On Trade in a Two Country World

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UNIVERSJDADE NOVA DE LISBOA

F acuidade de Economia

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liON TRADE IN A TWO COUNTRY WORLD" Vasco d'Or2Y Working Paper NQ 8p .; #/ /

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UNIVERSIDADE NOVA DE LISBOA Faculdade de Economia

Travessa Estevao Pinto

ON TRADE IN A TWO COUNTRY WORLD

by Vasco d·Orey·

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* Universidade Nova de Lisboa, Faculdade de Economia. Travessa Estevao Pinto, 1000 Llsboa, Portugal. I am grateful to Antonieta da Cunha e Sa, Luis Cunha, Diogo Lucena and tvlario Pascoa for their comments. Denis Cory must be ackno\vledged for a reading list relating to natural resources. Editorial . assistance of John Huffstot is also acknowledged. Any error

is

01Y

responsability. ,

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This paper sho'~"s that, under tf}e ·usual assumptions of the Hecksher­ Ohlin-S"amuelson nlodeL extensivaly used in the pure theory of international trade, free trade can be seen as the soiution to a Nash" c90perative game. The . question of free or fair trade is then' related to different admissible solution

concepts to a two person cooperative game.

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Ir{rRODUCTI()t~

The standard approach to the pure theory of international trade

assunles~ in its utmost sinlplicity and beauty, that countries can be described by

endowments of factors of production! usually assumed to be capital and labor,

technology rel-ating the use of factors' and the output of two goods, and the

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preferences of nationals of each country. these being defined over different

.consumption bundles. Under autarchy each country produces efficiently, and

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.any difference in factor endo"~m'ents, technologies or' tastes will result in

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different _. autarchy relative prices for the goods produced. Any such difference

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can. ~hen be exploited, and as the factors of production, capital and labor are assumed to be mobile \vithin eacQ country. but not between both countries. the exchange of goods produced (international trade). is t~e on~}' avenue left in thi~

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.\ endeavor. Once it is established. and a world

relativ:e-/

price. for the goods

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produced is found. one g'oes\on to prove that under the same conditions. both

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II trading partners benefit from this situation. Trade is not usually caused by a

strategic beh,~vior, but rather, by the casual observation that there are

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differences in both partners that can be exploited for their mutual benefit. As a·

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result, when t\VO countries of equal dimension are considered. in the absence of tariffs and quotas. the outcome of this process is usually called a free trade solution.

In the traditional approach. the cooperative behavior that a free trade solution fnay entail is totally ignored. After all, it has been long known within the profession that it takes two to tango, and the autarchy point is always available for any trading partner. One can then ask if free trade can be arrived -at after a strategic behavior is adopted by each country, a question that appears .

-never to have been explicity spelled out in the literature. Our main point here is that, under the usual assumptions that are made in the standard Hecksher­ Ohlin-Sanluelson modeL frl~e trade can be seen as a solution to a cooperative

ganle played by the t\"vO countries. Under this ~pproach' fr~e trade may be seen

as the consequence of SOOle differ;ence' bet\veen the t\yo countries, but is arriv~d at after some rules, (which nlust be' obeyed by t,he solution to their cooperative . approach to the joint welfare maximization), are agreed upon by the t\\'"o

players, (the 't\VO countl~ies).

The re!nainder of the paper will proceed as follows. Section 1 '\1lill show .that, under the usual assumptions of the Hecksher-Ohlin-S.amuelson model freeI

,trade can be seen as the solution to 'a Nash cooperative ga'me 1 _{that is played by}

both countries, Section I I will be devoted to the discussion of some other

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possible Qutcom.es of a cooperative game played by the two countries, and relate

them the ongoing arguments on free and fair trade. Section III will present a brief summary of the Inain conclusions presented here.

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L FREE TRADE AS THE OUTCOME OF A COOPERATIVE GAME.

