Published in International Economic Journal (1996), doi:10.1080/10168739600000018
Targeted Tariff Protection, Monopolistic Competition and Demand Interdependence
Hwan C. Lin∗ Final Version: March 1995 (Accepted March 22, 1995)
Abstract This paper examines how the welfare effects of tariff protection, either targeted or acrossthe-board, are dependent critically upon alternative patterns of demand linkages among several goods from monopolistically competitive sectors. It shows that when complementary possibilities are present, tariff protection may harm the import-complementing sector more than it assists the import-substituting sector, thereby aggravating consumption distortions and reducing national welfare. It also shows that across-the-board protection is much less feasible than is targeted protection because the former requires much more informational details of both demand linkages among noncompetitive markets and economic structure. Keywords: Targeted Tariff Protection, Across-the-Board Protection, Monopolitstic Competition, Duality, Small Economy, Welfare JEL Classification: F12, F13 ©All rights reserved. ∗ Department
of Economics, University of North Carolina at Charlotte, 9201 University City Blvd., Charlotte, NC 28223, USA. E-mail:
[email protected]
1
Introduction
Trade policy may be welfare enhancing under conditions of imperfect competition as shown by a series of papers over the past decade.1 These papers often assumed a single noncompetitive sector in the national economy. Yet little work examined their robustness in models with several sectors competing for resources and markets. As an exceptional work, Dixit and Grossman (1986) showed that targeted export promotion would be much less desirable in an environment with several oligopolistic industries competing for resources than in a simple one-firm economy (Brander and Spencer 1985). So noncompetitive models are still subject to the general equilibrium principle that promotion of one sector is implicitly taxation of another. But other than resource constraints, will some patterns of demand interdependence among several goods from noncompetitive sectors still work against the argument for trade policy under imperfect competition? This question cannot be easily answered in recent noncompetitive models including Dixit and Grossman (1986). These models, either oligopolistic or monopolistic, almost all assume a simplified pattern of demand linkages in which goods are substitutes with no complements in existence.2 Thus, for example, these models fail to tackle a troublesome problem with tariff policy that benefits the import-substituting sector at the expense of the import-complementing sector. This problem is particularly troublesome when each imported good is both a substitute for goods from a distorted, noncompetitive sector and a complement to goods from another distorted, noncompetitive sector at home. With such complex yet plausible demand linkages, tariffs are likely to work against the import-complementing sector via demand interdependence, thereby aggravating distortions and reducing national welfare. This paper seeks to make a contribution by examining this troublesome tariff policy problem in a general equilibrium framework. This examination builds on a small economy model with three goods: a homogenous good produced by competitive firms and two types of differentiated goods produced by monopolistically competitive firms. In this paper a general form of social welfare function allows each imported good to be both a substitute for goods from a noncompetitive sector and a complement to goods from another noncompetitive sector. In particular, we deal not only with protection of a targeted noncompetitive sector, 1 See,
for example, Flam and Helpman (1987), Gros (1987), Jones and Takemori (1989), Lancaster (1991) and Venables (1982, 1987) for models of monopolistic competition; Brander and Spencer (1985) and Eaton and Grossman (1986) for models of oligopoly; and Bagwell (1991) for models of monopoly. 2 Feenstra (1986) and Itoch and Kiyono (1987) did focus on the role of demand linkages in influencing the desirability of export subsidies, import quota and VER, but their models built on perfect competition.
but with protection of both noncompetitive sectors by imposing different tariff rates. The former is referred to as "targeted tariff protection" and the latter as "across-the-board tariff protection". In this paper we can clearly demonstrate how the welfare effects of tariff policy are dependent critically upon alternative patterns of demand linkages among noncompetitive markets. It is shown that both targeted and across-the-board tariff protection are welfare-enhancing when goods are all substitutes. Yet with complementary goods in existence, either type of protection may become a welfare- reducing policy if it harms the import-complementing sector more than it assists the import-substituting sector. Results from Flam and Helpman (1987), Venables (1982, 1987) and others, where tariff policy always benefits the importing country, are therefore not robust at all. Furthermore, the present paper shows that across-the-board tariff protection is much less feasible than targeted tariff protection, even though factor prices are independent of the reallocation of resources between the competitive sector and the noncompetitive sectors. This is because across-the-board protection requires much more informational details of both demand interdependence among noncompetitive markets and economic structure. Section 2 outlines a small-economy model. Section 3 establishes a general equilibrium for this economy using duality relations. Section 4 examines the welfare effects of small tariff protection for differentiated goods initially under free trade. There, we consider a symmetric case, in which noncompetitive sectors are symmetric, and a Ricardian case, in which both a free-entry regime and a no-entry regime are considered. Lastly, Section 5 concludes.
2
The Model
The paper considers a small open economy endowed with many productive factors, which are mobile across sectors but immobile internationally. This economy has three sectors: one competitive sector producing the numeraire homogeneous good (H) and two noncompetitive sectors producing two classes of differentiated goods (X and Y ) respectively under monopolistic competition. All goods are tradable in an integrated world market with no price discrimination. In this small economy framework, either X-type or Y -type products developed at home only represent a ’negligible’ subset of similar products on world markets.
2
2.1
Preferences
The economy has a well-defined social welfare function U[.] is differentiable, increasing, concave and homogeneous of degree one in [DH , DX , DY ],
U = U[DH , DX , DY ]
(1)
where DH = Domestic consumption quantity of good H; DX = Domestic consumption index of good X; DY = Domestic consumption index of good Y .
Following Spence (1976) and Dixit and Stiglitz (1977), the two consumption indexes DX and DY are determined by the CES functions: n+n∗
DX =
σX −1 σX Xi
! σσX−1 X
(2a)
d
∑
i=1 m+m∗
DY =
σY −1 σY Yj
! σσY−1 Y
d
∑
(2b)
j=1
where dXi (dY j ) = Domestic consumption guantity of good X(Y ) of variety i( j); σX (σY ) = The elasticity of substitution between similar products;3 n(n∗ ) = The number of domestic (foreign) varieties of good X; m(m∗ ) = The number of domestic (foreign) varieties of good Y . The CES functional forms of (2a) and (2b) ensure that (1) is also differentiable, increasing, concave and homogenous of degree one in [DH ; dX1 , ..., dXn+n∗ ; dY1 , ..., dYm+m∗ ]. Hence, the economy’s minimum expenditure E required to attain a given level of social welfare U can be determined by the expenditure function (Varian 1984): E = e[pH ; pX1 , ..., pXn+n∗ ; pY1 , ..., pYm+m∗ ;U] 3 The
(3)
elasticity of substitution (σ ) also represents the elasticity of demand facing a firm under large-group monopolistic competition, as in models with the Spence-Dixit-Stiglitz preferences.
