本國市場效果、比較利益以及生產力與加價分配

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科技部補助專題研究計畫成果報告

期末報告

本國市場效果、比較利益以及生產力與加價分配

計畫類別：個別型計畫

計畫編號： MOST

103-2410-H-004-011-執行期間： 103年08月01日至104年10月31日

執行單位：國立政治大學國際經營與貿易學系

計畫主持人：徐則謙

處理方式：

1.公開資訊：本計畫可公開查詢

2.「本研究」是否已有嚴重損及公共利益之發現：否

3.「本報告」是否建議提供政府單位施政參考：否

中　華　民　國　105　年　01　月　31　日

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中文摘要：過去認為，透過低生產力生產者的退出，貿易的自由化能提高產業

的生產力。然而最近的研究認為，如果工資固定不變，大國較無比

較利益的產業，或是技術相對落後的國家的產業，生產力反而因貿

易而下降。本文主張上述結果忽略工資變動的效果。將工資內生化

後，較小的國家的產業、有比較利益的產業，或是技術較進步國家

的產業，其生產力在貿易後可能會下降。此外，貿易自由化不一定

會增加產業的競爭程度，故加價有可能在貿易後上升。最後，我們

的數值例子顯示福利仍然隨貿易自由化上升。

中文關鍵詞：異質生產者、國際貿易、內生加價、產業生產力、李嘉圖模型、比

較利益

英文摘要： The conventional wisdom in modern international trade

theory is that trade liberalization improves sectoral

productivity by forcing low-productivity firms to exit.

However, recent research considering an invariable wage

shows that productivity decreases in sectors of a large

country where it has weak comparative advantage, or in

sectors of a technologically backward country. By

endogenizing wage, we show that sectoral productivity may

be reduced in small countries, in sectors with comparative

advantage, or in advanced countries. Furthermore, trade

liberalization does not necessarily reduce an endogenously

determined markup. Nevertheless, our numerical examples

demonstrate that welfare does increase with trade

liberalization.

英文關鍵詞： Heterogeneous Firms, International Trade, Endogenous

Markup, Sectoral Productivity, Ricardian Model, Comparative

Advantage

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1 Introduction

One important development in modern international trade theory involves analyses of the ways in which trade liberalization a↵ects sectoral productiv-ity by weeding out low-productivproductiv-ity firms. In a single-sector model, Melitz (2003) and Melitz and Ottaviano (2008) show that trade has a positive im-pact on industrial productivity through factor reallocation within an indus-try. The theory has been extended to consider whether the result is robust across sectors with di↵erent degrees of comparative advantage. The impact can di↵er in magnitude between sectors (Bernard et al., 2007). When two countries are di↵erent in terms of technological potential (Demidova, 2008), or in a Ricardian model with asymmetric country size (Fan et al., 2013), it is possible that the average sectoral productivity decreases with trade liberalization when there are no terms of trade consideration.

We show that the terms of trade are important. In a full-fledged gen-eral equilibrium framework that incorporates heterogeneous firms, endoge-nous markup, and Ricardian comparative advantage, we show that trade liberalization is not always productivity improving. We demonstrate that the patterns of impact are even more complicated than previously believed, particularly when wage is allowed to vary endogenously. Specifically, we combine Melitz and Ottaviano (2008)’s mechanism with Dornbusch et al. (1977). We adapt the translog utility system introduced by Feenstra (2003). The function is homothetic, and it implies a price elasticity that is monoton-ically increasing with the measure of entrants, decreasing with the average price in the industry, and decreasing with the productivity of producers. When the economy is more competitive, its elasticity increases, the markup decreases, the lowest survival productivity increases, and therefore average

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of Fan et al. (2013) and Demidova (2008), but with two distinguishing fea-tures: the wage and markup are variable. More interestingly, the patterns of impact we derive are not identical to those found in their research.

