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Comparative advantage under oligopoly
Journal of International Economics 43 (1997) 333–346
Comparative advantage under oligopoly a b, Tito Cordella , Jean J. Gabszewicz * a
Department of Economics, Universitat Pompeu Fabra, Barcelona, Spain; International Monetary Fund and CEPR b CORE, Universite´ Catholique de Louvain, Voie du Roman Pays 34, 1348 Louvain la Neuve, Belgium
Abstract We analyze the principle of comparative advantage when agents in the world market are aware of the influence their individual supply exerts on the equilibrium exchange rate of goods. We show that specialization following comparative disadvantage can be an oligopoly equilibrium in a Ricardian economy. Moreover, for a wide class of economies, it is the only one. Nonetheless, when the number of agents in each country increases without limit, the equilibrium in which specialization follows comparative advantage again obtains. Keywords: Comparative Advantage; Oligopoly JEL classification: F10; F12; D43
1. Introduction ‘‘Dostoevsky apparently once remarked that all of Russian literature emerged from under Gogol’s overcoat. It is at least as true that all of the pure theory of International Trade has emerged from chapter 7 of Ricardo’s Principles’’. This quotation from Findlay (1984), p. 18) beautifully illustrates how deeply the Ricardian principle of comparative advantage has impregnated the oldest subfield of Economics. Nonetheless, both the simplicity of the Ricardian model and the clarity of its conclusions have blocked from the view of economists the importance, for the principle of comparative advantage, of the implicit assumption *Corresponding author. Tel.: 32-10-474346; Fax: 32-10-474301; e-mail [email protected] 0022-1996 / 97 / $17.00 1997 Elsevier Science B.V. All rights reserved. PII S0022-1996( 96 )01479-1
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that the world market works competitively. In particular, and to our surprise, the question seems never to have been raised whether, and the extent to which, the use of market power by economic agents on the world market would alter the predictions of Ricardian theory. Implicit to the Ricardian model is, indeed, the assumption that the economic agents in both countries take the world price as given in the production and exchange of goods. What happens with the principle of comparative advantage when, on the contrary, the economic units perceive the influence of their individual supply on the world equilibrium exchange rate of goods and make a strategic use of it? The present paper is an attempt to shed some light on this question in the very framework proposed by Ricardo: The twocountry, two-commodity case with a single productive factor, labour, and linear technologies. It is well-known that the Ricardian model isolates differences in technology as the basis for trade. But the role of demands is not negligible since demand conditions select which particular outcome will obtain on the Pareto frontier under free trade. Of specific interest in this respect are country preferences according to which each country wants to consume only the good in which it has a comparative disadvantage in production. To illustrate, imagine that Spaniards were not interested in oranges and would want to consume only apples, while Belgians were not interested in apples and would want to consume only oranges. Under autarky, each country would fully specialize in the production of the good it likes, at a cost which can be exorbitant. On the contrary, under competitive free trade, Spain would specialize in the production of oranges and Belgium in the production of apples. Given the countries’ preferences, the whole production of oranges in Spain would be exchanged against the whole production of apples in Belgium, and both countries would enjoy considerable gains from trade. Accordingly, such demand conditions would create a priori the highest incentives for trade, since the mutual benefits to be expected from product specialization and trade would be the highest in both countries. The main outcome of our analysis is that, under such demand conditions, which guarantee the strongest incentives for trade, noncooperative strategic behaviour of economic agents in the world market may well fully destroy these potential benefits: In a wide class of Ricardian economies, and in spite of the possibility of free trade and huge comparative advantages, the autarkic outcome turns out to be the unique oligopoly equilibrium on the world market. In other words, without any trade restraint of any sort, the strategic interplay between agents leads spontaneously, as unique outcome, to the allocation which would follow from the most absolute protectionist edict! Still, this pessimistic result of our analysis must be tempered by a property which we also derive in the present paper: When the number of economic agents increases in the two countries, another oligopoly equilibrium appears along with the autarkic outcome, which Pareto-dominates it. When each country has an absolute advantage in the production of one good, this alternative oligopoly equilibrium coincides with the Ricardian competitive equilibrium. Accordingly,
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competitive forces restore the Ricardian outcome when the number of agents increases without limit, even if, however large this number, the autarkic outcome still remains as an alternative equilibrium. Our paper examines the extension of inter-industry trade to oligopoly situations in the light of some contributions to noncooperative general equilibrium analysis (Gabszewicz and Vial, 1972; Shapley and Shubik, 1977). We propose a Ricardian model in which all the agents act strategically. They possess a linear technology and manipulate the world price via the supply they send to the market for trade. Their payoffs are the utility levels which can be reached through the exchange mechanism operating under the market clearing conditions. While borrowing from Gabszewicz–Vial the Walrasian market clearing conditions as the institutional price mechanism, our approach shares with the Shapley and Shubik’s one that all agents are strategic and manipulate prices via their individual supply. It is well-known that, in a Shapley–Shubik exchange economy, ‘‘there is always the trivial, ‘‘null’’ equilibrium, at which nothing is bought or sold’’ (Shapley, 1976, p. 171), a proposition which is reminiscent of our next Proposition 1 in the Ricardian model. Our main result is that this outcome is the only non cooperative equilibrium for a large class of Ricardian economies. The fact that market imperfections may destroy incentives to production and trade, is also recognized in unemployment theory. Among the vast macroeconomic literature on coordination failures, Diamond (1982) presents a model in which the agents’ production decisions depend on the decisions of the other agents. This is so, because the agents do not consume their own production and need to exchange it with trading partners. Accordingly, if every agent expects no trading partner, production would not occur in this economy. While the intuition behind Diamond’s result and ours is quite similar, the very reason of no-trade is not. In Diamond’s paper, trade is coordinated by a stochastic matching process and the reason for the coordination failure is the lack of a Walrasian auctioneer. In our model, the reason for the coordination failure, and thus of the uniqueness of the no-trade equilibrium, is the ability of the agents to manipulate the Walrasian market clearing prices. In the next section, we present the model; Section 3 is devoted to analyze the oligopoly equilibria and Section 4 provides some concluding comments.
