.

In like manner, by drawing other lines from 0 we may see that the only points which need to be considered are the points lying on the boundary p'tR of the entire mass of points. These are, so to speak, the only eligible points. Among them, the one which represents the final choice will differ according to the rate 'of interest; but whatever ·the rate of·interest, the choice will 'always fallon a point in the desig nated perimeter. Therefore the perimeter p'tR, etc., the boun· dary of the swarm, alone represents the effectiverange of choice.

404 APPENDIX TO CHAPTER VIII The effective range of choice includes only the convex por tions of the boundary, the portions which would be points of tangency t of a straight line such as ab, touching and not cut ting the boundary, and revolving about the boundary as an It envelope." We Inay still further restrict the portions of the boundary to be included by limiting the position of the revolv ing ,line ab to the vertical position at the right and to the 45° position at the left. Any further rotation to the left would imply a negative rate of interest, which need not be considered. The configuration of this boundary line is the geometrical representation of those technical conditions which limit the income-stream available from capital. This boundary line representing the effective range of choice will be quite different for different times and places. It ,viII be different according to whether the capital of the individual considered consists largely of land, of machinery, or of other forms of wealth.

It is at this point, then, that the technical conditions of in dustry enter into our problem, and show their influence upon the rate of interest. In order to find what point on the boundary will be selected as the final choice, let us draw the line AB such that all points upon it will represent options possessing a fixed present value OA. AB will then be a straight line of which the slope is 1 +i', depending on the rate of interest. l We know that the present value of the income-stream represented by the point p is given by the formula "V = y' +1Y ., + constant terms,+t in which equation, as in those which follow, the subscripts "1", etc., are omitted for convenience, as it will readily be remem bered that the equations and diagrams always refer to a particu lar individual. The constant terms represent the discounted income of the years beyond the second year, the income-install ments of which are by hypothesis fixed. If now we give to V a fixed value, and transpose to the left-hand member the 1 It may be worth observing that, of its intercepts 0..4. and OB, OA is equal to the value of the income-stream Yl', Yl", as reckoned by discount in advance, and DB is equal to that value multiplied by the factor l+i'.

Hence OB is the value of the same income-stream reckoned by accumula tion next year. See The Nature of Capital and Income, Appendix to Ch. XIII, § 13.

SECOND APPROXIMATION H constant terms," the left-hand member may. be represented by a constant 11, and the equation becomes y" K=Y'+l+i'· This is evidently the equation of the straight line AB drawn so that OA is equal to K and OB is equal to K (1 +i').! We see, therefore, that all points on the line AB drawn in the manner described represent optional income-streams of equal present values. The line A'B', parallel to AB, drawn somewhat more remote from the origin 0, will in like manner represent the assemblage (or" locus") of all points whIch have a present value equal to OA' larger than OA.2 §4 We see at once that to select the point, among the entire swarm of points, which has the maximum present value,we need simply find that point which will be on a line parallel to AB and removed as far as possible from the origin O. Evi dently such a line is ab, tangent at t to the boundary line p'tR.

It is evident, therefore, that t is the point which possesses the maximum present value out of the entire mass. If the rate of interest rises, the slope of the line ab will be steeper and the point of tangency t will shift toward the right. In other words, the option now chosen will be one which has a larger y' but a smaller y"; that is, a larger income for the presen~ year and a smaller one for next year. On the other hand, if the rate of interest falls, the slope of the line ab will be more nearly horizontal and the point of tangency will rise, making y" larger and y' smaller. Not only is it true that t is the point at which the present value of the income is a maximuln, but it is also true that at this point the "marginal rate of return on sacrifice" will he equal to the rate of interest. We have seen that the slope of 1 See preceding footnote. 2 That .A' B' will be parallel to .AR is evident from the rule for con~ struction, for OA' must equal a constant X', and OB' must equal X' .(1 +i' ), therefore it is evident that, comparing the similar construc tion for .tiB, O.B 1 ., DB' .O.A = +~ =OA'· Consequently A'B' is parallel to AB.

406 APPENDIX TO CHAPTER VIII the line ab represents the ratio of exchange between next year's and this year's income, namely 1 +i'. In like manner the slope, at any point, of the boundary line p'tR represents the ratio of the return on sacrifice, or 1 + r'. This may be seen clearly from Fig. 28, where a slight variation from t" to t' produces in y' a small increase ktf, but in y" a small decrease kt". kt' may be designated by oy', and kt" by - oy", and we y/l o a-.. ..... __.......-..-.~~------""'!I'"--.y J FIG. 28. may state that 8y' represents a slight increase in this year's income, and - By" the consequent slight decrease in next year's -oy"income. The ratio of these two, namely ~,is what was called the marginal ratio of return to sacrifice, or 1 +r'; that is, -8y"BY'=l+r'. Returning to Fig. 27, it is evident that at the point of tangency t the slope of the straight line ab will be identical with that of the curve p'tR. In other words, l+i'=l+r'.

Whence it follows that i' = r'. We see, then, from the diagram: (1) that the point t for a SECOND APPROXIMATION 407 particular individual, with a particular rate of interest, is de terminate, and (2) that the point t is· the one which corresponds with the choice of maximum present value, where r' = i'. § 5 We shall now proceed to the consideration of the case of three years instead of two. .A. geometrical representation may still be used, by employing three dimensions and three mu tually perpendicular axes, OY', OY", OY'''. Any point P will indicate a possible income-stream for the three years; for its coordinates y', y", y'" may be taken to indicate the income installinents for the first, second, and third years respectively. Representing the various options by various points, we have a mass of points occupying three dimensions, like a swarm of bees, and we wish to select from this series of points that one which has the maximum present value. It is evident that we need not consider as eligible every point in the swarm, but need only consider its boundary, or outside surface. For, if any point P be joined to the origin 0 and prolonged beyond P to the remotest point P' in the mass, it is evident that P', hav ing all of its three coordinates larger than the coordinates of P, will have necessarily a larger present value. Therefore, the only point on the line OP which needs to be considered is P' farthest from the origin 0, or on the surface bounding the swarm.