3
where pH = Domestic consumer price of good H; pXi = Domestic consumer price of good X of variety i(= 1, ..., n + n∗ ); pY j = Domestic consumer price of good Y of variety j(= 1, ..., m + m∗ ). The expenditure function (3) satisfies regularity conditions; i.e. e[.] is differentiable, increasing, concave and homogenous of degree one in prices. In each sector, firms (either domestic or global) are symmetric initially under free trade,4 and firms are always ’nationally’ symmetric whether free or restricted trade takes place, for they presumably have the same technology and the same elasticity of demand for their output as usually seen in trade models of monopolistic competition. So, taking the homogeneous good as numeraire (i.e. pH = 1), we can rewrite (3) as: E = e[1, ˜ pX , pX ∗ + t, pY , pY ∗ + ψt;U, n, n∗ , m, m∗ ]
(3’)
where pZ (pZ ∗ ) is the representative price of good Z(= X,Y ) set by domestic (foreign) firms and t(ψt) is a uniform specific tariff rate imposed on the imports of X-type (Y -type) products with ψ being a nonnegative coefficient. As ψ equals zero, tariff protection is targeted to X-type products only. As ψ is positive and greater than one, a higher tariff rate is imposed on the imports of Y -type products than it is on the imports of X-type products; the reverse is true if ψ is smaller than one. Like (3), e[.] ˜ is differentiable, increasing, concave and homogeneous of degree one in prices. Lemma 1. Given (1), (2) and (3’), the Envelope theorem implies that (see Appendix A for proof):
e˜n ≡
∂ e[.] ˜ e˜1 [.]pX =− 1. The assumption of free entry and exit W ]) and qY j = y j (pY j − CY [W W ]) denote a firm’s profit of firms drives profits to zero. Let qXi = xi (pXi − CX [W margin respectively in each differentiated-good sector. Then a firm’s profit margin must be restricted to its 7 Our
cost structures are borrowed from Flam and Helpman (1987), which extends Venables (1982) by allowing differentiated goods to be profitably exportable.
6
fixed cost: W ], qXi = Cn [W W ], qY j = Cm [W
i = 1, ..., n
(6a)
j = 1, ..., m.
(6b)
Using conditions (5a) and (6a) together with the definition of qX , a firm’s zero-profit output in the X industry is: xi =
qXi σX pXi
(7a)
qY j σY pY j
(7b)
Similarly, the corresponding on in the Y industry is
yj =
Conditions (7) imply that the inverse of σ represents a firm’s profit margin (or fixed cost) share in its sales revenues (e.g. 1/σX = qXi /(pXi xi )) and this share increases with the firm’s monopoly power. Moreover, conditions (5) - (7) imply a symmetric equilibrium over domestic firms. Thus, as mentioned in getting (3’), we adopt the shorthand pX = pXi , pY = pY j , x = xi , y = y j , qX = qXi and qY = qY j , for i = 1, ..., n and j = 1, ..., m. Given a vector of production factors V , mobile between industries but immobile internationally, the economy’s full employment condition is:
H·
W] W] W] W] W] ∂CX [W ∂CY [W ∂Cn [W ∂Cm [W ∂CH [W +X · +Y · +n· +m· =V W W W W W ∂W ∂W ∂W ∂W ∂W
(8)
W ]/∂W W is a vector of unit input requirements for good Z(= H,Y or X), and where X = nx, Y = my, ∂CZ [W W ]/∂W W and ∂Cm [W W ]/∂W W can be seen respectively as a vector of unit input requirements for developing ∂Cn [W a differentiated product. With duality relations, conditions (4), (5), (6) and (8) can be summarized by the revenue function that determines the economy’s GDP, σX − 1 σY − 1 GDP = r 1, pX , pY , qX , qY ; V σX σY
(9)
V ] is a standard revenue function: differentiable, increasing, convex and homogeneous of Certainly, r[.;V degree one.8 Thus, (9) implies that there are five activities (H, X,Y, n, m), each subject to a linearly ho8 Here,
the integer problem with the number of product varieties n(m) is ignored as in the trade literature based on
7
mogeneous, concave cost function C[.]. By Hotelling’s lemma, the output level of each activity is given respectively by:
r1 [.] =
∂ r[.] =X ∂ pX (σX − 1)/σX
(10a)
r2 [.] =
∂ r[.] =Y ∂ pY (σY − 1)/σY
(10b)
r3 [.] =
∂ r[.] =n ∂ qX
(11a)
r4 [.] =
∂ r[.] =m ∂ qY
(11b)
Furthermore, the small economy’s supply elasticity matrix is positive semidefinite:
ε11 ε12 ε13 ε14
ε21 ε22 ε23 ε24 [εi j ] = ε31 ε32 ε33 ε34 ε41 ε42 ε43 ε44
where εhk ’s represent own or cross supply elasticities: ε11 = ∂ log r1 [.]/∂ log pX (σX −1)/σX , ε12 = ∂ log r1 [.] /∂ log pY (σY − 1)/σY , ε13 = ∂ log r1 [.]/∂ log qX , ε14 = ∂ log r1 [.]/∂ log qY , and the like. It holds that the own supply effects εhk for h = k are all positive. To focus on demand linkages, we assume all activities substitutable in production so that εhk is negative and is in absolute terms less than εhh for h 6= k, as in Flam V ],it holds that εh1 + εh2 + εh3 + εh4 = and Helpman (1987). Besides, due to the linear homogeneity of r[.;V −εh0 , the supply elasticity of activity h with respect to the price of the numeraire good. monopolistic competition.
8
3
Equilibrium
The small economy’s general equilibrium is determined by conditions (7a) and (7b), which ensure zero profits, together with the following: σX − 1 σY − 1 e[1, ˜ pX , pX ∗ + t, pY , pY ∗ + ψt;U, n, n , m, m ] =r 1, pX , pY , qX , qY ; V σX σY ∗
∗
(12)
+ t e˜2 [.] + ψt e˜4 [.] e˜1 [.] + ndX∗ = r1 [.]
(13a)
e˜3 [.] + mdY∗ = r2 [.]
