A Consumption-Loan Model of Interest (Paul Samuelson)

An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money

Paul A. Samuelson

The Journal of Political Economy, Vol. 66, No. 6. (Dec., 1958), pp. 467-482.

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http://www.jstor.org Mon Dec 3 14:23:36 2007 THE JOURNAL OF POLITICAL ECONOMY

Volume LXVl DECEMBER 1958 Number 6

AN EXACT CONSUMPTION-LOAl\j MODICL OF IN'L'EREST TYITII OR IYITHOUT THE SOCIAL CONTRIVA1XCEOF IlONICY*

PAUL A. SAblUELSON hfassachusetts Institute of Technology

Y FIRST published paper1 has market interest rates which each man come of age, and at a time had to accept parametrically as given to M when the subjects it dealt with him. have come back into fashion. It de- Now I should like to give a complete veloped the equilibrium conditions for a general equilibrium solution to the de- rational consumer's lifetime consump- termination of the time-shape of inter- tion-saving pattern, a problem more est rates. This sounds easy, but actually recently given by Harrod the useful it is very hard, so hard that I shall have name of "hump saving" but which to make drastic simplifications in order Landry, Bohm-Bawerk, Fisher, and to arrive at exact results. For while others had touched on long before my Bohm and Fisher have given us the time.2 It dealt only with a single indi- essential insights into the pure theory vidual and did not discuss the mutual of interest, neither they nor other writers determination by all individuals of the seem to have grappled with the following tough problem: in order to define an *Research aid from the Ford Foundation is equilibrium path of interest in a perfect gratefully acknowledged. capital market endowed with perfect ccr- ' "A Note on hleasurement of Utility," Rezliew oj Economic Studies, IV (1937), 155-61. tainty, you have to determine all interest As an undergraduate student of Paul Douglas rates between now and the end of time; at Chicago, I was struck by the fact that ne might, every finite time period points beyond from the marginal utility schedule of consumptions, itself! deduce saving behavior exactly in the same way that we might deduce gambling behavior. Realizing Some interesting mathematical bound- that, watching the consumer's gambling responses ary problems, a little like those in the to varying odds, ue could deduce his numerical modern theories of dynamic program- marginal utilities, it occurred to me that, by watching the consumer's saving responses to varying ming, result from this analysis. *And the interest rates, we might similarly measure his way is paved for a rigorous attack on a marginal utilities, and thus the paper was born. (I simple model involving money as a store knew and pointed out, p. 155, n. 2; p. 160, that such a cardinal measurement of utility hinged on a cer- of value and a medium of exchange. My tain refutable "independence" hypothesis.) essay concludes with some provocative 468 PAUL A. SAMUELSON remarks about the field of social col- Thiinen and others for assuming such a lusions, a subject of vital importance for fact, ends up with his own celebrated political economy and of great analytical third cause for interest, which also as- interest to the modern theorist. serts the fact of net productivity. Con- trary to much methodological discussion, THE PROBLEM STATED there is nothing circular about assuming Let us assume that men enter the brute facts-that is all we can do; we labor market at about the age of twenty. certainly cannot deduce them, although, They work for forty-five years or so and admittedly, we can hope by experience then live for fifteen years in retirement. to refute falsely alleged facts.) (As children they are part of their par- For the present purpose, I shall make ents' consumptions, and we take no the extreme assumption that nothing note of them.) Katurally, they want to will keep at all. Thus no intertemporal consume in their old age, and, in the ab- trade with Nature is possible (that is, for sence of comprehensive social security- all such exchanges we would have i = an institution which has important bear- -1!). If Crusoe were alone, he would ing on interest rates and saving-men obviously die at the beginning of his re- mill want to consume less than they tirement years. produce during their working years so But we live in a world where new gen- that they can consume something in the erations are always coming along. For- years when they produce nothing. merly we used to support our parents in If there were only Robinson Crusoe, their old age. That is now out of fashion. he would hope to put by some durable But cannot men during their productive goods which could be drawn on in his years give up some of their product to old age. He would, so to speak, want to bribe other men to support them in their trade with Mother Nature current con- retirement years? Thus, forty-year-old sumption goods in return for future con- A gives some of his product to twenty- sumption goods. And if goods kept per- year-old B, so that when =\. gets to bc fectly, he could at worst always make seventy-five he can receive some of the the trade through time on a one-to-one product that B is then producing. basis, and we could say that the interest Our problem, then, is this: In a sta- rate was zero (i = 0). If goods kept im- tionary population (or, alternatively, perfectly, like ice or radium, Crusoe one growing in any prescribed fashion) might have to face a negative real inter- what will be the intertemporal terms of est rate, i < 0. If goods were like rabbits trade or interest rates that will spring up or yeast, reproducing without super- spontaneously in ideally competitive vision at compound interest, he would markets? face a positive rate of interest, i > 0. This last case is usually considered to be SIhIPLIFYING ASSUJIPTIONS technologically the most realistic one: To make progress, let us make con- that is, machines and round-about proc- venient assumptions. Break each life up csses (rather than rabbits) are con- into thirds: men produce one unit of sidered to have a "net productivity," product in period 1and one unit in period and this is taken to be brute fact. (Rohm 2; in period 3 they retire and produce himself, after bitterly criticizing naive nothing. (No one dies in midstream.) productivity theorists and criticizing In specifying consumption preferences, AN EXACT CONSUMPTION-LOAN MODEL OF INTEREST 469 I suppose that each man's tastes can be producer produces 1 unit, total product summarized by an ordinal utility func- is B + B. Now, for convenience of sym- tion of the consumptions of the three bols, let Rt = 1/(1 + it) be the discount periods of his life: U = U(C1, Cz, C3). rate between goods (chocolates) of period This is the same in every generation and t traded for chocolates of the next period, has the usual regular indifference-curve t+ 1. Thus, if Rt = 0.5, you must concavities, but for much of the argu- promise me two chocolates tomorrow to ment nothing is said about whether, sub- get me to part with one chocolate today, jectively, men systematically discount the interest rate being 100 per cent per future consumptions or satisfactions. period. If Rt = 1, the interest rate is (Thus Bohm's second cause of interest zero, and tomorrow's chocolates cost 1.0 may or may not be operative; it could of today's. If Rt > 1, say Rt = 1.5, the even be reversed, men being supposed to interest rate is negative, and one future overvalue the future!) chocolate costs 1.5 of today's. [Clearly, In addition to ignoring Bohm's second Rt is the price of tomorrow's chocolates cause of systematic time preference, I am expressed in terms of today's chocolates in a sense also denying or reversing his as numeraire.) first cause of interest, in that we are not IT-e seek the equilibrium levels of . . . supposing that society is getting more Rt, Rt+],. . . , that will clear the corn- prosperous as time passes or that any petitive markets in which present and single man can expect to be more pros- future goods exchange against each other. perous at a later date in his life, since, on At time t each man who is beginning the contrary, during his years of retire- his life faces3 the budget equation, ment he must look forward to producing even less than during his working years. Finally, recall our assumption that no goods keep, no trade with Nature being possible, and hence Rohm7sthird techno- This merely says that the total dis- logical cause of interest is being denied. counted value of his life's consumptions Cnder these assumptions, what will be must equal the discounted value of his the equilibrium time path of interest productions. Subject to this constraint, rates? he will, for each given Rt and Rt+l, determine an optimal (CI, C2, CQ)to maxi- INDIVIDUAL SAVING FUNCTIONS mize LT(C1,CS, C3), which we can sum- The simplest case to tackle to answer marize by the "demand" functions, this question is that of a stationary population, which has always been stationary in numbers and will always be 1 rule out, as I did explicitly in my 1937 paper (p.160), the Ulysses-Strotz-Allais phenomenon stationary. This ideal case sidesteps the whereby time perspective distorts present decisions difficult "planning-until-infinity" aspect planned for the future from later actual decisions. of the problem. In it births are given by Thus, if at the end of period 1 his ordinal preference follows V(CI, Cz, C3) rather than U(C1. Cp, C3), I am = Bt B, the same constant for all posi- assuming (dV/aC,)/(aV/aC,) I (aU/aC ) /(aU/ tive and negative t. ac,). Hence all later decisions will ratify earlier Kow consider any time t. There are B plans. For a valuable discussion of this prol~lemsee R. H. Strotz, "M>opia and Inconsistency in Dy- men of age one, B men of age two, and namic Utility Maximization," Review qf Economic B retired men of age three. Since each Studies, XXIII (1956), 165-80. 170 PAUL A. SAMUELSON It might be convenient for us to work cel out to zero in every period. (Remem- with "net" or "excess demands" of each ber that no goods keep and that real man: these are the algebraic differences net investment is impossible, all loans between what a man consumes and what being "consumption" loans.) he produces. Net demands in this sense At any time t there exist Bt men of the are the negative of what men usually call first period, Bt-l men of the second ((saving," and, in deference to capital period, and Bt-2men of the third period. theory, I shall work with such "net sav- The sum of their savings gives us the ing" as defined by fundamental equilibrium condition:

