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Vertical trade and free trade agreements
J. Japanese Int. Economies 24 (2010) 569–585
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Vertical trade and free trade agreements Yasushi Kawabata a,*, Akihiko Yanase b, Hiroshi Kurata c a
Graduate School of Economics, Nagoya City University, 1 Yamanohata, Mizuho-cho, Mizuho-ku, Nagoya 467-8501, Japan Graduate School of International Cultural Studies, Tohoku University, Japan c Faculty of Economics, Tohoku Gakuin University, Japan b
a r t i c l e
i n f o
Article history: Received 25 September 2008 Revised 11 January 2010 Available online 2 April 2010 JEL classifications: F12 F13 F53 Keywords: Free trade agreements Vertical trade Cournot competition
a b s t r a c t Kawabata, Yasushi, Yanase, Akihiko, and Kurata, Hiroshi—Vertical trade and free trade agreements We investigate the effects of free trade agreements (FTAs) on tariffs and welfare in vertical trade. We consider a three-country model, where an FTA is formed between a country exporting a final good and a country exporting an intermediate good. The FTA unambiguously leads to a reduction in the member country’s tariff, but may cause the non-member country’s tariff level to increase. In the case, where FTA raises the non-member country’s tariff level, the FTA increases that country’s welfare. In contrast, the FTA may render its member countries better off. This result implies that the formation of an FTA may not always be Pareto-improving. J. Japanese Int. Economies 24 (4) (2010) 569–585. Graduate School of Economics, Nagoya City University, 1 Yamanohata, Mizuhocho, Mizuho-ku, Nagoya 467-8501, Japan; Graduate School of International Cultural Studies, Tohoku University, Japan; Faculty of Economics, Tohoku Gakuin University, Japan. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction Recently, the number of free trade agreements (FTAs) has been rapidly increasing across the world. According to Ministry of Economy, Trade and Industry (METI), Japan (2008), while there were 46 FTAs worldwide in 1990, the number increased to 194 in 2007. In particular, countries in Asia, the Middle East, and Africa, which previously lacked FTAs, are now accelerating the formation of FTAs.
* Corresponding author. Fax: +81 52 872 5014. E-mail address: [email protected] (Y. Kawabata). 0889-1583/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jjie.2010.03.002
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With an increase in their number, FTAs have become an important topic in international trade theory. One research stream on FTAs clarifies their effects on welfare (e.g., Kose and Riezman, 2000; Yi, 2000; Bond et al., 2004), and another focuses on multilateralism and FTA (e.g., Krishna, 1998; Freund, 2000; Ornelas, 2005; Saggi, 2006). The above studies on FTA address horizontal trade, i.e., international trade in which countries mutually export final goods. However, vertical trade, i.e., international trade in which one country exports intermediate goods to other countries that process the imported inputs into final goods for their exports, has gained significance in recent trade transactions. Such vertical trade structures are particularly notable in the Asian region. For example, Japan exports components such as semiconductors for computers to other Asian countries such as China, and imports computers mainly from China (JETRO, 2006). Further investigations are required to consider vertical trade structures. The purpose of this paper is to clarify how FTAs affect tariff rates and, in turn, welfare levels in the context of a vertical trade structure. We construct a simple three-country model, in which one ‘‘upstream” country exports an intermediate good to two other ‘‘downstream” countries and imports a final good from both of them. Before the FTA, the government in each country determines its tariff level in a noncooperative fashion. We then consider an FTA between the upstream country and one of the downstream countries. The other downstream country is regarded as a non-member country. Thus, tariffs between the FTA member countries are eliminated, while an external tariff of the upstream country and the tariff of the non-member downstream country are still endogenously determined. In this setting, we investigate how the FTA affects each country’s tariff levels and, in turn, its welfare. In our model, which considers Cournot competition with linear demands and constant marginal costs, each country’s optimal tariff is influenced by the other countries’ tariffs. This finding is in contrast to those of previous studies, including Freund (2000), Ornelas (2005), and Saggi (2006), which adopt a similar assumption in the analysis of FTAs with horizontal trade structures. In these studies, it is shown that the formation of an FTA induces its member countries to reduce their external tariffs, but the FTA does not affect the non-member country’s tariff. Thus, these studies do not completely illustrate the strategic relationship between FTA member and non-member. We propose an alternative framework, apart from the existing horizontal trade models, to describe the strategic relationship between member and non-member countries. We obtained the following results with regard to the FTA’s tariff effects. The formation of the FTA unambiguously induces its member country to reduce the tariff for the non-member country. This external tariff after the FTA may be negative (i.e., an import subsidy). In contrast, while the non-member country may or may not raise its tariff level, it certainly imposes a positive tariff after the FTA. In most cases, the FTA induces the non-member country to raise its tariff. With regard to the effect of the FTA on each country’s welfare, the following results are obtained. For the non-member country, raising the tariff after the FTA leads to an increase in its welfare. As regards the FTA members, the downstream country is better off after the FTA as long as its FTA partner (i.e., the upstream country) imposes a positive external tariff after the FTA. However, it may be worse off if the external tariff imposed by the FTA partner is negative. In contrast, the FTA may or may not make the upstream member better off. In addition, some numerical analyses indicate that the FTA may increase the members’ joint welfare as well as world welfare. As mentioned above, this paper focuses on tariff and welfare effects of FTAs; therefore, it is related to FTA studies for the most part. However, studies on vertical trade policies are also an important reference for this study (e.g., Bernhorfen, 1997; Ishikawa and Spencer, 1999; Chang and Sugeta, 2004; Yanase and Kawabata, 2008). In fact, this paper is structured on the basis of the literature on vertical trade policies. Thus, our study links these two research streams and provides a new insight into FTA studies. The rest of this paper is organized as follows. Section 2 presents the basic model. Section 3 derives the equilibrium and examines pre-FTA tariffs. Section 4 considers the effects of the FTA on the member country’s external tariff and the non-member country’s tariff levels. Section 5 analyzes the effects of the FTA on welfare. Section 6 concludes the paper.
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2. The model The world economy consists of three countries, A, B, and C, which trade among themselves in a final good and an intermediate good. The final good is produced by n P 1 identical firms in countries B and C, respectively, and l P 1 identical firms in country A. In contrast, the intermediate good is produced by m P 1 identical firms located only in country A. We further assume that the final good is consumed only in country A. Therefore, country A exports the intermediate good to countries B and C, and imports the final good from these countries. Note that some quantity of the final good is domestically supplied in country A. In the following paragraphs, we often use the terms ‘‘final-good firms” and ‘‘intermediate-good firms” to refer to firms producing the final good and the intermediate good, respectively. The government in country A imposes a specific tariff, T iA , on final-good imports from country i (i = B, C), whereas the government in country i imposes a specific tariff, ti, on intermediate-good imports from country A. The setting is briefly illustrated in Fig. 1. In the subsequent analysis, we consider an FTA between countries A and B, and regard country C as the non-member country. The FTA eliminates tariffs between the FTA members A and B. Therefore, the values of T BA and tB move toward zero after the FTA. We assume that production of one unit of the final good requires one unit of the intermediate good. Let us use indexes Ak and ij to represent the kth final-good firm in country A (k = 1, . . ., l), and the jth final-good firm in country i (i = B, C, j = 1, . . ., n), respectively. Let yAk and yij be the output of final-good firms Ak and ij, respectively, and Y be the total output of the final good. Then, the profits of the finalgood firms Ak and ij are given by
PAk ¼ ðpðYÞ r A ÞyAk ;
ð1aÞ
Pij ¼ ðpðYÞ ri T iA Þyij ;
ð1bÞ
and
respectively, where p(Y) is the inverse demand for the final good and ri is the price of the intermediate good in country i’s market (i = A, B, C).
Fig. 1. Model structure.
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We use the index h to denote intermediate-good firms in country A (h = 1,. . .,m). The marginal cost of production for the intermediate good is normalized at zero. Then, the intermediate-good firm h’s profit, Ph, is expressed by
Ph ¼ r A xAh þ ðr B t B ÞxBh þ ðrC t C ÞxCh ;
ð2Þ
xih
where is the intermediate-good firm’s supply to country i (i = A, B, C). In our setting, each country’s welfare is defined as follows:
WA ¼
m X
Ph þ
h¼1
WB ¼ WC ¼
n X
l X
PAk þ CS þ T BA
k¼1 m X
PBj þ tB
j¼1
h¼1
n X
m X
PCj þ tC
j¼1
n X
! yBj
j¼1
! xBh ;
þ T CA
n X
! yCj ;
ð3Þ
j¼1
ð4Þ
! xCh
ð5Þ
;
h¼1
RY where CS ¼ 0 pðsÞds pðYÞY is the consumer surplus in the final-good market in country A. The welfare of country A is the sum of the profits of intermediate- and final-good firms, the consumer surplus, and the country’s tariff revenue. In contrast, the welfare function of country i is the sum of the profits of final-good firms in the country and its tariff revenue (i = B, C). The model involves three stages of decision making. In the first stage, each government endogenously determines its own tariff levels. Under the FTA, although the tariffs between countries A and B are eliminated, the governments in countries A and C must set the tariff level. In the second stage, the intermediate-good firms in the intermediate-good market in countries A, B, and C are in Cournot competition. In the third stage, given intermediate-good prices, the final-good firms play Cournot competition in the final-good market in country A.1 We derive the subgame-perfect equilibrium of this model. In particular, we focus on endogenous tariff levels and welfare values before and after FTA. 3. Equilibrium 3.1. Equilibrium in the final-good market Using backward induction, we begin to consider the final-good market. Each final-good firm chooses its output so as to maximize its own profit, taking the outputs of rivals, the price of the intermediate good, and tariffs as given. From Eqs. (1a) and (1b), the first-order conditions for profit maximization of final-good firms Ak and ij are obtained as follows (i = B, C; j = 1, . . ., n, k = 1, . . ., l):
pðYÞ r A þ p0 ðYÞyAk ¼ 0; pðYÞ r i
T iA Pl
0
þ p ðYÞyij ¼ 0; Pn
ð6aÞ i ¼ B; C:
ð6bÞ
Pn
Note that Y ¼ k¼1 yAk þ j¼1 yBj þ j¼1 yCj . The Cournot-Nash equilibrium outputs in the final-good market are derived from conditions (6a) and (6b). In terms of symmetry, the outputs of all firms within ~Ak ¼ y ~A and y ~ij ¼ y ~i ði ¼ B; C; j ¼ 1; . . . ; a country are the same in an equilibrium. That is, y n; k ¼ 1; . . . ; lÞ, where the tilde is used to represent the equilibrium value. Thus, in an equilibrium, e ¼ ly ~A þ ny ~ B þ ny ~C . Y 3.2. Equilibrium in the intermediate-good market Before analyzing the Cournot competition among intermediate-good firms in country A, we focus on the inverse demand for intermediate goods. Since one unit of the final good is produced with one 1 This setting implies that the downstream firms recognize their market power in the final-good market but act as price-takers in the intermediate-good market. This setting is often adopted in literature on vertical trade (e.g., Bernhorfen, 1997; Hwang et al., 2007; Yanase and Kawabata, 2008). Ishikawa and Spencer (1999) provide a detailed discussion of the justification for this setting.
