The trade imbalance story

Journal of InternationalEconomics4 (1974)I W-137. @ Narth-Holland PublishingCompany

THE TRADE IMBALANCE STORY

David GALE Wntversityof Ca&fimia,Berkeley0Cal% 947m, U.S.A.

A few years ago the author presented an example of trade between two countries in which there is a steady state competitive equilibrium where one country exports a constant amount to the other in every period [Gale (1971)]. Such a situation may appear to be paradoxical on two levels which I will call the naive and the sophisticated levels. On the naive level, one might ask how such a situation can be budgetarily feasible. How can count.ry A continually receive goods from country B without giving anything in return or, otherwise stated, how is country A paying for its constant stream of imports? On the sophisticated level, a person familiar with the usual optimal&y properties of competitive equilibrium might wonder why the usually benevolent han competitive prices has led country B to behave in such a seemingly non-optimal way as to continually give away some of its goods instead of consuming them itself. The present work represents a reconsideration not of the particular model treated earlier but of the trade imbalance phenomenon in general. Working with two well known simple models, one is led to conclude the following: (1) ,for &a& models which a&nit steady state equilibriumpewuuzent trade in$alance fi the rule rather than the exception. Thus, it was not necessary to construct the rather intricate two-sector constant coefficient example of thiz previous paper. virtually any dynamic equilibrium model will provide illustrations of the same phenomenon. (2) (This was not contained in the earlier paper). The ‘seemingly non-optimal’behavior of country B described above is, under ‘rxwmal”conditiosts (the term will be made precise in a moment) actual& ed0nntqeou.s in the sense that country B is better ofboth in the lortgQtd short run than it ivouldbe ifit were to operate autarkically. (This will also be true for the importinz country, as on.e Iwouldexpect) A specific numerical example will perhaps illustrate these rde:ts better than further verbal description. We imagine that countries A and B are economies of the type introduced by Solow (1956) in which there is a single good which can either be consumed or invested to yield more of itself. The population of each country is growing at the rate of 2 % and in each country people save 15 % of their

120

D.We, Zhtik imbabwe

income and consume the rest. The production functions in both countries are Cobb_Douglas and the only differencebetween them is that x unitsof capital yields a flow of xos2sin country A and xoe3in country B. Under the assumption that goods are freely transferable between countries, one can now formulate the dynamic equilibrium-growth-trade model in a unique manner. The stoults are the following: starting from any initial values or capital stock j;rl the two Qountries,in the course of time a steady state will be reached in which country B exports approximately 1.476 ofits national output to country B on a continuing basis forever. The reason for this is not far to seek for one also finds that in the steady state the people of country A own about 10% of the capital stock if country B, and the continuous flow from B to A represents interest payments on these holdings (this clears up the ‘naive’ paradox mentioned in the opening paragraph). Viewed in this way the result seems natural enough. There is certainly nothing surprising aibout one country investing in another. The thing, however, which ‘fels wrong*to many of my respected colleagues is the perman,ence of the arrangement. Why must country B remain a debtor forever to country A? Cannot the original investment be somehow ‘paid off’s0 that eventually country B will be able to go it alone? Our analysis indicates that in fact the com$etitivemarket mechanismwill not bring this about. This illustrates our fW conchtsion. As to the second, it turns out in the above example that in the -1 steady states consumption in country A is up by about 0.15% over what it would be under long-run autark,y while that of country B is up by 0.02% cornpared to its autarkic steady state*(The reason these increases are so small is that the countries ar? w aearly identical to begin with). Having observed the s@ady state trade imbalance in the example one must explain the directionof imbalance - why is country A with the lower CobbDouglas exponent, the permanent importer, rather than the other way around? The reason is that country A is tecJuzoZogfcuZZy superiorto country B in the sense that if both countries produce so as to maximize profits at any interest ra.te p, then country A will always achieve more output per unit invested, as one easily shows, and this extra output gets investedin country B, The ejcampleis typical of the general situation. A complete analysis is carried out in the first half of the paper for general Solow-type economies and in the second half for the pure exchange consumption-loans economies introduced by Samuelson (1958).The ‘trade imbalatncestory’ can be described quite concisely for the case of any number of countries. First, assume each country C1operating autarkically has a steady state with an interest rate pt. Next suppose there exists a world steady state with free international t&e and investment with a world interest rate JL Then C1will be a permanent importer or exporter according as pi is less than or greater than jL A few asmarks concerning what 3[know of the previous literature on this question: in my earlier paper the analysis broke up naturally into two cases, the IWFI& case in which the steady state interest rate exceeded the po