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\Ve

will

assu me in this section that the usual assu mptions characterizing

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the Hecksher-Ohlin-Sanl uelson model are satisfied. There are two countries, A and B, t\VO factors of production, capital (K) and labor (L) and 1'\1l0 goods being produced in each country, good X and good Y, the technology of production being. the sanle in each country. Consu mers have preferences that are specified by an ordinal utility function, U

(C~

,

C~)

and V

(C~

,

~)

being, respectively, the utility

function of a As and Bs representative consumer, where

cl

is the consumption of good i-X,Y in country j-AIB.

1 Nash's solution of a· two-perS(ln cooperative game is consist~nt with four axions: i) group

rationality; ii) Vnn-Neuman-Mol'genstern invariance or-utility indicators; iii) Sy,mmetry

of players: iv) Independence of irrelevant alterna.tives. (See Nash.·1953. Friedma.n. 1986)

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Given the endO\Vn1ents of capital an~ labor,' each economy will be

producing on its production possibility frontier. \vith. the meaning that for a~y t

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feasible 'production of good X the maximum fe.a~ible anlount of good Y is being

- produced. Let t

t

OA _{'< X - •.. ',;} - v f ()- A) _{y /} V "'0 (.1 ) t I • < 0, y < (2) f

t

-be, respectively, the production possibility frontier of country A and country B,

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,

assuining that the technology of production and the factor en.dowments in each

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country are given: and where

Q~

IS the maximum amount of}ood X produced:

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given-that the amount

Q~7

of good Yis being prodl!.ced:

w¥~e

j-A.B .

.\

The possibility of trade allows domestic consumption of any good to be

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different [roal domestic production of the good. Thus, 'Vlhen trade is allowed for, Ii I we haye that

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(4)

CB

_x

= QB

x

+ !v!

(5)

(6)

where IV! stands for the amOiJnt traded of good X and E stands for the amount traded of good Y. E(~uutions (3), (4), (5) and (6) have t.he usual meaning that,

once trade is allo\Ved. consunlption of any good

in

any country is equal to dOll1estic

production

plus impor.ts or

minus

exports of the satne good. The requirenlcnt that, at the world relative price, trade must be balanced can be , expressed by

E p!vI == 0 (7)

meaning that, at \vorld relative prices, P, exports and inlports balance:

In autarchy each country. produces and consume's so that the domestic Inarginal rate of sU.bstitution in consumption equals the domestic marginal rate of s~bstitution in production, the autarchy price level prevailing in country A and B being deter D1ined by this relation. As autarchy is always a possibl~ reginle, the utility levels associeted with the no trade situation for countries A

.

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.\ and B'ure the natural threat points of the Nash cooperati,(;e game. Giver.. that we'

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are U,ssu ming cardinal utility\ functions, we take the utility levels associated "\ttith the autarchy situation

as

being zero.

It is \videly t:now that under the Nash: cooperative behavior the solution to the galne can be found by the cooperative maxinlization of the product of the utility gains accrued to each player, these being measured from its threat point. Once that both players adopt the Nash behavior for t11e cooperative trade ganlet

. the solution can be found by:

Max U (C;.

C~)

V

(C~ .~)

(8)

subject to the restrictions imposed by (l) to (7). As it is shown in the appendix, the solution to this problem can be characterized by the following conditions:

(9) (,

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t

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t

!

t

t

.,.

l . !

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.,

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\vhere IvIRSi and IvlRTj are., respect~velYt the marginal rate of substitution in consumption and the marginal rate of transformation in production in country j=A,B. \Vhat we have obtained in (9), taking into account that (7) is satisfied, are·, the conditions characterizing a free trade solution, meaning that the outcome of a cooperative Nash bargaining game between two countries, when balanced . trade is allo\ved, is the free trade solution. This res1:1lt can be most .easily

interpreted if we take into account the analysis developed 'by Roth (Roth, 1977).