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unit of the intermediate good, the total demand for the intermediate input in countries A, B, and C are ~ A ; ny ~B , and ny ~C , respectively. Letting Xi be the total supply of the intermediate good to country i (i = A, ly B, C), the market-clearing conditions in these markets are given by
~A ; X A ¼ ly
~B ; X B ¼ ny
~C : and X C ¼ ny
ð7Þ A
B
C
Eq. (7) yields the inverse demand of the intermediate good, ri = ri(X , X , X ), i = A, B, C. Each intermediate-good firm determines the outputs in markets A, B, and C so as to maximize its own profit, taking the output of rival intermediate-good firms and tariffs as given. From Eq. (2), the first-order conditions for profit maximization of intermediate-good firm h in country A (h = 1, . . ., m) are given by
rA þ @r A @X B @rA @X C
@r A @X
A
xAh þ
@r B @X
A
xBh þ
xAh þ rB t B þ xAh þ
@r B @X C
@r B @X B
@rC @X A
xCh ¼ 0;
xBh þ
xBh þ r C t C þ
@rC @X B @r C @X C
xCh ¼ 0;
ð8Þ
xCh ¼ 0:
The conditions in Eq. (8) determine the Cournot-Nash equilibrium outputs in the intermediategood market. From symmetry, all firms’ outputs are the same. That is, ~ xih ¼ ~ xi ði ¼ A; B; CÞ, and thus X i ¼ m~ xi in an equilibrium. Substituting these outputs into the equilibrium outputs and prices in the final-good market, we have the subgame-perfect equilibrium solutions, all of which are functions of the tariff rates, given the number of intermediate- and final-good firms. That is,
~xi ¼ ~xi ðT BA ; T CA ; t B ; tC ; l; m; nÞ; ~r i ¼ ~ri ðT BA ; T CA ; tB ; t C ; l; m; nÞ;
ð9Þ
~i ðT BA ; T CA ; t B ; tC ; l; m; nÞ; ~i ¼ y y ~¼p ~ðT BA ; T CA ; t B ; tC ; l; m; nÞ; p
i ¼ A; B; C:
In the subsequent analysis, we assume a linear demand: p(Y) = a Y, where a > 0 is large enough to ensure positive outputs. Then, the uniqueness and stability of the Cournot-Nash equilibrium are guaranteed, and the second-order conditions for profit maximization of any firm are satisfied in both the final- and intermediate-good markets. 3.3. Endogenous pre-FTA tariffs We now focus on the first stage, the endogenous tariff determination. Here, we consider the situation before FTA. First, we consider country i (i = B, C). Country i determines the import tariff ti to maximize Wi, given the other countries’ tariff levels. From Eqs. (4) and (5), country i’s first-order condition is given by
@W i @y @p @r i @xi þ mxi þ ti m ¼ nðp r i T iA Þ i þ nyi ¼ 0; @t i @t i @ti @t i @t i
ð10Þ
for i = B, C. The sum of the first and second terms gives the effect of the tariff ti on the profits of the final-good firm i. The tariff ti decreases the output, and therefore the profit, of the final-good firm i. The third and fourth terms are the changes in tariff revenues. An increase in the tariff ti increases (resp., decreases) the tariff revenue of country i for low (resp., high) tariffs. Substituting the equilibrium outputs into Eq. (10) and rearranging it, we obtain country i’s reaction function as
ti ¼
n o fl þ 2n þ 1 mðl þ 1Þg a ðl þ n þ 1ÞT iA þ nT jA þ nt j 2ðl þ 2n þ mn þ 1Þðl þ n þ 1Þ
;
ð11Þ
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for i, j = B, C; i – j. Notice that the tariff level depends on the other countries’ tariffs, unlike in the horizontal trade model (e.g., Freund, 2000), in which tariff levels do not depend on other countries’ tariffs. From Eq. (11), if l + 2n + 1 m(l + 1) > 0, then ti is related negatively to T iA and positively to T jA and tj. In other words, ti and T iA are strategic substitutes, whereas ti and T jA as well as ti and tj are strategic complements. Second, we consider country A. Country A determines the import tariffs T BA and T CA to maximize WA, given the other countries’ tariff levels. From Eq. (3), country A’s first-order condition for maximizing welfare with respect to T iA is given by
@W A @T iA
¼ mr A
@xA @T iA
þ T jA n
þ mðr j t j Þ
@yj @T iA
@xj @T iA
þ nyi þ T iA n
þ mðr i t i Þ
@yi @T iA
@xi @T iA
þ mxi
@r i @T iA
þ lðp r A Þ
@yA @T iA
þ lyA
¼ 0;
@p @T iA
Y
@p @T iA ð12Þ
for i, j = B, C; i – j. The sum of the first to fourth terms represents the effect of tariff T iA on the profits of the intermediate-good firms in country A. The tariff T iA lowers the price of the intermediate good in country i and decreases the supplies of intermediate-good firms in market i. On the other hand, it increases their supplies to country A and country j. Since the former effect dominates the latter, the tariff T iA decreases total output, xA + xB + xC, and in turn the profits of intermediate-good firms. The sum of the fifth and sixth terms gives the effect on the profits of final-good firms in country A; the tariff T iA increases the output and, in turn, the profits of final-good firms in country A. The seventh term expresses the effect on consumer surplus; the tariff T iA raises the price of the final good and decreases consumer surplus. The eighth term represents the effect on tariff revenue from country j; the tariff T iA increases the imports from country j, raising the tariff revenue from that country. Finally, the sum of the ninth and tenth terms represents the effect on tariff revenue from country i. The increase in the tariff T iA raises (resp., reduces) the tariff revenue from country i for low (resp., high) tariffs. Substituting the equilibrium outputs into Eq. (12) and rearranging the equation, we get country A’s reaction function as
T iA ¼
d1 a þ ð4l þ 4n þ 3ÞmnT jA d2 t i þ d3 nt j 4n2 þ ð4l þ 5Þn þ 2ðl þ 1Þ2
;
i; j ¼ B; C;
i – j;
ð13Þ
where d1 m(2l + 1) l 2n 1, d2 2(m 1)n2 + {(m 3)l + 2m 3}n + (l + 1)2(m 1), and d3 (3l + 2n + 2)m l 2n 1. Note that the tariff level depends on the other countries’ tariffs. This is a feature of the vertical trade model. Note also that in Eq. (13) T iA is positively related to T jA and tj. In other words, T iA and tj are strategic complements. In contrast, the relationship between T iA and ti is ambiguous because d2 can be either negative (e.g., m = 1) or positive (e.g., m P 2). Solving Eqs. (11) and (13), we obtain the following Nash equilibrium import tariffs (indicated by adding superscript ‘‘N”):
t NB ¼ t NC ¼ N
N
fl þ 2n þ 1 mðl þ 1Þgðl þ m þ 1Þa ðm þ
T BA ¼ T CA ¼
1Þf2mn2
þ 2ððm þ 1Þl þ 2m þ 1Þðl þ 1Þn þ ðm þ 1Þðl þ 1Þ3 g f2ðm þ 1Þn2 þ /1 n þ /2 ga
ð14Þ
;
ðm þ 1Þf2mn2 þ 2ððm þ 1Þl þ 2m þ 1Þðl þ 1Þn þ ðm þ 1Þðl þ 1Þ3 g
;
ð15Þ
where /1 (2m + 3)(m 1)l + (m + 1)(m 3) and /2 (m2 + 2m 1)l2 + 2(m2 + m 1)l + (m + 1)(m 1). Substituting Eqs. (14) and (15) into the equilibrium outputs and prices obtained in the second and third stages, all outputs, prices, and tariffs are functions of the number of firms. Therefore, the economy is described by parameters l, m, and n. The signs of the Nash equilibrium tariffs depend on a vector of parameters (l, m, n), which describes the industrial structure of the world economy. From Eqs. (14) and (15), we have the following lemma.2 2
All proofs of upcoming lemmas and propositions appear in the appendices.
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CN Fig. 2. Optimal policies and the number of firms (given m). (l, m, n) 2 H1: T BN A ¼ T A > 0;
tNB ¼ tNC > 0.
Lemma 1. If (l, m, n) 2 H1 {(l, m, n)jnt(l, m) 6 n 6 n1(l, m)}, all countries impose nonnegative tariffs BN CN N before the FTA; i.e., tN qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B ; t C ; T A ; T A P 0, where
nt ðl; mÞ
ðm 1Þðl þ 1Þ 2
and n1 ðl; mÞ
/21 þ 8ðm þ 1Þ/2
/1 þ
4ðm þ 1Þ
:
Lemma 1 requires that n, the number of final-good firms in each downstream country, be sufficiently large for positive optimal tariffs on the intermediate good and sufficiently small for positive optimal tariffs on the final good in the pre-FTA equilibrium. The explanation for this is as follows3: A smaller n provides higher overall profits in the final-good market, because final-good firms in countries B and C face fewer competitors. Thus, countries B and C may have a greater incentive to provide import subsidies for the intermediate good to enhance domestic firms’ competitiveness, rather than earn tariff revenues by imposing positive tariffs. Alternatively, a larger n leads to higher profits for intermediategood firms. Thus, country A may have a greater incentive to provide subsidies for final-good imports because more final-good imports lead to increased intermediate-good exports. Fig. 2 illustrates the relationship between optimal policies and the number of final-good firms, given the number of intermediate-good firms. We have the number of final-good firms in country A on the horizontal axis and the number of final-good firms in countries B and C on the vertical axis. N
N
On lines n = nt and n = n1, we have t NB ¼ tNC ¼ 0 and T BA ¼ T CA ¼ 0, respectively. Import tariffs on the intermediate good, t NB ¼ tNC , are positive above and negative below the n = nt line. In contrast, import N
N
tariffs on the final good, T BA ¼ T CA , are negative above and positive below the n = n1 line. Lemma 1 implies that if the difference between the number of final-good firms in upstream and downstream countries is extremely large, optimal policies for these countries can involve import subsidies. In the following analysis, we will assume (l, m, n) 2 H1. In other words, we will limit our attention to cases in which all the countries impose positive tariffs before FTA.
4. Effects of the FTA on tariff levels Next, we consider the situation after FTA. After the FTA is formed between countries A and B, tB and T BA become zero. However, country C still imposes an import tariff for the intermediate good, tC, and country A imposes an import tariff for the final good, T CA . 3
This is pointed out by one of the referees of this journal. The authors appreciate this crucial contribution.
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4.1. Endogenous tariffs under FTA Substituting tB ¼ T BA ¼ 0 into Eqs. (3) and (5) and partially differentiating them with respect to T CA and tC, respectively, we obtain country A’s and C’s reaction functions under FTA as
T CA ¼ tC ¼
d1 a d2 t C ; 4n2 þ ð4l þ 5Þn þ 2ðl þ 1Þ2 n o fn nt ðl; mÞg a ðl þ n þ 1ÞT CA ðl þ 2n þ mn þ 1Þðl þ n þ 1Þ
ð16Þ
ð17Þ
:
Solving Eqs. (16) and (17), we obtain the following Nash equilibrium import tariffs after FTA (indicated by adding superscript ‘‘F”):
2fn nt ðl; mÞgð2mn þ l þ m þ n þ 1Þa ; ðm þ 1Þðl þ n þ 1Þf2ð2m þ 1Þn2 þ ð3ðm þ 1Þl þ 4m þ 3Þn þ ðl þ 1Þ2 ðm þ 1Þg f2ð2m þ 1Þn2 þ /3 n þ /2 ga ¼ ; ðm þ 1Þðl þ n þ 1Þf2ð2m þ 1Þn2 þ ð3ðm þ 1Þl þ 4m þ 3Þn þ ðl þ 1Þ2 ðm þ 1Þg
t FC ¼ T CA
F
ð18Þ ð19Þ
where /3 3(m + 1)(m 1) l + 2m2 3m 3. Comparing Eqs. (14) to (18) and (15) to (19), we get the following results. Lemma 2. (1) For any ðl; m; nÞ 2 H1 ; tFC is nonnegative. F (2) If (l, m, n) 2 H2 {(l, m, n)jnt(l, m) 6 n 6 n2 (l, m)}, then T CA is nonnegative, and if (l, m, CF n) 2 H3 H1nH2, then T A is negative, where
n2 ðl; mÞ
/3 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /23 þ 8ð2m þ 1Þ/2 4ð2m þ 1Þ
:
Lemma 2 shows there are some possibilities that, as a result of the FTA, member country A could change its policy from tariff to subsidy. In contrast, non-member country C does not change its policy in response to the FTA. Fig. 3 illustrates the parameter regions shown by Lemma 2. In this figure, the F threshold line n = n2 is added into Fig. 2. On line n = n2, we have T CA ¼ 0. Import tariffs for the final CF good, T A , are negative above n = n2, and positive below the line. Lemma 2 shows that the parameter
CN Fig. 3. Country A’s optimal policy and the number of firms (given m). Region I ((l, m, n) 2 H2): T BN A ¼ T A > 0; CN ((l, m, n) 2 H3 H1nH2) : T BN T CF A ¼ T A > 0; A < 0.
T CF A > 0; Region II
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space H1 is divided into two regions. In region I, member country A imposes positive tariffs both before and after the FTA. In contrast, in region II, member country A sets positive tariffs before and negative tariffs (i.e., subsidies) after the FTA. Non-member country C imposes positive tariffs in both regions. 4.2. Comparisons of equilibrium tariffs We now focus on the effect of the FTA on the tariff levels. From Lemmas 1 and 2, it is clear that for some (l, m, n) 2 H1, country A may provide a subsidy after the FTA even when it chooses to impose a tariff before the FTA. This implies that country A’s tariff on final-good import is reduced, as formally demonstrated in the following proposition. Moreover, the following proposition indicates how country C’s tariff rate is affected by the FTA. Proposition 1 N
F
(1) For any ðl; m; nÞ 2 H1 ; T CA > T CA holds. (2) If (l, m, n) 2 H4 {(l, m, n)jnt(l, m) 6 n 6 n3 (l, m)}, then tNC 6 tFC holds, and if (l, m, n) 2 H5 H1nH4, then tNC > tFC holds, where
n3 ðl; mÞ
/4 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /24 þ 8ðl þ 1Þð2m þ 1Þ/5 4ðl þ 1Þð2m þ 1Þ
;
/4 (4m + 3)(m 1)l2 + (5m2 4m 6)l + 2m2 3m 3, and /5 (2m 1)(m + 1)l3 + (4m2 + 2m 3)l2 + (3m2 + m 3)l + (m + 1)(m 1). Proposition 1(1) shows that the FTA provides an incentive to member country A to reduce its tariff. It implies that the tariff complementarity effect (Bagwell and Staiger, 1999) holds in a vertical trade model. That is, when FTA members reduce their internal tariffs to zero, they find it advantageous to lower their external tariffs as well. The elimination of T BA decreases imports from country C, which in turn lowers country A’s tariff revenue and consumers surplus. Country A alleviates these negative effects by reducing T CA to expand imports from country C. Further, the reduction in T CA increases final-good output in country C, and thereby increases the profits of intermediate-good firms in country A. Therefore, country A has an incentive to reduce its external tariff after the FTA.
Fig. 4i. Country C’s tariff level and the number of firms (m < 2). Region I ((l, m, n) 2 H2): tNC 6 t FC if (l, m, n) 2 H4; tNC > t FC if (l, m, n) 2 H5; Region II ((l, m, n) 2 H3): tNC > tFC .
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Fig. 4ii. country C’s tariff level and the number of firms (m P 2). Region I ((l, m, n) 2 H2): t NC 6 t FC ; Region II ((l, m, n) 2 H3): tNC 6 tFC if (l, m, n) 2 H4; tNC > tFC if (l, m, n) 2 H5.
