Free trade equilibria to multi-country quota games

Journal of International

Economics 27 (1989) 319-333. North-Holland

Department of Economics, Queen’s University, Kingston, Canada K7L 3N6

Alice ENDERS* GATT Secretariat,

Geneva 1203, Switzerland and Department of Economics, Toronto, Canada

York Vvliversity,

Received December 1987, revised version received November 1988 We consider a symmetric N-country competitive general equilibrium endowment economy in which countries select quotas in a static Nash game to maximize the welfare of representative citizens. Unlike tariffs, quota revenues need not accrue solely to the quota-setting country. The only possible equilibrium outcome, other than autarky, is free trade. Free trade prevails unless the quota-setting country receives a fraction of quota rents greater than (N - 1)/N. We conclude that the prevalence of binding quotas, in contrast to tariffs, cannot be explained within the accepted rubric in which countries maximize the welfare of representative citizens.

With the restrictions placed on tariff levels by the General Agreement on Tariffs and Trade (GATT), the strategic use of import and export quotas is of interest. Tariffs and quotas, however, have different properties [ (~HZZ)j_ Pnncider s~saaj~ Nach I i ‘i_Z-;i;Githe s@m-iiyl ___i_ trgde i&-__ - gqsg ic urh;ph .. +=-3izes& -_-VP&rw~rrf+v G-w rr--.i J implements its policy instruments against its trading partners to maximize the welfare of a representative citizen, taking the competitive behavior of firms and consumer optimization as given. In tariff games, tariffs always bind on domestic prices so tariff revenues are continuous in tariff rates. Consequent!y, there is generally an interior equilibrium to th tariff game featurin positive tariffs an3 trade [Johnson (1954), Thursby and In contrast, export quotas, for example, do not bind when domestic net imports fall short. As domestic imports rrse to the export quota level, a *We are grateful for suggestiorbs of the two anonymous referees and the editor of the Journal. Steve Neston improved the paper. The first author acknowledges support from the Social Sciences and Humanities Research Council of Canada. We accept responsibility for any errors which remain. 00224996/89/U

50 @“l 1989.r~ Elsevier Science

3X

D. Bernhard! and A. Enders, Free trade equilibria

country’s quota revenues leap from zero to a positive level. Hence, quota revenue is a discontinuous function of quota levels. Consequently, with quotas, when all rents accrue to the imposing country, autarky is the unique librium outcome to the static quota game [Rodriguez (1974), Tower (i975)]; each country seeks to alter the terms of trade in its favor by tasking its quota be the sole binding quota. All gains to trade are eliminated.P To rationalize the existence of non-autarkic quotas within a context in which governments select (either import or export) quotas to maximize welfare, it is necessary to invoke an explicitiy dynamic setting. In a supergame structure Enders (1987) demonstrates that less restrictive quotas can be supported by threats to revert to harsher (perhaps autarkic) quotas if a country deviates from its specified quota level. In practice, tariffs and quotas also differ as revenue raising devices. While it is reasonable to view tariff revenue as appropriated and redistributed by the government, quota revenues typically accrue to the shareholders of the importing and exporting firms. In applied general equilibrium analysis, this distinction has been ignored as authors assume quota rents are appropriated by a government through an auctioning mechanism’. The assumption that all rents to a quota accrue to the imposing country is extreme and contrary to the practice of OECD countries. In this paper, intermediate divisions of rents are explicitly considered. Our intent is to rationalize the existence of less extreme quotas without resorting to a dynamic infinite horizon environment. The results we obtain remain stark: equilibrium quotas are generically either non-binding (i.e. free trade is an equilibrium outcome) or autarkic. Within a symmetric, N-country endowment economy in which consumers have identical Cobb-Douglas preferences, free trade is a Nash equilibrium outcome unless the quota-setting country receives a fraction of quota rents greater than (N - 1)/N. Essentially, as the number of countries composing the trading world rises, it is as if a country contemplating introducing quotas faces an ever-larger trading partner + so the gains to manipulating the terms of trade fall relative to the costs of the distortions generated by the quotas. When some of the quota revenues cannot be retained domestically, if the number of countries is sufficiently great, it may not be optimal for any country to restrict trade by quotas, To contrast, we examine the equivalent tariff game in which countries deploy tariffs, from which they collect all revenues. There, an interior -Nash equilibrium characicrized by positive tariffs and trade generically exists. If ‘The analog in oligopoly theory is Cournot and Bertrand competition. In Cournot, profits are continuous in quantities and an interior equilibrium typically exists. With Bertrand, there is a discontinuity in profits when prices are equal, as each firm seeks to undercut its competition to gain the entire market until all profits are competed away. ‘For criticisms of the auctionmg asstunption, see Falvey (1985).