for every t. Note that in Sz we have the interest rates of one earlier period than in In old age presumably Sg is negative, S1,and in S3 we have still earlier interest matched by positive youthful saving, rates (in fact, interest rates that are, at so as to satisfy for all (Xi, Rt~1)the time t, already history and no longer to budget identity, be determined.) \Ye have such an equation for every t, and if we take any finite stretch of time and write out the equilibrium conditions, we always find them containing discount Of course, these functions are subject rates from before the finite period and to all the restrictions of modern con- discount rates from afterward. iJTenever sumption theory of the ordinal utility or seem to get enough equations: lengthen- revealed preference type. Thus, with ing our time period turns out always to consunption in every period being a add as many new unknowns as it supplies "superior good," we can infer that equations, as will be spelled out later in dC3 dK t4 1 > 0 and dS3 dRt+l < 0. (This equations (14). says that lowering the interest rate THE STATIONARY CASE earned on savings carried over into retirement nus st increase retirement con- iVe can try to cut the Gordian knot sumption.) TT7e cannot unainbiguously by our special assumption of stationari- ness, namely, deduce the sign of dS1 dRt and other terms, for the reasons implicit in modern consumption theory. = B, a given constant for all time Ti-e can similarly work out the saving (6) functions for men born a period later, . . . RtPl= Rt = Rt+1= . . . which will be of the form S,(Rt+l,Rt+z), etc., containing, of course, the later inter- = R, the unknown discount rate. est rates they will face-likewise for 'I'he first of these is a demographic earlier interest rates facing men born datum; the second assumption of non- earlier. Finally, our fundamental condi- changing interest rates is a conjecture tion of clearing the market is this: Total whose consistency we must explore and net saving for the community must can- verify. AN EX.4CT CONSUMPTION-LOAN MODEL OF INTEREST 47 1 Now substituting relations (6) in a positive interest rate from the im- equation (3,we get one equilibrium patient consumers." equation to determine our one unknown R, namely, A BIOLOGICAL THEORY OF INTEREST AND POPULATION GRO\VTH A zero rate of population growth was seen to be consistent with a zero rate of interest for a consumption-loan world. By inspection, we recognize a solution I now turn to the case of a population of equation (7) to be I< = 1, or i = 0: growing exponentially or geometrically. that is, zero interest must be one equi- Now librium rate under our condition^.^ lThy? Because Bt = B(l + m)l, with by virtue of the budget identity (4). For m > 0, we have growth; for m < 0, Can a common-sense explanation of decay; for m = 0, our previous case of this somewhat striking result be given? a stationary population. As before, we Let me try. In a stationary system every- suppose one goes through the same life-cycle, albeit at different times. Giving over goods now to an older man is figuratively = R, a constant through time. giving over goods to yourselj when old. At what rate does one give over goods Now our clearing-of-the-market equa- to one's later self? At R > 1, or R < 1, tion is or R = I? To answer this, note that a chocolate today is a chocolate today, and when middle-aged A today gives over a chocolate to old B, there is a one-to-one physical transfer of chocolates, none melting in the transfer and none sticking to the hands of a broker. So, heuristical- or, cancelling B(l + m)', we have ly, we see that the hypothetical "transfer through time" of the chocolates must be at R = 1 with the interest rate i exactly zero. Recalling our budget identity (4), we Note that this result is quite inde- realize R = (1 + m)-' or i = nt is one pendent of whether or not people have root satisfying the equation, giving a systematic subjective preference for present consumption over future. \Thy? Because we have assumed that if anyone has such a systematic preference, every- lye have therefore established the fol- one has such a systematic preference. lowing paradoxical result: There is no one any different in the sys- If productive opportunities were to exist, Mother Sature xould operate as an important out- tem, no outsider-so to speak-to exact sider, ~vithwhom trade could take place, and our 4 We shall see that R = 1 is not the only root of conclusion would be modified. But recall our strong equation (7) and that there are multiple equilib- postulate that such technological opportunities are riums. non-existent. 472 PAUL A. SAMUELSON THEOREM:Every geometrically grow- case. The representative man is thought ing consumption-loan economy has an to maximize L7(C1, C2, C3), subject to equilibrium market rate of interest exactly equal to its biological percentage growth rate. Thus, if the net reproductive rate 1+ 1 being the lifetime product avail- gives a population growth of 15 per cent able to each man. The solution to this per period, i = 0.15 is the corresponding technocratic welfare problem (free in its market rate of interest. If, as in Sweden formulation and solution of all mention or Ireland, m < 0 and population de- of prices or interest rates) requires cays, the market rate of interest will be negative, with i < 0 and R > l!

