Sanghack Lee / Journal of Economic Research 12 (2007) 27–39

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Import Quotas in the Stackelberg Trade Model Sanghack Lee1 Kookmin University Received 22 February 2006 ; Accepted 5 May 2007 Abstract

This note examines the effects of import quotas in the Stackelberg duopoly model of Baye (1992) in which a home firm is a leader and a foreign firm is a follower in the home market. This note shows that import quotas can increase social welfare of the importing country, measured by the sum of consumer surplus, producer surplus and quota rents accruing to the importing country, if the quotas are set at a sufficiently high level. An increase in binding quotas is also shown to increase social welfare of the importing country if it can extract more than half of the quota rents. Quotas are also compared to tariffs. Unlike in the competitive markets, optimal quotas are shown to be more conducive to trade than optimal tariffs in the Stackelberg trade model. Keywords : Import Quota; Stackelberg Trade Model; Quota Rent; Tariff; Social Welfare JEL classification : F13, L13

1

School of Economics, Kookmin University, Seoul 136-702, South Korea. (E-mail) [email protected] (Phone) +82-2-910-4546. I wish to thank Michael R. Baye and two anonymous referees of this Journal for their valuable comments and suggestions on an earlier version of this note. This work was supported financially by research program 2007 of Kookmin University. The usual disclaimer applies.

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Import Quotas in the Stackelberg Trade Model

1

Introduction

One of central themes of research in international economics is to compare import quotas with tariffs. Much research has been focused on such comparison of tariffs and quotas in competitive markets, deriving (non)-equivalence of tariffs and quotas under different circumstances.2 Relatively little attention has been paid on the comparison of tariffs and quotas in imperfectly competitive markets, however.3 In his interesting paper, Michael R. Baye (1992) examines the effect of a quota in the Stackelberg duopoly model in which a home firm and a foreign firm compete in the home country. He shows that an import quota can increase profits of both firms. A binding quota is shown to decrease social welfare of the home country that is defined to be the sum of consumer surplus and producer surplus in the home country. However, his definition of social welfare excludes quota rents which may accrue to the home country. The purpose of this note is two-fold. First, this note examines welfare implications of import quotas when the home country can extract a portion of quota rent. It is shown that import quotas can increase social welfare, measured by the sum of consumer surplus, producer surplus and quota rents accruing to the importing country, if the quotas are set at a sufficiently high level. An increase in binding quotas is also shown to increase social welfare if the importing country can extract more than half of the quota rents. Secondly, this note compares quotas with tariffs in the Stackelberg trade model. Unlike in the competitive markets, optimal quotas are shown to be more conducive to trade than optimal tariffs. This result is in sharp contrast to the conventional wisdom that tariffs are more conducive to trade than quotas. The remainder of the note is organized as follows. Section 2 introduces the Stackelberg trade model of Baye (1992). Section 3 examines welfare implications of import quotas when a portion of quota rent is extracted by the home country. Section 4 analyzes the Stackelberg trade model with tariffs. The final section concludes. 2

See Bhagwati (1981, pp. xi - xii) for an overview of the literature. Rare exceptions are, among others, Bhagwati (1968), Shibata (1968), Rodriguez (1974) and Fung (1989). 3

Sanghack Lee / Journal of Economic Research 12 (2007) 27–39

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29

Stackelberg Trade Model of Baye (1992)

This section introduces Baye’s (1992) model in which international trade is modeled as a complete information Stackelberg duopoly. The home firm (firm 1) and the foreign firm (firm 2) compete in quantity in the home country. It is reasonable to assume that the home firm has more information on home demand conditions than the foreign firm. Thus, the home firm would naturally behave as a leader in the home market. The foreign firm observes the output of the home firm, and then determines how much to ship to the home country. The home firm is aware of this and maximizes profits, taking the foreign firm’s response into account. The two firms produce an identical product. Cost functions are given by C(xi ) = νxi ,

f or

i = 1, 2

(1)

where xi denotes firm i’s output and v is the constant (and identical) marginal cost for both firms. The (inverse) home demand for the product is given by P = α − β(x1 + x2 ).

(2)

Then, the profits of home and foreign firms are, respectively, given by Π1 (x1 , x2 ) = [α − β(x1 + x2 )]x1 − νx1 ,

(3)

Π2 (x1 , x2 ) = [α − β(x1 + x2 )]x2 − νx2 .

(4)

If the two firms were Cournot-Nash competitors, they would maximize (3) and (4), respectively, taking the other firm’s output as given. The reaction functions in such a case are given by x1 = R1 (x2 ) = (θ − βx2 )/2β,

(5)

x2 = R2 (x1 ) = (θ − βx1 )/2β,

(6)

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Import Quotas in the Stackelberg Trade Model

Figure 1

where θ = α − ν. The parameters α, β and θ are all assumed to be positive while v is non-negative. The free-trade Stackelberg equilibrium obtains when the home firm chooses x1 that maximizes Π1 (x1 , x2 ) subject to (6). Figure 1, adopted from Baye (1992), depicts the free-trade Stackelberg trade equilibrium. Isoprofit curves of home and foreign firms are, respectively, denoted π1 and π2 . R1 and R2 are reaction functions of home and foreign firms, respectively. The Stackelberg equilibrium corresponds to point xS in Figure 1, where 4 xS1 = θ/2β,

(7)

xS2

(8)

= θ/4β.