D. Gale, Trade im&aliame

121

grolwth rate, and the ‘abnormal’ case in which the inequality went the other way. Since in that paper we assumed a constant population the abnormal case corresponded to a negative interest rate world. For this case J. Green (1972) has provided some illuminating analysis. He observes that it is indeed non-optimal to be a permanent exporter and has shown how a country could extricate itself from this undesirable position. Such considerations, however, will not enter in this paper which is confined almost entirely to this normal, positive interest rate world. As to other writings on the subject there: seems to be a significant body of recent work in dynamic trade theory which treats the case in which trade betwees countries is unrestricted but international borrowing and lending is prohibited. Some of the initial work in this area is due to Oniki and Uzawa (1965)and Bardhan (1966). Of course, in such models trade will automatically he balanced not only in the long run but at every instant of time. Now there may be good reasons for studying models in which such restrictions are imposed, but it ~,XXQS important to point out that restricting the flee market by this trade balance constraint leads, as in the case of other interference with the free market, to (1) inefficient allocation of the world’s productive resources and (2) nonPareto-optimal distribution of the worlds goods. Thus, models of this type lead to theories of the so-called second best. Finally I come back to the matter alluded to in the beginning. Do the theorems proved here ‘explain away’ the apparent paradoxes of the earlier paper or do they just make them worse? Most economists to whom I have presented this material feel that a realistic model of a trading world should not come out this way, with all countries eventually locked in forever either as debtor-exporters or creditor-importers. My own feeling, for what it’s worth, is that the conclusions are in fact what one should expect and in some approximate sense what one actually sees. In fact the whole ‘story’ is presented in a few paragraphs in Samuelson’s elementary text (1972). ‘Different parts of the world have different amounts of resources: labor, minerals, climate, know-how. Were it not for ignorance or political boundaries, no one would push investment in North Am&a down to the point of 5 per cent returns if elsewhere there still existed 18 per cent opportunities. SOme capital would certainly be invested abroad. Thi.3 would give foreign labor higher wages, because now the foreign worker has Inore and better tools to work with. It would increase foreign production. By how much? Not only by enough to pay for the constint replacement of used-up capital goods but, in addition, by enough to pay us an interest or dividend return on our investment. This interest return would take the forin of goods and servica which we receive from abroad and which add to our standard of living. . . . When would we be repaid our principal? So long as we are earning a good return, there is no reason why we should ever wish to have it repaid. . . . Thus, there is no necessary reason why a country should ever be paid off for its past lending, uJess it has become relatively poorer.’

D. Gate, Da& imbdimce

122

This is exactly our first conclusion on the vrmanence of the trade imbalance. As to the second result a’bout mutual gains-to both :the importer and exporter, here is what Samuelson has to say about the century and a half during which England enjoyed its position as world creditor. ‘Mature c&& nation. England reached this stage some years ago, and as in such cases, her merchandise imports exceeded htz exports. Befort: we feel sorry for her because of her so-called ‘unfavorable” balance of trade, let us note what this really means. IIer citizens were living better because they were able to import much cheap food and in return did not have to part with much in the way of valuable export goods. The English were paying for their import surplus by ;the interest and dividend receipts they were receiving from past foreign lending. Fine for the English. But what about the rest of the world? Were they not worse off for having to send exports to England in excess of imp’orts? Not necesszirily.Normally, the capital goods that England had previously lent them permitted them to add to their domestic production - to add more than had to bz paid out to England in interest and dividends. Both parties were better OK Nineteenth-century foreign lending was twice blessed: it bfessed him who gave and him who received.’ The above quotations represent perhaps the best introduction to the material to be presented here, which is precisely an attempt to model the situation described so succinctly in the Samuelson text. If trade theorists still feel uneasy with the permanent imbalance result I would be most interested in knowing how they propose to get around it. I. A neoc~ical

production

model (Solow)

I.I. The one country model

In this section we will briefly reconstruct a variant of the Sol.ow model for a single country in a form which will extend easily to the n country case. We consider a labor force growing at the rate n so that at time t there are L@ workers. There is a single good and at a given instant of time we suppose &es’e:are x units per worker availabIe to the model. We call x the (per worker) ~utpitalslock.New goods or output can then be created at the ratef(.x) per worker i~r unit time wherefis some nonnegative increasing concave function. We may &ink of this as saying that one unit of labor working with a stock of x units of tloods produces aflow off(x) units of goods per unit time. We will now des&be a model of competitive equilibrium for this simp$z t~,hnology. We imagine agents consisting of producers and consumers. Consumers own the capital stock x and are prepared to rent it out to producers at any positive rental rate p. Given such a rental rate producers ~$1 ~ZPRXUU! that amount of capital 3 whiull maximizes their profit given by the expression