This a.uthor has shown that, under the conditions we have required for the

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bargaining game between the two countries, the result of the game will be Pareto optimal. The pure theory of international trade has been telling us that free' trade is, from the \vorld point of· view, a particular Pareto optimal (Bahg\vati and Srinivasan, 1986). Given the restrictions we are imposing on the

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ganle,. the outcC?me will be a Pareto optilllunl, with the ad/dftional property that

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trade \vill be balanced. T~le result follows in an obvious way.

II. TRADE - FREE OR FAIR?

The last section may shed some light upon a discussion that seems to· be in the order of the dllY. Should trade be seen as free trade or as fair trade? Once \ve have established that, under some very mild and usual assu mptions, free trade can be seen as the Nash solution to a cooperative game played by the two

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countries, \vhy should the Nash solution be the one agreed) upon by both trading partners? It is widely known that some other solutions have been proposed as a solution to a cooperative two person game, The Raiffa-Kalai-Smorodinsky_.

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solution (Kalai and Smorodinsky, 1975) and the minimal expectations solution (Friedman 1986) can be given as examples of different solution concepts -that are perfectly plausible to generate outcomes for the cooperative game that the

existence of

trade implies. Any solution

to the game

that

will result from the

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cOlnnl0n adop:ion of playing rules; that are different from. the ones characterizing the Nash solution, may viola~e some of- the conditions characterizing this one. meaning that some optimafity condition related to- a . world\vide Pareto optinlU111 situation, accompanied by balanced trade, may be violated. In this \vay both countries, -by common agreement over a set of bargaining rules, may be disposing of the free trade solutiOn, as the outcome of the trade game Inay be different from this situatio1).. The question of free or f.air trade is not passe (Krugnlan, 1987). Rather it can then be transposed to what

t

kind of zolutiGn concept is adopted as the 'Solution to the cooperative game

which is played by the two trading partners. In this

way

fair~ess will not be a

characteristic of the trading pattern that is arrived at, but rather, a property of

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the rules of the cooperative game \vhich

before the actual play has taken place, as

ganlt~.

IlL CONCLUSiONS

are agreed upon by both countries

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these determin~/the

outconle

of the

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I n this paper it was show~ that free trade can be seen as the result of a Nash cooperative game played by two countries, these being characterized as in the standard pure theory of international trade. The question of free or fair trade was also touched upon, and it was pointed out that, in the light of the previous result fairness should not be seen as a property of the trading pattern, but rather, as a property of the rules which the solution of the cooperative game nlust obey

APPENDIX

. The lagrangean corresponding to the non-linear optimization of (8) in the rnain t~xt, subject to the restriction (1)

to

(7), can be written as:

t'I 11 ( ... ACA) \' , ..BCB) ') (CA 'QA 'E)

~

(C' A 'OA M)

h "" ..) ,C X I y' l

ex

I Y' + ,. 1 '

x - x

+ , + f'. ~ Y -'I. Y - 1 +.

.Partial differentiation of (A 1 ) Jeads to

o"e

.

B B· aU A B. -A := V (CXlCyJ ~ (Cx.CyJ ~ 1'-1 = 0

ae

_x

.

ac

_x

a,,€

'.

B B

au

A A .

A

~ v (cx,e))

A

(Cx,C_{y )} -!- i.. 2 = 0 dey _oCy

aJ.

A A

av

B B

",B = U (C_xICy)

13

(ex ICy) + A3 =

o.

ac

'

'vC

x

x

a;.

~ A A "

a

V B B

13 ""

u (ex ICy)

B

(cx.e

_y) + A4- 0.

ac

oCy

y

a..e

9 (AI) .,I ~/ I I (A2)

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(A3) (A4) (AS) ! (A6)

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LA7)

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(AS) (A9) (AlO) . (A 11) a,~ - = - A7 rvl = 0 (A12) ap I

The partial derivatives of the lagrangean relative to _{. ,} the lagrange

nlultipliers are. just the restrictions. Assuming an interioyF/ solution, incomplete

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specialization and trade, .IV1 \vill be diffrent from zero so' that A 7-0, from tA 12).

. . . \ . .

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It follo"\vs from (A 1 0) and

LA

11 ).