Having restricted ourselves, therefore, to the bounding surface as including the effective range of choice, we next ask, what point on this surface has the maximum present value. To answer this question we observe that the assemblage or locus of all points or options which have a given present value V is found by drawing a plane cutting the three axes OY', OY", and 0 Y"'. The expression for the present value of the income stream is evidently -, y" y'" V - y + 1 + i' + (1 +i') (1 + i'~ + constant terms. If we transpose· "constant terIns" to the left, and remember that V is for the moment regarded as itself constant, we may call the·entil'e left-hand member a constant K, and have the equation y" y'"K-y'+ +. ·- 1 + i' (1 +i') (1 + iff) 408 APPENDIX TO CHAPTER VIII This is, in analytical geometry, the equation of a plane which cuts the Y' axis at a distance K from the origin, and cuts the Y" axis at a distance K(1 + i') from the origin, and the y'" axis at a distance K(l + i') (1 + iff) from the origin.

Similar considerations will show, just as in the case of the previous representation in two dimensions, that the farther the plane is from the origin 0, the larger the present value of the choices represented by points in this plane. Our problem, therefore, consists merely in finding that point on the bound ing surface which is also on the plane farthest from 0 among the parallel planes just drawn. This is evidently the point of tangency, and may be called t as before. It is also clear that a change in the rates of interest i' or iff will change the slope of the tangent plane, and therefore the point of tangency t. The algebraic interpretation of this case will be similar to the algebraic expression already given for two variables. When we proceed to consider four or more years instead of simply two or three years, the geometrical representation fails us, since the mind has difficulty in picturing spaces of 4, 5, ... and m dimensions.

§ 6 In order to show how the new equations which have just been expressed enter into the determination of the problem of interest, we construct Fig. 29, applying to the case only two unknown quantities y' and y". The incomes for the third and succeeding years are regarded for the moment as fixed. The diagram refers to a particular individual, and shows (1) the curve WPZ, giving the effective range of choice among different options open to him, and (2) a series of curves for total util ity or desirability, as explained in the Appendix to the pre ceding chapter. The line AB is drawn at a slope equal to 1 +i and tangent to the curve WZ at the point P. This line will be tangent to one of the family of desirability curves at some point Q. P represents, out of all the options, the par ticular income-stream chosen by the individual. This income stream P is, of course, as yet unmodified by borrowing and lending or buying and selling. The point Q represents the income-stream as finally thus modified. The coordinates of Pare y' and y", and the coordinates of Q are y' + x', y" +x", where x' and x" are the (algebraic) additions to the income y', SECOND APPROXIMATION' 409 y" by borrowing and lending or buying and selling. P andQ thus represent graphically the double choice explained in Chap~ ter VIII. We saw there that the individual first chooses, among the various eligible income-streams of different present values, that which is of maximum present value. WZ now represents the series of eligible income-streams, and P the income-stre,am of 'maximum present value. Also, we saw that after the indi vidual had chosen this income-stream he modified it by select ing another incolne-stream of the same present value but of maximum desirability. Q now represents this final choice.

It is worth our while in passing to emphasize that the indi vidual would not follow out this program unless the final step B· o 1020304050 69'10 z y' FIG. 29. A 70 60 50 40 80 20 ~-'-10 of modifying his income-stream by exchange were open to him,. For, if he were shut off from exchange (i.e. compelled to accept P, unmodified to Q), the income-stream of maximum present value (P) would not necessarily be that of maximum desira bility. In this case the maximum desirability would evidently be found at S, the point of tangency between WZ and a curve 410 APPENDIX TO CHAPTER VIII (not drawn) of desirability, andS would be chosen instead oIP. The choice of P is made only if there is freedom to replace it immediately by Q of higher desirability although of the same present value. The only difference between this deterlnination of the point Q and that shown in the Appendix to Chapter VII, where we assumed a fixed or rigid income-stream, is that there the point P was assumed as a fixed point, whereas here it is considered as a point of tangency to a fixed curve WZ. In the previous chapter, if the rate of interest changed, the line AB revolved about the point P. Under our new and more general hypoth esis, if the rate of interest changes, the line AB rolls upon the curve WZ. If the range of choice is reduced and the curve WZ is thereby restricted to smaller dimensions, the difference between the two cases is diminished, until, as a limiting case, we may suppose the curve WZ to shrink into a point, when the range of choice disappears entirely and the present diagram reverts to the one used in the previous Appendix.

If now we consider the case of three unknown quantities y', y", y''', it is only necessary to introduce three dimensions, replace the curve WZ by a surface, the line AB by a plane, and the curves of desirability by successive surfaces in concen tric layers, as shown in the Appendix to Chapter VII. The plane is now drawn tangent to the surface representing the effective range of choice, and the point of tangency P repre sents the income-stream chosen among all those eligible, while the point upon this plane Q, at which the plane is tangent to one of the desirability surfaces, represents the income-stream as finally modified by exchange. The previous discussion applies to one individual only. When we pass from the individual to society, we can no longer consider the plane (or in two dimensions, the line) as fixed in inclination. The problem of determining its inclination is the problem of determining the rate of interest. This is equiva lent to determining the inclination at which a series of parallel planes (or lines), for different individuals, must be passed, each tangent to its own surface (or curve) WZ, and such that the center of gravity of all the Q's coincides with that of all the P's.

This condition will evidently make the algebraic sum of all the x' 's zero, and likewise of the x" 's, etc. In other words, it will make the sums lent equal to those borrowed.

SECOND APPROXIMATION 411 This determination may be mentally represented by consid ering the set of parallel planes (or lines) to be first placed in any arbitrary inclination, corresponding .to an arbitrarily assumed rate of interest for each year. Each of these planes (or lines) will have a P and a Q; but unless the centers of gravity of the P's and of the Q's happen to coincide, the algebraic sum of the x"s, x'''s, etc., will not be zero, that is, the assumed rates of interest will not clear the market. We therefore now conceive the set of planes (or lines) to roll on their respective boundary surfaces (or curves) WZ, and to roll in unison, that is, so that they may be always mutually parallel. When such a position is found that the center of gravity of the P's coincides with that of the Q's, the market is cleared and the inclination of the planes (or lines) then found will represent the rate of interest. The rolling process here conceived simply visualizes the process given in Chapter VII, § 7, of finding tentatively the rate of interest which will clear the market.

We see, then, from our diagrams, how the different influences cooperate to determine the rate 0,£ interest, as represented by the common slope of the parallel planes (or lines). These planes (or lines) have the same slope as the two· curved sur faces (or lines) to which each is tangent. There is truth, therefore, both in the subjective and objective theory of interest. That the rate of interest is equal to the subjective rates of preference is indicated by the tangency of the plane (or line) to the desirability surfaces (or curves); that it is equal to the rate of return on sacrifice is indicated by its tangency to the surface (or curve) of effective range of choice. These two equalities are not incompatible, as has too often been assumed. Interest is determined partly by objective or tech nical factors which supply the range of opportunity (the boundary surfaces or curves); partly by subjective factors which determine individually the choice (the desirability curves).