2. The model We consider two countries * and ^ with n agents in each country. The set of agents is h1, . . . , n, n 1 1, . . . , 2nj with * 5 h1, . . . , nj the agents in country *, and ^ 5 hn 1 1, . . . , 2nj the agents in country ^. There are two consumption goods, x 1 and x 2 , produced out of labour; each agent is endowed with one unit of labour, irrespective of the country. All agents in country * have access to the
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HS
D
J
yi 1 2 yi ], ]] uy i [ [0, 1] a1 a2 where y i is the quantity of labour assigned by the agent i, i [*, to the production of good 1. Similarly, all agents in country ^ can produce any pair of the two 1 2 yi yi goods ]], ] , where y i now denotes the quantity of labour assigned by the a *1 a 2* agent i [^ to the production of good 2. Without loss of generality, we shall suppose that the agents in country 1 have a comparative advantage in the production of good 1, same production frontier in R 21 defined by the locus
S
D
a a *1 ]1 , ] . a 2 a *2
(1)
Furthermore, we suppose also that 0,a 1 ,a 1* and 0,a 2* ,a 2 , so that country * has an absolute advantage in the production of good 1 and country ^ in the production of good 2. Finally, we assume that the agents in each country are only interested in the good in which they have a comparative disadvantage in its production: The utility function u i of an agent i in country * is defined by u i (x 1 , x 2 ) 5 x 2 , i [ *, and by u i (x 1 , x 2 ) 5 x 1 , i [ ^. Clearly these preferences generate the largest incentives to organize trade between the two countries, since the citizens of each of them precisely prefer the commodity which the citizens of the other country produce more efficiently. With these preferences, at the Ricardian competitive equilibrium yˆ i , each country specializes in the production of the good in which it has a comparative advantage, i.e. yˆ i 51, i [*<^. At the Ricardian competitive equilibrium, the agents in both countries consider that the terms of trade are not under their control: They take the world price as given when they make their production decision. By contrast, we shall now assume that they perceive the influence of their individual supply on the equilibrium exchange rate of goods. Since their revenue, and thus their purchasing power as consumers, depends on this real exchange rate, they should take it into account when making their production decision. But the influence of each agent on the world price is only partial: All the other agents in both countries compete with him in the same manner on the world market. Accordingly, the equilibrium notion must reflect this complex system of rivalry among the producers of both countries: We shall be interested in a situation in which their supplies in both goods yield a noncooperative equilibrium in the game where each of them aims at influencing the world exchange rate to his own advantage. Let us now define precisely this
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game. A strategy of an agent i is a quantity of labour y i , y i [ [0, 1], agent i decides to assign to the production of the good in which he has a comparative advantage. Notice that such a quantity determines the quantity of both goods he supplies on the exchange market. Let each agent i choose a strategy y i and consider prices p1 and p2 on the world market, where p1 (respectively p2 ) denotes the price of commodity 1 (respectively commodity 2). The revenue of an agent i is given by yi 1 2 yi p1 ] 1 p2 ]], i [ * a1 a2 and by 1 2 yi yi ]] ] p1 1 p2 , i [ ^. a* a *2 1 Accordingly, at prices p1 and p2 on the world market, agent i [* maximizes his utility when spending his total income on good 2, so that the corresponding utility u i is given by p1 y i 1 2 y i i u (0, x 2 ) 5 ] ] 1 ]]. p2 a 1 a2
(2)
Similarly an agent i [^ maximizes his utility when spending his total income on good 1 only, and getting a utility level u i given by 1 2 y i p2 y i ]] u (x 1 , 0) 5 1 ] ]. a *1 p1 a *2 i
(3)
Given a 2n-tuple of strategies ( y 1 , . . . , y n , y n 11 , . . . , y 2 n), we denote by Y H the sumo i [ * y i and by Y F the sum o i [ ^ y i . Accordingly, assuming a 2n-tuple of strategies for which both Y H and Y F are different from 0, the resulting equilibrium world price clearing the market for good 1 must satisfy n 2YF p YF YH n 2YF ]] 1 ]2 ] 5 ] 1 ]], a *1 p1 a 2 a1 a 1* so that p1 a1Y F ] 5 ]] . p2 a *2 Y H
(4)
Substituting the equilibrium price (4) into (2) and (3), we are able to express the indirect utility function v i of each agent as a function of his and other agents’ strategies,
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YF 1 2 yi i v i ( y i , y 2i ) 5 ]] , i [* H y 1 ]] a2 a *2 Y
(5)
YH 1 2 yi v i ( y i , y 2i ) 5 ]]F y i 1 ]], i [ ^, a *1 a1Y
(6)
and
where y 2 i denotes the (2n21)-tuple of strategies of agents different from i. Nevertheless, when y i 50, for all i [*<^, no exchange can occur, and each 1 1 agent acts in autarky. Utilities are then simply given by ] in country * and ] in a2 a 1* country ^: This is the no-trade outcome. An oligopoly equilibrium is a Nash equilibrium of the game in which agent i has v i ( y i , y 2 i ) as payoff. More precisely,An oligopoly equilibrium is a 2n-tuple of strategies y˜ i such that, for all i, for all y i [[0, 1], v i (y˜ i , y˜ 2i ) > v i ( y i , y˜ 2i ). At an oligopoly equilibrium, each agent i chooses his production plan in such a way that, given the production plans chosen by the other agents and the resulting world exchange rate, no unilateral deviation from this choice can give him better market opportunities.
3. Equilibrium analysis Of particular interest in the analysis of noncooperative trade is the no-trade outcome envisaged above, namely, when each country specializes in the production of the good in which it has a comparative disadvantage, i.e. y˜ i 50, ; i [*<^. Indeed, we show that Proposition 1 The no-trade outcome is an oligopoly equilibrium. Proof First, assume on the contrary that an agent i [* can increase his utility 1 level ] obtained at the no-trade outcome. Then he must choose his strategy y i in a2 the semi-open interval ]0, 1], in which case he obtains a utility level given by (5), with y k 50, ; k[*<^, k±i, i.e. 1 2 yi 1 v i ( y i , y˜ 2i ) 5 ]] , ] 5 v i (y˜ i , y˜ 2i ), a2 a2 a contradiction. A similar reasoning applies to show that no deviation from y˜ i 50 increases v i for i [^. Proposition 1 derives from a simple intuition. Suppose that one of the agents in the home country decides to allocate part or all of its labor to the production of good 1 in which it has a comparative advantage, while taking the actions of all