(13b)
with
dX∗ =
X p−σ X E∗ PX∗ X
dY∗ =
pY−σY ∗ E PY∗ Y
where dZ∗ (Z = X or Y ) denotes per variety of foreign (uncompensated) demand for domestic output of good Z, given the foreign expenditure EZ∗ on good Z and the price index of good Z:9 X X PX∗ = np1−σ + n∗ p1−σ X X∗
Y PY∗ = mpY1−σY + m∗ pY1−σ ∗
Condition (12) represents the economy’s budget constraint with its domestic aggregate expenditure equal to its domestic aggregate (factor) income plus the rebated tariff revenues (= t e˜2 [.]+ψt e˜4 [.]). Condition (13a) is a market-clearing condition for good X and condition (13b) is that for good Y , each produced at home. The corresponding condition for good H is suppressed due to Walras law. In the smalleconomy framework, there are nine endogenous variables pX , pY , qX „ qY , x, y, n, m and U to be determined by solving conditions 9 The specifications of d ∗ X
and dY∗ are based on the implicit assumption that the rest of the world also has the SpenceDixit-Stiglitz preferences as does the small economy. The reason why foreign demands are specified in uncompensated (rather than compensated) form is to avoid the determination of foreign welfare level, which is irrelevant to the analysis and is ’asymptotically’ exogenous in the present small-economy framework.
9
(7a), (7b), (10a), (10b), (11a), (11b), (12), (13a) and (13b). The foreign variables including EX∗ , EY∗ , pX ∗ , pY ∗ , n∗ and m∗ are all exogenous. The price index of good Z, PZ∗ , (Z = X or Y ) which includes the prices of domestically produced differentiated goods can be taken as exogenous, since the small open economy only produces a negligible subset of similar products on world markets; i.e. n∗ and m∗ are infinitely large in relation to n and m.
4
Targeted Tariff Protection and Welfare
4.1
Differential form
In this section we begin to examine the welfare impact of tariff protection initially under free trade (t = 0), where firms are symmetric globally. Totally differentiating condition (12) and using Lemma 1 and conditions (7), one can decompose the welfare impact (in terms of good H) into sectoral gains/losses accruing to each noncompetitive sector: e˜U dU = (Gains/Losses)X + (Gains/Losses)Y
(14)
with 1 θX (Gains/Losses)X = GDP · SX (1 − θX ) pˆX + xˆ + nˆ σX σX − 1 1 θY mˆ (Gains/Losses)Y = GDP · SY (1 − θY ) pˆY + yˆ + σY σY − 1
(15a)
(15b)
where e˜U = ∂ e[.]/∂U, ˜ the inverse of marginal utility of income, is positive with normal goods; zˆ = dz/z, a percentage change in z throughout the paper; SX (SY ) = the output value of good X(Y ) as a fraction of the economy’s GDP; θX (θY ) = the economy’s domestic absorption share in domestic output of good X(Y ). According to (14), the sectoral gains/losses can be further decomposed into three effects in each noncompetitive sector. If they are all positive, the terms-of-trade effect pˆX ( pˆY ) – weighted by the export share 1 − θX (1 − θY ) – implies a shift in consumer surplus from foreign countries to the small economy, the scale-economy effect xˆ (y) ˆ – weighted by the profit-margin share 1/σX (1/σY ) – represents a rise of production efficiency, and the entrylvariety effect nˆ (m) ˆ – weighted by the domestic absorption share θX (θY )
10
over 1/(σX − 1) (1/(σY − 1)) – increases the diversity of choice.10 Of them, the scale-economy and variety effects (if positive) amount to drawing resources away from the competitive sector and into the noncompetitive sectors. They therefore help to lessen domestic consumption distortions (due to markup pricing), enhancing both home and foreign welfare. Yet, the terms of trade effect beggars thy neighbors. With Lemma 1 and appropriate exercises, one can derive the differential forms of conditions (13a) and (13b):11
(ε11 − θX δ11 + (1 − θX )σX ) pˆX + (ε12 − θY δ13 ) pˆY + ε13 qˆX + ε14 qˆY − nˆ = θX (δ12 + ψδ14 )(
dt ) p∗
(13a’)
(ε21 − θX δ31 ) pˆX + (ε22 − θY δ33 + (1 − θY )σY ) pˆY + ε23 qˆX + ε24 qˆY − mˆ = θY (δ32 + ψδ34 )(
dt ) p∗
(13b’)
where we let p∗ = p∗X = pY∗ for simplicity. The differential forms of the zero-profits conditions, (7a) and (7b), are given by
pˆX − qˆX + xˆ = 0
(7a’)
pˆY − qˆY + yˆ = 0
(7b’)
10 Note
that the parameter 1/(σ − 1) indicates the degree of monopoly power; as σ = 5, 1/(σ − 1) = .25 implying that price is 25% higher than marginal cost. 11 Note that when deriving (13a’), we use the result: (E for domestic expenditure on good X) X ! X np−σ ∂ e˜1 ∂ n ∂ EX /pX X e˜1U = e˜U = E e ˜ = e˜U ≈ 0 X U X X ∂E ∂ E np1−σ n + n∗ ∂E + n∗ p1−σ X X∗ as n/(n + n∗ ) approaches zero under the small-economy assumption. Similarly, when deriving (13b’), we use the result: (EY for domestic expenditure on good Y ) ∂ e˜2 m ∂ EY /pY e˜2U = e˜U = e˜U ≈ 0 ∂E m + m∗ ∂E as m/(m + m∗ ) approaches zero under the same assumption. Thus, in the present context the income effects of differential change in tariffs are negligible.
11
and those of (10a), (10b), (11a) and (11b) are given respectively by:
ε11 pˆX + ε12 pˆY + ε13 qˆX + ε14 qˆY − nˆ − xˆ = 0
(10a’)
ε21 pˆX + ε22 pˆY + ε23 qˆX + ε24 qˆY − mˆ − yˆ = 0
(10b’)
ε31 pˆX + ε32 pˆY + ε33 qˆX + ε34 qˆY − nˆ = 0
(11a’)
ε41 pˆX + ε42 pˆY + ε43 qˆX + ε44 qˆY − mˆ = 0.
(11b’)
To make the analysis tractable, we consider two simplified cases. In the first case, the two noncompetitive sectors are symmetric in the sense that they have the same (fixed) share in expenditure (αX = αY = α),12 W ] = CY [W W ] and the same (fixed) elasticity of demand (σX = σY = σ ), and the same cost functions (i.e., CX [W W ] = Cm [W W ], for all W > (0...0)). In the second case, by contrast, the model reduces to a Ricardian Cn [W model with one single factor of production (i.e. V and W collapse to a positive parameter, respectively).
4.2
The case with symmetric noncompetitive sectors
In this case, the Slutsky substitution matrix of Lemma 2 takes the specific form: [δi j ] ≈
−σX
σX − 1 + αX
0
αY
0
−1 + αX
0
0
αX
−σY
0
αX
0
αY σY − 1 + αY −1 + αY
under free trade.