§7 We have now seen how, on the simple hypothesis that the income-stream, or the group of optional income-streams, are foreknown, the problem of the rate of interest may be repre sented and solved, both algebraically and geometrically. We found that two of the interestdetermining conditions were based 412 APPENDIX TO CHAPTER VIII on the principle of finding a maximum. One of these two con ditions was that the income-stream selected should have the max imum present value,. the other was that this choice should be modified by exchange so as to secure the rnaximum desirability. We shall now proceed to show that these two conditions may be united into one, namely, that .both of these choices tend simply to secure the one end of maximum present desirability. It is true that maximum present value and maximum present de sirability are not interchangeable concepts, and we have seen that if an individual is, for any reason, not free to borrow or lend, his choice of income-streams will be determined in a dif ferent manner; the point of maximum desirability under these circumstances will have nothing to do with the straight line PQ, but will be at the point at which the curve WPZ of effec tive choice is tangent to a utility curve. But, given the freedom to interchange parts of the income-stream at the market rate of interest, the individual will, under these circumstances, gain the maximum desirability by first seeking that use of his capital which has the maximum present value, and then modifying the income thus obtained by the loan or sale market. This sub...

serviency of the principle of maximum present value to the principle of maximuln desirability was made evident in Chapter VIII. It becomes very clear geometrically. In Fig. 29 the individual is free to select any point on the line PQ, and to place this line at any distance from the origin, provided it passes through one of the swarm of points repre senting his total range of choice, and provided also that its slope always accords with the market rate of interest. It has already been made clear that, wherever he places the line, the point upon it of maximum desirability is Q, where it touches a desirability curve. It only remains to show that if the line were drawn, not through P, but through a different point in the swarm, while keeping the same slope, Q would be a point of lower desirability. This is evident, for if PQ is not drawn through P, its only other possible position must be parallel to that position, but not so far from the origin O. But in that case, Q would evidently be on a curve of lower desirability, since the family of such curves ascends as we recede from O.

§ 8 We may now summarize both the geometrical and algebraic determinations of the rate of interest: SECOND APPROXIMATION 413 1. We have the condition that, for each individual, the effective range of choice among income-streams is limited to a specific set of options, owing to the technical limitations of his capital, etc. Geometrically, this condition is represented by the surface (or curve) WZ. Algebraically, this condition is represented by n equations of the. type ~ (y', y", etc..) = 0, each relating to an individual. 2. We have the condition that the rate of preference for each individual for each year, as. estimated in the present, depends upon the future income-stream as indicated by its ann ual installinents. Geometrically, this condition is represented by the family of desirability surfaces (or curves). The slopes at each point correspond to the rates of preference, and the coordinates of the point correspond to the installments of the income-stream.

The fact that the slopes depend upon the position of that point represents the fact that the rate of preference depends on the income-stream. Algebraically, this condition is represented by n(m-1) equations of the type !=F(y'+x', y"+x", etc.), each relating to a single individual and a pair of consecutive years. 3. We have the condition that the market rate. of prefer ence of each individual is equal to the rate of interest and to each other; or, equivalently, that his total desirability is a maximum. Geometrically, this condition is represented by the fact that at Qthe inclination of the plane (or line) is the same as that of the desirability surface (or curve) at that point, - in short, th~t they are there tangentand further that the directions of the desirability curves at all the Q's are parallel, - in short, that the planes (or lines) drawn tangent to them are parallel to each other.

Algebraically, these conditions are represented by m-1 continuous equations of the type Ji=A=···=f",=i, 414 APPENDIX TO CHAPTER VIII each relating to two successive years, making n(m -1) equa tions in all. 4. We have the condition that out of the effective range of choice, that particular one of the income-streams is selected which possesses the maximum present value, Of, equivalently, that one is selected such that, when it is compared with its nearest neighbor, the marginal rate of return on sacrifice is equal to the rate of interest. Geometrically, this condition is represented by the fact that the plane (or straight line) AB is tangent to WZ for each individual. Algebraically, this condition is represented by 2 n(m -1) equations, consisting of n(m -1) pairs, of which the following is the type 1 +i = 1 +r = ",(y', y", etc.), there being one such double equation for each individual for each pair of successive years.

5. We have the condition that the sum added (by borrowing and lending, or buying and selling) in any year to the income of one individual is equal to that taken from others, or, equiv alently, that the algebraic sum of such lllodifications is zero. Geometrically, this condition is represented by the fact that, for each individual, the center of gravity of the Q's coincides with that of the P's. Algebraically, this condition is represented by m equations of the type one for each year. 6. We have the condition that for each individual the posi tive and negative modifications of his income-stream in dif ferent years mutually offset each other in present value, or, in more common language, what is borrowed is repayable with interest. Geometrically, this condition is represented by the fact that for each individual, P and Q lie in the same plane (or straight line) AB. Algebraically, this condition is represented by n equations of the type x" x'"

x' + 1 + i' + (1 +i') (1 + i") + etc. =0, one for each individual.

SECOND APPROXIMATION 415 Counting up the total number of equations thus indicated, we find:- ~ For the 1st condition, n equations, 2d condition, n(m-l) equations, 3d condition, n(m -1) equations, 4th condition, 2 n(m - 1) equations, 5th condition, m equations, 6th condition, n equations, making a total of 4mn + m - 2 n equations, of which, for reasons given in the Appendix to Chapter VII, only 4mn + m - 2.n -1 are independent equations. The number of unknowns is as follows:number of y's is mn, number of x's is mn, number of r's is n(m-1), number off's is n(m-1), number of i's is m -1, the total of which is also 4mn +m-2n -1. Hence the prob lem of interest is determinate. So much space has been devoted to stating these six condi tions in mathematical form, because, to those conversant with the mathematical tongue, the algebraic statement will ·show more definitely and clearly than is otherwise possible the determinateness of the problem, owing to the equality between the number of equations and the number of unknowns; while the geometrical method enables them to form a mental picture, clearer than would otherwise be possible, of the various factors at work, and especially of the manner in which the objective or "technical" conditions, as represented by WZ, cooperate with the subjective conditions which influence the rate of interest. It was, in fact, only through the geometrical repre· sentation that the writer was first enabled to grasp the signifi cance of,the "effective range of choice" in its general bearings.