OPTIRIURI PROPERTY OF THE BIO- But this formulation is seen to be LOGICAL INTEREST RATE identical with that of a single maximizing The equality of the market rate of man facing market discount rates R1 = interest in a pure consumption-loan R2 = 1. Hence the solution of equations world to the rate of population growth (10) and (11) is exactly that given was deduced solely from mechanically earlier by equation (3): that is, our finding a root of the supply-demand present welfare problem has, for its equations that clear the market. Ex- optimality solution, perience often confirms what faith avers: that competitive market relations achieve some kind of an optimum. Does the saving-consumption pattern given by Sl(R, R), S2(R, R), S3(R, R), where R = 1/(1 + m), represent some kind of a social optimum? One would Now that we have verified our con- guess that, if it does maximize something, jecture for the stationary m = 0 case, we this ecpilibrium pattern probably maxi- can prove it for population growing like mizes the "lifetime (ordinal) well-being B(l + m)', where mz>0. As before, we of a representative person, subject to the maximize U(C1, C2, C3) for the represent- resources available to him (and to every ative man. But what resources are now other representative man) over his life- available to him? Recall that in a grow- time." Or, what seems virtually the same ing population the age distribution is thing, consider a cross-sectional family permanently skewed in favor of the or clan that has an unchanging age dis- younger productive ages: society and tribution because the group remains in each clan has an age distribution pro- statistical equilibrium, though individu- portional to [I, 1/(1 + m), 1/(1 + w~)~] als are born and die. Such a clan will and has therefore a per capita output divide its available resources to maxi- to divide in consumption among the mize a welfare function differing only in three age classes satisfying scale from each man's utility function and will achieve the same result as the biological growth rate. To test this optimality conjecture, first stick to the stationary population AN EXACT CONSUMPTION-LOAN MODEI, OF INTEREST -1.73 By following a representative man interest to the life-expectancy of men of throughout his life and remembering means and their alleged propensity to go that there are always (1 + ?n)-' just from maintaining capital to the buying older than he and (1 + m)-2two periods of annuities at an allegedly critical posi- older, we derive this same "budget" or tive i. I seem to be the Grst, outside a availibility equation. Subject to equa- slave economy, to develop a biological tion (12), we maximize li(C1, C2, C3) theory of interest relating it to the re- and necessarily end up with the same productivity of human mothers. conditions as would a competitor facing Is there a common-sense market ex- the biological market interest rate R1 = planation of this (to me at least) as- Rz = 1/(1 + m): namely, tonishing result? I suppose it would go like this: in a growing population men of twenty outnumber men of forty; and retired men are outnumbered by workers more than in the ratio of the work span to the retirement span. 113th more workers to support them, the aged live better than in the stationary state-the Hence the identity of the social opti- excess being positive interest on their mality conditions and the biological savings. market interest theory has been demon- Such an explanation cannot be ~trated.~ deemed entirely convincing. Outside of social security and family altruism, the CODIDION-SENSE EXPLANATION OF aged have no claims on the young: cold BIOLOGICAL MARKET INTEREST and selfish coillpetitive markets will not RATE teleologically respect the old; the aged will get only what supply and demand Productivity theorists have always re- impute to them. lated interest to the biological habits of So we might try another more detailed rabbits and cows. And Gustav Cassel explanation. Recall that Inen of forty or long ago developed a striking (but rather of period 2 bargain with Inen of twenty nonsensical) biological theory relating or period 1, trying to bribe the latter to If C has the usual quasi-concavity, this social them with consumption in their optimum ~villbe unique-whether U does or does retirement. (Men of over sixty-five or of not have the time-symmetry that is sometimes (for concreteness) assumed in later arguments. A-ot only period 3 can make fresh bargains with no will the representative man's utility U be maxim- one: after retirement it is too late for ized, but so will the "total" of social utility en- them to try to provide for their old age.) joyed over a long period of time: specifically, the divergence from attainable bliss In a growing population there are more period 1men for period 2 men to bargain with; this presumably confers a competitive advantage on period 2 men, the manifestation of it being the positive over all time will be miminized, where U* is the utility achieved nhen RI = 1 = R?and Si = Si(1, interest rate. 1).This theorem may require that we use an ordinal So might go the explanation. It is at utility indicator that is concave in the Ci, as it is least superficialljr plausible, and it does always open to us to do. Of course, this entire footnote and the related qualitatively suggest a positive interest text need obvious modifications if m f 0. rate when population is growing, al- 474 PAUL '4. SAhIUELSON though perhaps it falls short of explain- \\'e think we know the right answer ing the remarkable quantitative identity just given in the two-period case. Let us between the growth rates of interest and test our previous mathematical methods. of population. Now our equations are much as before and can be summarized by: THE INFINITY P.4R4DOX REVEALED But will the explanation survive rigor- Maximize U(C1, Cz) = U(1 - S1, 0 - Sz) ous scrutiny? Is it true, in a growing or subject to S1+ RtSz= 0 . in a stationary population, that twenty- year-olds are, in fact, overconsuming so The resulting saving functions, S1(Rt) that the middle-aged can provide for and S2(Rt), are subject to the budget their retirement? Specifically, in the sta- identity, tionary case where R = 1, is it necessarily true that S1(l, 1) < O? Study of Sl(Rt)+ R,S2(Rt)= 0 for all Rt . (4') U(C1, Cz, CB)shows how doubtful such a general result would be; thus, if there is Clearing the market requires no systematic subjective time preference 0 = HtS1(Rt)+ Rt-lS2(Rt-l) for so that l; is a function symmetric in its 15') arguments, it would be easy to show that C1 = Cz = C3 = 3, with S1(l, 1) = Sz(l, 1) = +Q and S3(1, 1) = -3. Contrary If Bt = B(l +m)' and Rt = Rt+1= to our scenario, the middle-aged are not . . . = R, our final equation becomes turning over to the young what the young will later make good to them in retirement support. The budget equation (4') assures us that THE TXVO-PERIOD CASE equation (8') has a solution : The paradox is delineated more clearly if we suppose but two equal periods of life-work and retirement. Now it becomes impossible for any worker to find a worker younger than himself to be bribed to support him in old age. IThat- So the two-period matherllatics ap- ever the trend of births, there is but one pears to give us the same answer as be- equilibrium saving pattern possible: dur- fore--a biological rate of interest equal ing working years, consumption equals to the rate of population growth. product and saving is zero; the same Yet we earlier deduced that there can during the brutish years of retirement. be no ~~oluntarysaving in a kto-period \17hat equilibrium interest rate, or R, u~orld.Instead of S1 > 0, we must have will prevail? Since no transactions take S1 = O = Sz with R = + m . How can place, R = 010, so to speak, and ap- we reconcile this with the mathematics? pears rather indeterminate-and rather r\ later numerical example, where U = log academic. However, if men desperately C1 + log Cq + log Ca, shows that cases car] arise want some consumption at all times, only where n3 positive R, however large, will clear the R = m can be regarded as the (virtual) market. I adopt the harmless convention of setting R = rn in every case, even if the limit as R -+ m equilibrium rate, with interest equal to does not wipe out the discrepancy between supply -100 per cent per peri~d.~ and demand. AN EXACT CONSULlPTION-LO.lN AIODEL OF INTEREST' 475