To highlight the importance of endogenizing wage, we demonstrate that once wage is fixed, our model generates results similar to those of Fan et al. (2013). Trade liberalization may reduce the average productivity in sec-tors where the large country produces with a comparative disadvantage. The large country produces goods in those sectors even though it does not have a comparative advantage because of the home market e↵ect. How-ever, with an endogenously varying relative wage, the negative impact on average productivity in those sectors in the large country can be exacer-bated, eliminated, or even reversed to be positive. It is also possible that the sectors in the small country experience a reduction in their average pro-ductivity. Furthermore, because the relative wage may change nonmonoton-ically, the average productivity levels in the sectors where only one country has production activities, regardless of that country’s size, may also change nonmonotonically. Our results are di↵erent from those of Demidova (2008) who shows that trade liberalization increases productivity when countries are superior in technology, while the technological laggard countries su↵er from productivity decreases. In our model, whether a sector will experience productivity improvement depends on relative rather than absolute technol-ogy di↵erences. We also show that the markup increases when the average sectoral productivity decreases. Therefore, the markup does not necessar-ily decrease when trade is more liberalized. Although in some sectors it is possible that the markup increases and sectoral productivity decreases with trade liberalization, trade liberalization is always welfare enhancing for both countries in our numerical examples.

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In our model, trade liberalization changes sectoral productivity through two channels. First, trade liberalization reduces the price of foreign pro-ducers. Therefore, it decreases the markup of domestic firms and increases their productivity needed to survive. Second, trade liberalization also causes reallocation of labor across sectors. Whether the second channel increases productivity depends on how labor is reallocated. If more labor is allocated to a sector, the measure of entrants in that sector increases and therefore the competition level increases. In this case, both channels contribute to the decrease in the markup and the increase in sectoral productivity. If labor leaves the sector, the e↵ect of the second channel is negative. In this case, the markup increases and the sectoral productivity is reduced if the e↵ect of the second channel is dominant. Because the change in the relative wage also a↵ects the allocation of labor, we cannot capture the whole picture without endogenizing the relative wage.

This paper is also related to that of Bernard et al. (2007), who extend Melitz (2003)’s mechanism in the Heckscher–Ohlin model. They show that trade liberalization increases the average productivity of all sectors, but more so in the comparative advantage sectors. Okubo (2009) extends Melitz (2003) to a Ricardian model, and focuses on the changes in population and its e↵ects on the number of varieties. Huang et al. (2013) combine Krugman (1980) with the Ricardian model. They focus on how the interaction of Ricardo’s comparative advantage and the home market e↵ect shapes the patterns of specialization. They do not consider firm heterogeneity.

In the next section, we describe the setup of the model. Section 3 de-scribes how the equilibrium is determined given the relative wage and the allocation of labor. Section 4 analyzes how the patterns of specialization

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trade liberalization on patterns of specialization, on the lowest productivity level needed to survive, and on average sectoral productivity in both partial and general equilibrium frameworks. Numerical examples are provided in Section 6. The last section concludes this paper.

2 The Model

There are two countries, home and foreign. We use the home country as an example to describe the setup of the model. The foreign country has an analogous setup. We will highlight any di↵erences. We add an asterisk to represent variables related to the foreign country.

2.1 Demand

Consumers in each country share the same utility function:

U = Z i2[0,1] i ln Qidi, i > 0 Z i2[0,1] idi = 1,

where i is the index of sectors and Qi is the subutility. The consumption of

di↵erentiated goods comprises the subutility function. Qi is a homothetic

function.

Because both the utility and the subutility functions are homothetic, we follow the conventional two-stage optimization procedure to determine the optimal consumption. The two-stage optimization confers the following indirect utility function:

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ln V = Z i2[0,1] i ln Qi( iy,{pi(!)}) di (1) !2 ⌦i,

where y is the per capita income, ! is the index of varieties, ⌦i is the set

of varieties in sector i, and pi is the price of a variety. In line with Feenstra

(2003), Arkolakis et al. (2010), and Rodriguez-Lopez (2012), we assume the subutility function is translog:

ln Qi = ln iy 1 2 Mi 1 Mi Z !2⌦i ln pi(!)d! (2) 2Mi Z Z !,!0_2⌦_i ln pi(!)⇥ln pi(!0) ln pi(!)⇤d!0d!,

where Mi is a measure of ⌦i; and , as will be shown, is related to the

elasticity of demand.