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other agents as given. In order to have such a decision profitable, this agent must take his good 1-output to the market and exchange it for good 2. But, at the no trade outcome, all agents are already self sufficient, and they do not want to buy it either. There is no trade for the agent to make: he has wasted some of his labor allocation, as it would have been better to spend all of his labor in the production of good 2. It follows that autarky is an oligopoly equilibrium. The strength of Proposition 1 would be considerably enhanced if we could show that, for some Ricardian economies, the no-trade outcome would be the unique oligopoly equilibrium. Then, and in spite of, possibly huge, comparative advantages and strong incentives to organize advantageous exchange, the potential gains from trade would be inexorably wasted due to the noncooperative strategic interaction among the agents. That this is the case for large classes of Ricardian economies follows from the next analysis. First, the symmetry of the problem clearly implies that all the agents in the same country must play the same strategy at an oligopoly equilibrium, i.e. Y˜ H y˜ i 5 y˜ H 5 ], i [ * ; n and Y˜ F y˜ i 5 y˜ F 5 ], i [ ^. n The first derivatives ≠v i YF 1 H\hij ]i 5 ]]] 2 ], i [ *, H 2 Y a ≠y a 2* (Y ) 2 i
(7)
H
≠v Y 1 ]i 5 ]]] Y F\hij 2 ], i [ ^, a *1 ≠y a 1 (Y F )2
(8)
must be equal to zero at an interior maximum, with Y H\hi j (respectively Y F\hi j ) denoting the sum S k [ * y k (respectively S k [ ^ y k ). Accordingly, at an interior k ±i k ±i equilibrium, we must have a 2 (n 2 1)Y˜ F H y˜ 5 ]]]] a *2 n 2 and a *1 (n 2 1)Y˜ H F y˜ 5 ]]]] . a1n2
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Clearly, except in a very special case 1 , no values y˜ i []0,1[, ;i [*<^, can satisfy this system: No interior equilibrium exists. The following Lemma also excludes, as possible equilibria, situations in which one country specializes in the good he produces with a comparative disadvantage, while the other country produces both goods. Lemma 1 Excluding the no-trade outcome, the only remaining possibilities, for a 2 n-tuple of strategies (y˜ H , . . . , y˜ H , y˜ F , . . . ,y˜ F ) to be an oligopoly equilibrium are: 1. y˜ H 5y˜ F 51; 2. y˜ H 51; y˜ F []0,1[; 3. y˜ H []0,1[; y˜ F 51. Proof Since we have already shown that no interior equilibrium exists, the proof of the Lemma would be complete if we show that no pair ( y H , y F ) with y H 50, y F []0, 1], or y H []0, 1], y F 50 can be equilibria. Consider the first case. Suppose y˜ H 50, y˜ F []0, 1] is an equilibrium, then for all i in ^ we must have ≠v i ] ≠y i
U
y F 5 y˜ F y˜ H 50
>0
but from (8), it follows that ≠v i ] ≠y i
U
y F 5 y˜ F y˜ H 50
1 5 2 ] , 0, i [ ^, a 1*
a contradiction. Using (7), a similar reasoning applies for excluding the case y˜ H []0, 1], y˜ F 51. To identify the class of Ricardian economies in which the no-trade outcome is the unique oligopoly equilibrium, we may thus restrict ourselves to the three candidates identified in Lemma 1, and find conditions under which all of them are simultaneously excluded. Lemma 2 (i). y˜ H 5y˜ F 51 is an oligopoly equilibrium if, and only if a *2 a *1 (n 2 1)2 This exception occurs when ] 5 ]]] , in which case the economy has a continuum of 2 a2 a1n equilibria. This continuum includes, in particular, either y˜ H 51; y˜ F []0, 1[, or y˜ H []0, 1[; y˜ F 51. We assume, in this very particular case, that the agents select the corresponding equilibrium, namely, with one country specializing in the production of the good in which it has a comparative advantage and the other producing both goods. This selection is proposed in view of simplifying the presentation of the proofs, but the whole argument goes through without it.
1
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a 2* a1 n 2 1 n21 ] < ]] and ] < ]]; a2 n a 1* n
341
(9)
(ii). y˜ H 51; y˜ F []0, 1[is an oligopoly equilibrium if, and only if n 2 a *2 a *1 n ]]] < ] , ]]; 2 (n 2 1) a 2 a 1 n 2 1
(10)
H F (iii). y˜ []0, 1[; y˜ 51 is an oligopoly equilibrium if, and only if
2 n 2 1 a 2* (n 2 1) a 1* ]] , ] < ]]] . 2 n a2 n a1
(11)
i H F Proof First consider candidate (i). The necessary conditions for y˜ 5y˜ 5y˜ 51 to be an oligopoly equilibrium is that, for all i,
≠v i ]i ≠y
U
> 0, y 2i 51
which, by (7) and (8), implies that a *2 a1 n 2 1 n21 ] < ]] and ] < ]]. a2 n a *1 n ≠ 2v i Furthermore, it is easily checked that ]] i 2 , 0, so that these conditions are ≠( y ) also sufficient conditions for y˜ i 5y˜ H 5y˜ F 51 to be an oligopoly equilibrium. Now, consider candidate (ii). To be an oligopoly equilibrium, the necessary conditions ≠v i ]i 5 0, ;i [ ^ ≠y
(12)
≠v i ]i > 0, ;i [ * ≠y
(13)
and
must hold at equilibrium. With y˜ H 51, (8) and (12) imply that
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(n 2 1)a 1* F y˜ 5 ]]]; na 1
(14)
a necessary condition for y˜ H 51, y˜ F []0, 1[to be an oligopoly equilibrium is thus a 1* n ] , ]]. Furthermore, from (7) and (14), the inequality (13) can be rewritten a1 n 2 1 as a *1 a *2 n 2 ] > ]]]2 . a 1 a 2 (n 2 1) Moreover, by the concavity of the payoff function, these necessary conditions are also sufficient. A similar reasoning implies (iii). Finally it is readily seen from the necessary and sufficient conditions (10) and (11) that, given a 1 , a 1* , a 2 , a *2 and n, and excluding the no-trade outcome, at most one of the candidates above can be, indeed, an oligopoly equilibrium. According to Lemma 2 (i), an agent in the foreign country finds profitable to a1 n 2 1 deviate when ] > ]]. The first term of this inequality is a measure of the a 1* n absolute advantage of the agents in the home country in the production of good 1. Thus, the greater a *1 with respect to a 1 , the greater are the incentives of producing good 2, and exchanging it for good 1. The second term of the inequality is a measure of an agent’s market power 2 . The smaller n, the greater is his market power, and thus his incentive in manipulating the terms of trade by producing more of good 1. Accordingly, the decision to deviate from the competitive equilibrium depends on the relative magnitudes of these two effects. If both the inequalities in (9) are violated, then the agents in both countries have an incentive in deviating from the competitive equilibrium. If this is the case, then the only oligopoly equilibrium is the no-trade outcome. While, if only the agents in the foreign country have an incentive in deviating from the competitive equilibrium, (n 2 1)a 1* 3 F candidate (ii) can be an oligopoly equilibrium , with y˜ 5 ]]] (see (14)). na 1 This is the case, indeed, if for the agents in the home country it is still optimal to specialize following their comparative advantage, i.e. if the left inequality of (10) is satisfied. This happens when the gains from exchange, evaluated at y˜ F 5 (n 2 1)a 1* ]]], dominate the gains from manipulating the terms of trade. na 1 Now we are able to identify several classes of Ricardian economies for which 2
If we had considered a model with n 1 agents in country 2, the right inequality in (9) would have a1 n2 2 1 n21 written as: ] < ]]. Thus, ]] is a ‘‘concentration measure’’ of the foreign country industry. a *1 n2 n 3
Candidate (iii) can be an oligopoly equilibrium when the agents in the home country are those who have an incentive in deviating from the competitive equilibrium.