Here with identical noncompetitive sectors, all goods are substitutes; i.e., δi j > 0, i 6= j. The symmetry between the noncompetitive sectors enables us to rewrite (14): θ 1 ˆ + (nˆ + m) ˆ e˜U dU = GDP · S (1 − θ )( pˆX + pˆY ) + (xˆ + y) σ σ −1 12 That
is, (1) is reduced to be a Cobb Douglas utility function with αX = αY = α, X −αY αX αY U[DH , DX , DY ] = D1−α DX DY H
12
(14’)
where S = SX = SY , σ = σX = σY , θ = θX = θY , and it also holds that pˆX = pˆY , qˆX = qˆY and xˆ = yˆ due to conditions (5) - (7). Next, using the specific Slutsky matrix to simplify (13a’) and (13b’), we can express the differential system of (13a’) + (13b’), (7a’) + (7b’), (10a’) + (10b’) and (11a’) + (11b’) in the simple matrix form:
ε11 +ε12 +ε21 +ε22 +σX +σY 2
ε13 +ε14 +ε23 +ε24 2
1
−1
ε11 +ε12 +ε21 +ε22 2 ε31 +ε32 +ε41 +ε42 2
ε13 +ε14 +ε23 +ε24 2 ε33 +ε34 +ε43 +ε44 2
−1
0
pˆX + pˆY
0 1 qˆX + qˆY −1 −1 nˆ + mˆ −1 0 xˆ + yˆ
Γ
0 = 0 0
(16)
where Γ = θ (δ12 + δ32 + ψδ14 + ψδ34 )(dt/p∗ ) is positive, whether targeted (ψ = 0) or across-the-board (ψ > 0) tariff protection is adopted. By Cramer’s rule, the solutions to (16) are given by: ε13 + ε14 + ε23 + ε24 ε33 + ε34 + ε43 + ε44 1 1− + Γ > 0 pˆX + pˆY = 4 2 2 1 ε11 + ε12 + ε21 + ε22 ε31 + ε32 + ε41 + ε42 qˆX + qˆY = 1+ − Γ > 0 4 2 2 1 (ε10 + ε20 − 2)(ε30 + ε40 ) nˆ + mˆ = Γ 4 4 1 (ε10 + ε20 )(ε31 + ε32 + ε41 + ε42 ) + Γ 4 4 1 (ε30 + ε40 )(ε13 + ε14 + ε23 + ε24 ) + Γ > 0 4 4 1 (ε30 + ε40 ) − (ε10 + ε20 ) xˆ + yˆ = Γ 4 2
(17)
(18)
(19)
(20)
where 4 = (σX + σY )(ε33 + ε34 + ε43 + ε44 )/4 − (ε13 + ε14 + ε23 + ε24 )/2 > 0 because εhh > 0, εhk < 0 for h 6= k, and εhh > |εhk |, as we have mentioned earlier. For the same reason, the large-bracket terms [.] containing εhk ’s on the right-hand sides of (17), (18) and (19) are all positive. Accordingly, given Γ > 0, it holds that pˆX + pˆY > 0, qˆX + qˆY > 0 and nˆ + mˆ > 0. Yet, xˆ + yˆ is ambiguous in sign because in absolute terms ε30 + ε40 may be greater or less than ε10 + ε20 . Our results here are a particular generalization of Flam and Helpman (1987), since in this symmetric case the present model collapses into F-H’s two-sector model. Intuitive explanations for our results are provided as follows.
13
First of all, tariff protection – either targeted or across-the-board – raises domestic consumer prices of the tariff-ridden differentiated goods. Because all goods are substitutes in this symmetric case, domestic demand shifts to local production of both goods X and Y . Second, all else equal, the demand shifts lead domestic monopolists to raise their prices (pX and pY ) and thus their profit margins (qX and qY ) expand. As a result, even targeted tariff protection for X-type products (ψ = 0) translates into across-the-board protection for domestic noncompetitive sectors as a whole, turning the combined terms of trade ( pˆX + pˆY > 0) to the advantage of the small home economy and raising the combined profit margin (qˆX + qˆY > 0) at home. Third, with all activities substitutable in production, improved profitability draws domestic resources away from the homogenous-good sector and into the two noncompetitive, differentiated-good sectors by inducing entry and/or output expansion. Thus, protection enhances the levels of domestic output and employment in noncompetitive sectors as a whole: (nˆ + m) ˆ + (xˆ + y) ˆ > 0 in terms of (19) and (20). In this sense, tariff policy counteracts the consumption distortion in each noncompetitive sector and is always welfareenhancing,13 despite the possibility of a negative combined scale-economy effect (xˆ + yˆ < 0) – which will be realized if the entry activity rather than the production of differentiated goods mainly competes for resources with the production of the homogenous good (i.e. |ε30 + ε40 | > |ε10 + ε20 |).14 Note that once resources cannot be reallocated between the competitive and noncompetitive sectors (i.e. εh0 = ε0h = 0 for h 6= 0), both nˆ + mˆ and xˆ + yˆ vanish into zero.15 Then the aforementioned welfare enhancement will stem exclusively from the combined terms-of-trade effect – i.e. tariff protection, either targeted or across-the-board, is merely a beggar-thy-neighbor policy. In what would follow we turn to the Ricardian case in which two regimes will be considered, respectively.