D. Bernhardt and A. Enders, Free trade equilibria

321

both commercial policy instruments are un never implemented, -while tariffs are. If tariffs GATT) to be less than their equilibrium levels retains the lion’s shares of quota revenues [m outcome is autarky, there is an equilibrium in selected. While these exact results clearly derive fr specification of preferences and endowments, the likely outcome when quota rents do no setting country is surely robust. In particular, not alter the outcome. Since casual observatio of quotas and a less extreme distribution prevalence of binding quotas cannot be expl in which each country maximizes the welfare must search for alternative explanations in w within a country is important and which invol rents within a country.

ricted, binding quotas are constrained (perhaps by n again, unless a country han (N- l)/Nj so that the ich non-binding quotas are

In our simple N country economy, let Xii representative consumer of country i of good endowment. The preferences of the representat are given by

resent consumption by the nd let %ijbe the associated onsumer of each country i

Ui=

nXij

[

e particular (but standard) plication that free trade is true solely to the quotaeducing production should dicates both the existence nts, we conclude that the within the accepted rubric epresentative citizen. One the distribution of rents barriers to redistribution of

9

1

i

and endowments are ‘anti-symmetric’ so that X,

=

2,

OQsl,

for i=j.

To normalize aggregate endowments of each good to one: 1-z qj=- N -

19

foriZj,

SO that Cy= 1 Xii= 1. Comparative advantage requires the country’s share of the world market for its export good, z, be greater than I/M3 3More precisely, endowments are sold by profit-maximizing firms whose shares are held by the representative consumer. Given the quotas (or tariffs) these firms allocate their endowment guods to the home and fureign markets to maximize profits. Since in our environment these profit-maximizing allocations correspond to those made when the representative consumer owns the endowment directly, we drop the formalism of firms to simplify exposition.

D. Bernhardt and A. Enders, Free trade equilibria

322

Each country

i

has as its commercial instrument a vector of export quotas:

which it may set against its (IV- 1) trading partners. Countries i and j receive fractions a and (1 -a), respectively, of the nonnegative export quota rents associated with its export of good i to each country j, where a is a common parameter representing the domestic share of total export quota rents.4 The unit revenue derived from the quota is equal to the difference between the price of its export good domestically and abroad. While we explicitly examine the use of export quotas as the sole instrument of commercial policy, the analysis turns out to be without loss of generality: for every set of export quotas there is an equivalent import quota vector, and conversely. A country that retains a proportion a of export quota rents will retain 1 - a of import quota rents, on the argument that one coufrtry’s export quota is exactly equivalent to its trading partner’s import quota when the levels are identical. Thus, the choice of export over import quotas as commercial policy instruments depends solely on a. For expositional clarity, without loss of generality, we restrict attention to export quotas (implicitly then a 2 0.5). Letting fi denote the vector of quotas on which equilibrium prices and income depend, the representative agent in country i chooses consumption levels Xij(fi) of the j = 1,. . . , N goods to maximize utility, given the budget constraint: max Wi Xij

S*t-C Pij( j

fi)Xij(V) =

Illi(f

where pi,(V) is the price in country i of good j, and rnA6) is country i’s endowment income given the quota vector 6. Country i then selects export quotas to maximize the welfare of its representative consumer taking as given the quota vector 6-i of the other (IV- 1) countries and competitive behavior by consumers and firms:

4At the end of this section we discuss a two-stage game in which LXis chosen endogenously.

D. Bernhardt and A. Enders, Free trade equilibria

323

where the competitive market-clearing prices and the represen;a2ve consumer’s income depend explicitly on the quotas selected. Cobbpreferences yield demands for good j in country i of

so country i’s optimization problem can be rewritten as

max n Bi (

j

mi(iri,fi_i) NPij(fii,

C-i)