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the only oligopoly equilibrium is the no-trade outcome. To prove this statement, we introduce restrictions on the parameters (see (15) below) such that the necessary and sufficient conditions (9), (10), and (11) are all simultaneously violated. Proposition 2 For any number of competitors in either country, there are Ricardian economies for which the no-trade equilibrium is the unique oligopoly equilibrium. Proof Assume that the Ricardian economy (a 1 , a *1 , a 2 , a *2 , n) satisfies the inequalities a *1 n 2 1 a *2 n ]] , ] , ] , ]]. n a2 a1 n 2 1
(15)
First, the left and the right inequalities in (15) contradict the fact that y˜ H 5y˜ F 51 is an oligopoly equilibrium (Lemma 2). Now, assume that y˜ H 51, y˜ F []0, 1[is an oligopoly equilibrium. Then, by Lemma 2, the left inequality of (10) must hold, a *2 a1 n2 ] ]]]2 < ] , a 2 (n 2 1) a 1* which, with the right inequality of (15), implies that a *2 n2 n ] ]]]2 , ]], a 2 (n 2 1) n21 which, in turn, implies that a *2 n21 ] , ]], a2 n an inequality which contradicts the left inequality of (15). Consequently, y˜ H 51, y˜ F []0, 1[is not an oligopoly equilibrium. By a similar argument and using (15) and the right inequality of (11) in Lemma 2, y˜ H []0, 1[, y˜ F 51 is not an oligopoly equilibrium either. This completes the proof of Proposition 2. To illustrate the above proposition, it is worthwhile to consider the following example of a Ricardian economy: 1 1 a 1 5 a *2 5 ]; a *1 5 a 2 5 ]; n 5 2. 3 2 By Proposition 2, the no-trade equilibrium is the only oligopoly equilibrium, and y˜ H 50, y F 51, so that the world production of the two goods is (4, 4). At the competitive equilibrium, we have yˆ H 51, yˆ F 50, so that the world output is (6, 6): Strategic behaviour causes a reduction of world production of one third. Proposition 2 shows that a noncooperative behaviour of the economic agents can have devastating effects on the efficiency of trade when there is ‘‘competition among the few’’. In spite of the existence of, possibly huge, comparative
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advantages, each country may prefer to remain in autarky, wasting the advantageous opportunities opened by the possibility of free trade. Proposition 2 reveals that the set of Ricardian economies where this waste is observed, is non empty, however large the number n of competitors in each country. Nevertheless, as revealed by the following proposition, this set tends to vanish when n tends to infinity. Proposition 3 There exists a number n¯ such that, if the number of competitors in both countries exceeds n¯ , the set of oligopoly equilibria consists exactly of the no-trade equilibrium and the competitive equilibrium. Proof Since a 1 ,a *1 and a 2 .a 2* we have a *1 a *2 ] . 1 . ]. a1 a2 ¯ Consequently, there exists a number n¯ such that, ;n>n, a *1 n n 2 1 a 2* ] > ]] and ]] > ] a1 n 2 1 n a2 which are the inequalities (9). Propositions 2 and 3 have been demonstrated when country * has an absolute advantage in the production of good 1 and country ^ in the production of good 2. But analogous properties can be proved when one country has an absolute advantage in the production of both goods (for a detailed analysis, see Cordella and Gabszewicz, 1993). Furthermore, the whole analysis above goes through without assuming an equal number of agents in each country. Accordingly, the class of situations in which the comparative advantages are not exploited in the international exchange of goods is, by far, wider than the one resulting from (15). Nonetheless, in these other classes also, competition is restored when the number of agents increases in both countries. In the example analyzed above, we have assumed somewhat extreme preferences of the agents, and one might wonder whether our propositions do not rely heavily on this assumption. Proposition 1 holds under much more general conditions than those in the corpus of the paper. In particular, it holds for any ‘‘well behaved’’ utility functions of the agents. The argument is the following. First notice that, in a Ricardian framework, where all agents inside each country have the same constant returns to scale technology, there are no gains from trade at the autarky equilibrium. This implies that, before trade is opened, if agents act in a completely autarkic way inside their countries, they reach the same utility level as at the competitive equilibrium. From the Pareto optimality of the competitive equilibrium, it follows directly that, under autarky, the no-trade outcome is an oligopoly equilibrium. It is now easy to show that when trade is opened between the two countries, the no-trade outcome is still an oligopoly equilibrium. Suppose not, and assume that,
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after the opening of trade, some agent can increase his utility level obtained at the no-trade outcome, by producing more of good 1 (respectively 2) and supplying it to the market. This deviation increases his utility only if he can get more of good 2 (respectively 1) through the market, than he obtained under autarky. But, at the no-trade outcome, there is no supply of good 2 (respectively 1) since, by assumption, everybody acts optimally in an autarkic manner. A contradiction. Once proved that, in a Ricardian economy, the no trade outcome is an oligopoly equilibrium under very general conditions, it remains to check that the noncooperative behaviour of the agents may well destroy any incentive to trade, when agents also want to consume the good in which they have a comparative advantage. To check this, we have considered a variation of our example in which agents have strictly monotone preferences in both goods. More precisely, we assume that the utility of the agents in country * (respectively country ^) is given by u i (x 1 , x 2 )5 ´ ln x 1 1x 2 (respectively u i (x 1 , x 2 )5x 1 1 ´ ln x 2 ), with ´ .0 small. As for the technological coefficients and the number of agents, we choose, as in 1 1 the example provided above, a 1 5 a *2 5 ]; a *1 5 a 2 5 ]; n 5 2. For ´ small, the 3 2 preferences of the agents in this example are close to the extreme preferences 3 considered in the above analysis. Furthermore, when 0 , ´ , ], it can be checked 5 that, in the world market, there exists a unique oligopoly equilibrium (with the 5´ exception of the autarkic one), which is interior, and given by y˜ i 5 ], ;i [ H < 3 ^. Accordingly, when ´ tends to zero, the unique oligopoly equilibrium on the world market tends to the no-trade equilibrium. Thus we conclude that our findings, in the extreme case in which agents are only interested in one good (´ 50), are robust to a slight perturbation of preferences implying strict monotonicity in both goods.
4. Concluding remarks ‘‘Of the four possible [Ricardian competitive] equilibria in which at least one country specializes in production, the principle of comparative costs rules out only one: The inefficient case where both countries specialize according to comparative disadvantage’’ (Jones and Neary, 1984, p. 11). As seen in the above pages, this inefficient case is the only one to actualise in a wide class of Ricardian economies when the world market is oligopolistic. No doubt, in many situations, the loss to be incurred by interacting agents exploiting their market power can be considerable, compared with the gains which would accrue from cooperation. Nevertheless, it really came as a surprise to us that noncooperation in production and trade can be so harmful, since it fully destroys advantageous exchange. Ricardo asserted that ‘‘under a system of perfectly free commerce, each country naturally devotes its capital and labour to such employments as are most beneficial
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to each’’ (Ricardo, 1951, p. 133). Our example shows that this assertion must sometimes be qualified: it is only when sufficient competitive forces are restored, i.e., when the number of agents involved in trade increases, that this system of perfectly free commerce starts again to operate.
Acknowledgements We are indebted to B. De Meyer, V. Ginsburgh and H. Polemarchakis for helpful discussion. We are also grateful to our referees for interesting comments which improved considerably the last version of the paper. This text presents research results of the Belgian program on Interuniversity Poles of Attraction.