In contrast, Proposition 1(2) states that the FTA may or may not provide such an incentive to nonmember country C. An intuition is as follows. After the formation of the FTA between countries A and B, country A sets T BA ¼ 0, which has a negative effect for country C.4 At the same time, from Proposition 1 (1), country A reduces T CA , which is a positive effect of the FTA. If the positive effect dominates the negative one, the final-good output in country C increases. Then, intermediate-good imports from country A must increase. This increases country C’s incentive to impose a higher tariff on country A. If the negative effect is dominant, the result is just the opposite. Fig. 4 explains the changes in non-member country C’s tariff levels. The threshold line n = n3 is included in Fig. 3. On the n = n3 line, we have t NC ¼ tFC . Above n = n3, t NC > tFC , but below the line, t NC < tFC . Fig. 4i shows that line n = n3 appears below the n = n2 line. That is, in region I (i.e., the region where country A imposes a positive tariff both before and after the FTA), country C may or may not raise its tariff, while in region II (i.e., the region where country A sets a positive tariff before and a negative tariff after the FTA), country C reduces its tariff. In contrast, Fig. 4ii shows that line n = n3 appears above the n = n2 line. That is, country C raises its tariff in region I, but may or may not do so in region II. In light of Lemma 2, part (2) of Proposition 1 can be restated as follows. Corollary 1 (1) Suppose that (l, m, n) 2 H2, where all countries impose positive tariffs after the FTA between countries A and B. Then, for m P 2, non-member country C raises its tariff after the FTA. For m < 2, it is possible that country C would reduce its tariff after the FTA. (2) Suppose that (l, m, n) 2 H3, where country A provides subsidies after the FTA. Then, for m < 2, nonmember country C reduces its tariff after the FTA. For m P 2, it is possible that country C would raise its tariff after the FTA. 5. Effects of the FTA on Welfare Finally, we investigate the effects of the FTA on welfare. Before starting the analysis, let us define N N F ~i ðT BA ; T CA ; tNB ; tNC Þ, yFi y ~i ð0; T CA ; 0; tFC Þ, i = A, B, C, Y N lyNA þ nyNB þ nyNC , and Y F lyFA þ nyFB þ nyFC . yNi y The following result, which clarifies the effects of the FTA on the equilibrium output levels, is useful for the subsequent welfare analysis. 4
Further, country B sets tB = 0 after the FTA, which increases the above negative effect.
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Proposition 2 (1) (2) (3) (4)
For any ðl; m; nÞ 2 H1 ; For any ðl; m; nÞ 2 H1 ; If (l, m, n) 2 H4 (resp., For any (l, m, n) 2 H1,
yNA > yFA . yNB < yFB . (l, m, n) 2 H1nH4), yNC 6 yFC (resp., yNC > yFC ). YN < YF.
Proposition 2(1)–(3) say that the FTA decreases domestic final-good output in country A, increases final-good imports from country B, and may or may not increase final-good imports from country C. These results are obtained by the following mechanisms: From Proposition 1 (1), the FTA reduces country A’s tariff as well as eliminating tariff on imports from country B. This reduction in country A’s tariff decreases domestic final-good production and increases final-good imports from country B. At the same time, the reduction may increase final-good imports from country C. If so, country C has an incentive to raise its tariff on intermediategood imports. Of course, raising country C’s tariff has a negative effect on final-good production in country C. However, if the reduction in country A’s tariff is sufficiently large, final-good imports from country C still increase even if country C raises its tariff on intermediate-good imports. Proposition 2(4) shows that, irrespective of the imports from country C, the FTA increases the total output in country A. It implies that the direct effect of the FTA, i.e., the effect from the elimination of tariffs between countries A and B, is sufficiently larger than the others. In the following analysis, we use numerical analysis, partly, to clarify the result. In the numerical analysis that follows, we assume that a = 10. 5.1. Non-member country First, we focus on non-member country C. Country C’s welfare consists of the profits of the finalgood firms and tariff revenue. The welfare levels before and after the FTA, indicated by W NC and W FC , respectively, are given by the following expression:
W iC ¼ nðyiC Þ2 þ mt iC xiC ¼ nyiC ðyiC þ t iC Þ;
i ¼ N; F:
ð20Þ
Note that we use Eq. (7) to obtain the second part of Eq. (20). As shown in Lemma 2 and Proposition 1, country C imposes positive tariffs after the FTA, and its tariff level may or may not be higher than that before the FTA. We find that this change in tariff levels before and after the FTA is also critical to country C’s welfare. That is, the following proposition is evident from Proposition 2 and Eq. (20). N F F N F Proposition 3. If (l, m, n) 2 H4, where t N C 6 t C holds, then W C 6 W C , and if (l, m, n) 2 H5, where t C > t C N F holds, then W C > W C .
Proposition 3 shows that whenever the FTA raises (resp., lowers) country C’s tariff tC, it also increases (resp., decreases) country C’s welfare WC. From Proposition 1, as a result of the FTA, country A’s tariff is reduced but country C’s tariff may or may not decrease. As we observe in Proposition 2, if country A’s tariff reduction is relatively substantial, final-good output is increased. Furthermore, a higher tC causes country C’s tariff revenue to increase. Thus, the FTA results in an increase in country C’s welfare. However, the final-good output can decrease, especially when only a few intermediategood firms exist. In this case, country C’s tariff, and consequently its tariff revenue, decreases. In sum, the FTA causes country C’s welfare to decrease. Table 1 shows changes in tariff and welfare for country C before and after the FTA, under different combinations of (l, m, n). It clearly illustrates the statement in Proposition 2. Whenever the tariff imposed by country C increases (resp., decreases) after the FTA, country C’s welfare increases (resp., decreases).
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Table 1 Changes in tariff and welfare. (l, m, n)
Region
T CF A
DtC
DWA
DWB
DWC
P
(2, (3, (6, (5, (6, (1, (3, (4, (5, (4,
I I I II II II II I I II
0.058 0.222 0.315 0.014 0.035 0.036 0.018 0.048 0.094 0.028
0.001 0.006 0.004 0.004 0.007 0.003 0.0003 0.007 0.008 0.002
0.372 0.145 0.0003 0.277 0.277 0.557 0.315 0.103 0.023 0.292
0.096 0.226 0.176 0.0001 0.046 0.059 0.087 0.359 0.473 0.0002
0.006 0.028 0.016 0.019 0.035 0.057 0.005 0.132 0.167 0.029
0.468 0.371 0.175 0.277 0.231 0.615 0.403 0.462 0.451 0.291
1, 1, 1, 1, 1, 2, 2, 2, 2, 2,
1) 1) 1) 3) 4) 2) 5) 5) 5) 7)
DtC ¼ tFC tNC ;
i¼A;B DW i
P
i¼A;B;C DW i
0.462 0.399 0.192 0.258 0.195 0.558 0.408 0.594 0.617 0.262
DW i ¼ W Fi W Ni
5.2. FTA member countries 5.2.1. Country B’s welfare Second, we consider FTA member country B. Country B’s welfare levels before and after the FTA are shown as follows:
W NB ¼ nðyNB Þ2 þ mt NB xNB ¼ nyNB ðyNB þ t NB Þ;
ð21Þ
W FB ¼ nðyFB Þ2 :
ð22Þ
and
Before the FTA, country B’s welfare consists of the sum of the profits of final-good firms in the country and its tariff revenue. After the FTA, eliminating T BA and tB increases the final-good firm’s outputs and profits, whereas the tariff revenue disappears. Therefore, the effect of FTA on country B’s welfare depends on whether the positive effect of the increase in profits outweighs the negative effect of the lost tariff revenue. F Proposition 4. If (l, m, n) 2 H2, then W N B < W B. F
In light of Lemma 2, Proposition 4 states that the FTA makes country B better off if T CA P 0. HowF ever, if (l, m, n) 2 H3, where T CA < 0, we cannot rule out the possibility that the FTA makes country B worse off (i.e., W NB > W FB ). The intuitive explanation is as follows: In most cases, the increase in output (and therefore profits) of the final-good firms in country B outweighs the loss of its tariff revenue. However, if country A offers subsidies to non-member country C and country C reduces its tariff after the FTA, the increase in output (and profits) of final-good firms in country B would not be significant, and the profit increase could be dominated by the loss of tariff revenue. This result can be verified by the numerical analysis reported in Table 1, which shows the change in welfare levels of country B before and after the FTA under different combinations of (l, m, n). In region F F I, where T CA P 0; W NB < W FB always holds, while in region II, where T CA < 0, there are some cases in which W NB > W FB . 5.2.2. Country A’s welfare Third, we focus on country A. Country A’s welfare levels before and after the FTA are organized as follows: N
N
W NA ¼ mPN ðxN Þ þ lPNA ðyNA Þ þ CSN ðY N Þ þ nT BA yNB þ nT CA yNC ;
ð23Þ
and F
W FA ¼ mPF ðxF Þ þ lPFA ðyFA Þ þ CSF ðY F Þ þ nT CA yFC :
ð24Þ
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Country A’s welfare before the FTA consists of the profits of intermediate- and final-good firms in the country, the consumer surplus, and the tariff revenue from countries B and C. In light of Propositions 1 and 2, let us examine how country A’s welfare would be affected. First, the FTA eliminates country A’s tariffs on imports from country B, T BA . Furthermore, from Proposition 1 (1), country A’s tariffs on imports from country C, T CA , and therefore its tariff income, decrease as a result of the FTA. This is a negative effect for country A. Second, from Proposition 2 (1), outputs and profits of the final-good firms in country A decrease because of the FTA. This is a negative effect for country A. Third, from Proposition 2 (4), the final-good supply in country A increases as a result of the FTA, and therefore its price decreases. This enhances consumer surplus in country A, which is a positive effect for the country. Fourth and finally, the increase in the total supply of the final good, from Proposition 2 (4), corresponds to an increase in the total supply of the intermediate good. This increases the profits of the intermediate-good firms in country A, which is a positive effect for the country. Thus, country A faces two negative effects (a decrease in the country’s tariff revenue and final-good firms’ profits) and two positive effects (an increase in consumer surplus and intermediate-good firms’ profits). Depending on the relative magnitude of these effects, country A’s welfare may or may not increase as a result of the FTA. Table 1 shows the change in country A’s welfare levels before and after the FTA under different combinations of (l, m, n). In region I, where country A imposes a positive external tariff after the FTA, W NA > W FA holds in some cases. However, in region II, where country A provides a negative external tariff (i.e., a subsidy) after the FTA, W NA < W FA always holds. This implies that the positive effects are stronger, especially in region II.
5.3. World welfare Finally, we discuss on the effect of FTA on world welfare. Table 1 shows the changes in the joint welfare of FTA members, and world welfare under different combinations of (l, m, n). We can see that the FTA increases joint welfare for any parameter combinations in Table 1. Furthermore, from Proposition 3, non-member country C’s welfare increases when its tariff level is raised as a result of the FTA. We can summarize these findings as follows: When a non-member country raises its tariff level after the FTA, world welfare increases. In addition, according to numerical simulation, even when country C’s tariff level and welfare are lowered as a result of the FTA, world welfare increases (see Table 1). This implies that the increase in total output caused by the FTA may be critical to this result. Proposition 2 (4) shows that irrespective of country C’s policy, the FTA causes the total final-good output to increase. If the effect of the output increase is strong enough, the FTA may enhance world welfare. Note that our result implies that the formation of an FTA may not always be Pareto-improving. From Table 1, we observe that there are some cases in which the FTA increases welfare in all countries – (l, m, n) = (3, 1, 1) and (4, 2, 5), for example. This is a Pareto-improving FTA scenario. However, in most cases, the FTA would decrease the welfare of either of the three countries. Although transferring welfare may be an alternative, it is practically difficult to compensate for welfare loss. Therefore, the FTA is not Pareto-improving in such cases.