D. Gab, Trdk imbalrutcc

123

f(3)-p3, this being output minus rent of capital. To balance supply and demand p will adjust itself so that 2 = x which means, assuming f is differgzntiablethat, p =f(x), the usual marginal product equation. Finally, the profits7achieved by production must be turned back to the consumer-workers as wages o. So we have the usual relations

and we see that rent and wages are determined by the stock of capital, the former being a decreasing, the latter an increasing function of x. To complete the description of the model we specify the behavior of consumers as follows: consumers have two sources of income, their rest income given by xf’(x) and their wage incomef(x)-@"(x). We assume that they save the fraction crPof the former and a, of the latter where cP 2 cm, which is the usual assumption that capitalists save more than workers. The Solow model corresponds to the special case bP = co. For present purposes it costs us no more to make this simple generalization. Denoting per capita savings by s we now have s = o$(x)x+q@(f(x)-xf’(x)). (1.3) Then aggregate savings S is just te”% and everything which is saved goes increase the aggregate stock of capital X = XL@*so we have

to

and dividing through by LtP gives 3 = (a$‘(x)-n)x+cr,(f(x)-~~(~)),

. (14)

which is the differential equation whose solution gives the time path of capital and hence of all other quantities of the model. For the Solow case (1.4) simplifies to* = @f(X)-rax. We will make the assumptions (a)fandr are positives andf’ decreases montonically to zero. (b) b$‘(O) > it.

.

Condition (b) is necessary in order for the economy to grew at all. If it were not satisfied it would Jrneanthat people’s propensity to save is so low that it is impossible to maintain any positive level of per capita capital. Theorem 1. Eq. (1.4) has Q uniqueposithe stationary solution Z, aad all other s&tiow approueh2 monotonicallyin t. m Denoting the right hand side of (1.4) by 4(x) we must show that there is ia unique Z such that d(Z) = a. Wlehave MM

= (Q -cr,lf’(x)+GJ(x)/x-n

and the first term on the right decreases to zero monotonically as x becomes infinite from (a) (we here use bP 2; 6,). T”neterm j@)/~ is also decreasing because ,,fis positive and concave (proof is left as an exercise for the reader) and further ~‘(x)/.xapproaches zero, for given e > 0 there is an x8 such that f’(q) c &. But by concavity./&)-.&&) I< E(%-XJ so&)/x S Cf(xj-G&-t4 so lim,+ a$(xIIx 5 e. Thus, #(X)/Xdecreasesmonotonicallyto -n, On ‘de other hand, lim,, &(x)/x = t@(O)‘-n which is positive from (b). Therefore there is a unique positive x’such that #(%) = 0. Finally the stability of Xfollowsfrom the standard criterion from the fact that 4(x) is positive for x < XLand negative forx > 3. l 1.2. The c4se of several countries We now suppose there are PEdifferent countries CI, q SCm each with its own production function fr and savings propensities dfl and da. Population grows at the same rate it in all countries. We suppose that goods can be transported costlessly and instantaneously between countries. Let us denote by WI the we&h of Cr, this being the amount of goods owmed by consumersin Cd. The aggregate world wealth &IV, is denoted by x and is of course the world stock of cq@d. By the usual competitive condition this stock will be allocated among the m countries in such a way as to maximizeworld output (any other allocation would in fact be inefficient).This means that all countries Cd which are used for producing at all will have inputs xi > 0 such thatfi’(xl) = p where p is the WORM interest rate and marginal product of capital. Notice that WIand xctwill in general not be equal. As already noted wI is the amount of goods supptied by consumers in Cr while xl is the amount dkmtznded by producersin Giand only the aggregatequantities&v~ and &xl are equal. We can now set up the equations of motion as before. The rentals income of Cl is pw, and the wage income is wt = j&)-pxr so per worker savings is again 4 = +M++@!J&, l

l

and as in the previous section all saving goes to increase wealth in C1so just as in the derivationof (1.4) we have rirl= (c$p-n)~++~~c~~, ri= 1,. . .,m,

(1. 9

is is a system of difSerentia1 equations in the wi above for given the IQ this determines x which determines p and the x1 which in turn determines the q. Thus(1.5)determinesthe behaviorof all quantities in the model. We are especially interested in the exports et of CI and how they change with time. Now the flow of exports from Cris precisely the differencebetween goods j&J produced in Cl and the good; ‘used’ in Cr either for consumption or for building up the stock of capital. The equation is et = j&)--(1

-fl&wi-(l

-o,)o~-*~

-nxi

0 . 6)

but from (15) (~~~w+yJ)

= ti,+m# so we get finally the simple expression

Please observe that the omcial quantity in determining exports is wi-xi which represents the exoess of wealth over capital in Cl, thus, the amount of foreign capital owned by CT1 (this can be negative, of course, with the obvious interpretation). 1.3. Stdy

states

In seotion 6 we will show that for the case where ap and u,,,are the same in all countries there exists a unique world steady state which is globally stable. Presumably this will also hold if the de and c$ were nearIy the same in all countries. In this section we will simply assume the existence of such a steady state andobtainits proper&sas regardsthe patternof trade. A steady state by def&ion is a state in which all per capita quantities are constant in time, so, in particular, tit-& = 0 for all i. It follows from (1.8) that steady state exports of C#af@