References Cordella, T. and J.J Gabszewicz, 1993, Comparative advantage under oligopoly, CORE Discussion Paper 9307, Universite´ Catholique de Louvain, Louvain-la-Neuve. Diamond, P., 1982, Aggregate demand management in search equilibrium, Journal of Political Economy 90, 881–894. Findlay, R., 1984, Growth and development in trade models, in: R. Jones and P. Kenen, eds., Handbook of international economics (North-Holland, Amsterdam) 185–236. Gabszewicz, J.J. and J.P. Vial, 1972, Oligopoly a` la Cournot in a general equilibrium analysis, Journal of Economic Theory 4, 381–400. Jones, R. and J.P. Neary, 1984, The positive theory of international trade, in: R. Jones and P. Kenen, eds., Handbook of international economics (North-Holland, Amsterdam) 162–184. Ricardo, D., 1951, On the principle of political economy and taxation, in: P. Sraffa, ed., The work and correspondence of David Ricardo, Vol.1 (Cambridge University Press, Cambridge). Shapley, L., 1976, Non cooperative general exchange, in: S. Lin, ed., Theory and measurement of economic externalities (Academic Press, New-York) 155–175. Shapley, L. and M. Shubik, 1977, Trade using one commodity as a means of payment, Journal of Political Economy 85, 937–968.
Comparative advantage under oligopoly a b, Tito Cordella , Jean J. Gabszewicz * a
Department of Economics, Universitat Pompeu Fabra, Barcelona, Spain; International Monetary Fund and CEPR b CORE, Universite´ Catholique de Louvain, Voie du Roman Pays 34, 1348 Louvain la Neuve, Belgium
Abstract We analyze the principle of comparative advantage when agents in the world market are aware of the influence their individual supply exerts on the equilibrium exchange rate of goods. We show that specialization following comparative disadvantage can be an oligopoly equilibrium in a Ricardian economy. Moreover, for a wide class of economies, it is the only one. Nonetheless, when the number of agents in each country increases without limit, the equilibrium in which specialization follows comparative advantage again obtains. Keywords: Comparative Advantage; Oligopoly JEL classification: F10; F12; D43
1. Introduction ‘‘Dostoevsky apparently once remarked that all of Russian literature emerged from under Gogol’s overcoat. It is at least as true that all of the pure theory of International Trade has emerged from chapter 7 of Ricardo’s Principles’’. This quotation from Findlay (1984), p. 18) beautifully illustrates how deeply the Ricardian principle of comparative advantage has impregnated the oldest subfield of Economics. Nonetheless, both the simplicity of the Ricardian model and the clarity of its conclusions have blocked from the view of economists the importance, for the principle of comparative advantage, of the implicit assumption *Corresponding author. Tel.: 32-10-474346; Fax: 32-10-474301; e-mail [email protected] 0022-1996 / 97 / $17.00 1997 Elsevier Science B.V. All rights reserved. PII S0022-1996( 96 )01479-1
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that the world market works competitively. In particular, and to our surprise, the question seems never to have been raised whether, and the extent to which, the use of market power by economic agents on the world market would alter the predictions of Ricardian theory. Implicit to the Ricardian model is, indeed, the assumption that the economic agents in both countries take the world price as given in the production and exchange of goods. What happens with the principle of comparative advantage when, on the contrary, the economic units perceive the influence of their individual supply on the world equilibrium exchange rate of goods and make a strategic use of it? The present paper is an attempt to shed some light on this question in the very framework proposed by Ricardo: The twocountry, two-commodity case with a single productive factor, labour, and linear technologies. It is well-known that the Ricardian model isolates differences in technology as the basis for trade. But the role of demands is not negligible since demand conditions select which particular outcome will obtain on the Pareto frontier under free trade. Of specific interest in this respect are country preferences according to which each country wants to consume only the good in which it has a comparative disadvantage in production. To illustrate, imagine that Spaniards were not interested in oranges and would want to consume only apples, while Belgians were not interested in apples and would want to consume only oranges. Under autarky, each country would fully specialize in the production of the good it likes, at a cost which can be exorbitant. On the contrary, under competitive free trade, Spain would specialize in the production of oranges and Belgium in the production of apples. Given the countries’ preferences, the whole production of oranges in Spain would be exchanged against the whole production of apples in Belgium, and both countries would enjoy considerable gains from trade. Accordingly, such demand conditions would create a priori the highest incentives for trade, since the mutual benefits to be expected from product specialization and trade would be the highest in both countries. The main outcome of our analysis is that, under such demand conditions, which guarantee the strongest incentives for trade, noncooperative strategic behaviour of economic agents in the world market may well fully destroy these potential benefits: In a wide class of Ricardian economies, and in spite of the possibility of free trade and huge comparative advantages, the autarkic outcome turns out to be the unique oligopoly equilibrium on the world market. In other words, without any trade restraint of any sort, the strategic interplay between agents leads spontaneously, as unique outcome, to the allocation which would follow from the most absolute protectionist edict! Still, this pessimistic result of our analysis must be tempered by a property which we also derive in the present paper: When the number of economic agents increases in the two countries, another oligopoly equilibrium appears along with the autarkic outcome, which Pareto-dominates it. When each country has an absolute advantage in the production of one good, this alternative oligopoly equilibrium coincides with the Ricardian competitive equilibrium. Accordingly,
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competitive forces restore the Ricardian outcome when the number of agents increases without limit, even if, however large this number, the autarkic outcome still remains as an alternative equilibrium. Our paper examines the extension of inter-industry trade to oligopoly situations in the light of some contributions to noncooperative general equilibrium analysis (Gabszewicz and Vial, 1972; Shapley and Shubik, 1977). We propose a Ricardian model in which all the agents act strategically. They possess a linear technology and manipulate the world price via the supply they send to the market for trade. Their payoffs are the utility levels which can be reached through the exchange mechanism operating under the market clearing conditions. While borrowing from Gabszewicz–Vial the Walrasian market clearing conditions as the institutional price mechanism, our approach shares with the Shapley and Shubik’s one that all agents are strategic and manipulate prices via their individual supply. It is well-known that, in a Shapley–Shubik exchange economy, ‘‘there is always the trivial, ‘‘null’’ equilibrium, at which nothing is bought or sold’’ (Shapley, 1976, p. 171), a proposition which is reminiscent of our next Proposition 1 in the Ricardian model. Our main result is that this outcome is the only non cooperative equilibrium for a large class of Ricardian economies. The fact that market imperfections may destroy incentives to production and trade, is also recognized in unemployment theory. Among the vast macroeconomic literature on coordination failures, Diamond (1982) presents a model in which the agents’ production decisions depend on the decisions of the other agents. This is so, because the agents do not consume their own production and need to exchange it with trading partners. Accordingly, if every agent expects no trading partner, production would not occur in this economy. While the intuition behind Diamond’s result and ours is quite similar, the very reason of no-trade is not. In Diamond’s paper, trade is coordinated by a stochastic matching process and the reason for the coordination failure is the lack of a Walrasian auctioneer. In our model, the reason for the coordination failure, and thus of the uniqueness of the no-trade equilibrium, is the ability of the agents to manipulate the Walrasian market clearing prices. In the next section, we present the model; Section 3 is devoted to analyze the oligopoly equilibria and Section 4 provides some concluding comments.