6. Concluding remarks We have investigated the effects of an FTA on tariff levels and welfare in the context of vertical trade. We have constructed a three-country model with an upstream country that exports an intermediate good and two downstream countries that export a final good. We consider an FTA between the upstream country and one of the downstream countries, treating the other downstream country as a non-member. In terms of the effects of the FTA on tariff levels, we show that the FTA unambiguously induces its member country to reduce its tariff for the non-member country. In some cases, the tariff level after
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the FTA becomes negative. In contrast, the non-member country may or may not raise its tariff after the FTA. In terms of the effects of the FTA on welfare, we demonstrate that the FTA is beneficial to the nonmember country when its tariff level is increased. Some numerical simulations report that the FTA may or may not make the member countries better off, while it may lead to an increase in the joint welfare of the member countries, which in turn causes an increase in world welfare. Some of the above results regarding the FTA’s welfare effects are obtained by using numerical simulations. In fact, we have not explicitly obtained the conditions for an increase in the upstream country’s welfare, the joint welfare of member countries, or world welfare. We intend to derive these conditions explicitly in a future study. Furthermore, our model offers several possibilities for further studies. For example, including firm entry or determining of the trade pattern would enrich our analysis and provide other interesting results. Considering a customs union between the downstream countries, which sets a common external tariff against the upstream country, is also an interesting topic. We would like to extend our model to include these topics in future studies. Acknowledgments We thank Kenzo Abe, Ichiroh Daitoh, Taiji Furusawa, Yoichi Hizen, Jun-ichi Itaya, Yukio Karasawa, Kazuharu Kiyono, Noritaka Kudoh, Kazuo Machino, Ryoichi Nomura, Masao Oda, Takao Ohkawa, Makoto Okamura, Masayuki Okawa, Chisato Shibayama, Nobuhito Suga, Yasunobu Tomoda, Keiichi Umada, Shigemi Yabuuchi, Chisato Yoshida, and two anonymous referees for their constructive comments and suggestions. The usual disclaimer applies. This work has been financially supported by the Japan Economic Research Foundation and Grants-in-Aid for Scientific Research (C) (No. 21530217). Appendix A. Proof of Lemma 1 First, t NB ¼ t NC P 0 holds if l + 2n + 1 m(l + 1) P 0 (see Eq. (14)). This condition corresponds to t n P nt ðm1Þðlþ1Þ . It is clear that @n > 0 for m > 1; that is, nt is increasing in l, given m. 2 @l N N Next, T BA ¼ T CA P 0 holds if 2(m + 1) n2 + /1n + /2 > 0, where /1 (2m + 3)(m 1)l + (m + 1) (m 3) and /2 (m2 + 2m 1) l2 + 2(m2 + m 1)l + (m + 1) (m 1). This condition corresponds to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi / þ /21 þ8ðmþ1Þ/2 1 2 n 6 n1 1 4ðmþ1Þ . Since @/ > 0 and @/ > 0, we have @n@l1 > 0, that is, n1 is increasing in l, given m. @l @l
. We determine the slope of nt and n1 at l = 0. Thus, we have At l = 0, nt ¼ n1 ¼ m1 2
@nt m 1 @n1 2m 1 < : ¼ ¼ 2 2 @l l¼0 @l l¼0
ðA:1Þ N
N
Therefore, if (l, m, n) 2 H1 {(l, m, n)jnt (l, m) 6 n 6 n1 (l, m)}, then all tariffs, t NB ; t NC ; T BA , and T CA , are nonnegative. h Appendix B. Proof of Lemma 2 Proof of (1): t FC P 0 if l + 2n + 1 m(l + 1) P 0 from Eq. (18). Since we only consider parameters such that (l, m, n) 2 H1, we have t FC P 0. h F Proof of (2): T CA P 0 if 2(2m + 1)n2 + /3 n + /2 P 0 from Eq. (19), where 2 2 /2 (m + 2m 1)l + 2(m2 + m 1) l + (m + 1)(m 1) (see Eq. (15)) and /3 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /3 þ /23 þ8ð2mþ1Þ/2 @/2 2 (m + 1)(m 1)l + 2m 3m 3. This condition corresponds to n 6 n2 . Since @l > 0 4ð2mþ1Þ 3 and @/ > 0, we have @n@l2 > 0, that is, n2 is increasing in l, given m. @l At l = 0, n2 ¼ m1 ¼ nt ¼ n1 . We then compare the slope of n2 with nt and n1 at l = 0. We have 2
@nt m 1 @n2 3m3 þ m2 þ m 1 @n1 2m 1 < : ¼ ¼ ¼ < 2 2 2ð2m þ m þ 1Þ 2 @l l¼0 @l l¼0 @l l¼0
ðB:1Þ
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583
Therefore, if (l, m, n) 2 H2 {(l, m, n)jnt(l, m) 6 n 6 n2 (l, m)} (resp., (l, m, n) 2 H3 {(l, m, n)jn2(l, F m) < n 6 n1(l, m)}), T CA is nonnegative (resp., negative). h Appendix C. Proofs of Proposition 1 and Corollary 1 Proof of (1): Let us define
f1 2ðm þ 1Þn2 þ /1 n þ /2 ; f2 2ð2m þ 1Þn2 þ /3 n þ /2 ; g 1 2mn2 þ 2ðml þ 2m þ l þ 1Þðl þ 1Þn þ ðm þ 1Þðl þ 1Þ3 ; and n o g 2 ðl þ n þ 1Þ 2ð2m þ 1Þn2 þ f3ðm þ 1Þl þ 4m þ 3gn þ ðl þ 1Þ2 ðm þ 1Þ : Using them, Eqs. (15) and (19) are rewritten as N
T CA ¼
af1 ðm þ 1Þg 1
F
and T CA ¼
af2 : ðm þ 1Þg 2
ðC:1Þ
Since we consider the vector of parameters such that (l, m, n) 2 H1, we have f1 f2 = mn{l + 2n + 1 m(l + 1)} > 0, and g1 g2 = n[2(2m + 1)n2 + {(7m + 5)l + 6m + 5}n + (l + 1){2(m + 1) l + N F m + 2}] < 0. Thus, f1 > f2and g1 < g2. Therefore, from (C.1), we have T CA > T CA . h N F Proofs of (2) and Corollary 1: From Eqs. (14) and (18), t C < tC holds if 2(l + 1)(2m + 1)n2 /4 n /5 < 0, where /4 (4m + 3) (m 1)l2 + (5m2 4m 6) l + 2m2 3m 3, and 3 2 2 2 /5 (2m 1)(m + 1) l + (4m + 2m 3) l + (3m + m 3) l + (m + 1)(m 1). This condition correpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi / þ /24 þ8ðlþ1Þð2mþ1Þ/5 4 5 sponds to n 6 n3 4 4ðlþ1Þð2mþ1Þ . Since @/ > 0 and @/ > 0, we have @n@l3 > 0, that is, n3 is increas@l @l ing in l, given m. At l = 0, n3 ¼ m1 ¼ nt ¼ n1 ¼ n2 . We investigate the slope of nt, n1, n2, and n3 at l = 0. Then, we have 2
@nt m 1 @n3 3m3 þ 2m2 3 @n2 3m3 þ m2 þ m 1 @n1 2m 1 < ; ¼ ¼ ¼ ¼ < < 2 2ð2m2 þ m þ 1Þ 2 @l l¼0 @l l¼0 2ð2m2 þ m þ 1Þ @l l¼0 @l l¼0
ðC:2Þ for m < 2, and
@nt m 1 @n2 3m3 þ m2 þ m 1 @n3 3m3 þ 2m2 3 @n1 2m 1 ¼ ¼ ¼ ¼ < ; 6 < 2 2ð2m2 þ m þ 1Þ 2 @l l¼0 @l l¼0 @l l¼0 2ð2m2 þ m þ 1Þ @l l¼0 ðC:3Þ for m P 2 (equality holds at m = 2). Therefore, if (l, m, n) 2 H4 {(l, m, n)jnt (l, m) 6 n 6 n3(l, m)} (resp., (l, m, n) 2 H5 {(l, m, n)jn3 (l, m) < n 6 n1(l, m)}), tNC is greater than (resp., less than) t FC . From (C.2), (C.3), and Lemma 2, the threshold n3 line passes through the parameter region H2 (resp., H3) for m < 2 (resp., m P 2). Therefore, if (l, m, n) 2 H2 and m P 2, t NC < tFC . If (l, m, n) 2 H2 and m < 2, there exists a parameter such that t NC < tFC (see Fig. 4i). If (l, m, n) 2 H3 and m < 2, t NC > tFC . If (l, m, n) 2 H3 and m P 2, there exists a parameter such that t NC > t FC (see Fig. 4ii). h Appendix D. Proof of Proposition 2 Proof of (1): Subtracting the per-firm output of the final good in country A under the FTA from the pre-FTA output, we have N
yNA
yFA
F
mn½2ðT CA þ tNC Þ ðT CA þ t FC Þ ; ¼ ðm þ 1Þðl þ 2n þ 1Þ
ðD:1Þ
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Y. Kawabata et al. / J. Japanese Int. Economies 24 (2010) 569–585 N
N
F
N
N
where we used the fact that T BA ¼ T CA and t NB ¼ tNC . Since (l, m, n) 2 H1, it holds that T CA < T CA < 2T CA . In addition, we have
2t NC tFC ¼
½l þ 2n þ 1 mðl þ 1Þa ðm þ 1Þðl þ n þ 1ÞM N M F
X;
ðD:2Þ
where
M N 2mn2 þ 2fðm þ 1Þl þ 2m þ 1gðl þ 1Þn þ ðl þ 1Þ3 ðm þ 1Þ > 0; M F 2ð2m þ 1Þn2 þ f3ðm þ 1Þl þ 4m þ 3gn þ ðl þ 1Þ2 ðm þ 1Þ > 0;
X ðl þ 1Þðl þ n þ 1Þðl þ 2n þ 1Þ2 þ ½ðl þ 2Þðl þ 1Þ3 þ ðl þ 2Þ2 ðl þ 9Þn þ 16ðl þ 1Þn2 þ 10n3 m þ ½ðl þ 1Þ2 þ 2nðn þ 1Þðl þ 2n þ 1Þm2 þ 2mnlðl þ 2n þ 1Þðl þ 2n þ 1 mlÞ: The assumption (l, m, n) 2 H1, i.e., l + 2n + 1 m(l + 1) > 0, implies that l + 2n + 1 ml > 0. ThereN
F
fore, X > 0 holds, and hence, 2t NC > t FC . In sum, we have 2ðT CA þ tNC Þ ðT CA þ t FC Þ > 0. h Proof of (2): The difference between yNB and yFB is N
yNB yFB ¼
F
m½ðl þ 1ÞðT CA þ tNC Þ þ nðT CA þ t FC Þ : ðm þ 1Þðl þ 2n þ 1Þ
ðD:3Þ N
F
If (l, m, n) 2 H2, it is clear that yNB < yFB because the equilibrium tariffs T CA ; t NC ; T CA , and t FC are all F
F
nonnegative. If ðl; m; nÞ 2 H3 ; T CA is negative. Nevertheless, since T CA þ tFC ¼ mla=M F > 0, we obtain N
F
ðl þ 1ÞðT CA þ tNC Þ þ nðT CA þ t FC Þ > 0, and hence yNB < yFB . h Proof of (3): Substituting the pre-FTA tariff rates (14) and (15) into the equilibrium output of the final good in country C, we have
yNC ¼
ðl þ m þ 1Þðl þ n þ 1Þma ðm þ 1ÞMN
ðD:4Þ
:
In light of (14), (D.4) is rewritten as
yNC ¼
ðl þ n þ 1Þm tN : l þ 2n þ 1 mðl þ 1Þ C
ðD:5Þ
Substituting the tariff rates in (18) and (19) under the FTA into the equilibrium output in country C, we have
yFC ¼
ð2mn þ l þ m þ n þ 1Þma ðm þ 1ÞM F
ðD:6Þ
:
In light of (18), (D.6) is rewritten as
yFC ¼
ðl þ n þ 1Þm tF : l þ 2n þ 1 mðl þ 1Þ C
ðD:7Þ
Subtracting (D.7) from (D.5) yields
yFC yNC ¼
ðl þ n þ 1Þm ðt F t NC Þ: l þ 2n þ 1 mðl þ 1Þ C
ðD:8Þ
Since we assume l + 2n + 1 m(l + 1) > 0, the sign of Eq. (D.8) depends on the sign of tFC tNC . Proof of (4): The difference between YN and YF is N
YN YF ¼
h
F
mn½2ðT CA þ tNC Þ ðT CA þ t FC Þ ¼ yFA yNA ; ðm þ 1Þðl þ 2n þ 1Þ
which is unambiguously negative, as previously shown, if (l, m, n) 2 H1.
ðD:9Þ h
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Appendix E. Proof of Proposition 4 Subtracting country B’s welfare after the FTA from that before the FTA, we have
tN tN nðtNB Þ2 W NB W FB ¼ n½ðyNB Þ2 þ t NB yNB ðyFB Þ2 ¼ n yNB þ B þ yFB yNB þ B yFB 2 2 4 ) ( N BN CF N N F t t B m½ðl þ 1Þðt B þ T A Þ þ nðt C þ T A Þ nðtNB Þ2 ¼ n yNB þ B þ yFB : ðm þ 1Þðl þ 2n þ 1Þ 2 2 4
ðE:1Þ
F
We have shown that t FC þ T CA > 0. Moreover, N
N
tNB mðl þ 1ÞðtNB þ T BA Þ ½2mn þ l þ 2n þ 1 mðl þ 1Þt NB 2mðl þ 1ÞT BA ¼ ; 2 ðm þ 1Þðl þ 2n þ 1Þ 2ðm þ 1Þðl þ 2n þ 1Þ N
ðE:2Þ
N
which, by substituting the values of t NB ¼ tNC and T BA ¼ T CA , is nonpositive if
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðl þ 1Þ ðm2 1Þl 1 þ m½m 1 þ m2 þ ðm þ 1Þ2 l þ ð2m þ 1Þðm þ 1Þl
n6
nB ðl; mÞ:
2½ðl þ 2Þm þ l þ 1
ðE:3Þ By computation, we can show that n2(l, m) < nB (l, m) < n1(l, m). This means that, if we define
HB {(l, m, n)jnt (l, m) 6 n 6 nB(l, m)}, H2 HB H1 holds. Therefore, it follows that if (l, m, n) 2 H2, we have W NB < W FB . Note that if (l, m, n) 2 H3 H1nH2, Eq. (E.2) may be positive. Therefore, the term in the brace brackets in (E.1) does not necessarily become negative.
h
References Bagwell, K., Staiger, R.W., 1999. Regionalism and multilateral tariff co-operation. In: Pigott, J., Woodland, A. (Eds.), International Trade Policy and the Pacific Rim. Macmillan, London, pp. 157–185. Bernhorfen, D.M., 1997. Strategic trade policy in a vertical related industry. Review of International Economics 5, 429–433. Bond, E.W., Riezman, R.G., Syropoulos, C., 2004. A strategic and welfare theoretic analysis of free trade areas. Journal of International Economics 64, 1–27. Chang, W.W., Sugeta, H., 2004. Conjectural variation, market power, and optimal trade policy in a vertical related industry. Review of International Economics 12, 12–26. Freund, C., 2000. Multilateralism and the endogenous formation of preferential trade agreements. Journal of International Economics 52, 359–376. Hwang, H., Lin, Y.S., Yang, Y.P., 2007. Optimal trade policies and production technology in vertically related markets. Review of International Economics 15, 823–835. Ishikawa, J., Spencer, B.J., 1999. Rent-shifting export subsidies with an imported intermediate product. Journal of International Economics 48, 199–232. JETRO, 2006. JETRO White Paper on Foreign Direct Investment. Japan External Trade Organization, Tokyo (in Japanese). Kose, M.A., Riezman, R.G., 2000. Understanding the welfare implications of preferential trade agreements. Review of International Economics 8, 619–633. Krishna, P., 1998. Regionalism and multilateralism: a political economy approach. Quarterly Journal of Economics 113, 227– 251. Ministry of Economy, Trade and Industry, Japan, 2008. Report on Compliance by Major Trading Partners with Trade Agreements – WTO, FTA/EPA, and BIT.. Ornelas, E., 2005. Trade creating free trade areas and the undermining of multilateralism. European Economic Review 49, 1717– 1735. Saggi, K., 2006. Preferential trade agreements and multilateral tariff cooperation. International Economic Review 47, 29–57. Yanase, A., Kawabata, Y., 2008. Strategic import policies in a three-country model with vertically related industries. Economics Bulletin 6, 1–7. Yi, S.S., 2000. Free-trade areas and welfare: an equilibrium analysis. Review of International Economics 8, 336–347.