' 1

A Nash equilibrium is a set of quota vectors (@, . . . , PN) such that each country i is maximizing the welfare of its representative consumer, taking the optimal quota selections of its trading partners pi as given. It is immediately apparen t that one equilibrium outcome is autarky which occurs when each country selects the zero vector as its export quota vector. Given that no country exports to country i, i maximizes the welfare of its representative consumer by not exporting anything in return. In addition, if a country receives a fraction of rents such that each country would unilaterally deviate from free trade by setting a binding quota, then some country would deviate from any given set of hypothetical equilibrium binding, non-auiarkic quotas. Since a country’s net imports are bounded from above by fi-i, it maximizes consumption by maximizing wealth and minimizing net exports. At any potential equilibrium with binding quotas which is characterized by positive trade, some country deviates to make its quota bind and the quotas imposed against it slack by a further restriction of trade. See Enders (1987) for a more complete de4opment. Consequently, to find equilibria in which ccuntries set non-autarkic quotas in a static quota game it is clear one wants to find the conditions under which countries find it optimal not to set binding quotas at all. This, in turn, Depends on the share, a, a country retains of the quota revenues, which determines the real income the country derives from the imposition of the quota. The question then can be posed as follows: Given that countries 2 through N do not set binding quotas, how small need country l’s share of the quota revenues be for it to find it optimal to set non-binding quotas, so that free trade emerges as a Nash equilibrium? To address these issues first recognize that we can simplify notation so that the only country setting quotas is country 1 sets export quotas

D. Bernhardt and A. Enders, Free trade equilibria

324

against its (N- 1) trading partners.’ An export quota level may be binding on a country’s net imports, in which case country 1 receives nonnegative export quota rents, the unit revenue equal to the difference between the price of good 1 in the (N- 1) countries with which it trades and its domestic price (normalized to unity), according to a E [0, 11. Note that a nonbinding export quota generates no rents, and does not affect equilibrium prices. As a result, export quota levels are assumed either binding or (without loss of generality) without slack so that they are equal to actual levels of net exports. For each export quota vi for i= 1,. . . , N, country i receives a share (1 -a) of export quota rents. With non-binding quotas, prices of country j’s export good in each country have a common world price pij=pj, i= 1,. . . , N; j=2,. . . , N. Incomes are then Q,V3,

l

l

l

9vN

m,=z+ fi

CPi+gC(Pil-1)vi9 i i

1-Z mi=PN-l Pjl +zPj+N2

i

&Pi+(l-a)(Pjl-1)s. .*

for j=2,...,N,

where, for convenience, we have dropped the formal dependence of the variables on country l’s quota vector. General equilibrium requires the markets for good 1 across all (N - 1) countries balance? Vi=Xil--

1-Z N - 1, fori=

,..., N.

Also net exports must sum to zero in the world markets for goods 2,. . . , N so that N

c xij=l,

i= 1

for j=2 ,..., N.

(2)

3n our forma 1 analysis markets are bilateral and a world market for goods which are quota constrained does not exist. I! country 1 sets N- 1 different binding quotas, there are N- 1 different country specific prices for its export good. If countries resold their quota-constrained imports on a world market, prices would equalize in each country and each effective export quota would equal the average export quota. wever, the existence of a world market is irrelevant since the indirect payoffs of a represen e citizen in a country considering deviating from free trade are concave in the quotas its government sets. Consequently, its government optimally treats its (M- 1) trafling partners symmetrically, so its optimal behavior wtrh and without world markets for quota-restricted goods coincide. Free trade is an equilibrium outcome in the one iron t if and o if it be supported in the second. Wsing ras’ , the mar for 1 in country 1 is ignored.

D. Bernhardt and A Enders, Free trade equilibria

Lemma 1.

325

World prices of goods 2 through IV, are equal:

pj=p,

j=2,...,lV.

for

Proof

Symmetric CobbDouglas preferences yield demand functions’ Xij=V2JIVpij* Since Pij=Pj, i= 1,. . .,N; j=2,. . . ,A!, we obtain:

Summing across countries yields: N

N

C p2Xi2 i= 1

C

=

N

=* = C

paXi

l

l

PjXij*

;= 1

i= 1

But from (2), it then follows that p2 = p3 = Lemma

2. Country

2’s

l

l

l

=pj

-

= p.

income is given by ml = IY[z-cf=,

vi].

Proof. From the demand function for the numeraire good, we obtain x11 =mJV. Domestic consumption of its market good is also given by country l’s endowment minus its exports: N X11=2-

c i=2

Vi*

Substituting for xl 1 and re-arranging yields: ml=N

Z-iVi [

i=1 .

0

2

Consequently, the utility of the representative citizen in country rewritten as

Lemma 3. Country I sets identical export quotas, v2 =* uN = v on each of its (N - 1) trading part rateful to a refereefor ~~~~~i~yi~

l

l

1

can be

= vi = vi+ I =-

2.

l

l

=

D. Bernhardt and A. Enders, Free trade equilibria

326

Proof.

See the appendix.