a-adthis equation tells a story. There are three cases according as p is greater than, equal to or less than n. If p = n we have the familiar ‘golden rule’ steady state in which world output is at its greatest sustainable value. In this case all countries will have a balance of trade, but of course the equality of p and n will come about only by coincidence. In the case p > n which we will call the nodcase, CT1 is a net importer or exporter according as x1is less than or greater than wl. The interpretation here is the natural one. If, say, wi excee& ;rr this means that the rep? amed by oonsumers in Cr exceeds the rent paid out by producersof CP In other words, Cr is earning rent, income on capital stocks mter than that of its own country and this income from ‘foreign capital’ accounts for its net imports. Finally the case p < n is the paradoxicalnon-Pareto optimal csse in which the rental rate is so low that people have to pays relative to the population growth rate, for holding capital. In this case countries whose wealth exceeds their capital must pay for the extra savings through exports. From now on we will restrict our analysis to the normal case, as the other two correspond to savings propensities which are apparently much higher than those encountered in the real world. Now eq. (1.9) does not by itself prove that steady states must be unbalanced, turn out to be the equal in for it could be the case

world steady state. In order to settle this question observe from ec;l.(1.5) that for a steady state since6~ = 0 we have (d,p-n)w,f~~Q$ = 0.

(1.10)

We will assume that lrf, r 0 so that at least some fraction of wages 8& 88ved. We also suppose that country Cs is an active producer in the world steady state so that xf is positive.Then mt > 0 so n 3 Q$ and (1.11) Substituting this in (1.9)gives (1.12)

but now we note that the second factor above is precisely the function 44~) on tie right hand side of eq. (1.4) since p = f;(&) for all i (marginal product of capital is the same in all countries). Denote by Xl the steady state capital for Ct operating autarkically and let X1be the capital stock of Cr in the world steady state. From Theorem 1 we know that (b(x) has the same sign as x-E& It follows from (1.l2) that Ct is a net importer or exporter in the world steady state according as jsi is greater than or less than XI.Finally sinceff(x) is decreasing for all i we see that Cj is an importer or exporter according as its au98tkjc stiady state interest rate pi is less than or greater than the world steady state interest rate p”.T&se results may -besummarizedas follows: Theorem 2. In tr normal worldsteady state every countrywillhave CIbalmce

of trade on!” in t/x q~c&l case irewhich all countries have the same autatkic steadystute rirtteredrt rate. In any other c4se the world stemfy sfute infetesf tote will be between the v~ious mtutkic rutes and countries whose autatkictate is h&et (lower)than the wotldtatewillbe mf expottets(importers).

As special cases of the general result we have the situation where all countries are technologically identical, that is they have identical production functions, but different savings propensities, Then the countries with the higher savings rate will be the lmg run importers as one would expect since being bigger savers they will end up owning foreign capital. On the other hand, in a CobbDouglas world in which all production functions are of the form C,x;il@ and all qi are the same, those countries with the lower values of aSwill end up being the net importers. 1.4+Gainsfrom trade Having observed that the normal situation among trading Solow countries is a perpetual trade!imbalance we now turn to the question of welfm and ask is this situation ‘good or bad’ and if so for whom. In traditional static trade theory one can show that it is always possible for everyone to benefit or at

for no one to be hurt; by opening up trade. In the present dynamic case this is no longer at all obvious, especially for the permanent exporter, the coun*xywith the so4led “trade surpIus*,which appears to be giving some of its produce away and getting nothing in return. We will show, however, that the traditional results do carry over and both permanent importers and exporters arxzbetter off in a sense to be defined,under trade than in autarky. of being better off in this model is not immediately obvious sina there ate no utili@ functions around to measure such things. However, there does seem to be a simple criterion for determining the national welfare of a country and that is the size of nuti~1~~2 3zcomeI# which is the sum of wage i&o= q and rent lincomepwl.l *Please note that in this model we have not split up the population into workers and capitalists. We are assuming that each member of a country is a consumer-renter-worker so that all share equally in the national income. We then have Ieast

(1.13) Note that in the autarkic case where wI = x#we get Ii = fix3 so that income is the same as output as it should be. Now suppose we look at a country at the moment w&n t&e is initiated and suppose at that moment its wealth is wi. In order to equalize marginal productivity in all countries x4 will in general change to a value different from wI, possibly smaller, possibly greater. However in either case I#will increase for by strict concavity offwe have

so

and wehaveshow~:

Short Run Dade Gains l%eutem. National income i’nevery country will be . igh at the onset of trade as it was in autarky. In general countries will expetience cIIIupwardjump of pncome.(The &continuity arises from the somewhat uBrea!&t&aw’amptiunof instantaneousand costless transpttution.) sting question concernsthe long run. We will show that each country has at least as high a national income in a world steady state as it would in its own autarkic steady state. We are able to prove this only in the Solow case where G: = c&for all Cd. Let jcand 3 be the capita1stock for eountfy C (we omit the subscript i in the subsequent analysis) in its autarkic a;ld world steady states respectively.