4.3
The Ricardian case: free entry
In the Ricardian case there is one single factor of production. We firstly consider a regime where firms can freely enter the X (Y ) industry as assumed in the symmetric case. In this regime arbitrage in the numeraire 13 See
Appendix C for proof. result is similar to that of Flam and Helpman (1987) where only one monopolistically competitive sector competes with one competitive sector for resources. Intuitively, the scale-economy effect is more likely to be positive because the production factors allocated to develop new products (i.e. the entry activity) should be quite dissimilar to those engaging in direct production activities. 15 In terms of (20), it is straightforward that xˆ + yˆ ≈ 0 if ε = 0 for h 6= 0. Next, let us consider (19). Due to h0 linear homogeneity of r[.; v], it holds that ε10 + ε11 + ε12 + ε13 + ε14 = 0. Thus, as ε10 ≈ 0, it follows that ε11 + ε12 ≈ −(ε13 + ε14 ), ensuring nˆ + mˆ ≈ 0. 14 This
14
homogeneous good makes the vector of factor rewards W collapse into a parameter in terms of pH = 1 = W ], thereby locking in pX and pY , due to markup pricing conditions (5) as well as qX and qY , due CH [W to zero profits conditions (6). These results in turn tie both x and y to their steady-state levels due to V ] is conditions (7). The economy’s gross domestic product r[1, pX (σX − 1)/σX , pY (σY − 1)/σY , qX , qY ;V therefore predetermined by its fixed factor endowments so that (10a’), (10b’), (11a’) and (11b’) cannot apply. In this Ricardian system with free entry, only n, m, and U are subject to differential change in tariffs and can be obtained by solving conditions (12), (13a) and (13b). Now taking total differentials of (13a) and (13b) yields (see Appendix D) nˆ = (σX − 1)(δ12 + ψδ14 )(
dt ) p∗
(13a")
mˆ = (σY − 1)(δ32 + ψδ34 )(
dt ). p∗
(13b")
Using these results and setting pˆX = pˆY = xˆ = yˆ = 0 in (15a) and (15b), (14) reduces to e˜U dU = GDP
SX θX SY θY nˆ + mˆ σX − 1 σY − 1 (14")
dt = GDP(SX θX [δ12 + ψδ14 ] + SY θY [δ32 + ψδ34 ])( ∗ ) p This implies that tariff protection influences national welfare only through its impact on entry activities. According to Lemma 2, each one of δ12 , δ14 , δ32 , and δ34 can be positive or negative in general. From (13a") and (13b"), it holds that nˆ > 0 and mˆ > 0 if δi j > 0 for i 6= j as in the symmetric case, but the reverse is true if δi j < 0, regardless of whether ψ = 0 or ψ > 0. Hence combining these results and (14"), we get the following clear-cut results: (i) if differentiated goods (X, X ∗ , Y , Y ∗ ) are all substitutes, both targeted and across-the-board tariff protection remain welfare-enhancing in the Ricardian case with free entry, as in the symmetric case; (ii) if these goods are all complements to one another, however, either type of protection is welfare reducing. For the latter, protection redirects resources from the production of goods X and Y to the production of good H, thereby aggravating consumption distortions. Demand interdependence among noncompetitive markets, however, is often more complicated than the analysis above. To explain, let us consider a plausible structure of demand interdependence: a) δ12 > 0 (imported X-type products are a substitute for domestic X-type products); b) δ32 < 0 (imported X-type products are a complement to domestic Y -type products);
15
c) δ14 < 0 (imported Y -type products are a complement to domestic X-type products); d) δ34 > 0 (imported Y -type products are a substitute for domestic Y -type products). When tariff protection is targeted to domestic X-type products (i.e. ψ = 0), only the demand linkages (a) and (b) matter to welfare analysis in terms of (14"). There,the targeted protection increases domestic demand for good X produced at home (due to δ12 > 0) and thus induces entry of firms into the protected import-substituting sector (X) according to (13a"); but at the same time it decreases domestic demand for good Y produced at home (due to δ32 < 0) and thus induces exit from the unprotected import-complementing sector. As a result, with nˆ > 0 and mˆ < 0, the welfare effect is generally ambiguous without looking into structural variables (SX , θX ; SY , θY ). For targeted tariff protection for good X to be welfare improving, therefore, the unprotected import-complementing sector Y must be less ’structurally significant’ than the protected import-substituting sector (X) with respect to the relative output-value share SY /SX and the relative domestic absorption share θY /θX . Using (14"), this conditional desirability of targeted tariff protection is shown as follows: dU > δ12 > − 0 iff − dt < |δ32 |
0, δ32 < 0.
(21)
Can across-the-board tariff protection (ψ > 0) be more desirable and feasibile than targeted tariff protection (ψ = 0) under the same demand-interdependence structure ((a)-(d))? The answer is negative. With the across-the-board protection in place, even entry activities become much less predictable than those under targeted protection. For example, n may increase or decrease due to (13a"), depending on whether δ12 is greater or less than ψ|δ14 |. Similarly, m may increase or decrease due to (13b"), depending on whether ψδ34 is greater or less than |δ32 |. Further, adjusting the relative tariff rate (via ψ) between goods X and Y cannot reduce this uncertainty. If ψ is raised, for example, it reinforces the substitution effect (δ34 ) in favor of domestic Y -type products, while aggravating the complementary effect (δ14 ) against domestic X-type products. Thus to ensure welfare enhancement, across-the-board tariff protection is much harder to enforce than is targeted tariff protection, since the former requires much more informational details of both demand interdependence among noncompetitive markets and economic structure.
16
4.4
The Ricardian case: no entry
We now analyze the welfare impact of tariff protection combined with the so-called "structural policy" that inhibits entry of firms. We assume that initially under free trade all existing firms break even. In this regime with the policy of entry inhibition (i.e. nˆ = mˆ = 0) in place, however, existing firms may make profits (losses) by increasing (decreasing) output once import tariffs are imposed, as opposed to the freeentry regime. Zero-profit conditions (7a) and (7b) therefore do not apply to this no-entry regime, where an individual firm’s output is purely demand-determined: x = e1 [.]/n − dX∗ and y = e˜3 [.]/m − dY∗ . Since profits (losses) are rebated to (paid by) consumers in lump sum, the representative consumer’s budget constraint (12) is modified as e[.] ˜ = GDP + t e˜2 [.] + ψt e˜4 [.] + n(qX −Cn ) + m(qY −Cm ) where n(qX − Cn ) and m(qY − CM ) are industry-wide profits (if positive) accruing respectively to goods X and Y . As defined earlier, q depends on an individual firm’s output according to qX = x(pX − CX ) and W ] = 1 exogenizes all variables but U, qY = y(pY −CY ). In this Ricardian case the normalization of pY = CY [W x (or qX ) and y (or qY ). Now totally differentiating the modified budget constraint, (13a), and (13b) yields, respectively, xˆ = θX (δ12 + ψδ14 )(
dt ) p∗
yˆ = θY (δ32 + ψδ34 )(
dt ) p∗
e˜U dU = nxˆ + my. ˆ Comparing these results with (13a"), (13b") and (14"), we note that any given structure of demand interdependence plays the same role both in the free-entry and no-entry regimes. Hence, if the effects of protection on n and m are replaced by those on x and y, the preceding analysis of the free-entry regime remains applicable to this no-entry regime. Under the same structure of demand interdependence ((a)-(d)), the imposition of tariffs (ψ = 0) targeted to X-type products leads individual firms to increase output (xˆ > 0) and make profits in the protected import-substituting sector (X), while forcing individual firms to decrease output (yˆ < 0) and make losses in the unprotected import-complementing sector (Y ). As a result, targeted tariff protection may increase or decrease national welfare as in the free-entry regime. But as analyzed in
17
the free-entry regime, the welfare effects become much more uncertain when ψ is positive (i.e. the economy enforces across-the-board tariff protection). Indeed, the argument of profit shifting either under oligopoly (Brander and Spencer 1985 and Eaton and Grossman 1986) or under monopolistic competition (Jones and Takemori 1989) applies to the present work. When goods from noncompetitive sectors are all substitutes, both targeted and across-the-board protection can extract profits from foreign to home firms, thereby benefiting the domestic economy. But this work shows that when the tariff-ridden goods are both a substitute for goods from a domestic noncompetitive sector and a complement to goods from another noncompetitive sector, profit shifting takes place both within and across national boundaries, and thereby national welfare may rise or fall.