\Ye substitute S1 = 0 = Sz in equa- problem, we must drop the assumption of tion (5') or equation (8'), and indeed a population that is, always has been, this does satisfy the clearing-of-the- and always will be stationarj- (or ex- market equation. -4pparently our one ponentially growing or exponentially de- equilibrium equation in our one un- caying). For within that ambiguous con- known R has more than a single solution ! text R = 1(R < 1, R > 1) was indeecl And the relevant one for a free market is an impeccable solution, in the sense that not that given by our biological or no one can point to a violated equi- demographic theorj- of interest, even librium condition. (Exactly the same though our earlier social optimality can be said of the two-period case, even argument does perfectly iit the two- though we "know" the impeccable solu- period case. tion is economically nonsense.) IYe must give mankind a beginning. TIIE PARADOX CONTEhIPLATED So, once upon a time, B rnen were born The transparent two-period case alerts into the labor force. Then B more. Then us to the possibility that in the three- B more. Until what? Lntil . . . ? Or until period (or n-period) case, the funda- no more men are born? Must we give mental equation of supply and demand mankind an end as well as a beginning? Inay have multiple solutions. .And, in- Even the Lord rested after the beginning. deed, it does.8 \Ye see that so let us tackle one problem at a time and keep births forever constant. Our equilibrium equations, with the constant is indeed a valid mathematical solution. H's omitted, now become This raises the following questions : Is a condition of no saving with dismal retirement consumption and interest rate of - 100 per cent per period think- able as the economically correct equilibrium for a free market? Surely, the non-myopic middle-aged will do almost anything to make retirement consumption, C3 non-zero?g One might conjecture that the fact that, in the three-period model, ~vorkers can always find younger workers to bargain with is a crucial difference from the two-period case.1° To investigate the There is nothing surprising ahout multiple solutions in economics: not infrequently income effects make possil~leother intersections, including the possibility of an infinite numt~erwhere demand and \Ye feel that S1- 0 - Sz- S3, while supply curves coincide. a mathematical solution, is not the eco- Before answering these questions, it would be nomically relevant one. Since SI(1, I), well to decide what the word "surely" in the previ- S2(1,I), and S3(1, I) do satisfy the last ous sentence means. Surely, no sentence heginning bcith the word "surely" can validly contain a ques- By introducing overlap hetween workers of tion mark at its end2 However, one paradox is different ages, the three-period model is essentially enough for one article, and I shall stick to my equivalent to a general n-period model or to the economist's last. continuous-time model of real life. 4-76 PAUL A. SAMUELSON of the written equations, we dare hope" often better than many people's theo- that the Invisible Hand will ultimately rems, has suggested to me that in the work its way to the socially optimal bio- three-period or n-period case I am taking logical-interest configuration-or that too bilateral a view of trade. We might the solution to equation (14) satisfies end up with S1 > 0 and encounter no contradictions to voluntary trade by lim Rt = 1, S,(Rt,Rt+4 I-+ m virtue of the fact that young men trade (15) with anyone in the market: they do not = S%(l,I), (i = 1, 2, 3) . know or care that all or part of the mo- tive for trade with them comes from the TIIE 13IPOSSIBILITY TIiEOREhI desire of the middle-aged to provide for But have we any right to hope that retirement. The present young are con- the free market will even ultimately ap- tent to be trading with the present old proach the specified social optimum? (or, for that matter, with the unborn or Does not the two-period case rob us of dead): all they care about is that their hope? \\?11 not all the trade that the trades take place at the quoted market three-period case makes possible consist prices; and, if some kind of triangular or of middle-aged period 2 people giving multilateral offsetting among the genera- consumption to young period 1 people in tions can take place and result in S1(Rt, return for getting consumption back Rt+l)positive and becoming closer and from them one period later? no not such closer to Sl(1, 1) > 0, why cannot this voluntary mutual-aid compacts suggest happen? that, if Kt does approach a limit x, it I, too, found the multilateral notion must be such as to make Sl(x, x) < O? appealing. But the following considera- \'\'hereas, for many men12 not too subject tions-of a type I do not recall seeing to systematic preference for the present treated anbwhere-suggest to me that over the future (not too affected by the ultimate approach to R = 1 and Bohm's second cause of interest), we SI(1, 1) > 0 is quite impossible. expect S1(l, 1) > 0. List all men from the beginning to A colleague, whose conjectures are time t. All the voluntary trades ever made must be mutually advantageous. " Our confidence in this viould be enhanced if the linear diiference equation relating small devia- If A gives something to B and B does tions rt = Rt - 1 had characteristic roots all less nothing for A directly in return, we than 1 in al~solutevalue. Thus a0r1+3+ airt+z + know B must be doing something for alrill+ njrt = 0, where the a, are given in terms of the S'(Rt. Rt+l) functions and their partial deriva- some C, who does do something good for tives, e\alusted at Rt = 1 Kt+]. Logically, this A. (Of course, C might be more than one w011ld be neither quite necessary nor sufficient: not man, and there might be many-linked sufficient, sincc the initial Ro,XI,I<,might be so far from 1 as to make the linear approximations connections within C.) irrelevant; not necesszry, since, with one root less Now consider a time when S1(Rt, than unity in a1)solute \slue, we might ride in Rt+l) has become positive, with S2(Rt-l, toward R = 1 on a razor's edge. In any case, as our later numerical example shows, our hope is a vain Rt) also positive. Young man A is then one. giving goods to old man B. Young man A