2. The model We consider two countries * and ^ with n agents in each country. The set of agents is h1, . . . , n, n 1 1, . . . , 2nj with * 5 h1, . . . , nj the agents in country *, and ^ 5 hn 1 1, . . . , 2nj the agents in country ^. There are two consumption goods, x 1 and x 2 , produced out of labour; each agent is endowed with one unit of labour, irrespective of the country. All agents in country * have access to the
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HS
D
J
yi 1 2 yi ], ]] uy i [ [0, 1] a1 a2 where y i is the quantity of labour assigned by the agent i, i [*, to the production of good 1. Similarly, all agents in country ^ can produce any pair of the two 1 2 yi yi goods ]], ] , where y i now denotes the quantity of labour assigned by the a *1 a 2* agent i [^ to the production of good 2. Without loss of generality, we shall suppose that the agents in country 1 have a comparative advantage in the production of good 1, same production frontier in R 21 defined by the locus
S
D
a a *1 ]1 , ] . a 2 a *2
(1)
Furthermore, we suppose also that 0,a 1 ,a 1* and 0,a 2* ,a 2 , so that country * has an absolute advantage in the production of good 1 and country ^ in the production of good 2. Finally, we assume that the agents in each country are only interested in the good in which they have a comparative disadvantage in its production: The utility function u i of an agent i in country * is defined by u i (x 1 , x 2 ) 5 x 2 , i [ *, and by u i (x 1 , x 2 ) 5 x 1 , i [ ^. Clearly these preferences generate the largest incentives to organize trade between the two countries, since the citizens of each of them precisely prefer the commodity which the citizens of the other country produce more efficiently. With these preferences, at the Ricardian competitive equilibrium yˆ i , each country specializes in the production of the good in which it has a comparative advantage, i.e. yˆ i 51, i [*<^. At the Ricardian competitive equilibrium, the agents in both countries consider that the terms of trade are not under their control: They take the world price as given when they make their production decision. By contrast, we shall now assume that they perceive the influence of their individual supply on the equilibrium exchange rate of goods. Since their revenue, and thus their purchasing power as consumers, depends on this real exchange rate, they should take it into account when making their production decision. But the influence of each agent on the world price is only partial: All the other agents in both countries compete with him in the same manner on the world market. Accordingly, the equilibrium notion must reflect this complex system of rivalry among the producers of both countries: We shall be interested in a situation in which their supplies in both goods yield a noncooperative equilibrium in the game where each of them aims at influencing the world exchange rate to his own advantage. Let us now define precisely this
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game. A strategy of an agent i is a quantity of labour y i , y i [ [0, 1], agent i decides to assign to the production of the good in which he has a comparative advantage. Notice that such a quantity determines the quantity of both goods he supplies on the exchange market. Let each agent i choose a strategy y i and consider prices p1 and p2 on the world market, where p1 (respectively p2 ) denotes the price of commodity 1 (respectively commodity 2). The revenue of an agent i is given by yi 1 2 yi p1 ] 1 p2 ]], i [ * a1 a2 and by 1 2 yi yi ]] ] p1 1 p2 , i [ ^. a* a *2 1 Accordingly, at prices p1 and p2 on the world market, agent i [* maximizes his utility when spending his total income on good 2, so that the corresponding utility u i is given by p1 y i 1 2 y i i u (0, x 2 ) 5 ] ] 1 ]]. p2 a 1 a2
(2)
Similarly an agent i [^ maximizes his utility when spending his total income on good 1 only, and getting a utility level u i given by 1 2 y i p2 y i ]] u (x 1 , 0) 5 1 ] ]. a *1 p1 a *2 i
(3)
Given a 2n-tuple of strategies ( y 1 , . . . , y n , y n 11 , . . . , y 2 n), we denote by Y H the sumo i [ * y i and by Y F the sum o i [ ^ y i . Accordingly, assuming a 2n-tuple of strategies for which both Y H and Y F are different from 0, the resulting equilibrium world price clearing the market for good 1 must satisfy n 2YF p YF YH n 2YF ]] 1 ]2 ] 5 ] 1 ]], a *1 p1 a 2 a1 a 1* so that p1 a1Y F ] 5 ]] . p2 a *2 Y H
(4)
Substituting the equilibrium price (4) into (2) and (3), we are able to express the indirect utility function v i of each agent as a function of his and other agents’ strategies,
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YF 1 2 yi i v i ( y i , y 2i ) 5 ]] , i [* H y 1 ]] a2 a *2 Y
(5)
YH 1 2 yi v i ( y i , y 2i ) 5 ]]F y i 1 ]], i [ ^, a *1 a1Y
(6)
and
where y 2 i denotes the (2n21)-tuple of strategies of agents different from i. Nevertheless, when y i 50, for all i [*<^, no exchange can occur, and each 1 1 agent acts in autarky. Utilities are then simply given by ] in country * and ] in a2 a 1* country ^: This is the no-trade outcome. An oligopoly equilibrium is a Nash equilibrium of the game in which agent i has v i ( y i , y 2 i ) as payoff. More precisely,An oligopoly equilibrium is a 2n-tuple of strategies y˜ i such that, for all i, for all y i [[0, 1], v i (y˜ i , y˜ 2i ) > v i ( y i , y˜ 2i ). At an oligopoly equilibrium, each agent i chooses his production plan in such a way that, given the production plans chosen by the other agents and the resulting world exchange rate, no unilateral deviation from this choice can give him better market opportunities.