4 Note that eq. (8) corrects the typographic error in eq. (8) of Baye (1992) which states that xS 2 = θ/2β. To my best knowledge, all the other mathematical derivations in Baye (1992) are correct, however.

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At xS in Figure 1, the home firm’s isoprofit curve (π1 ) is tangent to the foreign firm’s reaction function. The market price in the free-trade Stackelberg equilibrium is P S = α − β(xS1 + xS2 ) = α − 3θ/4,

(9)

and the profits of the two firms are ΠS1 = θ2 /8β,

(10)

ΠS2

(11)

2

= θ /16β.

At point xH in Figure 1, the isoprofit curve π1 , passing through xS , intersects the home firm’s reaction function. Since xS and xH lie on the same isoprofit curve, it follows that H S Π1 (xH 1 , x2 ) = Π1

(12)

Also, xH lies on the reaction function of the home firm. Thus, xH 1 is the best response output of the home firm to the foreign firms output xH 2 , should it behave as a Cournot-Nash competitor. That is, H H xH 1 = R1 (x2 ) = (θ − βx2 )/2β.

(13)

Simultaneous solution of eqs. (12) and (13) yields xH 1 xH 2

√ = θ/2 2β, √ = (1 − 1/ 2)θ/β.

(14) (15)

At point xL the foreign firm’s isoprofit curve π2 , passing through intersects the home firm’s reaction function. As in the case with H x , the following two equations are satisfied. xS ,

L S Π2 (xL 1 , x2 ) = Π 2

xL 1

=

R1 (xL 2)

= (θ −

(16) βxL 2 )/2β

(17)

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Import Quotas in the Stackelberg Trade Model

Simultaneous solution of eqs.(16) and (17) yields √ xL = (1 + 1/ 2)θ/4β, 1 √ xL = (1 − 1/ 2)θ/2β. 2

(18) (19)

The effects of quotas are now examined. Suppose that a quota of Q is imposed on import from the foreign firm. Given the quota, the effective reaction correspondence of the foreign firm is given by R2 (x1 , Q) = min[Q, (θ − βx1 /2β)].

(20)

The horizontal line x2 = Q intersects R2 , given by (6), at ((θ − 2βQ)/β, Q). For xS1 belonging to the interval [0, (θ − 2βQ)/β], the reaction function of the foreign firm is given by the horizontal line R2 (x1 , Q) = Q. For xS1 > (θ − 2βQ)/β, the reaction function of the foreign firm is R2 (x1 , Q) = (θ − βx1 )/2β. It is easy to find that the H import quota set above xH 2 , Q > x2 , is not binding. In such a case, the home firm produces the same amount xS1 as in the free-trade Stackelberg equilibrium. Firm 2 also produces the same amount xS2 . In this case the import quota does not have any effect on the Stackelberg trade equilibrium. The only effect of the non-binding quota is that the quota rents now accrue to quota holders. The quota set below xH 2 is binding, however. The home firm knows this and reduces its output down to the point where its isoprofit curve is tangent to the horizontal line Q denoting the level of the import quota. This point is in fact the intersection of the horizontal line Q with the reaction function of the home firm, R1 . The market equilibrium is given by the point ((θ − βQ)/2β, Q). The quota enables the home firm to reduce output without worrying about the increase in sales of the foreign firm in response to home firm’s output reduction. It is easy to find that the binding quota increases the profits of the home firm. What is surprising is that the binding quota can increase the profits of the foreign firm as well. Baye (1992) shows that if the level of quota lies in the S H range between xs2 and xH 2 , i.e., x2 < Q < x2 , the quota increases the 5 profits of both firms. 5

For a detailed discussion, see Baye (1992).

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Social Welfare with Quota Rents in the Stackelberg Trade Model

This section examines the effect of quotas on social welfare. Once quotas are imposed, quota-holders can enjoy quota rents. For example, with voluntary export restraints (VERs), the exporters obtain the quota rents. There are many mechanisms whereby governments allocate import quotas. One possible way is to auction off the quotas. The home government obtains revenues in such a case. As Hranaiova and de Gorter (2005) note, the existence of quota rents provides incentives for importers and exporters to engage in socially wasteful rent-seeking activities to obtain the quota rents. In the limiting case the quota rents are fully exhausted, (Tullock 1980). However, in many cases, the importers and the home government can extract a portion of quota rents in the form of government revenues or profits of importers. Let d(0 ≤ d ≤ 1) denote the portion of quota rent accruing to the home country in the form of quota rents or government revenues. The value of d would be determined by institutional factors such as quota allocation mechanism. It is zero with VERs. However, it may have a positive value when the home government allocates quotas. This note simply assumes that the value of d is given exogenously by institutional setting. Social welfare of the home country is defined to be the sum of consumer surplus, profits of the home firm and a portion of quota rents accruing to the home country. As shown in Section 2, in the free-trade Stackelberg equilibrium, the foreign firm exports xS2 (= θ/4β) and obtains profits in the amount of θ2 /16β. With non-binding quotas, i.e., Q ≥ xH 2 , the firms produce the same amounts as in free-trade equilibrium. Then, social welfare W (Q) with quota Q, given by the sum of consumer surplus, producer surplus and a portion of quota rents, is calculated as 2

W (Q) = β(xS1 + xS2 ) /2 + ΠS1 + dΠS2 = 13θ2 /32β + dθ2 /16β, 2 xS2 ) /2

(21)

where β(xS1 + is the consumer surplus and dΠS2 is the portion of the foreign profit accruing to the quota-holders of the home country.