If the role of the curve WZ is grasped, the most difficult part of the theory of interest is mastered.

APPENDIX TO CHAPTER XI THIRoD ApPROXIMATION § 1 (TO CR. 11, § 8) To attempt to formulate in mathematical language, in any useful manner, the complete laws determining the rate of in terest under the sway of chance, would be like attempting to express the complete laws which determine the path of a pro jectile when affected by random gusts of wind. Such formulre would need to be either too general or too empirical to be of much value. In science, the most useful formuloo are those which apply to the simplest cases. For instance, in the study of projectiles, the formula of most importance is that which ap plies to the path of a projectile in ideal vacuum. Next come the formulre which apply to a projectile in still air. It is seldom that the luathematician attempts to go beyond this, and take into account the effect of wind currents; and if he does so, he still falls short of actual conditions, by assuming the wind to be constant in direction and velocity. The "truth is that science always stops short of the final approximation necessary to reach reality. This is due to the nature of science itself, which is a study of what would happen under assumed condi tions, and only an approximate application of what does hap pen under actual conditions.! The consequence is that, in order to reach the final goal of real conditions, we usually cut the Gordian knots which remain; and for such summary solu tion, especially when the solution is general instead of numer ical, ordinary language is usually better than mathematical formulae. Accordingly, in treatises on projectiles we do not find any attempt to state their trajectories in general formulre which include the effects of gusts of wind. Still less is there any attempt to construct a formula for the path of a boomer ang or of a feather thrown out of a window.

To apply the analogy to the problem in hand, we have already stated the laws determining interest under the simpler 1 See the ,vriter's "Economics as a Science," in 8cienee, Aug. 31, BlOB. 416 THIRD APPROXIMATION 417 conditions, - first, when it was assumed that the income streams of individuals were both certain and fixed, and, secondly, when it was assumed that the income-streams were certain, but flexible. When we introduce the element ·of un certainty, our formulre cease to have the characteristics of simple clarifying shorthand which justify their use, and take on the characteristics of what Marshall calls the "lengthy translations of political economy into mathematics." While, therefore, it is not difficult to make these translations, they add little or nothing to our understanding of the problem. Inasmuch as it is our aim. to employ mathematics only 'when they add sOlll.ething which cannot be conveyed without their use, we refrain from wearying· the reader with cumbersome equations.

2E APPENDIX TO CIM.PTER XIV STA.TISTICAL DATA §1 The writer has found so much difficulty in securing a long series of yearly averages for rates of interest that the results are here presented in the hope that they may be of use to others. YEARLY AVERAGE RATES OF INTERESTl LONDON BERLIN PARIS NEW YORK CALCUTTA TOKYO SHA.NGHAI --- --- -~;---I---:----=----J----t----I---1824 .5 4.0 1825 .9 4.0 ..... 1826 .S S.O . 1827 ~~\ 4.5 1828~4.0 1829 3.4 4.0 1830 2.8 4.0 . . 1831 3.7 4.0 . . 1832 3.1 4.0 1833 2.7 4.0 1834 3.4 4.0 1835 3.7 4.0 1836 4.2 4.4 . . . . . . 1837 4.5 5.0 . . 1838 3.0 4.1 1839 5.1 5.1 1840 5.0 5.1 . . . . . . 1841 4.9 5.0 . 1842 3.314~3 . . 1843 2.2 4.0 . .; 1 The London, Berlin and Paris market rates are on first class merchants' bills. The figures for 1824-68 are from the evidence of D. B. Chapman before the Committee on the Bank Act, 1857, Sess. 2, X, pt. I, p. 463 (also reprinted in Hunt's Merchants' Magazine, Vol. 41 (1859), p.95). The remaining figures are compiled from the Economist. For those for 1884-94, the writer is indebted to Professor F. M. Taylor of Michigan University, who had collected them from the Economist for a different purpose. The Bank of England rates for 1824-43 are reduced from" Burdett's Official Intelligencer" (1894), p. 1771. The remaining ones for England, Germany, and France are reduced from those given in the Report of the Royal Commission on Depression of Trade, 1886, p. 373, and the Economist. They represent the bank "minimum." The rates 1896-1903 are from A. Sauerbeck's tables, Journal Royal Statistical Society, Vol. LXVII, Part I, p. 89. The New York rates are taken, the 418 STATISTICAL DATA YEARLY AVERAGE RATES OF INTEREST-Continued 419 LONDON BERLIN PARIS.NEW YORK CA.LCUTTA TOKYO SHANGHAI --- --- -~-·I-_:-----.,..--I----I--......,.--I---1844 2.1 2.52 • 4.3 4.0 1845 3.0 2.7 4.4 4.0 1846 3.8 3.3 4.7 4.0 1847 5.9 5.2 4.8 4.9 1848 3.2 3.7 4.7 4.0 1849-2.3 2.9 4.0 4.0 4.5 7.8 1850 2.2 2.5 4.0 4.0 4.8 7.2 1851 _3.1 3.0 4.0 4.0 5.9 8.3 1852 1.9 2.2 4.0 3.2 5.1 7.3 1853 3.7 2.7 4.2 3.2 6.9 10.1 1854 4.9 2.1 4.3 4.3 7.7 12.5 3.9 first two columns, from· a table by E. B. Elliott (afterward government actuary) in the (New York) Banker's Magazine, 1874. The quotations given as "60 days" apparently included single name paper. The third column to 1890 is compiled from a diagram of highest and lowest monthly rates prepared at Yale College by Mr. G. P. Robbins of the class of 1891, and has been completed from the Financial Review, by averaging the highest and lowest weekly rates. It has been found impossible to extend the New York table back beyond 1849, as the, rates are not systemati cally reported. The Calcutta rates are the minimum of the Bank of Bengal and have been kindly furnished by Messrs. Place, Siddons, and Gough, brokers, of Calcutta. The market rates of Tokyo are averages of the highest and lowest rates of each year, furnished by Mr. lchi Hara of the Bank of Japan, Tokyo. 'rhe bank rates are for the Tokyo and Yokoharna Cooperative Bank and were translated by Mr. Sakata, student at Yale, from a history of Japan by Zenshiro Tsuboya. The continuation of the table after 1895 has been supplied by Mr. Hitomi, one of my students, and is based upon: Financial Report of the De partment of the Treasury, Reports Tokyo Economic Magazine Pub,lishing Co., Reports Tokyo Bankers' Association, and Reports of Department of Agriculture and Commerce. (The last three sources are in Japanese only.) The tables for Shanghai have been procured through the kindness of Prof. F. W. Williams of the department of' Oriental History of Yale College, who obtained them from Mr. J. F.