3. Equilibrium analysis Of particular interest in the analysis of noncooperative trade is the no-trade outcome envisaged above, namely, when each country specializes in the production of the good in which it has a comparative disadvantage, i.e. y˜ i 50, ; i [*<^. Indeed, we show that Proposition 1 The no-trade outcome is an oligopoly equilibrium. Proof First, assume on the contrary that an agent i [* can increase his utility 1 level ] obtained at the no-trade outcome. Then he must choose his strategy y i in a2 the semi-open interval ]0, 1], in which case he obtains a utility level given by (5), with y k 50, ; k[*<^, k±i, i.e. 1 2 yi 1 v i ( y i , y˜ 2i ) 5 ]] , ] 5 v i (y˜ i , y˜ 2i ), a2 a2 a contradiction. A similar reasoning applies to show that no deviation from y˜ i 50 increases v i for i [^. Proposition 1 derives from a simple intuition. Suppose that one of the agents in the home country decides to allocate part or all of its labor to the production of good 1 in which it has a comparative advantage, while taking the actions of all
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other agents as given. In order to have such a decision profitable, this agent must take his good 1-output to the market and exchange it for good 2. But, at the no trade outcome, all agents are already self sufficient, and they do not want to buy it either. There is no trade for the agent to make: he has wasted some of his labor allocation, as it would have been better to spend all of his labor in the production of good 2. It follows that autarky is an oligopoly equilibrium. The strength of Proposition 1 would be considerably enhanced if we could show that, for some Ricardian economies, the no-trade outcome would be the unique oligopoly equilibrium. Then, and in spite of, possibly huge, comparative advantages and strong incentives to organize advantageous exchange, the potential gains from trade would be inexorably wasted due to the noncooperative strategic interaction among the agents. That this is the case for large classes of Ricardian economies follows from the next analysis. First, the symmetry of the problem clearly implies that all the agents in the same country must play the same strategy at an oligopoly equilibrium, i.e. Y˜ H y˜ i 5 y˜ H 5 ], i [ * ; n and Y˜ F y˜ i 5 y˜ F 5 ], i [ ^. n The first derivatives ≠v i YF 1 H\hij ]i 5 ]]] 2 ], i [ *, H 2 Y a ≠y a 2* (Y ) 2 i
(7)
H
≠v Y 1 ]i 5 ]]] Y F\hij 2 ], i [ ^, a *1 ≠y a 1 (Y F )2
(8)
must be equal to zero at an interior maximum, with Y H\hi j (respectively Y F\hi j ) denoting the sum S k [ * y k (respectively S k [ ^ y k ). Accordingly, at an interior k ±i k ±i equilibrium, we must have a 2 (n 2 1)Y˜ F H y˜ 5 ]]]] a *2 n 2 and a *1 (n 2 1)Y˜ H F y˜ 5 ]]]] . a1n2
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Clearly, except in a very special case 1 , no values y˜ i []0,1[, ;i [*<^, can satisfy this system: No interior equilibrium exists. The following Lemma also excludes, as possible equilibria, situations in which one country specializes in the good he produces with a comparative disadvantage, while the other country produces both goods. Lemma 1 Excluding the no-trade outcome, the only remaining possibilities, for a 2 n-tuple of strategies (y˜ H , . . . , y˜ H , y˜ F , . . . ,y˜ F ) to be an oligopoly equilibrium are: 1. y˜ H 5y˜ F 51; 2. y˜ H 51; y˜ F []0,1[; 3. y˜ H []0,1[; y˜ F 51. Proof Since we have already shown that no interior equilibrium exists, the proof of the Lemma would be complete if we show that no pair ( y H , y F ) with y H 50, y F []0, 1], or y H []0, 1], y F 50 can be equilibria. Consider the first case. Suppose y˜ H 50, y˜ F []0, 1] is an equilibrium, then for all i in ^ we must have ≠v i ] ≠y i
U
y F 5 y˜ F y˜ H 50
>0
but from (8), it follows that ≠v i ] ≠y i
U
y F 5 y˜ F y˜ H 50
1 5 2 ] , 0, i [ ^, a 1*
a contradiction. Using (7), a similar reasoning applies for excluding the case y˜ H []0, 1], y˜ F 51. To identify the class of Ricardian economies in which the no-trade outcome is the unique oligopoly equilibrium, we may thus restrict ourselves to the three candidates identified in Lemma 1, and find conditions under which all of them are simultaneously excluded. Lemma 2 (i). y˜ H 5y˜ F 51 is an oligopoly equilibrium if, and only if a *2 a *1 (n 2 1)2 This exception occurs when ] 5 ]]] , in which case the economy has a continuum of 2 a2 a1n equilibria. This continuum includes, in particular, either y˜ H 51; y˜ F []0, 1[, or y˜ H []0, 1[; y˜ F 51. We assume, in this very particular case, that the agents select the corresponding equilibrium, namely, with one country specializing in the production of the good in which it has a comparative advantage and the other producing both goods. This selection is proposed in view of simplifying the presentation of the proofs, but the whole argument goes through without it.
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a 2* a1 n 2 1 n21 ] < ]] and ] < ]]; a2 n a 1* n
341
(9)
(ii). y˜ H 51; y˜ F []0, 1[is an oligopoly equilibrium if, and only if n 2 a *2 a *1 n ]]] < ] , ]]; 2 (n 2 1) a 2 a 1 n 2 1
(10)
H F (iii). y˜ []0, 1[; y˜ 51 is an oligopoly equilibrium if, and only if
2 n 2 1 a 2* (n 2 1) a 1* ]] , ] < ]]] . 2 n a2 n a1
(11)
i H F Proof First consider candidate (i). The necessary conditions for y˜ 5y˜ 5y˜ 51 to be an oligopoly equilibrium is that, for all i,
≠v i ]i ≠y
U
> 0, y 2i 51
which, by (7) and (8), implies that a *2 a1 n 2 1 n21 ] < ]] and ] < ]]. a2 n a *1 n ≠ 2v i Furthermore, it is easily checked that ]] i 2 , 0, so that these conditions are ≠( y ) also sufficient conditions for y˜ i 5y˜ H 5y˜ F 51 to be an oligopoly equilibrium. Now, consider candidate (ii). To be an oligopoly equilibrium, the necessary conditions ≠v i ]i 5 0, ;i [ ^ ≠y
(12)
≠v i ]i > 0, ;i [ * ≠y
(13)
and
must hold at equilibrium. With y˜ H 51, (8) and (12) imply that
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(n 2 1)a 1* F y˜ 5 ]]]; na 1
(14)
a necessary condition for y˜ H 51, y˜ F []0, 1[to be an oligopoly equilibrium is thus a 1* n ] , ]]. Furthermore, from (7) and (14), the inequality (13) can be rewritten a1 n 2 1 as a *1 a *2 n 2 ] > ]]]2 . a 1 a 2 (n 2 1) Moreover, by the concavity of the payoff function, these necessary conditions are also sufficient. A similar reasoning implies (iii). Finally it is readily seen from the necessary and sufficient conditions (10) and (11) that, given a 1 , a 1* , a 2 , a *2 and n, and excluding the no-trade outcome, at most one of the candidates above can be, indeed, an oligopoly equilibrium. According to Lemma 2 (i), an agent in the foreign country finds profitable to a1 n 2 1 deviate when ] > ]]. The first term of this inequality is a measure of the a 1* n absolute advantage of the agents in the home country in the production of good 1. Thus, the greater a *1 with respect to a 1 , the greater are the incentives of producing good 2, and exchanging it for good 1. The second term of the inequality is a measure of an agent’s market power 2 . The smaller n, the greater is his market power, and thus his incentive in manipulating the terms of trade by producing more of good 1. Accordingly, the decision to deviate from the competitive equilibrium depends on the relative magnitudes of these two effects. If both the inequalities in (9) are violated, then the agents in both countries have an incentive in deviating from the competitive equilibrium. If this is the case, then the only oligopoly equilibrium is the no-trade outcome. While, if only the agents in the foreign country have an incentive in deviating from the competitive equilibrium, (n 2 1)a 1* 3 F candidate (ii) can be an oligopoly equilibrium , with y˜ 5 ]]] (see (14)). na 1 This is the case, indeed, if for the agents in the home country it is still optimal to specialize following their comparative advantage, i.e. if the left inequality of (10) is satisfied. This happens when the gains from exchange, evaluated at y˜ F 5 (n 2 1)a 1* ]]], dominate the gains from manipulating the terms of trade. na 1 Now we are able to identify several classes of Ricardian economies for which 2
If we had considered a model with n 1 agents in country 2, the right inequality in (9) would have a1 n2 2 1 n21 written as: ] < ]]. Thus, ]] is a ‘‘concentration measure’’ of the foreign country industry. a *1 n2 n 3
Candidate (iii) can be an oligopoly equilibrium when the agents in the home country are those who have an incentive in deviating from the competitive equilibrium.