l2 There is admittedly some econometric evi- expects something in return and will dence that many young adults do dissave, to ac- actually two periods later be getting quire assets and for other reasons. Some modifications of exposition would have to be made to allow goods from someone. From whom? It cer- for this. tainly cannot be directly from B: B will AN EXACT CONSUMPTION-LO.4N MODEL OF INTEREST 47 7 be dead then. Let it be from someoce Equations (14) now take the form called C. Can B ever do anything good for such a C, or have in the past done so? KO.B only has produce during his first two periods of life, and all the good he can do anyone must be to people who were born before him or just after him. That ftezler includes C. So the postulated pattern of S1> 0 is logically impossible in a free market: and hence 2it = 1 = &,-I, as an exact or approximate relation, is impossible. (Kote that, for some special pattern of time preference, the competitive solution ?night coincide with the "biological optimum. ")

X iSU1IEKICAL EXAMPLE ,1 concrete case will illustrate all this. 'I'he purest RIarshallian case of unitary price and income elasticities can be char- acterized by 2: = log C1 + log Cz + log CS,where all systematic time preference is replaced by symvzetry. .'i ~nasilnurnof Alsidefrom initial conditions, this can he written in the recursive form, log C, subject to (16) I C1+R1C2 +R1R2C7a = 1 +R1 implies Sote that dS1(RI, Rz) = 0 made our third-order difference equation degener- ate into a second-order difference equation. If we expand the last equation around Rt-2 = 1 = Rt-l, retaining only linear terms and working in terms of deviations and, after combining this with the bud- from the equilibrium level, rt = R, - 1, get equation, we end up with saving we get the recursive system, functions,

J J which obviously explodes away from 2 1 r = 0 and R = 1 for all small perturba- Sr(R1,R2) =---- , '7) 3 3R1 tions from such an equilibrium. This con- 1 1 firms our proof that the social optinzuf?~ S3(RI, Rr) = 0 -3gx-3~~.configuration can never here be reached by 478 PAUL A. SAMUELSON the competitive market, or even be ap- which implies that consumptioil loans proached in ever so long a time. lose about two-thirds of their principal IYhere does the solution to (18) in one period. This is here the competi- eventually go? Its first few R's are nu- tive price to avoid retirement starva- merically calculated to be [RI, Rz, R3, tion.14 . . . ] = [2, 33, 3+, . . .]. It is plain that the limiting Rt exceeds 1; hence a nega- RECAPITULATION tive interest rate i is being asymptotical- The task of giving an exact description ly approached. Substituting RtrS = of a pure consumption-loan interest Rt&l= R, = x in equation (19), we get model is finished. IVe end up, in the sta- the following cubic equation to solve for tionaiy population case, with a negative possible equilibrium levels :I3 market interest rate, rather than with the biological zero interest rate cor- %=4---- 1 2 or x2 x responding to the social optimum for the (2 1) representative man. This was proved by the impossibility theorem and verified IYe know that x = 1, the irrelevant by an arithmetic example. optimal level, is one root; so, dividing it A corresponding result will hold for out, we end up with changing population where m $ 0. The actual competitive market rate i, will always be negative and always less than Solving the quadratic, we have the biological optimality rate m.'" And

l4In other examples, this competitive solution would not deviate so much from the i = m hio- logical optimum. But it is important to realize that solutions to equations (14) that come from quasi- concave utility functions-with or without systematic time preference-cannot he counted on to approach as!mptotically the biological optimum configuration of equation (13). for the asymptote approached by the free In this case the linear approximation gives for ur = R1- 3.297 the recursion relation competitive market. The other root, A (3 - dG)/2, corresponds to a negative R, which is economically meaningless, in that it implies that the more we give up of This difference equation has roots easily shown to be less than 1 in absolute value, so the local stabil- today's consumption, the more we must ity of our competitive equilibrium is assured. give up of tomorrow's. l5 LtTriting A = 1/(1 m), our recursion rela- = + Our meaningful positive root, R tion (14) beconles 3.297, corresponds to an ultimate negative interest rate, 2.297 z=--=. 1-R --- R 3.297 ' For the case where U = B log C;, our recursion relation (18) becomes l3Martin J. Bailey has pointed out to me that the budget equation and the clearing-of-the-market equations do, in the stationary state, imply SI= RS3whenever R g 1, a fact which can be used to give an alternative demonstration of possible Then x = Rt = Kl., = R,-:! gives a cuhic equation equilibrium values. with hioln~icalroot corresponding to x = A and AN EXACT CONSUMPTION-LOAN MODEL OF INTEREST 479 increasing the productive years relative pirical hypotheses that entail a zero or to the retirement years of zero product negative rate. would undoubtedly still leave us with a 3. It may help us a little to isolate the negative interest rate, albeit one that effects of adding one by one, or together, climbs ever closer to zero. (a) technological investment possibili- Is this negative interest rate a hard- ties, (b) innovations that secularly raise to-believe result? Not, I think, when one productivity and real incomes, (c) strong recalls our extreme and purposely un- biases toward present goods and against realistic assumptions. 12'ith Bohm's third future goods, (d) governmental laws and technological reason for interest ruled more general collusions than are en- out by assumption, with his second visaged in simple laissez faire markets, reason involving systematic preference or (e) various aspects of uncertainty. To for the present soft-pedaled, and with be sure, other orderings of analysis would his first reason reversed (that is, with also be possible; and these separate people expecting to be poorer in the processes interact, with the whole not future), we should perhaps have been sur- the simple sum of its parts. prised if the market rate had not turned -2. It points up a fundamental and in- out negative. trinsic deficiency in a free pricing system, Yet, aside from giving the general bio- namely, that free pricing gets you on the logical optimum interest rate, our model Pareto-efficiency frontier but by itself is an instructive one for a number of has no tendency to get you to positions reasons. on the frontier that are ethically optimal 1. It shows us what interest rates in terms of a social welfare function; only would be implied if the "hump saving" by social collusions-of tax, expenditure, process were acting alone in a world fiat, or other type-can an ethical ob- devoid of systematic time preference.16 server hope to end up where he wants 2. It incidentally confirms what mod- to be. (This obvious and ancient point is ern theorists showed long ago but what related to 3d above.) is still occasionally denied in the litera- 5. The present model enables us to ture, that a zero or negative interest rate see one "function" of money from a new is in no sense a logically contradictory slant-as a social compact that can pro- thing, however bizarre may be the em- vide optimal old age social security. (This is also related to 3d above.) i = m. The relevant competitive market root is For the rest of this essay, I shall de- given hy velop aspects of the last two of these themes.