Seaman of Shanghai. The first column contains the rates ruling in the ~ative market, and the second, those of the Hong Kong. and Shanghai bank (under English control) on overdrawn current accounts, a species of demand loans and the ordinary form of lending in Shanghai. Mr. Seaman was told that the market rates'cannot be extended back beyond 1885, as the books of the Chinese banks for previous years are burnt. ~ This rate is only from September, when the operation of the Bank Act began. Previous to this the custom of the bank was to have a uni orm rate for all loans.

420 THE RATE OF INTEREST YEARLY AVERAGE RATES OF· INTEREST - Concluded LONDON BERLIN PARIS NEW YORK CALCUTTA TOKYO SHANGHAI ___ --- ---,..--1---,---..,.---1---- --,..---1--..,..--1855 4.7 2.9 1856 5.9 6.1 1857 7.1 6.7 1858 3.1 3.2 1859 2.5 3.7 1860 4.1 4.2 1861 5.5 5.3 3.0 1862 2.4 2.5 3.0 1863 4.3 4.4 3.5 1864 7.4 7.4 5.1 1865 4.6 4.8 4.6 1866 6.7 6.9 6.2 1867 2.3 2.6 2.9 1868 1.8 2.1 2.5 1869 3.0 3.2 3.2 1870 3.1 3.1 4.5 1871 2.7 2.9 3.8 1872 3.8 4.1 4.0 1873 4.5 4.8 4.5 1874 3.5 3.7 3.3 1875 3.0 3.2 3.7 1876 2.2 2.6 3.1 1877 2.3 2.9 3.3 1878 3.5 3.8 3.4 1879 1.8 2.5 2.7 1880 2.2 2.8 3.1 1881 2.9 3.5 3.4 1882 3.4 4.1 3.9 1883 3.0 3.6 3.1 1884 2.6 3.0 2.9 1885 2.0 2.9 2.9 1886 2.1 3.0 2.1 1887 2.4 3.3 2.3 1888 2.4 3.3 2.1 1889 2.7 3.6 2.7 1890 3.7 4.5 3.7 1891 2.5 3.3 3.0 1892 '1.5 2.5 1.8 1893 2.1 3.1 3.2 1894 1.0 2.1 1.7 1895 .8 2.0 2.0 1800 1.4 2.5 3.0 1897 1.8 2.6 3.1 1898 2.6 3.3 3.6 1899 3.3 3.8 4.5 1900 3.7 4.0 4.4 1901 ,3.2 3.8 3.1 1902 ~3.0 3.3 2.2 1903 .2 3.8 3.0 1904 3.3 3.1 1905 2.6 3.0 2.9 1906 4.0 ,,4:Z 4.0 4.1 4.5 4.9 5.5 5.8 6.1 4.5 3.7 4.2 3.5 4.0 3.6 4.0 5.5 4.0 3.8 4.1-. 4.6 B.3 6.5 5.0 3.7 6.2 3.7 4.0 2.7 4.0 2.5 4.1 2.5 4.8 4.0 4.2 5.7 4.3 4.2 5.1 5.0 5.0 5.2 4.4 4.0 4.3 4.7 3.2 4.0 4.1 2.3 3.4 4.4 1.8 2.3 4.3 2.0 2.2 3.7 2.2 2.6 4.2 2.5 2.8 4.4 3.7 3.9 4.5 3.4 3.8 4.1 2.6 3.1 4.0 2.4 3.0 4.1 2.5 3.0 3.3 2.2 3.0 3.4 2.4 3.0 3.3 2.8 3.3 3.7 2.6 3.1 4.5 2.6 3.0 3.9 2.6 3.0 3.2 1.8 2.7 4.1 2.2 2.5 3.1 1.8 2.5 3.1 1.6 2.1 3.7 1.8 2.0 3.8 1.8 2.0 4.3 2.1 2.2 5.0 3.0 3.1 5.3 3.2 3.3 4.1 2.5 3.0 3.3 2.4 3.0 3.8 2.8 3.0 4.2 2.2 3.0 3.8 2.1 3.0 5.1 2.7 3.0 6.6 9.3 7.0 9.9 6.9 10.4 3.8 6.7 5.0 7.2 6.1 8.4 5.4 9.0 5.6 6.8 5.0 6.7 7.2 9.3 6.1 10.2 5.9 7.8 5.9 8.7 5.8 8.8 6.0 10.8 5.1' 8.1 4.5 6.0 5.6 9.0 6.3 9.8 3.9 6.4 7.7 6.6 5.4 5.8 8.0 8.2 6.3 7.2 7.3.

9.1 7.2 6.1 8.0 10.3 6.0 5.5 5.2 5.2 4.8 5.0 5.2 5.2 5.7 5.5 5.2 4.1 4.7 5.7 4.9 4.8 6.0 5.7 4.3 7.1 3.4 3.8 5.8 3.4 3.8 4.2 4.4 4.4 4.9 5.5 4.2 4.3 5.7 9.7 6.5 7.0 6.1 4.9 4.2 4.2 5.1 5.5 8.7 6.9 9.1 5.1 5.8 6.0 5.7 4.7 5.0 3.9 6.2 5.7 6.8 8.4 5.3 6.3 4.6 5.3 6.6 6.8 6.4 5.4 6.0 5.6 5.5 7.0 5.8 3.1 3.5 4.9 5.4 4.3 •b; :: :: ~ ~~ . ~~ , ~ ~ . 9~ •.••t5.. .. •• ~o . "" .o ..•• 15. 18. •• 18. 18. •• 18. 18. •• 14. 18. •• 14. 18. •. 14. 14. .. 12. 14. .. 12. 14. .. 12. 15. .. 10. 13. .. 11. 15. .. 11. 16. .• 13. 17. .. 14. 17. •. 10. 17. .. 7.9 11. •. 12. 16. .. 13. 11. 4.6 8.8 9.2 6.1 9.0 8.8 6.5 10. 9.7 6.0 10. 10. 6.2 11. 11. 7.2 9.4 9.4 3.5 8.3 8.4 5.0 7.8 7.8 6.9 9.3 9.4 3.8 9.6 9.6 2.8 11.0 9.3 .. 11.4 9.9 .. 12.3 11.4 .• 10.4 8.8 .• 12.1 10.8 .. 13.1 11.9 .. 12.110.4 .. 10.7 •. .. 10.8 .... 13. 11. 12. 12. 11. 12. 10. 10. 10. 9. 9.5 9.5 9. 9. 8.5 8. 8. 8. 8. 7.5 7. 7. 7. 7. 7. 7. 7.