From eq. (1.4)we then get that the autarkic steady state income I,, is given by (1.14)

IA = j&) = (Pz/@)Z. In the worldsteady state income & from (1.l 1) is

Iw = pfl+& = pt&/(n-ap)+tD = n&/(n-crp)

(1.15)

and we must compare 1 and fw. Again from strict concavitywe have f(Z) -J(2) < (Z-3)~(2) & --f(3)-Zf(%) multiplying (1.16)by n/(n-ap)

for 2 # z, so

>f(x3-3f@)

= (n/c+p)z9 so

(1.16)

gives

n&/(n-trp) > (?l/tF)Zor Iw > A?=*, and we have: Long ncn trade gains theorem, Every country is at teast as well of in the world steady state as it i$ on its autarchicsteady state and is strictly better ~fl except fpor the coincidentalcase in which its autarkic sterrdystate Merest rate is the sameas the wotldsteady state intetestrate. Notice that this result does not require p > n and will hold also in the ‘abnormal’ case. 1.5. Existence,uniquenessand st&iIity of a w&d steady state We will here show that a Solow world has the same desirable properties regarding steady states as a single Solow country under the assumption that the savings constants Q and cc0are the same in all countries. I believe the results to hold under more general conditions but the proofs seem particularly simple for this case. We return to the system of eq. (1.5) which gives the dynamic behavior of the world model. Multiplying the ith equation by L1and summing g&s

and recalling that z .Ltwi= z L,q = x we have

Recall now that each x1 is a function of x so that the expression CL&J is also a function of x, th.e worldproductionfwrction,f(x). For convenience at this point we will choose the unit of labor so that c Lt = 1, and at the time t = 0 tire was one unit of labor iin the world, and L1 represents the fraction of the

D. a&,

mz& tndvrjaKc

129

world labor force belongingto Cfi Then sincef;(xl) = p for all i we have 9 = f’(x) and (1.17) becomes precisely eq. (2.4) in section 2. In order to prove the desired steady state properties therefore we need only show &a@(x) satisfies assumptions (a) and (b). Now (b) follows at once from the corresponding assumptionfor each of the ft. As for (a), f and/’ are clearly positive sincefr andfi are positivemdf’ decreases as x increasesbecause as x increasesso do all the x8 as we ha-le alreadyobserved.To show that lim,, J’(X) = 0 note that for any e > 0 there exists x0 such that if x z+x0 then&(x) < e for all & but then also f’(x) < e for, since x = CLtx,, at least one X~exceeds x0, hence fr#t) < 8for that x1but/‘&) = p = f’(x). Having verifiedassumptions(a) and (b) it follows from Theorem 1 that x converges to a unique steady state value. This implies immediatelythat each of the x1 convergeto constantvalues RI.It is still conceivablehowever that the w1for some reason fails to settle down even though the sum &PV, does. To see that this does not happen we return to qs. (1.5)which are of the form Ii+ = a(f)w1+b&)

(1.18)

whcxeG@)+ d and b&) + 6. We note that ii must be negative (and 6, positive) for otlzxwise all wf would continue to grow whereas we know that ~L#M+ stabilk*. Now if &) and b&) were actually constant at the values a and 6, then tie solution of (1.18) would be H+(#)= ke%&J converging to 6&L To show *&atthis happens even when a(t) and b&) are nonconstant but converge to d and lit is a technical matter of obtaining estimates in terms of the rates of convergence of a(t) and b&) for the solution w,&). We omit these somewhat laborious details. 2. A pare exchangemodel (Samdqon) 2.1. The modeland steadystateequiltirbn We consider a country consisting of people who live for I+ 1 time periods 0, l,..., l, and who subsiston a single good called irtcome.In the kth period of his life each person receives an endowment of jk units of income. Income is perishable so it must be consumed at once. In general it will therefore be mutually advantageous for people of different ages, with ditrerent endowments, to trade with each other in order to achieve a more satisfactory distribution of consumption over their lifetimes. We now set up an equilibrium model for this situation in the standard way and will be concerned in this section only with steady state quilibria. For simplicity we will assume a constant population though everything carries over in an obvious way to the case of a constantly growing population. Suppose then that the agents in the model are confronted with an interest rate r which they believe will remain constant throughoslrt their lives. They will then plan a &fetime

constunptionkhedkle c = (co, . to the budgetequation,

. cs cl)

their satisfaction which maximize (2* 1)

c c&l +r)k = c ik/(l Mk, where all sums will be taken to run from 0 to I u Since c is determined by t we will write c@)a c&) - &. Then eq. (2.1)becomessimply

oy

c 8&)/(1 +r)) - 0.