5
Concluding Remarks
This paper has shown that the welfare effects of tariff policy, either targeted or across-the-board, are sensitive to alternative patterns of demand interdependence among several goods from monopolistically competitive sectors. It highlights the importance of investigating whether there exist domestic goods (from noncompetitive sectors) complementary to the imports on which domestic tariffs are to be imposed. To ensure welfare enhancement, this paper also suggests the necessity of assessing the size and the domestic absorption share of the import-complementing sector relative to the import-substituting sector. If the two structural indicators are sufficiently significant, tariff policy tends to reduce welfare, and vise versa. Hence this work supplements Flam and Helpman (1987), Venables (1982, 1987) and others where tariff policy for differentiated goods are always welfare enhancing. As to the feasibility of targeted tariff protection relative to across-the-board protection, the present work has shown that it is much harder for a small open economy to enforce the across-the-board protection, which requires much more informational details of both demand linkages among noncompetitive markets and economic structure. This work has also established that unless goods are all substitutes, profits can also be shifted domestically from an import-complementing sector to an import-substituting sector under monopolistic competition, provided tariff policy is combined with a policy of inhibiting entry of firms. In short, profit shifting can occur both across and within national bounadries under a general structure of demand interdependence among noncompetitive markets. Yet the role of demand interdependence has been
18
negelected by recent oligopolistic models that emphasize the use of strategic trade policy to extract foreign rents.
Acknowledegments The author is grateful to Earl L. Grinols and an anonymous referee for valuable comments. This work was supported in part by funds provided by the University of North Carolina at Charlotte.
References Bagwell, K. (1991). Optimal export policy for a new-product monopoly. American Economic Review 81, 1156–1169. Brander, J. A. and B. J. Spencer (1985). Export subsidies and international market share rivalry. Journal of International Economics 18. Dixit, A. K. and G. M. Grossman (1986). Targeted export promotion with several oligopolistic industries. Journal of International Economics 21, 233–249. Dixit, A. K. and J. E. Stiglitz (1977). Monopolistic competition and optimal product diversity. American Economic Review 67, 297–308. Eaton, J. and G. M. Grossman (1986). Optimal trade and industrial policy under oligopoly. Quarterly Journal of Economics 101, 383–406. Feenstra, R. C. (1986). Trade policy with several goods and ’market linkages’. Journal of International Economics 20, 249–267. Flam, H. and E. Helpman (1987). Industrial policy under monopolistic competition. Journal of International Economics 79-102., 79–102. Gros, D. (1987). A note on the optimal tariff, retaliation and the welfare loss from tariff wars in a framework with intra-industry trade. Journal of International Economics 23, 357–367.
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Helpman, E. (1981). International trade in the presence of product differentiation, economies of scale and monopolistic competition: A chamberlin-heckscher-ohlin approach. Journal of International Economics 11, 305–340. Helpman, E. (1984). Increasing returns, imperfect markets, and trade theory. In R. W. Jones and P. B. Kenen (Eds.), Handbook of International Economics. Amsterdam: North-Holland. Helpman, E. and P. R. Krugman (1985). Market structure and foreign trade: Increasing returns, imperfect competition, and the international economy. MIT press. Itoch, M. and K. Kiyono (1987). Welfare-enhancing export subsidies. Journal of Politcal Economy 95, 115–137. Jones, R. W. and S. Takemori (1989). Foreign monopoly and optimal tariffs for the small open economy. European Economic Review 33, 1691–1707. Lancaster, K. J. (1991). The ’product variety’ case for protection. Journal of International Economics 31, 1–26. Spence, A. M. (1976). Product selection, fixed costs, and monopolistic competition. Review of Economic Studies 43, 217–235. Varian, H. R. (1984). Microeconomic Analysis. New York: W.W. Norton & Company. Venables, A. J. (1982). Optimal tariffs for trade in monopolistically competitive commodities. Journal of International Economics 12, 225–241. Venables, A. J. (1987). Trade and trade policy with differentiated products: A chamberlinian- ricardian mode. Economic Journal 97, 700–717.
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Appendix A
Proof for Lemma 1:
By definition, e[1, ˜ pX , pX ∗ , pY , pY ∗ ;U, n, n∗ , m, m∗ ] = min[DH + pX ndX + pX ∗ n∗ dX ∗ + pY mdY + pY ∗ m∗ dY ∗ : U[DH , DX , DY ] > U] where DX and DY are given by (2a) and (2b). The Lagrangian function is: L [.] = DH + pX ndX + pX ∗ n∗ dX ∗ + pY mdY + pY ∗ m∗ dY ∗ − λ (U[DH , DX , DY ] −U) The first order condition, ∂ L [.]/∂ dX = 0, can be written as: 1 σX −1 σX −1 σ −1 X − σ1 ∂U[.] ∂U[.] ∂ DX σX ∗ σX npX = λ · ndX X · =λ · ndX + n dX ∗ ∂ DX ∂ dX ∂ DX
so,
1 σX −1 σX −1 σ −1 X − σ1 ∂U[.] σX ∗ σX · ndX + n dX ∗ pX = λ · dX X ∂ DX
By the Envelope theorem and the first order condition above, e˜n =
∂L ∂n
1 σX −1 σX −1 σ −1 X − σ1 ∂U[.] σX σX ∗ σX X · dX · = pX dX − λ · ndX + n dX ∗ dX ∂ DX σX − 1 σX −1 e˜1 pX = pX dX − pX dX = pX dX =− σX − 1 σX − 1 (σX − 1)n
where e˜1 = ndX . In the same manner, one can easily derive e˜n∗ , e˜m , e˜m∗ .
21
B
Proof for Lemma 2:
First, based on the Slutsky equations (Varian 1984), it holds that ∂ ndX ∂ e˜1 [.] ∂ ndX = + ndX ∂ pX ∂ pX ∂E
(B.1a)
e˜12 ≡
∂ e˜1 [.] ∂ ndX ∂ ndX = + n∗ dX ∗ ∂ pX ∗ ∂ pX ∗ ∂E
(B.1b)
e˜13 ≡
∂ e˜1 [.] ∂ ndX ∂ ndX = + mdY ∂ pY ∂ pY ∂E
(B.1c)
e˜11 ≡
where E = e[1, ˜ pX , pX ∗ + t, pY , pY ∗ + ψt;U, n, n∗ , m, m∗ ] is household’s aggregate spending (see eq.(3’)): E = DH + pX ndX + (pX ∗ + t)n∗ dX ∗ + pY mdY + (pY ∗ + ψt)m∗ dY ∗
(B.2)
with X p−σ dX = X EX , PX
dY =
pY−σY EY , PY
(pX ∗ + t)−σX EX PX
(B.3a)
(pY ∗ + ψt)−σY EY PY
(B.3b)
dX ∗ = dY ∗ =
In equations (B.3), PX and PX are domestic price indexs of goods X and Y, respectively: X PX = np1−σ + n∗ (p∗X + t)1−σX X
(B.4a)
PY = mpY1−σY + m∗ (pY∗ + ψt)1−σY
(B.4b)
Next, we define the following spending shares:
βX = pX ndX /EX
(B.5a)
βY = pY mdY /EY
(B.5b)
where EX = pX ndX + (pX ∗ + t)n∗ dX ∗ is domestic spending on good X and EY = pY mdY + (pY ∗ + ψt)m∗ dY ∗ is domestic spending on good Y .