7. 7. 6.5 STATISTICAL DATA 421 All the rates in the fOfegoing table a~e entered as rates of "'interest," though the rates· for the Banks of England, Ger many, and France are rates of discount. Although the two are not quite equivalent, for the purposes of the foregoing work the distinction between them is, unnecessary, because, in a continuous series, -the' error, if any, affects all items nearly alike and thus cancels itself out in thecomparisons~ Had it been necessary, some of the tables could have been extended backward. Thus the Bank of England rate could be given as far back as 1696, but it was too inflexible to be of use. The Berlin and Paris. bank notes could also be ex tended a,nd the Paris market rate could be given back to 1~61 (except for· 1810 and 1811) from data in the El?onom'ist. Many of the sources from which the table has heen drawn also contain other information such as the rates for other money centers than those named, the weekly or monthly rates, the variation with the seasons, etc. § 2 Of sources not mentioned in the above note, the chief which the writer has encountered are: - .

Eleventh Census of the United States, Bulletin 71 (on re~l estate mortgages, 1880-89). . This is probably the most elaborate series of interest averages ever constructed. · Twelfth Census of the United States, Special Reports, wealth, debt, and taxation, pp. 143, 147, 394. Rates of interest on public debts. c, Commercial Valuation of Rail way Operating Property in the United States, 1904," United States Census Bulletin 21 (1905). Gives rates of return on railway securltiesto investor for 1904. Reports of the" Secretary of the Treasury. ".Reports of the Comptroller of Currency. The last two references contain statistics -of rates of interest realized on some United States Government bo:p.ds. R. A. Bayley, "National Loans of the United States" (Government Printing Office, Washington, 1882). Gives rates of interest and price of issue of all United States loans from July 4, 1776, to June'SO, 1880.. . .

Report of the New England Mutual Life Insurance Company, Boston, 1890. Gives rates realized by twenty representative in8~rance. companies for 1869-88, an~ for Massachusetts savings banks for 1877-89,a,nd bank diTi..

422 THE RATE OF INTEREST dends in Boston, New York and Philadelphia. The rates realized by the insurance companies for the twenty years, 1869-88, inclusive, were 6.0, 5.9, 6.1, 6.2, 6.5, 6.2, 6.5, 6.1, 5.6, 5.1, 5.0, 4.8, 4.8, 5.1, 5.1, 4.7,4.7, 4.9, 4.7, 4.6, respectively. These represent (if the writer mistakes not) the average rates earned on the par value of investments of all ages, some old, some new, some terminable soon and others having many years to run. For this reason they are of little or no use for the purposes of Chapter XIV. Lester W. Zartman, The Investments of Life insurance Oompanies, New York (Henry Holt & Co.), 1906. Gives earning rate of real estate, mortgage loans and bonds and stocks of the principal life insurance companies of the United States 1860-1904, and similar data for companies of England, Canada, Australia and other countries. W. B. Hedge, "On the Rates of Interest for the Use of Money in Ancient and Modern Times. Part I." Association Magazine, Vol. 6 (1857), pp. 301-333.

H. W. Farnam, "Some Effects of Falling Prices," Yale Review, August, 1895. F. M. Taylor, "Do we want an Elastic Currency?" Political Science Quarterly, March, 1896, pp. 133-157. Gives diagram showing the relation of surplus reserves and rates of dis count; also seasonal variation of rate of discount. Carl C. Plehn, "Notes concerning the Rates of Interest in California," Publications .A~nerican Statistical .Assoclation, Vol. VI (September 1899), p. 350. R. M. Breckenridge, "Discount Rates in the United States," Political Science Quarterly, Vol. XIII (March, 1898), p. 119. R. H. Inglis Palgrave, Analysis of the Transactions of the Bank of Eng land (London, 1874). Gives rates, 1844-72, and seasonal variation, 1844-56 and 1857-72. Shows dependence of rate on ratio of reserve to liabilities. R. H. Inglis Palgrave, Bank Rate and the Money Market in England, France, Germany, Holland and Belgium, 1844-1900. New York (Dutton), 1903.

Gives bank and market rates, with special reference to variability in England, France, Germany, Holland and Belgium. W. Stanley Jevons, Investigations in Ourrency and Finance (London, 1884). Contains diagram for prices of consols and 3 per cent. stock from 1731, and minimum rate of interest in London from 1824 ; also monthly varia tion in rate of interest, p. 10. The diagram for the price of consols shows that during the middle and first half of the eighteenth century the interest realized was almost as low as in the present generation. It is interesting to note that this was a period of falling prices. Robert Giffen, Essays in Finance, second series (London, 1886), p. 37. Seasonal variations of interest in connection with bank reserves, etc.

STATISTICAL DATA 423 Arthur Crump, English Manual of Banking (4th ed., London, 1879), pp. 141-144. Gives Bank of England rates for 1694-1876. George Clare, Money·Market Primer, 2d ed. (London, 1905). Diagrams for seasonal variations of interest, bank reserves, etc. M. G. Mullhall, Dictionary of Statistics (London, 1892), .pp. 76, 607. Gives rates for countries of Europe by five and ten year periods sinee 1850. William Farr, " On the Valuation of Railroads, Telegraphs," etc., JOU'f''»IIJ of the Royal Statistical Society, September, 1876, pp.464-530. . Rates of Discount and Exchange, Banks of England, FraU~J··.~Prussia, Vienna,1851-1885. Final Report Gold and Silver~Q~mission, Par liamentary Blue Book, 1888, appendix, p. 207. Commercial and Financial Statistics of·~$ish India. (Government Printing Office, Calcutta.) MonthlyDise0nnt,:~k,ef'Bengal, from 1861,and average quotations of ~ent8ec'Urities held in London.