Lemma 3 is a consequence of the underlying symmetry of preferences and endowments, and the fact that the indirect utility of the representative citizen in country 1 is concave in the quotas its government sets. Consequently, country 1 treats all its (N - 1) trading partners symmetrically when it chooses its export quota vector. We noted above this’tthe only possible condition under which a country chooses a non-autarkic quota as a Nash equilibrium strategy is when real income falls as a result of the imposition of a quota. Real income depends in turn on the fraction of rents obtained, so the condition sought is one on the parameter a. Repeating the question originally posed: Given countries 2 through N do not set binding quotas, under what circumstances will country 1 find it optimal to set non-binding quotas, so that free trade emerges as a Nash equilibrium? Theorem 1. Free trade is a Nash equilibrium of the static export quota game if the fraction of quota rents accruing to a quota-imposing country does not exceed a=(N- 1)JN. Proof:

Free trade is a Nash equilibrium outcome if, given the ‘free trade’ non-binding quota strategies selected by its (N - 1) trading partners, count! j 1 also selects ‘free trade’, non-binding qukrtas. At free trade? each country consumes l/N of each good. In consequence, a free trade export quota strategy for country 1 solves: v

i

+(I-4

1

RT_1=N’

so that

(Nz-l)

Vi=irtN

_

i l,v

2 =

,**-,N-

Lemma 3 allows us to treat all countries symmetrically in country l’s problem of selecting the optimal quota: maxU,== L’z-(N- l)V]Np-‘N-? First-order conditions require

dU1 -~ dv

p4

e~--w-ml N

dpO.S,though, it is always optimal to deviate: from free trade, j69since Obf aiways cuts the free trade in ifference curve ci at 5 ence, there always rateful

to a referee for su

D. Bernhardt and A. Enders, Free trade equilibria

329

exists a higher trade indifference curve tangent to 06-J at some binding quota less than K This reflects the fact that when N = 2, except when a =O.s, autarky, not free trade, is the unique equilibrium outcome. Since the indirect payoffs of a representative citizen in a country are concave in the quotas its government sets, its government optimally treats its (N- 1) trading partners symmetrically, so its optimal behavior with and without world markets for quota-restricted goods coincide. Thus, were quota revenues derived from i’s quota on exports to country j to dissipate to countries other than i and j, then provided the dissipation is symmetric across countries, the analysis still holds. This follows because again country 1 will set identical export quotas against each country at free trade if it finds it optimal to deviate from free trade and select binding quotas, so that diffusion of rents to other countries is the same, independent of the particular symmetric process. If ai#ai, then one can let a=max{cx,,..., aN}. If it is not in the interest of the country with the greatest share of the quota rents to deviate from free trade, it is in no country’s interest. Furthermore, provided min {al,. . . , aN} is sufficiently great, if one country wishes to set a binding quota, autarky is the unique Nash outcome. If the share of quota rents depends upon both the setting country i and the country upon whom the quota is imposed and hence is given by aij, then let ai*= max {ail, . . . , aiN}. One can show that a country would impose a binding quota, given others do not, if and only if it would were its vector of quota rent shares given by (+, . . . ,ais). If free trade is not an equilibrium outcome, then as well as the autarky equilibrium, equilibria involving unrestricted trade in some bilateral markets, ‘optimal’ quotas by one country and non-binding quotas by the second party in some markets and autarky in others may exist, depending on the particular shares of the quota rents. This last result depends on markets being bilateral, so that resale through third parties does not circumvent binding quotas. From simulation of the three-country case in which each country iteratively chooses its optimal quota of the other two countries, the free trade equilibrium appears stable for a ‘+‘~~zjf’2, -

i,j =l,...,N;

j#i.

As a country’s share of world market of its export good, z, increases beyond l/N, the Nash equilibrium tariff increases and exports fall (in the limit to zero) as each country seeks to exploit its increasingly monopolistic position over its export good. Also, as the number of countries grows so each country becomes relatively small, the Nash tariff declines accordingly: the distortion from the zero (free trade) tariff, (Nz - l)/( l-z) is of a lower order than (Iv/?)’ in the square root, so the Nash tariffs, although positive for zi l/N, approach zero as the number of countries gets large. Note that when the equilibrium tariffs are selected, no country has an incentive to alter the terms of trade further by introducing additional binding quotas. If quotas and tariffs are chosen simultaneously in a Nash game, this symmetric interior Nash equilibrium with positive, non-autarkic tariffs reasonably emer s. Copeland (1988) also obtains this result. Ne finds equilibria -. essentially the closed convex set between the countries’ tariff exist when tari and quotas are chosen simultane-

D. Bemhardt and A. Enders, Free trade equilibria

331

a free trade agreement such as G T, to be less than the Nash tariffs, additional binding quotas will not be introduced if l/N $d(N1)/N. For more extreme divisions of the quota rents, countries have an incentive to alter the terms of trade further by introducing binding quotas so autarky again emerges as the unique equilibrium outcome.