22
Using equations (B.1), (B.2), (B.3), (B.4) and (B.5), we can derive the following demand elasticities:
Derivation of δ11 ∂ ndX e˜11 pX pX ∂ ndX δ11 ≡ = + ndX e˜1 e˜1 ∂ pX ∂E X X ∂ np−σ ∂ np−σ pX X X ( EX ) + ndX ( EX ) = ndX ∂ pX PX ∂ E PX " # −σX −σX ∗d ∗ X −1 ∗ np−σ np np pX ∂ D ∂ p md (p + ψt)m H Y Y Y Y = −σX X EX − X 2 (1 − σX )npX−σX EX − X ( + + ) ndX PX PX ∂ pX pX pX PX np−σX pX ∂ DH ∂ pY mdY ∂ (pY ∗ + ψt)m∗ dY ∗ ndX X (1 − − − ) ndX PX ∂E ∂E ∂E pX ndX · ndX pX ndX ∂ DH ∂ pY mdY ∂ (pY ∗ + ψt)m∗ dY ∗ ndX = −σX − (1 − σX ) − ( + + ) ndX pX EX ndX EX ∂ pX ∂ pX ∂ pX pX ndX ∂ DH ∂ pY mdY ∂ (pY ∗ + ψt)m∗ dY ∗ + ndX (1 − − − ) ndX EX ∂E ∂E ∂E βX ∂ DH ∂ pY mdY ∂ (pY ∗ + ψt)m∗ dY ∗ = [−σX − βX (1 − σX )] − + + ) ndX ∂ pX ∂ pX ∂ pX ∂ DH ∂ pY mdY ∂ (pY ∗ + ψt)m∗ dY ∗ + βX 1 − − − ∂E ∂E ∂E ∂ DH ∂ DH ∂ mdY ∂ mdY βX ( + ndX ) + pY ( + ndX ) = −σX − βX (1 − σX ) + βX − ndX ∂ pX ∂E ∂ pX ∂E βX ∂ m∗ dY ∗ ∂ m∗ dY ∗ − (pY ∗ + ψt)( + ndX ) ndX ∂ pX ∂E e˜10 e˜13 pY e˜14 (pY ∗ + ψt) + + = −(1 − βX )σX − βX e˜1 e˜1 e˜1 +
= −(1 − βX )σX − βX (δ10 + δ13 + δ14 )
23
Derivation of δ12 pX ∗ + t ∂ ndX e˜12 (p∗X + t) ∗ ∗ ∂ ndX = + n dX δ12 ≡ e˜1 e˜1 ∂ pX ∗ ∂E −σX X ∂ np−σ pX ∗ + t ∗ ∗ ∂ npX X = ( EX ) + n dX EX ) ( ndX ∂ pX ∗ PX ∂ E PX X np−σ npX−σX ∂ DH ∂ pY mdY ∂ (pY ∗ + ψt)m∗ dY ∗ pX ∗ + t ∗ −σ X X = ( + + ) − (1 − σX )n (pX ∗ + t) EX − ndX PX ∂ pX ∗ ∂ pX ∗ ∂ pX ∗ PX2 X np−σ pX ∗ + t ∗ ∂ DH ∂ pY mdY ∂ (pY ∗ + ψt)m∗ dY ∗ X ∗ + n dX (1 − − − ) ndX PX ∂E ∂E ∂E pX ∗ + t ndX · n∗ dX ∗ pX ∗ + t ndX ∂ DH ∂ pY mdY ∂ (pY ∗ + ψt)m∗ dY ∗ ( + + ) = −(1 − σX ) − ndX EX ndX EX ∂ pX ∗ ∂ pX ∗ ∂ pX ∗ pX ∗ + t ndX ∗ ∂ DH ∂ pY mdY ∂ (pY ∗ + ψt)m∗ dY ∗ + ndX (1 − − − ) ndX EX ∂E ∂E ∂E pX ∗ + t ∂ DH ∂ pY mdY ∂ (pY ∗ + ψt)m∗ dY ∗ + + = −(1 − βX )(1 − σX ) − EX ∂ pX ∗ ∂ pX ∗ ∂ pX ∗ ∂ DH ∂ pY mdY ∂ (pY ∗ + ψt)m∗ dY ∗ + (1 − βX ) 1 − − − ∂E ∂E ∂E 1 − βX ∂ DH ∗ ∗ ∂ DH = −(1 − βX )(1 − σX ) + (1 − βX ) − ∗ ( + n dX ) n dX ∗ ∂ pX ∗ ∂E 1 − βX ∂ mdY ∂ m∗ dY ∗ ∂ m∗ dY ∗ ∗ ∗ ∂ mdY ∗ − ∗ pY ( + n dX ) + (pY ∗ + ψt)( + n dX ∗ ) n dX ∗ ∂ pX ∗ ∂E ∂ pX ∗ ∂E e˜20 e˜23 pY e˜24 (pY ∗ + ψt) + + = (1 − βX )σX − (1 − βX ) e˜2 e˜2 e˜2 = (1 − βX )σX − (1 − βX )(δ20 + δ23 + δ24 )
24
Derivation of δ13 pY ∂ ndX ∂ ndX e˜13 pY = + mdY δ13 ≡ e˜1 e˜1 ∂ pY ∂E X X ∂ np−σ ∂ np−σ pY X X = ( EX ) + mdY EX ) ( ndX ∂ pY PX ∂ E PX ∂ mdY pY ndX ∂ DH ∂ (pY ∗ + ψt)m∗ dY ∗ = − ( + pY + mdY + ) ndX EX ∂ pY ∂ pY ∂ pY ∂ DH ∂ pY mdY ∂ (pY ∗ + ψt)m∗ dY ∗ pY ndX (1 − − − )mdY + ndX EX ∂E ∂E ∂E ∂ mdY pY ∂ DH ∂ (pY ∗ + ψt)m∗ dY ∗ =− + pY + mdY + EX ∂ pY ∂ pY ∂ pY pY ∂ DH ∂ pY mdY ∂ (pY ∗ + ψt)m∗ dY ∗ + (1 − − − )mdY EX ∂E ∂E ∂E ∂ DH ∂ mdY ∂ mdY ∂ m∗ dY ∗ ∂ m∗ dY ∗ pY ∂ DH ( + mdY ) + pY ( + mdY ) + (pY ∗ + ψt)( + mdY ) =− EX ∂ pY ∂E ∂ pY ∂E ∂ pY ∂E pY ∂ DH ∂ mdY ∂ m∗ dY ∗ ∂ DH ∂ mdY ∂ m∗ dY ∗ =− ( ) + pY ( ) + (pY ∗ + ψt)( ) + mdY + mdY + ndY EX ∂ pY ∂E ∂ pY ∂E ∂ pY ∂E e˜3 pY e˜30 e˜33 pY e˜34 (pY ∗ + ψt) + + =− EX e˜3 e˜3 e˜3 =−
pY mdY (δ30 + δ33 + δ34 ) EX
=−
EY /E pY mdY · (δ30 + δ33 + δ34 ) EX /E EY
=−
αY βX (δ30 + δ33 + δ34 ) αX
Using the same algebraic exercise as used above, one can obtain all other δi j ’s. All these results can be summarized in the following Slutsky matrix:
25
Slutsky matrix
δ11 δ12 δ13 δ14