Tooke, History of Prices, and Tooke and Newmarch, History of Prices from 1793 to 185(J. J. Liegeois, Essai sur l'histoire et la lI~glslation de l'usure (Paris, 1863). Saugrain (Gaston), La ba'isse du taux de l'inte'ret-causes et consequences (Paris, Larose, 1890). Boucher, P. B., :Histoire de l'usure ches 1es Egyptiens, les Grecs, 1(38 Romains, nos au cetres et les Chinois (Paris, 1806, 1819). Alph. Courtois, fils, Histoire des Banques en France (Paris, 1881). Gives rate of interest at the Bank of France, 1800. Viscomte G. D' Avenel, Histoireeconomique de la propriete, des Salaires, des Denrees et de tous les Prix en general depuit~ l'an 1200 jusqu'en l'an 1800 (Paris, 1894), vol. II, p. 882. This work contains also tables of the purchasing power of money. Dictionaire des Finances, Article "Inter@t." Gives rates at which France has borrowed. Jahrbucher fur Nationa.lQkonomie und 8tatistik, ]'ebruary, 1896, pp.

282-83. Gives bank and market rates for London, Paris, Berlin, Amsterdam;, Brussels, Vienna and St. Petersburg, 1841-80 by decades, and 1881-95 by years. HandtOorterbuch der Btaatswissenschaften, Articles "Banken" and H Zinsfuss." Gives rates for Bank of Prussia and Germany, 1847-89; also for Bank of Austria, 1878-89; Switzerland, 1883-88. Adolf Soetbeer, Materiallen zur Wiihrungsjrage (Berlin, 1886), p. 78. Covers 1851-85for Banks of England, France and Germany and market rates of Hamburg and Vienna.

424 THE RATE OF INTEREST Gustav Schmoller, Grundriss der Allgemeinen Volkswirtschajtslehre, Leipsic (Duncker und Humblot), 1904, PP. 206-208. Gives resume of course of interest rates from ancient to modern times. Also gives interest rates, London, Paris, Berlin, Amsterdam, Vienna, New 'York, and St. Petersburg. Billeter, Geschichte des Zinsfusses im Griechisch-Romischen .i1ltertum bis auf Justinian (Leipsic, February, 1898). "According to the recent researches of Biileter, the normal rate of inter est on good security during the period of greatest prosperity in Athens was about 12 per cent.; while in Rome at the close of the Republic it had fallen to between 4 and 6 per cent. Starting in again during the early middle ages at a rate of 20 per cent. and 15 per cent., it gradually fell, until in the great financial centers of Holland towards the close of the eighteenth cen tury it reached a rate of between 2 per cent. and 3 per cent." -From Principles Of Economics, by Seligman, N.Y. (Longmans), 1905, p.404.

E. Laspeyres, Geschichte der 'Volkswirthschaftlichen Ansichten der Nieder lander, Leipsic, 1863. (Preisschriften der f. Jablonowskischen Gesell schaft, Bd. XI.) [Contains Zins oder Wucher; pp. 256 fi.] Austrian Government, Tabellen zur Wahrungsfrage (Vienna, 1892), pp. 204-206. (Second edition, 1896, and third edition, 1903, 4.) Covers rates since 1861 for banks of Italy, England, France, Germany, Austria, Belgium, and Holland, and market rates in Vienna since 1869. Wilhelm von Lucam, Die Oesterreichische Nationalbank wahrend der Dauer des dritten Privilegiums (Vienna, 1876). Gives rates for Bank of Austria, 1817-75. Theodor Hertzka, Wiihrung und Handel (Vienna, 1876). Gives the number of weeks each rate lasted for the Banks of England, France, Germany, and Austria during 1844-73. G. Winter, "Zur Geschichte des Zinsfusses in Mittelalter," Zeitschrift fur Social und Wirtschajtsgeschichte (Weimar), 1875, IV, 2; 1896, IV, 161.

W. J. Streuber, Der Zin!uss bei den Romern, eine historischantiquarische Abhandlung (Basel, 1857). J. Kahn, Geschichte des Zinju8ses in Deutschland seit 1815 und die Ursachen seiner Veranderung. (Stuttgart, J. A. Cotta, 1893.) M. Newmann, Geschichte des Wuchers in Deutschland bis sur Begrun dung der heutigen Zinsgesetze (1664). Halle, 1865. Sombart: Der Moderne Kapitalismus, I, 219. Gives the following table showing the rise in price of a rent of one mark (in rent purchase) in Frankfurt a. M.: in 1304, 14-15 l\larks; 1314-1315, 16-17 Marks; 1323-1327, 18 Marks; 1333, 19 Marks; 1358, 24: Marks. Rodbertus, "Ein Versuch, die Rohe des antiken Zinsfusses zu erkHi,ren." Jahrb. f. Nat. Oek., Bd. XLII (Jena, 1884). J. Conrad, Politische Oekonornie, Jena (Gustav Fischer), 1905, p. 175. Gives rates of discount 1871-1904 in London, Paris, Berlin, Vienna, and St. Petersburg.

STATISTICAL DATA 425 A. N. Kiaer, Om seddelbanker (Kristiania, 1877). Contains diagram of bank rates at Kristiania, Stockholm, and Kjoben havn, 185~76. J. P. Norton, BtatisticaZ Bt1,f,dies in the New York Money Market, New Haven (Tuttle, Morehouse, and Taylor), 1903. Financial and Economical Annual of Japan, Tokyo, Government Print ing Office. §3 The following tables of index numbers are appended in order that the reader may verify the periods of rising and falling prices which have been discussed in Chapter XIV and for the reason that many of the tables, notably those for In dia, Japan and Chi.na, have not been easily accessible to most readers. INDEX NUMBERS OF PRICES IN' SEVEN COUNTRIES 1 ENGLAND GERMANY FRANCE UNITED STATES INDIA. JAPAN CHINA 1824 105 1825 124 1826 108 1827 108 1828 97 ..... 1829 95 1830 97 ..... 1831 98 1832 93 1833 90 .1834 93 1835 96 1836 103 1837 101 1838 101 1839 110 1840 104 98 1841 102 98 1842 90 ... '.... 90 1843 85 84 1844 83 85 1845 89 88 1846 89 95 1847 94 95 1848 82 88 1849 77 83 1850 77 89 1851 79 100 99 1852 781 102 98 1853 95 114 105 1854 102 121 105 '1855 101 124 109 426 THE RATE OF INTEREST INDEX NUMBERS OF PRICES IN SEVEN COUNTRIES - Concluded ENGLAND GERMANY FRANCE UNITED INDIA JAPAN CHINASTATES --- --1856 · .. 101 123 · . · .. 112 · . · .. · .... · ...