Contents lists available at ScienceDirect
Journal of The Japanese and International Economies journal homepage: www.elsevier.com/locate/jjie
Vertical trade and free trade agreements Yasushi Kawabata a,*, Akihiko Yanase b, Hiroshi Kurata c a
Graduate School of Economics, Nagoya City University, 1 Yamanohata, Mizuho-cho, Mizuho-ku, Nagoya 467-8501, Japan Graduate School of International Cultural Studies, Tohoku University, Japan c Faculty of Economics, Tohoku Gakuin University, Japan b
a r t i c l e
i n f o
Article history: Received 25 September 2008 Revised 11 January 2010 Available online 2 April 2010 JEL classifications: F12 F13 F53 Keywords: Free trade agreements Vertical trade Cournot competition
a b s t r a c t Kawabata, Yasushi, Yanase, Akihiko, and Kurata, Hiroshi—Vertical trade and free trade agreements We investigate the effects of free trade agreements (FTAs) on tariffs and welfare in vertical trade. We consider a three-country model, where an FTA is formed between a country exporting a final good and a country exporting an intermediate good. The FTA unambiguously leads to a reduction in the member country’s tariff, but may cause the non-member country’s tariff level to increase. In the case, where FTA raises the non-member country’s tariff level, the FTA increases that country’s welfare. In contrast, the FTA may render its member countries better off. This result implies that the formation of an FTA may not always be Pareto-improving. J. Japanese Int. Economies 24 (4) (2010) 569–585. Graduate School of Economics, Nagoya City University, 1 Yamanohata, Mizuhocho, Mizuho-ku, Nagoya 467-8501, Japan; Graduate School of International Cultural Studies, Tohoku University, Japan; Faculty of Economics, Tohoku Gakuin University, Japan. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction Recently, the number of free trade agreements (FTAs) has been rapidly increasing across the world. According to Ministry of Economy, Trade and Industry (METI), Japan (2008), while there were 46 FTAs worldwide in 1990, the number increased to 194 in 2007. In particular, countries in Asia, the Middle East, and Africa, which previously lacked FTAs, are now accelerating the formation of FTAs.
* Corresponding author. Fax: +81 52 872 5014. E-mail address: [email protected] (Y. Kawabata). 0889-1583/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jjie.2010.03.002
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With an increase in their number, FTAs have become an important topic in international trade theory. One research stream on FTAs clarifies their effects on welfare (e.g., Kose and Riezman, 2000; Yi, 2000; Bond et al., 2004), and another focuses on multilateralism and FTA (e.g., Krishna, 1998; Freund, 2000; Ornelas, 2005; Saggi, 2006). The above studies on FTA address horizontal trade, i.e., international trade in which countries mutually export final goods. However, vertical trade, i.e., international trade in which one country exports intermediate goods to other countries that process the imported inputs into final goods for their exports, has gained significance in recent trade transactions. Such vertical trade structures are particularly notable in the Asian region. For example, Japan exports components such as semiconductors for computers to other Asian countries such as China, and imports computers mainly from China (JETRO, 2006). Further investigations are required to consider vertical trade structures. The purpose of this paper is to clarify how FTAs affect tariff rates and, in turn, welfare levels in the context of a vertical trade structure. We construct a simple three-country model, in which one ‘‘upstream” country exports an intermediate good to two other ‘‘downstream” countries and imports a final good from both of them. Before the FTA, the government in each country determines its tariff level in a noncooperative fashion. We then consider an FTA between the upstream country and one of the downstream countries. The other downstream country is regarded as a non-member country. Thus, tariffs between the FTA member countries are eliminated, while an external tariff of the upstream country and the tariff of the non-member downstream country are still endogenously determined. In this setting, we investigate how the FTA affects each country’s tariff levels and, in turn, its welfare. In our model, which considers Cournot competition with linear demands and constant marginal costs, each country’s optimal tariff is influenced by the other countries’ tariffs. This finding is in contrast to those of previous studies, including Freund (2000), Ornelas (2005), and Saggi (2006), which adopt a similar assumption in the analysis of FTAs with horizontal trade structures. In these studies, it is shown that the formation of an FTA induces its member countries to reduce their external tariffs, but the FTA does not affect the non-member country’s tariff. Thus, these studies do not completely illustrate the strategic relationship between FTA member and non-member. We propose an alternative framework, apart from the existing horizontal trade models, to describe the strategic relationship between member and non-member countries. We obtained the following results with regard to the FTA’s tariff effects. The formation of the FTA unambiguously induces its member country to reduce the tariff for the non-member country. This external tariff after the FTA may be negative (i.e., an import subsidy). In contrast, while the non-member country may or may not raise its tariff level, it certainly imposes a positive tariff after the FTA. In most cases, the FTA induces the non-member country to raise its tariff. With regard to the effect of the FTA on each country’s welfare, the following results are obtained. For the non-member country, raising the tariff after the FTA leads to an increase in its welfare. As regards the FTA members, the downstream country is better off after the FTA as long as its FTA partner (i.e., the upstream country) imposes a positive external tariff after the FTA. However, it may be worse off if the external tariff imposed by the FTA partner is negative. In contrast, the FTA may or may not make the upstream member better off. In addition, some numerical analyses indicate that the FTA may increase the members’ joint welfare as well as world welfare. As mentioned above, this paper focuses on tariff and welfare effects of FTAs; therefore, it is related to FTA studies for the most part. However, studies on vertical trade policies are also an important reference for this study (e.g., Bernhorfen, 1997; Ishikawa and Spencer, 1999; Chang and Sugeta, 2004; Yanase and Kawabata, 2008). In fact, this paper is structured on the basis of the literature on vertical trade policies. Thus, our study links these two research streams and provides a new insight into FTA studies. The rest of this paper is organized as follows. Section 2 presents the basic model. Section 3 derives the equilibrium and examines pre-FTA tariffs. Section 4 considers the effects of the FTA on the member country’s external tariff and the non-member country’s tariff levels. Section 5 analyzes the effects of the FTA on welfare. Section 6 concludes the paper.
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2. The model The world economy consists of three countries, A, B, and C, which trade among themselves in a final good and an intermediate good. The final good is produced by n P 1 identical firms in countries B and C, respectively, and l P 1 identical firms in country A. In contrast, the intermediate good is produced by m P 1 identical firms located only in country A. We further assume that the final good is consumed only in country A. Therefore, country A exports the intermediate good to countries B and C, and imports the final good from these countries. Note that some quantity of the final good is domestically supplied in country A. In the following paragraphs, we often use the terms ‘‘final-good firms” and ‘‘intermediate-good firms” to refer to firms producing the final good and the intermediate good, respectively. The government in country A imposes a specific tariff, T iA , on final-good imports from country i (i = B, C), whereas the government in country i imposes a specific tariff, ti, on intermediate-good imports from country A. The setting is briefly illustrated in Fig. 1. In the subsequent analysis, we consider an FTA between countries A and B, and regard country C as the non-member country. The FTA eliminates tariffs between the FTA members A and B. Therefore, the values of T BA and tB move toward zero after the FTA. We assume that production of one unit of the final good requires one unit of the intermediate good. Let us use indexes Ak and ij to represent the kth final-good firm in country A (k = 1, . . ., l), and the jth final-good firm in country i (i = B, C, j = 1, . . ., n), respectively. Let yAk and yij be the output of final-good firms Ak and ij, respectively, and Y be the total output of the final good. Then, the profits of the finalgood firms Ak and ij are given by
PAk ¼ ðpðYÞ r A ÞyAk ;
ð1aÞ
Pij ¼ ðpðYÞ ri T iA Þyij ;
ð1bÞ
and
respectively, where p(Y) is the inverse demand for the final good and ri is the price of the intermediate good in country i’s market (i = A, B, C).
Fig. 1. Model structure.
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We use the index h to denote intermediate-good firms in country A (h = 1,. . .,m). The marginal cost of production for the intermediate good is normalized at zero. Then, the intermediate-good firm h’s profit, Ph, is expressed by
Ph ¼ r A xAh þ ðr B t B ÞxBh þ ðrC t C ÞxCh ;
ð2Þ
xih
where is the intermediate-good firm’s supply to country i (i = A, B, C). In our setting, each country’s welfare is defined as follows:
WA ¼
m X
Ph þ
h¼1
WB ¼ WC ¼
n X
l X
PAk þ CS þ T BA
k¼1 m X
PBj þ tB
j¼1
h¼1
n X
m X
PCj þ tC
j¼1
n X
! yBj
j¼1
! xBh ;
þ T CA
n X
! yCj ;
ð3Þ
j¼1
ð4Þ
! xCh
ð5Þ
;
h¼1
RY where CS ¼ 0 pðsÞds pðYÞY is the consumer surplus in the final-good market in country A. The welfare of country A is the sum of the profits of intermediate- and final-good firms, the consumer surplus, and the country’s tariff revenue. In contrast, the welfare function of country i is the sum of the profits of final-good firms in the country and its tariff revenue (i = B, C). The model involves three stages of decision making. In the first stage, each government endogenously determines its own tariff levels. Under the FTA, although the tariffs between countries A and B are eliminated, the governments in countries A and C must set the tariff level. In the second stage, the intermediate-good firms in the intermediate-good market in countries A, B, and C are in Cournot competition. In the third stage, given intermediate-good prices, the final-good firms play Cournot competition in the final-good market in country A.1 We derive the subgame-perfect equilibrium of this model. In particular, we focus on endogenous tariff levels and welfare values before and after FTA. 3. Equilibrium 3.1. Equilibrium in the final-good market Using backward induction, we begin to consider the final-good market. Each final-good firm chooses its output so as to maximize its own profit, taking the outputs of rivals, the price of the intermediate good, and tariffs as given. From Eqs. (1a) and (1b), the first-order conditions for profit maximization of final-good firms Ak and ij are obtained as follows (i = B, C; j = 1, . . ., n, k = 1, . . ., l):
pðYÞ r A þ p0 ðYÞyAk ¼ 0; pðYÞ r i
T iA Pl
0
þ p ðYÞyij ¼ 0; Pn
ð6aÞ i ¼ B; C:
ð6bÞ
Pn
Note that Y ¼ k¼1 yAk þ j¼1 yBj þ j¼1 yCj . The Cournot-Nash equilibrium outputs in the final-good market are derived from conditions (6a) and (6b). In terms of symmetry, the outputs of all firms within ~Ak ¼ y ~A and y ~ij ¼ y ~i ði ¼ B; C; j ¼ 1; . . . ; a country are the same in an equilibrium. That is, y n; k ¼ 1; . . . ; lÞ, where the tilde is used to represent the equilibrium value. Thus, in an equilibrium, e ¼ ly ~A þ ny ~ B þ ny ~C . Y 3.2. Equilibrium in the intermediate-good market Before analyzing the Cournot competition among intermediate-good firms in country A, we focus on the inverse demand for intermediate goods. Since one unit of the final good is produced with one 1 This setting implies that the downstream firms recognize their market power in the final-good market but act as price-takers in the intermediate-good market. This setting is often adopted in literature on vertical trade (e.g., Bernhorfen, 1997; Hwang et al., 2007; Yanase and Kawabata, 2008). Ishikawa and Spencer (1999) provide a detailed discussion of the justification for this setting.