.

Conclusion

We expect that the qualitative free trade outcome that emerges when countries select quotas in this very standard economy carries over to more general economies: unless a country retains the lion’s share of the quota revenues, if it maximizes the welfare of its representative consumer, it will not set binding quotas. If countries do retain the preponderance of quota revenues, autarky emerges. The conclusion drawn is that the prevalence of binding quotas, in contrast to tariffs, cannot be explained within the accepted rubric in which countries maximize the welfare of representative citizens. Alternative explanations in which the distribution of rents within a country is important and involve barriers to redistribution are required.

Appendix

Proof of Lemma 3.

Solving for p from (5) and (4’) we find 3 equal to

+~(N_2+Z)CiYinisi[(~-l

for i, j=2,...,

N. Also, solving for dpldoi from (5) and (4’):

(1 =-Z)g = -(IV--a)--api, i

where

+a)Uj+(l-z)]

-=-afIid$ -a,~~U~d$, i

i

D. Bernhardt and A. Enders, Free trade equilibria

332

(N-2+z)$ .

db -

l)[(N-

(N-

CiVi

1 +cr)ojt(l

-Z)]’

jzi9

yields: [(N- 1 +a)Vj+(l X [N(N-l)Vif(N-a)(l-Z)+a(l-Zjpil] (N-l)(!-Z){nj[(N-l+a)vj+(l-Z)]} j.#i

--w-N

a

-= dvi

-Z)]j

’

1+a)vk+(l-81

+a(N_2+Z)CjVjnk,j[(N--

Welfare of the representative agent in country 1 depends on consumption of goods j=l,..., N, which in turn depends on income, ml, and prices, pi* Differentiating !$

ul=

L 1 N

N p-w),~l=

z_

i

p-(N-1)

Vi

i=2

[I

with respect to vi and manipulating yields (N - 1) first-order conditions:

w

-=Np+(N-1)

Z-CVj

dvi

[

dp 0 1 do=

j

i=2,...,N.

)

i

Consider any arbitrary pair of export quotas vk, vl, for k #Z. First-order conditions require:

dU, d”l

dv,= dv,’

so

that

dp_dp dvk-dv-,’

y substitution for dp/dvk and dp/dvl, we obtain by eliminating common terms: [N(N- l)v,+(N-a)(1 -@+a(1 [(N - 1 +a)o,+(l -z)]

-z)p,,]

[JV(N-- l)vl+(N-a)(1 -z)+a(1 = -__.-[(N- i +ajuli+ -z)]

-z)pJ -*

One obvious solution is vk=-vi; to fee the solution is unique, collect terms

above to yield:

D. Bernhardt and A. Enders, Free trade equilibria

ut VI> vkimplies Pk1 >

pII

333

as

1 +a)p+(L-u)(N - l)](vt--v,J l)[(AT AV-:+oC)v,+(l--z)][(N-l+Ci)V~+(l--211’

[(w-2+z)(MPkl -Prl

=@_

But vl>vk then implies all the terms on the left-hand side of (5) are positive, a contradiction, so vk=vz. 1 eferences Copeland, RR., 1988, Tariffs and quotas: Retaliation and negotiation with two instruments of protection, Journal of International Economics (forthcoming). Enders, A., 1987, Tariffs and quotas in strategic multi-country interactions, Mimeo, Que:.‘cr University Economics Department. Johnson, 1LG., 1934, Optimum tariffs and rctsliaticn, Reyie:v of Economic Studies 21, 142-153; Falvey, R.E., 1985, Quotas and retaliation, Economics Letters 17, 373-377. --- -A-- AIII~L~~~~I A-eAewnwa~-rr~uwnn~.w Cnrrmnm;r. @P&-W Melvin, J. (l986), The non_equim.nlmmem mP,aACf:. ,**&~Ia- V. CULI‘IJtiiuA Z----r rrrryvc L yUOlQ3, .rw..w 76, 1131-l 134. Rodriguez, C.A., 1974, The non-equivalence of tariffs and quotas under retaliation, Journal of International Economics 4, 294-298. Thursby, M. and R. Jensen, 1983, A conjectural variation approach to strategic tariff equilibria, Journal of International Economics 14, 145-162. Tower, E., 1975, The optimum quota and retaliation, Review of Economic Studies 42, 623-630.