1857 · .. 105 130 · .. 114 · .... · . · .. · . 1858 • 0. 91 114 · . o. 113 .... · ... · ... 1859 · .. 94 116 · .. . . 103 · .... . . · .. 1860 · .. 99 121 · .. 100 · .. · . · .. · .. 1861 · .. 98 118 100 94 o •• · . · ... · ... 1862 · .. 101 123 118 104 · ... · . · .. .0 1863 · .. 103 125 127 132 · .... · . · . · .. 1864 00. 105 129 129 172 · .... · .. · ... 1865 o •• 101 123 112 232 · .. ·. · . · . · ... :.I866 000 102 126 115 188 ·. · .. · .. · ... 1867 · .. 100 124 100 166 ... · .. · ... 1868 · .. 99 122 95 174 · . · .. · ... .0. 1869 · .. 98 123 97 152 ·.... · .... · . 1870 · .. 96 123 94 144 · .... · ... .0 1871 · .. 100 127 '94 136 O. ·. · .... · ... 1872 · .. 109 136 105 132 00 · .:. · ... ' . .. . -. 1873 · .. 111 138 103 129 100 104 · ... 1874 • • 0 102 136 94 130 98 104 100 1875 .0. 96 130 87 129 95 105 103 1876 · .. 95 128 85 123 99 102 111 1877 o. 94 128 82 114 121 105 101 1878 · .. 87 121 78 105 125 114 106 1879 000 83 117 76 95 119 145 111 1880 00. 88 122 79 105 112 160 105 1881 00. 85 121 76 108 99 175 110 1882 .0. 84 122 73 109 98 159 108 1883 · .. 82 122 73 107 96 130 103 1884 o •• 76 114 72 103 97 116 104 1885 • • 0 72 109 70 93 95 116 105 1886 .00 69 104 69 93 99 107 107 1887 · .. 68 102 71 94 101 109 105 1888 O •• 70 102 74 96 104 112 100 1889 O •• 72 106 80 98 107 116 105 1890 · .. 72 108 83 94 103 124 104 1891 · .. 72 1091 79 94 104 123 104 1892 · .. 68 106 · ... 89 115 124 108 1893 O •• 68 102 O. 00. 89 119 129 109 1894 · .. 63 92 · . · .. 81 · .... 132 · ..

1895 · .. 62 91 · . . . 79 · .... 145 · ... 1896 · .. 61 91 . ... 76 · .... 133 · . . . 1897 · .. 62 92 · .... 76 · .... 173 · .. 1898 · .. 64 93 · .... 78 · .... 159 · ... 1899 · .. 68 111 · .... 86 · . · .. 175 · ... 1900 · .. 75 110 · . · .. 92 · .... 165 · ... 1901 · .. 70 103 · .... 91 · .... 160 · . .. 1902 · .. 69 99 · .... 95 · .... 165 ·... 1903 · .. 69 104 · .... 96 · ... · .... · ... 1904 · .. 70 104 · .... 95 · .... ·.... ·.. 1905 · .. 72 107 · .... 98 · .... · .... · ... 1906 · .. 77 . . . . . · .... 103 · .... · .... ·...

STATISTICAL DATA 427 1 For England, the figures of prices are from J evons and Sau9"~ck. Those from Sauerbeck begin in 1852. They are taken from the AICtm.til 'Senate report on Wholesale Prices, 1893 (I, 247), and from· the Journal 01 the ROllal8tatistical 8ociety. Those from Jevons are from 1824 to 1852 inclusive, and are taken from his" Investigations in Currency and Fi nance." In order to· make the tables of Jevons andSauerbeck continu ous, Jevons's number for 1852 is called 78 (i.e., Sauerbeck's for that year) instead of 66, as given in the" Investigations," and all the other numbers are raised in the ratio of 78 to 65. J evons's figures are for forty commodities; Sauerbeck's are for forty-five. The German numbers are from Soetbeer, Heinz, and Conrad. Those for 1851-91 inclusive, are from Soetbeer, continued by Heinz, and given in the Aldrich report (I, 294) ; those for 1891-1906 inclusive, are from Conrad, as given in his Jahrbucher, 1894-1906, but are all magnified in the ratio of 109 to 98 in order to make the series continuous, since Heinz's figure for 1891 is 109, and Conrad's, 98. The statistics of Soetbeer and Heinz cover 114 commodities.

The French numbers are from the Aldrich report (I, 385) founded on the figures of the Oom'lnissionpermanente des 1Jaleurs. They cover only sixteen articles. The figures for the United States are those of Professor Falkner in the Aldrich report (I, 9, 13), the weighted averages (last method) being em ployed. They have been continued after 1891 by using the figures of tlle Bulletin of the United States Department of Labor, March, 1907, p. 260. The figures of this report have all been reduced ina fixed ratio in order to bring the initial figure for 1891 into coincidence with the figure, for that year, of the Aldrich report, 94. Those for India, Japan, and China are from the Japanese report of the Commission for investigation of monetary systems, 1895. The writer is under great obligations to Mr. lchi Hara, of Tokyo, for a copy of the re port, and to Mr. Sakata, of Yale University, for translating the tables.

That for India is an average of three tables which cover respectively twenty -one articles of export, sixteen articles of export priced at Calcutta and Bombay, and eight grains at Bombay. That for Japan is an average of three tables, of forty ...two articles at Tokyo, sixteen at Osaka, and thirty -one articles of export. The continuation of the table after 1896 has been supplied to me by Mr. Hitomi, one of my students, and is based on Reports of the Tokyo Eco nonlic Magazine Publishing Company. That for China is an average of three tables, of twenty inland commod ities, seventeen articles of export, and fifteen food-stuffs in Shanghai. The tables for India were based on official statistics, those for Japan on information from guilds and merchants, and those for China on the reports of the consuls of Japan and England (Mr. Jameson) in China. In the Japanese report the prices for Japan are reduced to a silver basis. As silver was at a premium up to 1885, it has been necessary in constructing the above table to reconvert into currency by applying the premium for 1873-86, viz., 4, 4, 3, 1, 3, 10, 32, 48, 70, 67, 26, 9, 5 per cent respectively.