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the only oligopoly equilibrium is the no-trade outcome. To prove this statement, we introduce restrictions on the parameters (see (15) below) such that the necessary and sufficient conditions (9), (10), and (11) are all simultaneously violated. Proposition 2 For any number of competitors in either country, there are Ricardian economies for which the no-trade equilibrium is the unique oligopoly equilibrium. Proof Assume that the Ricardian economy (a 1 , a *1 , a 2 , a *2 , n) satisfies the inequalities a *1 n 2 1 a *2 n ]] , ] , ] , ]]. n a2 a1 n 2 1
(15)
First, the left and the right inequalities in (15) contradict the fact that y˜ H 5y˜ F 51 is an oligopoly equilibrium (Lemma 2). Now, assume that y˜ H 51, y˜ F []0, 1[is an oligopoly equilibrium. Then, by Lemma 2, the left inequality of (10) must hold, a *2 a1 n2 ] ]]]2 < ] , a 2 (n 2 1) a 1* which, with the right inequality of (15), implies that a *2 n2 n ] ]]]2 , ]], a 2 (n 2 1) n21 which, in turn, implies that a *2 n21 ] , ]], a2 n an inequality which contradicts the left inequality of (15). Consequently, y˜ H 51, y˜ F []0, 1[is not an oligopoly equilibrium. By a similar argument and using (15) and the right inequality of (11) in Lemma 2, y˜ H []0, 1[, y˜ F 51 is not an oligopoly equilibrium either. This completes the proof of Proposition 2. To illustrate the above proposition, it is worthwhile to consider the following example of a Ricardian economy: 1 1 a 1 5 a *2 5 ]; a *1 5 a 2 5 ]; n 5 2. 3 2 By Proposition 2, the no-trade equilibrium is the only oligopoly equilibrium, and y˜ H 50, y F 51, so that the world production of the two goods is (4, 4). At the competitive equilibrium, we have yˆ H 51, yˆ F 50, so that the world output is (6, 6): Strategic behaviour causes a reduction of world production of one third. Proposition 2 shows that a noncooperative behaviour of the economic agents can have devastating effects on the efficiency of trade when there is ‘‘competition among the few’’. In spite of the existence of, possibly huge, comparative
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advantages, each country may prefer to remain in autarky, wasting the advantageous opportunities opened by the possibility of free trade. Proposition 2 reveals that the set of Ricardian economies where this waste is observed, is non empty, however large the number n of competitors in each country. Nevertheless, as revealed by the following proposition, this set tends to vanish when n tends to infinity. Proposition 3 There exists a number n¯ such that, if the number of competitors in both countries exceeds n¯ , the set of oligopoly equilibria consists exactly of the no-trade equilibrium and the competitive equilibrium. Proof Since a 1 ,a *1 and a 2 .a 2* we have a *1 a *2 ] . 1 . ]. a1 a2 ¯ Consequently, there exists a number n¯ such that, ;n>n, a *1 n n 2 1 a 2* ] > ]] and ]] > ] a1 n 2 1 n a2 which are the inequalities (9). Propositions 2 and 3 have been demonstrated when country * has an absolute advantage in the production of good 1 and country ^ in the production of good 2. But analogous properties can be proved when one country has an absolute advantage in the production of both goods (for a detailed analysis, see Cordella and Gabszewicz, 1993). Furthermore, the whole analysis above goes through without assuming an equal number of agents in each country. Accordingly, the class of situations in which the comparative advantages are not exploited in the international exchange of goods is, by far, wider than the one resulting from (15). Nonetheless, in these other classes also, competition is restored when the number of agents increases in both countries. In the example analyzed above, we have assumed somewhat extreme preferences of the agents, and one might wonder whether our propositions do not rely heavily on this assumption. Proposition 1 holds under much more general conditions than those in the corpus of the paper. In particular, it holds for any ‘‘well behaved’’ utility functions of the agents. The argument is the following. First notice that, in a Ricardian framework, where all agents inside each country have the same constant returns to scale technology, there are no gains from trade at the autarky equilibrium. This implies that, before trade is opened, if agents act in a completely autarkic way inside their countries, they reach the same utility level as at the competitive equilibrium. From the Pareto optimality of the competitive equilibrium, it follows directly that, under autarky, the no-trade outcome is an oligopoly equilibrium. It is now easy to show that when trade is opened between the two countries, the no-trade outcome is still an oligopoly equilibrium. Suppose not, and assume that,
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after the opening of trade, some agent can increase his utility level obtained at the no-trade outcome, by producing more of good 1 (respectively 2) and supplying it to the market. This deviation increases his utility only if he can get more of good 2 (respectively 1) through the market, than he obtained under autarky. But, at the no-trade outcome, there is no supply of good 2 (respectively 1) since, by assumption, everybody acts optimally in an autarkic manner. A contradiction. Once proved that, in a Ricardian economy, the no trade outcome is an oligopoly equilibrium under very general conditions, it remains to check that the noncooperative behaviour of the agents may well destroy any incentive to trade, when agents also want to consume the good in which they have a comparative advantage. To check this, we have considered a variation of our example in which agents have strictly monotone preferences in both goods. More precisely, we assume that the utility of the agents in country * (respectively country ^) is given by u i (x 1 , x 2 )5 ´ ln x 1 1x 2 (respectively u i (x 1 , x 2 )5x 1 1 ´ ln x 2 ), with ´ .0 small. As for the technological coefficients and the number of agents, we choose, as in 1 1 the example provided above, a 1 5 a *2 5 ]; a *1 5 a 2 5 ]; n 5 2. For ´ small, the 3 2 preferences of the agents in this example are close to the extreme preferences 3 considered in the above analysis. Furthermore, when 0 , ´ , ], it can be checked 5 that, in the world market, there exists a unique oligopoly equilibrium (with the 5´ exception of the autarkic one), which is interior, and given by y˜ i 5 ], ;i [ H < 3 ^. Accordingly, when ´ tends to zero, the unique oligopoly equilibrium on the world market tends to the no-trade equilibrium. Thus we conclude that our findings, in the extreme case in which agents are only interested in one good (´ 50), are robust to a slight perturbation of preferences implying strict monotonicity in both goods.
4. Concluding remarks ‘‘Of the four possible [Ricardian competitive] equilibria in which at least one country specializes in production, the principle of comparative costs rules out only one: The inefficient case where both countries specialize according to comparative disadvantage’’ (Jones and Neary, 1984, p. 11). As seen in the above pages, this inefficient case is the only one to actualise in a wide class of Ricardian economies when the world market is oligopolistic. No doubt, in many situations, the loss to be incurred by interacting agents exploiting their market power can be considerable, compared with the gains which would accrue from cooperation. Nevertheless, it really came as a surprise to us that noncooperation in production and trade can be so harmful, since it fully destroys advantageous exchange. Ricardo asserted that ‘‘under a system of perfectly free commerce, each country naturally devotes its capital and labour to such employments as are most beneficial
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to each’’ (Ricardo, 1951, p. 133). Our example shows that this assertion must sometimes be qualified: it is only when sufficient competitive forces are restored, i.e., when the number of agents involved in trade increases, that this system of perfectly free commerce starts again to operate.
Acknowledgements We are indebted to B. De Meyer, V. Ginsburgh and H. Polemarchakis for helpful discussion. We are also grateful to our referees for interesting comments which improved considerably the last version of the paper. This text presents research results of the Belgian program on Interuniversity Poles of Attraction.
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