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f

. (A13)

. f

Dividing. (A3) by (A2). (AS) by (A4). (A7) by (A6) and (A9) by (A8). it

can be found that

I

- - - - ­ =

= -

y'

=

'1'1

proving (9) in the nlain texL

REFERENCES

Bhagwati, Jagdish N. and Srinivasan, T.N., "Lectures on International Trade", Cahlbridge, l\lassachussetts; The _. ~1.I.T. Press, -1983.

,

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'Friedl11all, Jiltlles \V., "Garne Theory \vlth Applications to ?conomics", New York: Oxford University Press, 1986.

Kalal, Ehud and Smorodinsky, K1eir, "Other Solutions to Nash's Bargaining' Proble nl" EC01101netrica. 1982, 43, 513-18.

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Kruglnan. Paul R .. "ls Free Trade Passe ?",The Journal of Economic Perspectives. Fa111987. vol. 1, 2, 131-44.

Nash,

J,

"T\vo Person Cooperative Ganlcs", Econometri~a, January 1953, 2 L 128­

40.

Roth, A. E., "Individual Ration.ality and Nash's Solution to the Bargaining

Problem", 1\1athenlatics of Operations Research, 1977,2,64 - 66.

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nQ 73 - ANTUNES, Ant6nio :Pais 'e GASPAR, I Vitor : II Tributacao ,

In(81)ti'?OS e In";testimentos:' A:nalise Qualitativ"S,".

(Novembl'o, 1987).

Jo~e Pedro: 1I0ptiIilal, Wage Rigi~ity: A Suggested

t,

.~

!1ethclIjology to Test the Theory and an Application".

(Outubro, 1987) .

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nQ 76 - CARPAL, L.tl. B.; "Three ~totes on Syu.etrlc Gam.es with

A3}l'12lJlletric Equilibris'l _{(Novembro, 19,87).} •

n~ 76 - LtTCE!l8., Diogo: liTo Search 'or Not to Search". (Fevereiro, 1987)

nQ 77 - SA,. ,Jorge ':lasconcelos e: itA Theory of Synergy". (Fevere1ro,

1987) .

n9 78 - VILARES I l1anuel .Jose : "OS Bens Intermedios Importe:dos Como

Factor de Produc~o". (JulhO, 1987).

Jorge Vasconcelos e: "How To Com.pete And COlll]jl1.micate in

. rIa t.ure Industrj.a.l Product;:". (l1aio, 1.~-68).

,\

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r;,9. 80 - POE, Rafe.el: ."Learninl;J and Capacit}.. ,Expansion

IN

A. New !Iarket. Under Uncertaintyll. (Fevereiro, 1988).

PEREIRA, AI! redo l1arv";~o: II Sut~...ey ,of Dynamic Com.putat.ional

Gen:;:;f.3.1 Equilibrium. l10dels For Tax Policy , E'It8.11.~3.t.iollu. (Outubro, 1987). .

112 82 - SA, Jorge Vasconcelos e: "Everything IS Important: But Some

Thinl;Js Are l10re Im.portant TrJal'). OthersII. (l1alo",

1988) .

n2 83 - BAHeIA, Pa.ulo, GASPAR, Vitor e PEREIRA, Pedro T.: "Consumer

" Tnecl"Y Wi th NonConvex COll!iUDlptic.n 5e tsII •

(llaio ~ 1988) .

nQ 81 - CJ...EiF:AL... Luis l1a.rtins Bara ta: "Politica de Precos do Sector

das TelecolJl1.Ulic:a 90e3 : Uma Analise de Equilibrio Geral". (Junho de, 1985; Revisto em. l1aio de 1988).

11£ a5 - FEREIPA, Alt redo l1aryao: "DAGEN- A Dyne,Dlic Applied General Ecp.tilibri.UlA tlodel For Tax Policy Evaluation".

(.Abril, 1988).

nQ 86 - D' OHEY, Vasco: liOn Trade in a Two Country World u, (JU11I.'10 ~

1988) .

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