SOCIAL COMPACTS AND THE OPTIMUM Where m = 0, X = 1, we have x = 3.297; for m-+ m, X+O, x-+2 and i-+ -4; for m-+ -1, If each man insists on a quid pro quo, X + m , x -+ m and i +-1. Thus the market rate of interest is always hetween -1 and -f, growing we apparently continue until the end of as m grows, in agreement with the small husk of time, with each worse of: than in the truth in our earlier "common-sense explanation." social optimum, biological interest case. 16T. Ophir, of the Massachusetts Institute of Yet how easy it is by a simple change in Technology and Hebrew University, Jerusalem, has the rules of the game to get to the opti- done unpublished work showing how systematic time preference will tend to alter the equilibrium mum. Let mankind enter into a Hobbes- interest rate pattern. Rousseau social contract in which the 480 PAUL A. SAMUELSON young are assured of their retirement The economics of social collusions is a subsistence if they will today support rich field for analysis, involving fascinat- the aged, such support to be guaranteed ing predictive and normative properties. by a draft on the yet-unborn. Then the Thus, when society acts as if it were social optimum can be achieved within maximizing certain functions, we can one lifetime, and our equations (14) will predict the effect upon equilibrium of become specified exogenous disburbances. And certain patterns of thought appropriate to a single mind become appropriate,

from t = 3 on. l8NOW,admittedly, there is usually lacking in IVe economists have been told17 that the real world the axes of symmetry needed to make all this an easy process. In a formulation elsewhere, what we are to economize on is love or I have shoxn some of the requirements for an altruism, this being a scarce good in our optimal theory of public expenditure of the Sax- imperfect world. True enough, in the Wicksell-Lindahl-Musgrave-Bowen type, and the failure of the usual voting and signaling mechan- sense that we want what there is to go as isms to converge to an optimum solution (see &'The far as possible. But it is also the task of Pure Theory of Public Expenditare," Reviea of political economy to point out where Econonzics end Statistics, XXXVI [Sovember, 19541, 387-89, and "Diagrammatic Exposition of common rules in the form of self-imposed Public Expenditure," ibid., XXXVII [Sovemher, fiats can attain higher positions on the 19551, 350-56). Such a model is poles apart from social welfare functions prescribed for us the pure case in which Walrasian laissez faire hap- pens to be optimal. I should be prepared to argue by ethical observers. that a good deal of shat is important and interesting The Golden Rule or Kant's Cate- in the real world lies betrieen these extreme poles, gorical Imperative (enjoining like people perhaps in hetween in the sense of displaying properties that are a blending of the polar properties. to follow the common pattern that makes But such discussion must await another time. each best off) are often not self-enforc- l9 How can the competitive coniiguration with ing: if all but one obey, the one may gain negative interest rates be altered to everyone's ad- selfish advantage by disobeying-which vantage? Does not this deny the Pareto optimality is where the sheriff comes in: we po- of perfect competition, which is the least (and most) Tve can expect from it? Here we encounter one msre litically invoke force on ourselves, at- paradox, which no d~btarises from the "infinity" tempting to make an unstable equi- aspect of our model. If we assume a large finite span librium a stable one.Is to the human race-say 1 million generations- then the iinal few generations face the equations Once social coercion or contracting is admitted into the picture, the present problem disappears. The reluctance of the young to give to the old what the old can never themselves directly or indi- rectly repay is overcome. Yet the young never suffer, since their successors come under the same requirement. Everybody ends better off. It is as simple as that.Ig where T = 1,000,000.