Y. Kawabata et al. / J. Japanese Int. Economies 24 (2010) 569–585
573
unit of the intermediate good, the total demand for the intermediate input in countries A, B, and C are ~ A ; ny ~B , and ny ~C , respectively. Letting Xi be the total supply of the intermediate good to country i (i = A, ly B, C), the market-clearing conditions in these markets are given by
~A ; X A ¼ ly
~B ; X B ¼ ny
~C : and X C ¼ ny
ð7Þ A
B
C
Eq. (7) yields the inverse demand of the intermediate good, ri = ri(X , X , X ), i = A, B, C. Each intermediate-good firm determines the outputs in markets A, B, and C so as to maximize its own profit, taking the output of rival intermediate-good firms and tariffs as given. From Eq. (2), the first-order conditions for profit maximization of intermediate-good firm h in country A (h = 1, . . ., m) are given by
rA þ @r A @X B @rA @X C
@r A @X
A
xAh þ
@r B @X
A
xBh þ
xAh þ rB t B þ xAh þ
@r B @X C
@r B @X B
@rC @X A
xCh ¼ 0;
xBh þ
xBh þ r C t C þ
@rC @X B @r C @X C
xCh ¼ 0;
ð8Þ
xCh ¼ 0:
The conditions in Eq. (8) determine the Cournot-Nash equilibrium outputs in the intermediategood market. From symmetry, all firms’ outputs are the same. That is, ~ xih ¼ ~ xi ði ¼ A; B; CÞ, and thus X i ¼ m~ xi in an equilibrium. Substituting these outputs into the equilibrium outputs and prices in the final-good market, we have the subgame-perfect equilibrium solutions, all of which are functions of the tariff rates, given the number of intermediate- and final-good firms. That is,
~xi ¼ ~xi ðT BA ; T CA ; t B ; tC ; l; m; nÞ; ~r i ¼ ~ri ðT BA ; T CA ; tB ; t C ; l; m; nÞ;
ð9Þ
~i ðT BA ; T CA ; t B ; tC ; l; m; nÞ; ~i ¼ y y ~¼p ~ðT BA ; T CA ; t B ; tC ; l; m; nÞ; p
i ¼ A; B; C:
In the subsequent analysis, we assume a linear demand: p(Y) = a Y, where a > 0 is large enough to ensure positive outputs. Then, the uniqueness and stability of the Cournot-Nash equilibrium are guaranteed, and the second-order conditions for profit maximization of any firm are satisfied in both the final- and intermediate-good markets. 3.3. Endogenous pre-FTA tariffs We now focus on the first stage, the endogenous tariff determination. Here, we consider the situation before FTA. First, we consider country i (i = B, C). Country i determines the import tariff ti to maximize Wi, given the other countries’ tariff levels. From Eqs. (4) and (5), country i’s first-order condition is given by
@W i @y @p @r i @xi þ mxi þ ti m ¼ nðp r i T iA Þ i þ nyi ¼ 0; @t i @t i @ti @t i @t i
ð10Þ
for i = B, C. The sum of the first and second terms gives the effect of the tariff ti on the profits of the final-good firm i. The tariff ti decreases the output, and therefore the profit, of the final-good firm i. The third and fourth terms are the changes in tariff revenues. An increase in the tariff ti increases (resp., decreases) the tariff revenue of country i for low (resp., high) tariffs. Substituting the equilibrium outputs into Eq. (10) and rearranging it, we obtain country i’s reaction function as
ti ¼
n o fl þ 2n þ 1 mðl þ 1Þg a ðl þ n þ 1ÞT iA þ nT jA þ nt j 2ðl þ 2n þ mn þ 1Þðl þ n þ 1Þ
;
ð11Þ
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for i, j = B, C; i – j. Notice that the tariff level depends on the other countries’ tariffs, unlike in the horizontal trade model (e.g., Freund, 2000), in which tariff levels do not depend on other countries’ tariffs. From Eq. (11), if l + 2n + 1 m(l + 1) > 0, then ti is related negatively to T iA and positively to T jA and tj. In other words, ti and T iA are strategic substitutes, whereas ti and T jA as well as ti and tj are strategic complements. Second, we consider country A. Country A determines the import tariffs T BA and T CA to maximize WA, given the other countries’ tariff levels. From Eq. (3), country A’s first-order condition for maximizing welfare with respect to T iA is given by
@W A @T iA
¼ mr A
@xA @T iA
þ T jA n
þ mðr j t j Þ
@yj @T iA
@xj @T iA
þ nyi þ T iA n
þ mðr i t i Þ
@yi @T iA
@xi @T iA
þ mxi
@r i @T iA
þ lðp r A Þ
@yA @T iA
þ lyA
¼ 0;
@p @T iA
Y
@p @T iA ð12Þ
for i, j = B, C; i – j. The sum of the first to fourth terms represents the effect of tariff T iA on the profits of the intermediate-good firms in country A. The tariff T iA lowers the price of the intermediate good in country i and decreases the supplies of intermediate-good firms in market i. On the other hand, it increases their supplies to country A and country j. Since the former effect dominates the latter, the tariff T iA decreases total output, xA + xB + xC, and in turn the profits of intermediate-good firms. The sum of the fifth and sixth terms gives the effect on the profits of final-good firms in country A; the tariff T iA increases the output and, in turn, the profits of final-good firms in country A. The seventh term expresses the effect on consumer surplus; the tariff T iA raises the price of the final good and decreases consumer surplus. The eighth term represents the effect on tariff revenue from country j; the tariff T iA increases the imports from country j, raising the tariff revenue from that country. Finally, the sum of the ninth and tenth terms represents the effect on tariff revenue from country i. The increase in the tariff T iA raises (resp., reduces) the tariff revenue from country i for low (resp., high) tariffs. Substituting the equilibrium outputs into Eq. (12) and rearranging the equation, we get country A’s reaction function as
T iA ¼
d1 a þ ð4l þ 4n þ 3ÞmnT jA d2 t i þ d3 nt j 4n2 þ ð4l þ 5Þn þ 2ðl þ 1Þ2
;
i; j ¼ B; C;
i – j;
ð13Þ
where d1 m(2l + 1) l 2n 1, d2 2(m 1)n2 + {(m 3)l + 2m 3}n + (l + 1)2(m 1), and d3 (3l + 2n + 2)m l 2n 1. Note that the tariff level depends on the other countries’ tariffs. This is a feature of the vertical trade model. Note also that in Eq. (13) T iA is positively related to T jA and tj. In other words, T iA and tj are strategic complements. In contrast, the relationship between T iA and ti is ambiguous because d2 can be either negative (e.g., m = 1) or positive (e.g., m P 2). Solving Eqs. (11) and (13), we obtain the following Nash equilibrium import tariffs (indicated by adding superscript ‘‘N”):
t NB ¼ t NC ¼ N
N
fl þ 2n þ 1 mðl þ 1Þgðl þ m þ 1Þa ðm þ
T BA ¼ T CA ¼
1Þf2mn2
þ 2ððm þ 1Þl þ 2m þ 1Þðl þ 1Þn þ ðm þ 1Þðl þ 1Þ3 g f2ðm þ 1Þn2 þ /1 n þ /2 ga
ð14Þ
;
ðm þ 1Þf2mn2 þ 2ððm þ 1Þl þ 2m þ 1Þðl þ 1Þn þ ðm þ 1Þðl þ 1Þ3 g
;
ð15Þ
where /1 (2m + 3)(m 1)l + (m + 1)(m 3) and /2 (m2 + 2m 1)l2 + 2(m2 + m 1)l + (m + 1)(m 1). Substituting Eqs. (14) and (15) into the equilibrium outputs and prices obtained in the second and third stages, all outputs, prices, and tariffs are functions of the number of firms. Therefore, the economy is described by parameters l, m, and n. The signs of the Nash equilibrium tariffs depend on a vector of parameters (l, m, n), which describes the industrial structure of the world economy. From Eqs. (14) and (15), we have the following lemma.2 2
All proofs of upcoming lemmas and propositions appear in the appendices.
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Y. Kawabata et al. / J. Japanese Int. Economies 24 (2010) 569–585
CN Fig. 2. Optimal policies and the number of firms (given m). (l, m, n) 2 H1: T BN A ¼ T A > 0;
tNB ¼ tNC > 0.
Lemma 1. If (l, m, n) 2 H1 {(l, m, n)jnt(l, m) 6 n 6 n1(l, m)}, all countries impose nonnegative tariffs BN CN N before the FTA; i.e., tN qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B ; t C ; T A ; T A P 0, where
nt ðl; mÞ
ðm 1Þðl þ 1Þ 2
and n1 ðl; mÞ
/21 þ 8ðm þ 1Þ/2
/1 þ
4ðm þ 1Þ
:
Lemma 1 requires that n, the number of final-good firms in each downstream country, be sufficiently large for positive optimal tariffs on the intermediate good and sufficiently small for positive optimal tariffs on the final good in the pre-FTA equilibrium. The explanation for this is as follows3: A smaller n provides higher overall profits in the final-good market, because final-good firms in countries B and C face fewer competitors. Thus, countries B and C may have a greater incentive to provide import subsidies for the intermediate good to enhance domestic firms’ competitiveness, rather than earn tariff revenues by imposing positive tariffs. Alternatively, a larger n leads to higher profits for intermediategood firms. Thus, country A may have a greater incentive to provide subsidies for final-good imports because more final-good imports lead to increased intermediate-good exports. Fig. 2 illustrates the relationship between optimal policies and the number of final-good firms, given the number of intermediate-good firms. We have the number of final-good firms in country A on the horizontal axis and the number of final-good firms in countries B and C on the vertical axis. N
N
On lines n = nt and n = n1, we have t NB ¼ tNC ¼ 0 and T BA ¼ T CA ¼ 0, respectively. Import tariffs on the intermediate good, t NB ¼ tNC , are positive above and negative below the n = nt line. In contrast, import N
N
tariffs on the final good, T BA ¼ T CA , are negative above and positive below the n = n1 line. Lemma 1 implies that if the difference between the number of final-good firms in upstream and downstream countries is extremely large, optimal policies for these countries can involve import subsidies. In the following analysis, we will assume (l, m, n) 2 H1. In other words, we will limit our attention to cases in which all the countries impose positive tariffs before FTA.
4. Effects of the FTA on tariff levels Next, we consider the situation after FTA. After the FTA is formed between countries A and B, tB and T BA become zero. However, country C still imposes an import tariff for the intermediate good, tC, and country A imposes an import tariff for the final good, T CA . 3
This is pointed out by one of the referees of this journal. The authors appreciate this crucial contribution.
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4.1. Endogenous tariffs under FTA Substituting tB ¼ T BA ¼ 0 into Eqs. (3) and (5) and partially differentiating them with respect to T CA and tC, respectively, we obtain country A’s and C’s reaction functions under FTA as
T CA ¼ tC ¼
d1 a d2 t C ; 4n2 þ ð4l þ 5Þn þ 2ðl þ 1Þ2 n o fn nt ðl; mÞg a ðl þ n þ 1ÞT CA ðl þ 2n þ mn þ 1Þðl þ n þ 1Þ
ð16Þ
ð17Þ
:
Solving Eqs. (16) and (17), we obtain the following Nash equilibrium import tariffs after FTA (indicated by adding superscript ‘‘F”):
2fn nt ðl; mÞgð2mn þ l þ m þ n þ 1Þa ; ðm þ 1Þðl þ n þ 1Þf2ð2m þ 1Þn2 þ ð3ðm þ 1Þl þ 4m þ 3Þn þ ðl þ 1Þ2 ðm þ 1Þg f2ð2m þ 1Þn2 þ /3 n þ /2 ga ¼ ; ðm þ 1Þðl þ n þ 1Þf2ð2m þ 1Þn2 þ ð3ðm þ 1Þl þ 4m þ 3Þn þ ðl þ 1Þ2 ðm þ 1Þg
t FC ¼ T CA
F
ð18Þ ð19Þ
where /3 3(m + 1)(m 1) l + 2m2 3m 3. Comparing Eqs. (14) to (18) and (15) to (19), we get the following results. Lemma 2. (1) For any ðl; m; nÞ 2 H1 ; tFC is nonnegative. F (2) If (l, m, n) 2 H2 {(l, m, n)jnt(l, m) 6 n 6 n2 (l, m)}, then T CA is nonnegative, and if (l, m, CF n) 2 H3 H1nH2, then T A is negative, where
n2 ðl; mÞ
/3 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /23 þ 8ð2m þ 1Þ/2 4ð2m þ 1Þ
:
Lemma 2 shows there are some possibilities that, as a result of the FTA, member country A could change its policy from tariff to subsidy. In contrast, non-member country C does not change its policy in response to the FTA. Fig. 3 illustrates the parameter regions shown by Lemma 2. In this figure, the F threshold line n = n2 is added into Fig. 2. On line n = n2, we have T CA ¼ 0. Import tariffs for the final CF good, T A , are negative above n = n2, and positive below the line. Lemma 2 shows that the parameter
CN Fig. 3. Country A’s optimal policy and the number of firms (given m). Region I ((l, m, n) 2 H2): T BN A ¼ T A > 0; CN ((l, m, n) 2 H3 H1nH2) : T BN T CF A ¼ T A > 0; A < 0.
T CF A > 0; Region II
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space H1 is divided into two regions. In region I, member country A imposes positive tariffs both before and after the FTA. In contrast, in region II, member country A sets positive tariffs before and negative tariffs (i.e., subsidies) after the FTA. Non-member country C imposes positive tariffs in both regions. 4.2. Comparisons of equilibrium tariffs We now focus on the effect of the FTA on the tariff levels. From Lemmas 1 and 2, it is clear that for some (l, m, n) 2 H1, country A may provide a subsidy after the FTA even when it chooses to impose a tariff before the FTA. This implies that country A’s tariff on final-good import is reduced, as formally demonstrated in the following proposition. Moreover, the following proposition indicates how country C’s tariff rate is affected by the FTA. Proposition 1 N
F
(1) For any ðl; m; nÞ 2 H1 ; T CA > T CA holds. (2) If (l, m, n) 2 H4 {(l, m, n)jnt(l, m) 6 n 6 n3 (l, m)}, then tNC 6 tFC holds, and if (l, m, n) 2 H5 H1nH4, then tNC > tFC holds, where
n3 ðl; mÞ
/4 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /24 þ 8ðl þ 1Þð2m þ 1Þ/5 4ðl þ 1Þð2m þ 1Þ
;
/4 (4m + 3)(m 1)l2 + (5m2 4m 6)l + 2m2 3m 3, and /5 (2m 1)(m + 1)l3 + (4m2 + 2m 3)l2 + (3m2 + m 3)l + (m + 1)(m 1). Proposition 1(1) shows that the FTA provides an incentive to member country A to reduce its tariff. It implies that the tariff complementarity effect (Bagwell and Staiger, 1999) holds in a vertical trade model. That is, when FTA members reduce their internal tariffs to zero, they find it advantageous to lower their external tariffs as well. The elimination of T BA decreases imports from country C, which in turn lowers country A’s tariff revenue and consumers surplus. Country A alleviates these negative effects by reducing T CA to expand imports from country C. Further, the reduction in T CA increases final-good output in country C, and thereby increases the profits of intermediate-good firms in country A. Therefore, country A has an incentive to reduce its external tariff after the FTA.