(23

Further the aggregate excess demandS(r) at interest rate r ti given by s(r) = c

W) a

6k@),

and clearly for equilibrium this quantity must be zero. Howeverthis is not sufficientfor equilibrium. In addition one must have that the indebtecikss of societynets out to zero. Specifically,the indebtechessof a person of age k at the interest rate r is given by d&(r)= &$(l+r)L-‘, I-1

W)

where of course dk(r) will be positive or negative according as the person is 43 debtor or creditor in the kth period of his life, The condition for equilibrium is then simply

which is the condition that the market for debts shall clear in every period. It is perhaps not immediatelyobvious that ciearing of the debt marketimpliesthat twess demand will be zero. In fact we have, Lemma 1. Ifthe budgetequation (2.2)2swt&&d

then 6(r)

=

-d(r).

I From qs. (2.4) and (2.5) we have

the case r = 0 a where the last equation follows from eqs. (2.2) and (2.3). separate argument is needed, but in that case 6(r) - zial 61(r) directly from q. (23.1

the Lemma shotvs that if d(t) = 0 then S(r) = 0. On the other = 0 without d(r) = 0 exactly in the case where r = 0 is tk familiar *golden-n&’steady state for this mod& ad as to whether stady state equilibriawill in general

undo the fouowiag weak assumptions:

(ii9theaatilityof consumption C&is continuous and increasing at all ages k.

m Let us choose units of income so that Ci, < 1. Then S(r) will surely be positivewheneverc&) 2 I for any k. Now letj be the smallest subscript k such that ik is positive, and choose consumptions c,+¶, . . . c, > 1 so that C= (0,... O,I/,C~+~~. . .. q)>(O,O ,... QJ,..., 1) where > means ‘is prefemd to’. By continuity of preferences, for e positive but sufficiently small we have, 3 = (0; e . . 0, ii-s, cl+!, . . ., q) >e (e, . . ., G i,+q I, . . ., 1) = E. Nowletuschoosersokugethat

’

e > c,+J(l+r)+.

. .+q/(l+r)*+‘,

8 > i,+*/(l

a . +iil(f+r)f’J

+r)+.

(2 .6)

and r8 > i’.

(2 7) l

that the consumption E lies within the budget set at intemt rater and hencethe consumption c(r) actually chosen must be preferred to CT. Condition(2) on the other hand shows that the first j- 1 coordinates of c(r) must be kss than 8 and c,(r) < ii+43 in order to lie in the budget set. It follows fmm the increasingpropertyof preferences therefore that at least one 1 and henceb(r) is positive. t&(r), k > I, must ex Symmetrically let j be the largest subscript such that lj > 0 and choose a IfgO- .O) which is preferred to (I, 1, Condition

(1)

assure%

l

E=(co

,...

+&43,0

- I that 8 * (1 +r)+a+. a .+(I

,...,

O)>(I,...,l,il+e,e

,...,

l

l

8).

Now choose I so closeto

‘and -a P (hr>c,. argument is now the same as for the previous case. e > (1 fr)c,_~+(l

The

+r)‘c,,

-tr)ic,

( 2*8) (2.9)

Theotem

1.

&u&r assumptions (i) and (ii) a steady state equilsibriumwilt

always exist.

m From Lemma 1, S(r) = -rd(r). Hence if S(r) = 0, for some r + 0, tha d(r) = 0 and we have equilibrium. If not then since S(r) is positive for r sa791and r large from Lemma 2 it follows that 6(r) 2 0 for all r (otherwise by cxrsltinuity it would have to cross the r-axis at least twice). In this case however we must have d(0) = 0 for ifd(0) # 0 then by choosingr close to 0 and with the opposite sign from w(O)we would have rd(r) = S(r) < 0. n There are really three cases possible according as the equilibrium value of r is positive, negative or zero. We will refer to these as hormal, abnormal and coincidentalcwzs respectively.0f course there is no reason from what has been said so far to assume that the equilibrium r is unique. However, I con,btiv:re that for most of ths popular utility function uniqueness will indeed hold. This is easy to show for examplein the CobbDouglas case. I;llheotem2.

Let

the utility function u(c) be given by

u(c)=~a,logck,ak~O,~aL=

1.

(2.10)

Zkn there is a uniquestea& state equilibrium.

m We compute the excessdemand function S(r) by maximizingu(c) subject to the budget constraint (1). Solving the simplecalculus problem gives c,(r) = ~(1 + r)’ C ij/(l

+r)j,

(2.11)

so

S(r) = C (ck(+jJ

= C (ak(l+r))‘C iJ(l -t+Qk-_Cik,

and (1 + r)‘Ei(r)= (zak( 1+ r)")(z

ik(1 + r)*-9 - (x Q(1 + r)‘,

which is a polynominal of degree 2n in (1 +r) and all coefficientsare positive except that of (1 +t)“. Therefore by Descartes’ rule of signs it can have at most two roots. One OCR them is of course r = 0 and the other gives the unique equilibrium.H 2.3.