δ21 δ22 δ23 δ24 [δi j ] = δ31 δ32 δ33 δ34 δ41 δ42 δ43 δ44
≈
−βY ( ααYX )h3
δ14
−βX σX − (1 − βX )h2
−βY ( ααYX )h3
−(1 − βX )( ααYX )h5
−(1 − βY )σY − βY h6
−(1 − βX )( ααYX )h5
βY σY − βY h6
δ24 δ34 δ44
−(1 − βX )σX − βX h1 (1 − βX )σX − (1 − βX )h2
−βX (1 − σX ) − βX h1 = −βX ( ααYX )h4 −βX ( ααYX )h4
−σX
σX − (δ20 + δ23 + δ24 )
0
0
−(δ20 + δ23 + δ24 )
0
δ14
0
−( ααYX )(δ20 + δ21 + δ22 ) −σY
0
−( ααYX )(δ20 + δ21 + δ22 )
0
δ24 δ34 δ44
under free trade,
(B.6)
where h1 = δ10 + δ13 + δ14 ,
h2 = δ20 + δ23 + δ24 ,
h3 = δ30 + δ33 + δ34 ,
h4 = δ10 + δ11 + δ12 ,
h5 = δ20 + δ21 + δ22 ,
h6 = δ30 + δ31 + δ32 ,
βX ≡
pX ndX pX ndX = , EX pX ndX + (pX ∗ + t)n∗ dX ∗
βY ≡
pY mdY pY mdY = . EY pY mdY + (pY ∗ + ψt)m∗ dY ∗
Note that initially under free trade (t = 0), the model features the symmetric equilibrium of pX = pX ∗ and pY = pY ∗ coupled with dX = dX ∗ and dY = dY ∗ so that
βX =
n , n + n∗
βY =
m m + m∗
Thus under free trade (B.6) is obtained (see Lemma 2) due to βX = βY ≈ 0 under the small-economy assumption.
26
C
Proof for dU/dt > 0 in terms of (14"), (17), (19) and (20)
Without loss of generality, let us first ignore the multiplier factors of (14"), (17), (19) and (20), e˜U , GDP, S, 1/4, and θ (δ12 + δ32 )(dt/pX∗ ), for the sake of convenience. Then by progressively eliminating some positive terms in the large brackets of (17), (19) and (20), we can obtain the following results: dU ε33 + ε34 + ε43 + ε44 1 ε30 + ε40 θ ε30 + ε40 > (1 − θ ) + + − dt 2 σ 2 σ −1 2 ε30 + ε31 + ε32 + ε40 + ε41 + ε42 1 ε30 + ε40 θ ε30 + ε40 = (1 − θ ) − + + − 2 σ 2 σ −1 2 ε30 + ε40 1 ε30 + ε40 θ ε30 + ε40 > (1 − θ ) − + + − 2 σ 2 σ −1 2 1−θ ε30 + ε40 1 ε30 + ε40 θ ε30 + ε40 > − + + − =0 σ 2 σ 2 σ 2
D
Derivation of (13a") and (13b") for the Reicrdian case
As noted in the text, in the Ricardian case, the normalization of pH to one exogenizes all variables but n, m and U. This result thus makes di∗ (i = X,Y ) predetermined under the small economy assumption. Noting these results, we can derive the differential forms of (13a) and (13b) as follows:
e˜12 dt + ψ e˜14 dt + e˜1n dn + e˜1m dm = (x − dX∗ )dn
(D.1)
e˜32 dt + ψ e˜34 dt + e˜3n dn + e˜3m dm = (y − dY∗ )dm
(D.2)
where x − dX∗ = dX = e˜1 [.]/n and y − dY∗ = dY = e˜3 [.]/m, while e˜in and e˜im , i=1,2, due to Lemma 1, are given by ∂ e˜n ∂ e˜1 [.]pX = − ∂ pX ∂ pX (σX − 1)n ∂ (x − dX∗ )pX x − dX∗ = − =− , e˜1 [.] = n(x − dX∗ ) ∂ pX (σX − 1) σX − 1 ∂ e˜m ∂ e˜3 [.]pY e˜1m = = − ∂ pX ∂ pX (σY − 1)m ∂ (y − dY∗ )pY = − = 0, e˜3 [.] = m(y − dY∗ ) ∂ pX (σY − 1)
e˜1n =
27
(D.3)
(D.4)
∂ e˜1 [.]pX ∂ e˜n = − ∂ pY ∂ pY (σX − 1)n ∂ (x − dX∗ )pX = − = 0, e˜1 [.] = n(x − dX∗ ) ∂ pY (σX − 1) ∂ e˜3 [.]pY ∂ e˜m = − e˜3m = ∂ pY ∂ pY (σY − 1)m ∂ (y − dY∗ )pY y − dY∗ = − =− , e˜3 [.] = m(y − dY∗ ) ∂ pY (σY − 1) σY − 1 e˜3n =
(D.5)
(D.6)
Substituting (D.3)-(D.6) into (D.1) and (D.2), using the relations e˜1 [.] = n(x − dX∗ ) and e˜3 [.] = m(y − dY∗ ), and rewriting the resulting equations in elasticity form, we can obtain
δ12 (
dt dt 1 ) + ψδ14 ( ∗ ) = ( )nˆ ∗ pX pY σX − 1
(D.7)
δ32 (
dt 1 dt ) + ψδ34 ( ∗ ) = ( )mˆ ∗ pX pY σY − 1
(D.8)
Rearranging (D.7) and (D.8) and letting p∗X = pY∗ = p∗ initially under free trade, both (13a") and (13b") obtain.
28