Fig. 4i. Country C’s tariff level and the number of firms (m < 2). Region I ((l, m, n) 2 H2): tNC 6 t FC if (l, m, n) 2 H4; tNC > t FC if (l, m, n) 2 H5; Region II ((l, m, n) 2 H3): tNC > tFC .
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Fig. 4ii. country C’s tariff level and the number of firms (m P 2). Region I ((l, m, n) 2 H2): t NC 6 t FC ; Region II ((l, m, n) 2 H3): tNC 6 tFC if (l, m, n) 2 H4; tNC > tFC if (l, m, n) 2 H5.
In contrast, Proposition 1(2) states that the FTA may or may not provide such an incentive to nonmember country C. An intuition is as follows. After the formation of the FTA between countries A and B, country A sets T BA ¼ 0, which has a negative effect for country C.4 At the same time, from Proposition 1 (1), country A reduces T CA , which is a positive effect of the FTA. If the positive effect dominates the negative one, the final-good output in country C increases. Then, intermediate-good imports from country A must increase. This increases country C’s incentive to impose a higher tariff on country A. If the negative effect is dominant, the result is just the opposite. Fig. 4 explains the changes in non-member country C’s tariff levels. The threshold line n = n3 is included in Fig. 3. On the n = n3 line, we have t NC ¼ tFC . Above n = n3, t NC > tFC , but below the line, t NC < tFC . Fig. 4i shows that line n = n3 appears below the n = n2 line. That is, in region I (i.e., the region where country A imposes a positive tariff both before and after the FTA), country C may or may not raise its tariff, while in region II (i.e., the region where country A sets a positive tariff before and a negative tariff after the FTA), country C reduces its tariff. In contrast, Fig. 4ii shows that line n = n3 appears above the n = n2 line. That is, country C raises its tariff in region I, but may or may not do so in region II. In light of Lemma 2, part (2) of Proposition 1 can be restated as follows. Corollary 1 (1) Suppose that (l, m, n) 2 H2, where all countries impose positive tariffs after the FTA between countries A and B. Then, for m P 2, non-member country C raises its tariff after the FTA. For m < 2, it is possible that country C would reduce its tariff after the FTA. (2) Suppose that (l, m, n) 2 H3, where country A provides subsidies after the FTA. Then, for m < 2, nonmember country C reduces its tariff after the FTA. For m P 2, it is possible that country C would raise its tariff after the FTA. 5. Effects of the FTA on Welfare Finally, we investigate the effects of the FTA on welfare. Before starting the analysis, let us define N N F ~i ðT BA ; T CA ; tNB ; tNC Þ, yFi y ~i ð0; T CA ; 0; tFC Þ, i = A, B, C, Y N lyNA þ nyNB þ nyNC , and Y F lyFA þ nyFB þ nyFC . yNi y The following result, which clarifies the effects of the FTA on the equilibrium output levels, is useful for the subsequent welfare analysis. 4
Further, country B sets tB = 0 after the FTA, which increases the above negative effect.
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Proposition 2 (1) (2) (3) (4)
For any ðl; m; nÞ 2 H1 ; For any ðl; m; nÞ 2 H1 ; If (l, m, n) 2 H4 (resp., For any (l, m, n) 2 H1,
yNA > yFA . yNB < yFB . (l, m, n) 2 H1nH4), yNC 6 yFC (resp., yNC > yFC ). YN < YF.
Proposition 2(1)–(3) say that the FTA decreases domestic final-good output in country A, increases final-good imports from country B, and may or may not increase final-good imports from country C. These results are obtained by the following mechanisms: From Proposition 1 (1), the FTA reduces country A’s tariff as well as eliminating tariff on imports from country B. This reduction in country A’s tariff decreases domestic final-good production and increases final-good imports from country B. At the same time, the reduction may increase final-good imports from country C. If so, country C has an incentive to raise its tariff on intermediategood imports. Of course, raising country C’s tariff has a negative effect on final-good production in country C. However, if the reduction in country A’s tariff is sufficiently large, final-good imports from country C still increase even if country C raises its tariff on intermediate-good imports. Proposition 2(4) shows that, irrespective of the imports from country C, the FTA increases the total output in country A. It implies that the direct effect of the FTA, i.e., the effect from the elimination of tariffs between countries A and B, is sufficiently larger than the others. In the following analysis, we use numerical analysis, partly, to clarify the result. In the numerical analysis that follows, we assume that a = 10. 5.1. Non-member country First, we focus on non-member country C. Country C’s welfare consists of the profits of the finalgood firms and tariff revenue. The welfare levels before and after the FTA, indicated by W NC and W FC , respectively, are given by the following expression:
W iC ¼ nðyiC Þ2 þ mt iC xiC ¼ nyiC ðyiC þ t iC Þ;
i ¼ N; F:
ð20Þ
Note that we use Eq. (7) to obtain the second part of Eq. (20). As shown in Lemma 2 and Proposition 1, country C imposes positive tariffs after the FTA, and its tariff level may or may not be higher than that before the FTA. We find that this change in tariff levels before and after the FTA is also critical to country C’s welfare. That is, the following proposition is evident from Proposition 2 and Eq. (20). N F F N F Proposition 3. If (l, m, n) 2 H4, where t N C 6 t C holds, then W C 6 W C , and if (l, m, n) 2 H5, where t C > t C N F holds, then W C > W C .
Proposition 3 shows that whenever the FTA raises (resp., lowers) country C’s tariff tC, it also increases (resp., decreases) country C’s welfare WC. From Proposition 1, as a result of the FTA, country A’s tariff is reduced but country C’s tariff may or may not decrease. As we observe in Proposition 2, if country A’s tariff reduction is relatively substantial, final-good output is increased. Furthermore, a higher tC causes country C’s tariff revenue to increase. Thus, the FTA results in an increase in country C’s welfare. However, the final-good output can decrease, especially when only a few intermediategood firms exist. In this case, country C’s tariff, and consequently its tariff revenue, decreases. In sum, the FTA causes country C’s welfare to decrease. Table 1 shows changes in tariff and welfare for country C before and after the FTA, under different combinations of (l, m, n). It clearly illustrates the statement in Proposition 2. Whenever the tariff imposed by country C increases (resp., decreases) after the FTA, country C’s welfare increases (resp., decreases).
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Table 1 Changes in tariff and welfare. (l, m, n)
Region
T CF A
DtC
DWA
DWB
DWC
P
(2, (3, (6, (5, (6, (1, (3, (4, (5, (4,
I I I II II II II I I II
0.058 0.222 0.315 0.014 0.035 0.036 0.018 0.048 0.094 0.028
0.001 0.006 0.004 0.004 0.007 0.003 0.0003 0.007 0.008 0.002
0.372 0.145 0.0003 0.277 0.277 0.557 0.315 0.103 0.023 0.292
0.096 0.226 0.176 0.0001 0.046 0.059 0.087 0.359 0.473 0.0002
0.006 0.028 0.016 0.019 0.035 0.057 0.005 0.132 0.167 0.029
0.468 0.371 0.175 0.277 0.231 0.615 0.403 0.462 0.451 0.291
1, 1, 1, 1, 1, 2, 2, 2, 2, 2,
1) 1) 1) 3) 4) 2) 5) 5) 5) 7)
DtC ¼ tFC tNC ;
i¼A;B DW i
P
i¼A;B;C DW i
0.462 0.399 0.192 0.258 0.195 0.558 0.408 0.594 0.617 0.262
DW i ¼ W Fi W Ni
5.2. FTA member countries 5.2.1. Country B’s welfare Second, we consider FTA member country B. Country B’s welfare levels before and after the FTA are shown as follows:
W NB ¼ nðyNB Þ2 þ mt NB xNB ¼ nyNB ðyNB þ t NB Þ;
ð21Þ
W FB ¼ nðyFB Þ2 :
ð22Þ
and
Before the FTA, country B’s welfare consists of the sum of the profits of final-good firms in the country and its tariff revenue. After the FTA, eliminating T BA and tB increases the final-good firm’s outputs and profits, whereas the tariff revenue disappears. Therefore, the effect of FTA on country B’s welfare depends on whether the positive effect of the increase in profits outweighs the negative effect of the lost tariff revenue. F Proposition 4. If (l, m, n) 2 H2, then W N B < W B. F
In light of Lemma 2, Proposition 4 states that the FTA makes country B better off if T CA P 0. HowF ever, if (l, m, n) 2 H3, where T CA < 0, we cannot rule out the possibility that the FTA makes country B worse off (i.e., W NB > W FB ). The intuitive explanation is as follows: In most cases, the increase in output (and therefore profits) of the final-good firms in country B outweighs the loss of its tariff revenue. However, if country A offers subsidies to non-member country C and country C reduces its tariff after the FTA, the increase in output (and profits) of final-good firms in country B would not be significant, and the profit increase could be dominated by the loss of tariff revenue. This result can be verified by the numerical analysis reported in Table 1, which shows the change in welfare levels of country B before and after the FTA under different combinations of (l, m, n). In region F F I, where T CA P 0; W NB < W FB always holds, while in region II, where T CA < 0, there are some cases in which W NB > W FB . 5.2.2. Country A’s welfare Third, we focus on country A. Country A’s welfare levels before and after the FTA are organized as follows: N
N
W NA ¼ mPN ðxN Þ þ lPNA ðyNA Þ þ CSN ðY N Þ þ nT BA yNB þ nT CA yNC ;
ð23Þ
and F
W FA ¼ mPF ðxF Þ þ lPFA ðyFA Þ þ CSF ðY F Þ þ nT CA yFC :
ð24Þ
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Country A’s welfare before the FTA consists of the profits of intermediate- and final-good firms in the country, the consumer surplus, and the tariff revenue from countries B and C. In light of Propositions 1 and 2, let us examine how country A’s welfare would be affected. First, the FTA eliminates country A’s tariffs on imports from country B, T BA . Furthermore, from Proposition 1 (1), country A’s tariffs on imports from country C, T CA , and therefore its tariff income, decrease as a result of the FTA. This is a negative effect for country A. Second, from Proposition 2 (1), outputs and profits of the final-good firms in country A decrease because of the FTA. This is a negative effect for country A. Third, from Proposition 2 (4), the final-good supply in country A increases as a result of the FTA, and therefore its price decreases. This enhances consumer surplus in country A, which is a positive effect for the country. Fourth and finally, the increase in the total supply of the final good, from Proposition 2 (4), corresponds to an increase in the total supply of the intermediate good. This increases the profits of the intermediate-good firms in country A, which is a positive effect for the country. Thus, country A faces two negative effects (a decrease in the country’s tariff revenue and final-good firms’ profits) and two positive effects (an increase in consumer surplus and intermediate-good firms’ profits). Depending on the relative magnitude of these effects, country A’s welfare may or may not increase as a result of the FTA. Table 1 shows the change in country A’s welfare levels before and after the FTA under different combinations of (l, m, n). In region I, where country A imposes a positive external tariff after the FTA, W NA > W FA holds in some cases. However, in region II, where country A provides a negative external tariff (i.e., a subsidy) after the FTA, W NA < W FA always holds. This implies that the positive effects are stronger, especially in region II.
5.3. World welfare Finally, we discuss on the effect of FTA on world welfare. Table 1 shows the changes in the joint welfare of FTA members, and world welfare under different combinations of (l, m, n). We can see that the FTA increases joint welfare for any parameter combinations in Table 1. Furthermore, from Proposition 3, non-member country C’s welfare increases when its tariff level is raised as a result of the FTA. We can summarize these findings as follows: When a non-member country raises its tariff level after the FTA, world welfare increases. In addition, according to numerical simulation, even when country C’s tariff level and welfare are lowered as a result of the FTA, world welfare increases (see Table 1). This implies that the increase in total output caused by the FTA may be critical to this result. Proposition 2 (4) shows that irrespective of country C’s policy, the FTA causes the total final-good output to increase. If the effect of the output increase is strong enough, the FTA may enhance world welfare. Note that our result implies that the formation of an FTA may not always be Pareto-improving. From Table 1, we observe that there are some cases in which the FTA increases welfare in all countries – (l, m, n) = (3, 1, 1) and (4, 2, 5), for example. This is a Pareto-improving FTA scenario. However, in most cases, the FTA would decrease the welfare of either of the three countries. Although transferring welfare may be an alternative, it is practically difficult to compensate for welfare loss. Therefore, the FTA is not Pareto-improving in such cases.