The

trade imbalance

theorem

We now suppose there are m countries each with its own population, endowment and preferences satisfying assumptions (i) and (ii). Let +(r) = rdj(r) be the excess demand function and wealth of Cj as a function of r. Then the world excess demand function is S(t) = &(r) and the world wealth fwcbon is

d(r) = Et&(r) so we have the world idknti?y-8(r)

= d(r). A worlde@ltirkm

corresponds to a value of r such that d(r) = 0 [implying, of course, 5(r) = 01. Such an equilibrium will always exist under our assumptions since S(r),being the sum of the Gl(r),mustsatisfyLemma2. Of courseat equilibrium the excess demands of the individualcountries CJ will except by co&denoe not equal zero. Let r’ be a worldsteady stute equihWuminterest rate. &Wry Cl will be a net importer or exporter at this equilibrium according as 6(F) is positive or negative. Let us now make the assumption that each country Cl in the autarkic state has a unique steady state equilibrium with interest rate ?“. The imbalance result is then the following: Theorem3.

Let

r’ be

a worlijsteadystate interestrate. Then:

(a) Ifr’ > 0 (normakme) the@C’ willbe m importerat eqzdibriumsf r’> ?”and anexporter 5/r’ < Fl. (b) If i < 0 @dno~ if r’c q.

case) then Cj will be m exporter r2 > rI an&m exporter

(c) In the coikidenttalcare ii = 0 al! countrid willhme a balmce of trade. m Part (c) is obvious for if r’ = 0 then &(F) = Od’r) = 0 for all j. Let us consider (a) with r’ > FI. There are two possibilities: either f, is positive or negative. If ?j < 0 then 6&r)> 0 for r > 0 for if S(r) < 0 for some positive r then, since S(r) is positive for r large,there would have to be a second root r, of i&(r) hence of d’,(r),contrary to assumption of a unique tiutarkic steady sate. If P > 0 aben by the same reasoningSj(r) > 0 for r > Fj [since sj(r) > 0 for r large] so in either case aj(F) > 0 and C, is an importer. If r’ < F)then since we are in the normal case f’ > 0, and we see that si(r) <: 0 for ‘0 < r < r, for if 6&r)were positive somewhereon this interval it would have to have an addition4 root. [If 6hr) 3 0 throughout the interval then d(O) = 0 then 8 has a ‘double root’ at 0 which is considered a violation of uniqueness f*.] Hence S(F) < 0 and CJ is an exporter. We leave to the reader the proof for case (b) which is completely symmetric to the onejust given. B Please note that we have not assumed that the world steady state wits unique. Even in the Cobb-Douglas case in which all countries have unique autarkic steady states it is posrible for there to be different world steady states at different interest rates. This does not affect of our theorem. Also the dichotomy between normal and abnormal world Headystates depends only on whether the world interest rate is positive or negative, so, for instance, in a normal world steady state countries with i;J< r’will be importers even if their au&xl& steady states are abnormal, that is, negative.

D. Gale, Tr&

134

imbahce

2.4. Goinsfrom trade IIaving observed that in the pure exch;Lagemodel the operation of the market will lead to permanent imbalance we again ask, as in the case of our earlier model,, whether both importer and exporter are better off than they w~ulfi have been on their own. We confine ourselves to the steady state only. Let each c;ountry compare the lifetime satisfaction of a typical citizen in its autarkic steady state with this same quantity in the world steady state. Will it always be the case, as in the Solow model, that this second quantity will be at least as great as the first? Unfortunately this will not always be true. On the other hand, there is a simple device bywhich a country can assure that if it enters into trade with the rest of the world its citizens will be no worse off at any time than they would be in their own autarkic steady state. Namely, the country Cr calculates its autarkic steady state consumption program c’j = (& . . .P @. It then redistributes endowments between people of different ages so that a person of age k remives the amount 8: instead of $. We now give the familiar argument. Each individual in planning his lifetime consumption always has the option of consuming his own endowment, and it therefore follows that the consumption schedule he actually chooses will be at least as satisfactory and in general strictly better than the schedule 8. The fact that some form of transfers are necessary in order to assure this should not surprise us, for even in the static case it is generally necessary to make some redistribution of goods to insure that no individual trader is hurt when his country decides to engage in trade with the !outsideworld. 2S. Stability, an example

In this section we will go as far as we can at present in deriving stability results similar to those obtained for the Solow model. I expe& rather general stability results hold. We content ourselves here with a single example, a generalization of one used by Samuelson (1958). We suppose that people live for three periods and that their lifetime utility function is given by d+q 9 c2 9 c3) = log Cl +log

c2 +log c3 e

(2.12)

Thus, people’s desire for consumption remains the same throughout their lives. (We could have put in “impatience’ by introducing coefficients al, a2, a3 as in the previous section. This would complicate the analysis somewhat but would not aB&;;:t *he ha1 result.? Now in order to describe the non-steady state behavior of this model it is necessary to specify some rule as to how people make comsumption decisions so that we can define a demand function and a competitive program. Among the various possibilities I will choose what appears to be the simplest one; people obsenkng the interest factor p = l+r today go on the assumption that this