17 D. H. Robertson, What Does the Economist If xe depart from the negative interest rate pattern, Xaximize? (a keynote address at the Columbia the final young will be cheated by the demise of the bicentennial celebrations, May, 1954), published by human race. Should sach a cheating of one genera- the Trustees of the University in the Proceedings of tion 30 million years from now perpetually condemn the Conference, 1955 (New York: Doubleday & Co.) society to a suboptimal coniiguration? Perfect and reprinted as chap. ix in D. H. Robertson, competition shrugs its shoulders at such a question EconMnic Commentaries (London: Staples, 1956). and (not improperly) sticks to its Pareto optimality. AN EXACT CONSUMPTION-LOAN MODEL OF INTEREST 481 even though we reject the notion of a vations for higher living standards that a group mind. (Example : developed social free market channels into \iTalrasian security could give rise to the same bias equilibrium when the special conditions toward increasing population that exists for that pattern happen to be favorable among farmers and close family groups, -these same motivations often lead to where children are wanted as a means of social collusions and myriad uses of the old age support.) apparatus of the state. For good or evil, The economics of collusion provides these may not be aberrations from laisser. an important field of study for the faire, but theorems entailed by its in- theorist. Such collusions can be im- trinsic axioms. portant elements of strength in the struggle for existence. Reverence for life, CONCLUSION: MONEY AS A SOCIAL in the Schweitzer sense of respecting ants CONTRIVANCE and flowers, might be a handicap in the Let me conclude by applying all these Darwinian struggle for existence. (And, considerations to an analysis of the role since the reverencer tends to disappear, of money in our consumption-loan world. the ants may not be helped much in the In it nothing kept. All ice melted, and so long run.) But culture in which altruism did all chocolates. (If non-depletable land abounds-because men do not think to existed, it must have been superabun- behave like atomistic competitors or be- dant.) iyorkers could not carry goods cause men have by custom and law over into their retirement years. entered into binding social contracts- There is no arguing with Nature. But may have great survival and expansion what is to stop man-or rather men- powers. from printing oblongs of paper or stamp- An essay could be written on the wel- ing circles of shell. These units of money fare state as a complicated device for can keep.21(Even if ink fades, this could self- or reinsurance. (From this view, the be true.) With ideal clearing arrange- graduated income tax becomes in part a ments, money as a medium of exchange device for reducing ex ante variance.) might have little function. But remem- That the Protestant Ethic should have ber that a money medium of exchange is been instrumental in creating- individual- itself a rather efficient clearing arrange- istic capitalism one may accept; but that ment. So suppose men officially through the it should stop there is not necessarily state, or unofficially through custom, plausible.20 I17hat made Jeremy Ben- make a grand consensus on the use of tham a Benthamite in 1800, one suspects, these greenbacks as a money of exchange. might in 1900 have made him a Fabian Now the young and middle-aged do have (and do we not see a lot in common in the something to hold and to carry over into personalities of James Mill and Friedrich their retirement years. And note this: as Engels?). long as the new current generations of Much as you and I may dislike govern- ment "interferences" in economic life, we 2' I have been asked whether introducing durable money does not violate my fiat against durable must face the positive fact that the moti- goods and trades with Nature. All that I must insist on is that the new durable moneys (or records) be 20Recall the Myrdal thesis that the austere themselves quite worthless for consumption. The planned economies of Europe are Protestant, the essence of them as money is that they are valued Catholic countries being individualistic. only for what they will fetch in exchange. 382 PAUL A. SAMUELSON workers do not repudiate the old money, In short, the use of money can itself be this gives workers of one epoch a claim on regarded as a social compact.22 IThen workers of a later epoch, even though economists say that one of the functions no real quid pro quo (other than money) of money is to act as a store of wealth and is possible. that one of money's desirable properties tVe then find this remarkable fact: is constancy of value (as measured by without legislating social security or constancy of average prices), we are en- entering into elaborate social compacts, titled to ask: How do you know this? society by using money will go from the IYhy should prices be stable? On what non-optimal negative-interest-rate con- tablets is that injunction written? Per- figuration to the optimal biological-inter- haps the function of money, if it is to est-rate configuration. How does this serve as an optimal store of wealth, is so happen? I shall try to give only a to change in its value as to create that sketchy account that does not pretend optimal pattern of lifetime saving which to be rigorous. could otherwise be established only by Take the stationary population case alternative social contrivance^.^^ with m = 0. IVith total money M con- I do not pretend to pass judgment on stant and the flow of goods constant, the the policies related to all this. But I do price level can be espected very soon to suggest for economists' further research level off and be constant. The productive the difficult analysis of capital models invest their hump savings in currency; in which grapple with the fact that each their old age they disinvest this cur- and every today is followed by a to- rency, turning it over to the productive morrow. workers in return for sustenance. With population growing like (1 In terms of immediate self-interest the existing + productive workers should perhaps unilaterally m)t, output will come to grow at that repudiate the money upon which the aged hope to rate. Fixed M will come to mean prices live in retirement. (Compare the Russian and Belgium calling-in of currencies.) So a continuing falling like 1/(1 + m) t. Each dollar saved social compact is required. (Compare, too, current today will thus yield a real rate of inter- inflationary trends which do give the old less purest of exactly m per period-just what chasing power than many of them had counted on.) the biological social-optimality configu- 23 Conversely, with satisfactory social security ration calls for. Similarly, when m < 0 programs, the necessity for having secular stable and population falls, rising prices will prices so that the retired are taken care of can be lightened. Even after extreme inflations, social create the desired negative real rate of security programs can re-create themselves anew interest equal to m. astride the community's indestructible real tax base. http://www.jstor.org

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You have printed the following article: An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money Paul A. Samuelson The Journal of Political Economy, Vol. 66, No. 6. (Dec., 1958), pp. 467-482. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28195812%2966%3A6%3C467%3AAECMOI%3E2.0.CO%3B2-Z

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[Footnotes]

1 A Note on Measurement of Utility Paul A. Samuelson The Review of Economic Studies, Vol. 4, No. 2. (Feb., 1937), pp. 155-161. Stable URL: http://links.jstor.org/sici?sici=0034-6527%28193702%294%3A2%3C155%3AANOMOU%3E2.0.CO%3B2-4

18 The Pure Theory of Public Expenditure Paul A. Samuelson The Review of Economics and Statistics, Vol. 36, No. 4. (Nov., 1954), pp. 387-389. Stable URL: http://links.jstor.org/sici?sici=0034-6535%28195411%2936%3A4%3C387%3ATPTOPE%3E2.0.CO%3B2-A

18 Diagrammatic Exposition of a Theory of Public Expenditure Paul A. Samuelson The Review of Economics and Statistics, Vol. 37, No. 4. (Nov., 1955), pp. 350-356. Stable URL: http://links.jstor.org/sici?sici=0034-6535%28195511%2937%3A4%3C350%3ADEOATO%3E2.0.CO%3B2-A

NOTE: The reference numbering from the original has been maintained in this citation list.