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Import Quotas in the Stackelberg Trade Model

With non-binding quotas, the foreign firm exports the same amount xS2 as in free trade. However, the profits θ2 /16β should be divided among quota holders, the foreign firm and the home government. Note that the portion of quota rents accruing to the home country is assumed to be exogenously given and denoted d in eq.(21). When a binding quota Q is imposed, i.e., Q < xH 2 , the foreign firm can export up to Q. The home firm takes this into account when determining its own output. The home firm produces (θ − βQ)/2β and the foreign firm delivers Q to the home market. Consumer surplus of the home country is given by (θ + βQ)2 /8β. The home firm’s profit is (θ − βQ)2 /4β. The home country reaps d proportion of the quota rent. Social welfare is then given by W (Q) = (θ + βQ)2 /8β + (θ − βQ)2 /4β + d(θQ − βQ2 )/2 = (3 − 4d)βQ2 /8 + (2d − 1)θQ/4 + 3θ2 /8β,

(22)

where d(θQ − βQ2 )/2 denotes the profit of the foreign firm when the import quota is set at Q < xH 2 . We now compare (21) with (22). Depending on the value of d, we have three cases to examine. Case 1.

3 4

≤d≤1

It follows that dW/dQ > 0 when Q ≤ xH 2 . That is, an increase in binding quota increases social welfare. This result is in sharp contrast to Baye’s (1992) result that dW/dQ < 0 when Q ≤ xH 2 . The difference comes from the assumption that d proportion of the quota rents now accrue to the home country. Moreover, W (xH 2 ) < 2 2 13θ /32β + dθ /16β, where the term in the right hand side of the inequality, 13θ2 /32β + dθ2 /16β, is the social welfare with non-binding 2 2 quota. Thus, for any Q ≤ xH 2 , W (Q) < 13θ /32β + dθ /16β. That is, social welfare with binding quota is always smaller than that with non-binding quota. Case 2.

1 2

≤d≤

3 4

It is easy to find that dW/dQ > 0 when Q ≤ xH 2 . Again, an increase in binding quota increases social welfare. Moreover, W (xH 2 )
W (t∗ ) if d > d∗ (≈ 0.66) and vice versa. In other words, social welfare under optimal quota is greater than that under optimal tariff if the importing country can absorb a substantial portion of quota rents. Otherwise, social welfare under optimal quota is smaller than that under optimal tariffs. When VERs are imposed, all the quota rents accrue to the foreign firms, and thus d = 0. It follows that W (t∗ ) > W (0). In other words, optimal tariffs are always better than VERs since d is zero with VERs.

5

Concluding Remarks

This note has examined the effects of quotas in the Stackelberg trade model when the importing country can extract a portion of quota rents. Optimal quotas are shown to increase social welfare if the importing country can reap half or more of quota rents. However, this result should be interpreted with caution. In the real world, rent-seeking activities are likely to be associated with allocation of quota rents, thereby reducing the social welfare with quotas. Optimal quotas are shown to be more conducive to trade than optimal tariffs in the Stackelberg trade model. This result is in sharp contrast to the conventional wisdom that tariffs are more conducive to trade than quotas.

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References Baye, M.R., “Quotas as Commitment in Stackelberg Trade Equilibrium,” Jahrb¨ ucher f¨ ur National¨okonomie und Statistik, 209, 1992, 20-30. Bhagwati, J. (ed.), International Trade: Selected Readings, Cambridge: Massachusetts, MIT Press, 1981. Bhagwati, J., “More on the equivalence of tariffs and quotas,” American Economic Review, 58, 1968, 142-146. Fung, F.C., “Tariffs, quotas, and international oligopoly,” Oxford Economic Papers, 41, 1989, 749-757. Hranaiova, J. and H. de Gorter, “Rent Seeking with Politically contestable Rights to Tariff-rate Import Quotas,” Review of International Economics, 14, 2005, 805-821. Rodriguez, C. A., “The Non-equivalence of tariffs and quotas under retaliation,” Journal of International Economics, 4, 1974, 295-298. Shibata, H., “Note on the equivalence of tariffs and quotas,” American Economic Review, 58, 1968, 137-142. Tullock, G., “Efficient Rent Seeking,” In J.M. Buchanan, R.D. Tollison and G. Tullock (eds.) Toward a Theory of the Rent-seeking Society, College Station, TX: Texas A&M University Press, 1980.