6. Concluding remarks We have investigated the effects of an FTA on tariff levels and welfare in the context of vertical trade. We have constructed a three-country model with an upstream country that exports an intermediate good and two downstream countries that export a final good. We consider an FTA between the upstream country and one of the downstream countries, treating the other downstream country as a non-member. In terms of the effects of the FTA on tariff levels, we show that the FTA unambiguously induces its member country to reduce its tariff for the non-member country. In some cases, the tariff level after
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the FTA becomes negative. In contrast, the non-member country may or may not raise its tariff after the FTA. In terms of the effects of the FTA on welfare, we demonstrate that the FTA is beneficial to the nonmember country when its tariff level is increased. Some numerical simulations report that the FTA may or may not make the member countries better off, while it may lead to an increase in the joint welfare of the member countries, which in turn causes an increase in world welfare. Some of the above results regarding the FTA’s welfare effects are obtained by using numerical simulations. In fact, we have not explicitly obtained the conditions for an increase in the upstream country’s welfare, the joint welfare of member countries, or world welfare. We intend to derive these conditions explicitly in a future study. Furthermore, our model offers several possibilities for further studies. For example, including firm entry or determining of the trade pattern would enrich our analysis and provide other interesting results. Considering a customs union between the downstream countries, which sets a common external tariff against the upstream country, is also an interesting topic. We would like to extend our model to include these topics in future studies. Acknowledgments We thank Kenzo Abe, Ichiroh Daitoh, Taiji Furusawa, Yoichi Hizen, Jun-ichi Itaya, Yukio Karasawa, Kazuharu Kiyono, Noritaka Kudoh, Kazuo Machino, Ryoichi Nomura, Masao Oda, Takao Ohkawa, Makoto Okamura, Masayuki Okawa, Chisato Shibayama, Nobuhito Suga, Yasunobu Tomoda, Keiichi Umada, Shigemi Yabuuchi, Chisato Yoshida, and two anonymous referees for their constructive comments and suggestions. The usual disclaimer applies. This work has been financially supported by the Japan Economic Research Foundation and Grants-in-Aid for Scientific Research (C) (No. 21530217). Appendix A. Proof of Lemma 1 First, t NB ¼ t NC P 0 holds if l + 2n + 1 m(l + 1) P 0 (see Eq. (14)). This condition corresponds to t n P nt ðm1Þðlþ1Þ . It is clear that @n > 0 for m > 1; that is, nt is increasing in l, given m. 2 @l N N Next, T BA ¼ T CA P 0 holds if 2(m + 1) n2 + /1n + /2 > 0, where /1 (2m + 3)(m 1)l + (m + 1) (m 3) and /2 (m2 + 2m 1) l2 + 2(m2 + m 1)l + (m + 1) (m 1). This condition corresponds to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi / þ /21 þ8ðmþ1Þ/2 1 2 n 6 n1 1 4ðmþ1Þ . Since @/ > 0 and @/ > 0, we have @n@l1 > 0, that is, n1 is increasing in l, given m. @l @l
. We determine the slope of nt and n1 at l = 0. Thus, we have At l = 0, nt ¼ n1 ¼ m1 2
@nt m 1 @n1 2m 1 < : ¼ ¼ 2 2 @l l¼0 @l l¼0
ðA:1Þ N
N
Therefore, if (l, m, n) 2 H1 {(l, m, n)jnt (l, m) 6 n 6 n1 (l, m)}, then all tariffs, t NB ; t NC ; T BA , and T CA , are nonnegative. h Appendix B. Proof of Lemma 2 Proof of (1): t FC P 0 if l + 2n + 1 m(l + 1) P 0 from Eq. (18). Since we only consider parameters such that (l, m, n) 2 H1, we have t FC P 0. h F Proof of (2): T CA P 0 if 2(2m + 1)n2 + /3 n + /2 P 0 from Eq. (19), where 2 2 /2 (m + 2m 1)l + 2(m2 + m 1) l + (m + 1)(m 1) (see Eq. (15)) and /3 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /3 þ /23 þ8ð2mþ1Þ/2 @/2 2 (m + 1)(m 1)l + 2m 3m 3. This condition corresponds to n 6 n2 . Since @l > 0 4ð2mþ1Þ 3 and @/ > 0, we have @n@l2 > 0, that is, n2 is increasing in l, given m. @l At l = 0, n2 ¼ m1 ¼ nt ¼ n1 . We then compare the slope of n2 with nt and n1 at l = 0. We have 2
@nt m 1 @n2 3m3 þ m2 þ m 1 @n1 2m 1 < : ¼ ¼ ¼ < 2 2 2ð2m þ m þ 1Þ 2 @l l¼0 @l l¼0 @l l¼0
ðB:1Þ
Y. Kawabata et al. / J. Japanese Int. Economies 24 (2010) 569–585
583
Therefore, if (l, m, n) 2 H2 {(l, m, n)jnt(l, m) 6 n 6 n2 (l, m)} (resp., (l, m, n) 2 H3 {(l, m, n)jn2(l, F m) < n 6 n1(l, m)}), T CA is nonnegative (resp., negative). h Appendix C. Proofs of Proposition 1 and Corollary 1 Proof of (1): Let us define
f1 2ðm þ 1Þn2 þ /1 n þ /2 ; f2 2ð2m þ 1Þn2 þ /3 n þ /2 ; g 1 2mn2 þ 2ðml þ 2m þ l þ 1Þðl þ 1Þn þ ðm þ 1Þðl þ 1Þ3 ; and n o g 2 ðl þ n þ 1Þ 2ð2m þ 1Þn2 þ f3ðm þ 1Þl þ 4m þ 3gn þ ðl þ 1Þ2 ðm þ 1Þ : Using them, Eqs. (15) and (19) are rewritten as N
T CA ¼
af1 ðm þ 1Þg 1
F
and T CA ¼
af2 : ðm þ 1Þg 2
ðC:1Þ
Since we consider the vector of parameters such that (l, m, n) 2 H1, we have f1 f2 = mn{l + 2n + 1 m(l + 1)} > 0, and g1 g2 = n[2(2m + 1)n2 + {(7m + 5)l + 6m + 5}n + (l + 1){2(m + 1) l + N F m + 2}] < 0. Thus, f1 > f2and g1 < g2. Therefore, from (C.1), we have T CA > T CA . h N F Proofs of (2) and Corollary 1: From Eqs. (14) and (18), t C < tC holds if 2(l + 1)(2m + 1)n2 /4 n /5 < 0, where /4 (4m + 3) (m 1)l2 + (5m2 4m 6) l + 2m2 3m 3, and 3 2 2 2 /5 (2m 1)(m + 1) l + (4m + 2m 3) l + (3m + m 3) l + (m + 1)(m 1). This condition correpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi / þ /24 þ8ðlþ1Þð2mþ1Þ/5 4 5 sponds to n 6 n3 4 4ðlþ1Þð2mþ1Þ . Since @/ > 0 and @/ > 0, we have @n@l3 > 0, that is, n3 is increas@l @l ing in l, given m. At l = 0, n3 ¼ m1 ¼ nt ¼ n1 ¼ n2 . We investigate the slope of nt, n1, n2, and n3 at l = 0. Then, we have 2
@nt m 1 @n3 3m3 þ 2m2 3 @n2 3m3 þ m2 þ m 1 @n1 2m 1 < ; ¼ ¼ ¼ ¼ < < 2 2ð2m2 þ m þ 1Þ 2 @l l¼0 @l l¼0 2ð2m2 þ m þ 1Þ @l l¼0 @l l¼0
ðC:2Þ for m < 2, and
@nt m 1 @n2 3m3 þ m2 þ m 1 @n3 3m3 þ 2m2 3 @n1 2m 1 ¼ ¼ ¼ ¼ < ; 6 < 2 2ð2m2 þ m þ 1Þ 2 @l l¼0 @l l¼0 @l l¼0 2ð2m2 þ m þ 1Þ @l l¼0 ðC:3Þ for m P 2 (equality holds at m = 2). Therefore, if (l, m, n) 2 H4 {(l, m, n)jnt (l, m) 6 n 6 n3(l, m)} (resp., (l, m, n) 2 H5 {(l, m, n)jn3 (l, m) < n 6 n1(l, m)}), tNC is greater than (resp., less than) t FC . From (C.2), (C.3), and Lemma 2, the threshold n3 line passes through the parameter region H2 (resp., H3) for m < 2 (resp., m P 2). Therefore, if (l, m, n) 2 H2 and m P 2, t NC < tFC . If (l, m, n) 2 H2 and m < 2, there exists a parameter such that t NC < tFC (see Fig. 4i). If (l, m, n) 2 H3 and m < 2, t NC > tFC . If (l, m, n) 2 H3 and m P 2, there exists a parameter such that t NC > t FC (see Fig. 4ii). h Appendix D. Proof of Proposition 2 Proof of (1): Subtracting the per-firm output of the final good in country A under the FTA from the pre-FTA output, we have N
yNA
yFA
F
mn½2ðT CA þ tNC Þ ðT CA þ t FC Þ ; ¼ ðm þ 1Þðl þ 2n þ 1Þ
ðD:1Þ
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Y. Kawabata et al. / J. Japanese Int. Economies 24 (2010) 569–585 N
N
F
N
N
where we used the fact that T BA ¼ T CA and t NB ¼ tNC . Since (l, m, n) 2 H1, it holds that T CA < T CA < 2T CA . In addition, we have
2t NC tFC ¼
½l þ 2n þ 1 mðl þ 1Þa ðm þ 1Þðl þ n þ 1ÞM N M F
X;
ðD:2Þ
where
M N 2mn2 þ 2fðm þ 1Þl þ 2m þ 1gðl þ 1Þn þ ðl þ 1Þ3 ðm þ 1Þ > 0; M F 2ð2m þ 1Þn2 þ f3ðm þ 1Þl þ 4m þ 3gn þ ðl þ 1Þ2 ðm þ 1Þ > 0;
X ðl þ 1Þðl þ n þ 1Þðl þ 2n þ 1Þ2 þ ½ðl þ 2Þðl þ 1Þ3 þ ðl þ 2Þ2 ðl þ 9Þn þ 16ðl þ 1Þn2 þ 10n3 m þ ½ðl þ 1Þ2 þ 2nðn þ 1Þðl þ 2n þ 1Þm2 þ 2mnlðl þ 2n þ 1Þðl þ 2n þ 1 mlÞ: The assumption (l, m, n) 2 H1, i.e., l + 2n + 1 m(l + 1) > 0, implies that l + 2n + 1 ml > 0. ThereN
F
fore, X > 0 holds, and hence, 2t NC > t FC . In sum, we have 2ðT CA þ tNC Þ ðT CA þ t FC Þ > 0. h Proof of (2): The difference between yNB and yFB is N
yNB yFB ¼
F
m½ðl þ 1ÞðT CA þ tNC Þ þ nðT CA þ t FC Þ : ðm þ 1Þðl þ 2n þ 1Þ
ðD:3Þ N
F
If (l, m, n) 2 H2, it is clear that yNB < yFB because the equilibrium tariffs T CA ; t NC ; T CA , and t FC are all F
F
nonnegative. If ðl; m; nÞ 2 H3 ; T CA is negative. Nevertheless, since T CA þ tFC ¼ mla=M F > 0, we obtain N
F
ðl þ 1ÞðT CA þ tNC Þ þ nðT CA þ t FC Þ > 0, and hence yNB < yFB . h Proof of (3): Substituting the pre-FTA tariff rates (14) and (15) into the equilibrium output of the final good in country C, we have
yNC ¼
ðl þ m þ 1Þðl þ n þ 1Þma ðm þ 1ÞMN
ðD:4Þ
:
In light of (14), (D.4) is rewritten as
yNC ¼
ðl þ n þ 1Þm tN : l þ 2n þ 1 mðl þ 1Þ C
ðD:5Þ
Substituting the tariff rates in (18) and (19) under the FTA into the equilibrium output in country C, we have
yFC ¼
ð2mn þ l þ m þ n þ 1Þma ðm þ 1ÞM F
ðD:6Þ
:
In light of (18), (D.6) is rewritten as
yFC ¼
ðl þ n þ 1Þm tF : l þ 2n þ 1 mðl þ 1Þ C
ðD:7Þ
Subtracting (D.7) from (D.5) yields
yFC yNC ¼
ðl þ n þ 1Þm ðt F t NC Þ: l þ 2n þ 1 mðl þ 1Þ C
ðD:8Þ
Since we assume l + 2n + 1 m(l + 1) > 0, the sign of Eq. (D.8) depends on the sign of tFC tNC . Proof of (4): The difference between YN and YF is N
YN YF ¼
h
F
mn½2ðT CA þ tNC Þ ðT CA þ t FC Þ ¼ yFA yNA ; ðm þ 1Þðl þ 2n þ 1Þ
which is unambiguously negative, as previously shown, if (l, m, n) 2 H1.
ðD:9Þ h
585
Y. Kawabata et al. / J. Japanese Int. Economies 24 (2010) 569–585
Appendix E. Proof of Proposition 4 Subtracting country B’s welfare after the FTA from that before the FTA, we have
tN tN nðtNB Þ2 W NB W FB ¼ n½ðyNB Þ2 þ t NB yNB ðyFB Þ2 ¼ n yNB þ B þ yFB yNB þ B yFB 2 2 4 ) ( N BN CF N N F t t B m½ðl þ 1Þðt B þ T A Þ þ nðt C þ T A Þ nðtNB Þ2 ¼ n yNB þ B þ yFB : ðm þ 1Þðl þ 2n þ 1Þ 2 2 4
ðE:1Þ
F
We have shown that t FC þ T CA > 0. Moreover, N
N
tNB mðl þ 1ÞðtNB þ T BA Þ ½2mn þ l þ 2n þ 1 mðl þ 1Þt NB 2mðl þ 1ÞT BA ¼ ; 2 ðm þ 1Þðl þ 2n þ 1Þ 2ðm þ 1Þðl þ 2n þ 1Þ N
ðE:2Þ
N
which, by substituting the values of t NB ¼ tNC and T BA ¼ T CA , is nonpositive if
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðl þ 1Þ ðm2 1Þl 1 þ m½m 1 þ m2 þ ðm þ 1Þ2 l þ ð2m þ 1Þðm þ 1Þl
n6
nB ðl; mÞ:
2½ðl þ 2Þm þ l þ 1
ðE:3Þ By computation, we can show that n2(l, m) < nB (l, m) < n1(l, m). This means that, if we define
HB {(l, m, n)jnt (l, m) 6 n 6 nB(l, m)}, H2 HB H1 holds. Therefore, it follows that if (l, m, n) 2 H2, we have W NB < W FB . Note that if (l, m, n) 2 H3 H1nH2, Eq. (E.2) may be positive. Therefore, the term in the brace brackets in (E.1) does not necessarily become negative.
h
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