135

D. Gale, 7kadk imbake

same interest factor will prevail for the rest of their lives. Of course, this naive forecasting rule turns out to be completely correct only if one has already arrived at a steady state. However for typically fluctuating quantities like interest rates it is not clear that more sophisticated forecasting methods would -bea great deal more reasonable than the naive one, Again we look first at the case of a single country where people have the income vector i = (i 1, i2 B i3), and we suppose that the interest fator in period t .is p1 = 1 +r,. The young people then calculate their expected lifetime wealth w. as the present value of their income stream and get

To maximize utility they will choose a East period consumption c1 to be Cl =

u1+ 12lPtf

h/Pm

*

(2.13)

N”owthe middle aged people enter period t with a certain amount of wealth wt, this being the amount they saved or dissaved in their youth. Their future expected wealth is fz + i3/pt so the total expected wealth w1is given by

and in order to maximize utility log c2 + log c3 they will choose their consumption c2 to be c2 =

(2.14)

(W?+i2+23/pd2

(again we leave to the reader the job of solving the constrained maximum problem). Finally, in period t the old people enter the last period of their lives with wealth - wt (this is again the balance condition which requires the aggregate wealth of the whole population to net out to zero). To this will be added endowment jS and the sum will be spent entirely on consumption giving c3 =

-wg+i3.

(2-W

if we

take the number wt to be given by past history then the interest factor pt must adjust so as to exactly clear the market, that is, so that cl + c2 + c3 = iI + i2 + i3.Adding eqs. (2.131,(2.14) and (2.15), and solving for w, gives Now

3w, = - 2iJ3 - iJ2 -t (i2/3+ i3/2)/p,

+

i3/3p,2=

#(pJ

.

(2.W

Note that #(pJ is a decreasing function of p so that pr is determined by w, provided wt > -2i,/3 -i,/2. But w,+~ is now determined by pt for IV,+1 is just the savings of the young people in period t multiplied by pt and we have 3w,+i = p,(i1/3-i,/6~,-i,l6~~)

= #(at)~

(2.17)

Eqs. (2.16) and (2.17) provide a first order recursion which describes the C

evdution of -the model over time. The condition for a steady state is that w, = %+1. Equa.ting (2.16) and (2.17) gives the equation for the steady state interest factDrp, i; +(24 +i,)/p-(i,+2i3)/p2

= is/p2 = 0.

(2.18)

Notice that the condition’for p to exceed 1 is exactly that ij exceed i1 The analysis of the stability of the steady state (2.17) is now a standard exercise in so-called ‘cob web theory’. The functions 4 and $ have been plotted in fig. 1. Notice that from eq. (2.15) # is convex and decreases from + 00 to -2i,/3 - i2/2, while from eq. (2.16) 9 is concave and increases from - co to + oo.

-

Fig. 1

me successive values of (p,, WJ will spiral around the point ($, fi) and will either converge to it or approach some limit cycle about it. The condition for convergence is precisely that the sum of the slopes of the two curves should be negative at @, @).Using eqs. (2.16) and (2.17) we get 3[4’(p) + #‘@)I = il - (i2+ Q/p2

-

24/p3.

(2.19)

As noted if p is positive then i3 exceeds iI. It follows that for values of p close to f the right side of eq. (2 19) will be negative. p don’t know whether there are any values of i,, it, i 3 which make eq. (2.19) positive. It could be conceivable for eno’;?rill~svalues of p.] It follows that the trade pattern will converge to (JF,@). The above analysis considers the case of a single country. However it applies as well to any number of countries cj each with the utility function (2.12) but with different income streams ii = (if, ii, ii). To see this simply observe that the dynamic equations (2.16) and (2.17) apply exactly as before if we interpret the quantities is, i2, 4 as aggregates, thus ik = X&p and W,= C,w{.As the numbers

pI convtzrgetoward the steady state value j5 the consumption in each of the countries will converge to steady state vailuesand the whole system will approach an equilibrium of the sort analyzed in the previous section. References Bardhan, P.K., 1966, On fwtor accumulation and the pattern of international specialization, Review of Economic Studies 33, no. 93,39-44. Gale, D., 1971, General equilibrium with imbalance of trade, Journal of International Economics 1, no. 2,141-158. Green, J., 1972, The question of collective rationality in Professor Gale’s model of trade imbalance, Journal of International Economics 2, no. 1,39-56. Oniki, H. and H. Uzawa, 1965, Patterns of trade and investment in a dynamic model of international trade, Review of Economic Studies 32 (l), no. 39,15-38. Samuelson, P.A., 1958, An exact consumption-loan model of interest with or without the social contrivance of money, Journal of Political Economy 66,467-482. Samuelson, PA., 1972, Economics: An introductory analysis, 9th ed. (McGraw-Hill, New York;& Solow, R.M., 1956, A contribution to the theory of economic growth, Quarterly Journal of Economics 70,65-94.