2.2 Supply

Producers use labor to produce. The labor endowment is L. Factors can freely move across sectors but cannot move between countries. In both countries, potential entrants first decide which sector to enter. They then pay the fixed cost

fi = f aiw,

where f > 0 is a constant, aiis the labor requirement, and w is the wage rate.

After paying the fixed cost, productivity is drawn from a Pareto distribution function with density

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where ' _{2 [b, 1) is the productivity level. Entrants then decide whether} to produce. If they produce, ai ' units of labor are needed to produce one

unit of a good. Otherwise, they exit. We choose an index of sectors so that the labor requirement in the home country relative to that in the foreign country increases with the sector index

a0 a⇤ 0 < . . .a1/2 a⇤ 1/2 · · · < a_a1_⇤ 1 .

In all sectors, when goods are shipped internationally, ⌧ > 1 units of a good must be shipped in order for one unit to arrive. ⌧ measures the degree of trade restriction. Trade is costless within a country.

3 Equilibrium

We first derive the equations governing the equilibrium of the home coun-try. Following the same procedure, the equations of the foreign country are analogously derived. We assume the market structure is monopolistically competitive. We choose foreign labor as the numeraire.

Each consumer in the home country spends iy on varieties produced in

sector i. By using equation (2) and Roy’s identity, the share of iy spending

on good ! in sector i can be derived as

(3) si(!) = @ ln Qi @ ln pi(!) @ ln Qi @ ln iy = ln  ˆ pi pi(!) ,

and therefore the revenue received by firm ! from sector i, denoted as ri(!),

is

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In equation (3): (5) pˆi = exp ✓ 1 Mi + 1 Mi Z !2⌦i ln pi(!)d! ◆

is the maximum price that a firm can charge because a price higher than ˆpi

generates negative market share. The first term in the parentheses strictly decreases in Mi and the second term is the average of the log price charged

by producers; therefore ˆpi is strictly decreasing in the competition level.

Firms choose a price to maximize their profit, ⇡i(!) = ri(!) c(!)qi(!),

where qi(!) = ri(!) pi(!) is the production level and c(!) is the marginal

cost. The first-order condition implies

(6) pi(c) =  1 1 "i(c) 1 c.

The marginal cost changes because of the di↵erent productivities of firms and because of the shipping cost. c = aiw ' if the good is produced by firms

at home with productivity ' and the good is sold at home. The marginal cost increases by ⌧ if the good is shipped abroad. "i(c) is the elasticity of

demand and can be written as

(7) "i(c) = 1 @ ln ri(c)

@ ln pi(c)

= 1 + si(c)

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To ensure that the elasticity is greater than one, we assume is greater than zero. The elasticity changes endogenously with the market share.

Let mi(c) = ⇥1 1 "i(c)⇤ 1 be the markup. The closed-form solution

of the markup can be derived by applying the Lambert_{W function. Using}

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equations (3), (6), and (7), given some rearrangement, yields mi(c)emi(c) = ˆ pi ce. 2

Then by applying the Lambert_{W function, we can solve}

mi(c) =W ✓ ˆ pi ce ◆ ,3

whereW(·) is the conventional symbol of the Lambert W function. There-fore (8) pi(c) =W ✓ ˆ pi ce ◆ c.

The LambertW function satisfies W0 _{> 0,}_W00_{< 0,}_{W(0) = 0, and W(e) = 1}

when it is defined on interval [0,_{1). Therefore, the markup is strictly} decreasing in c and strictly increasing in ˆpi. The latter indicates a

less-competitive market. This relationship is the key mechanism in this model. The same e↵ect is also reflected in the price elasticity. By combining equations (6) and (8), we obtain the elasticity faced by the firm as

(9) "i(c) = W⇣pˆi ce ⌘ W⇣pˆi ce ⌘ 1 .

The elasticity is strictly increasing in c and strictly decreasing in ˆpi.

Combining equations (7) and (9), we obtain the closed-form solution of

2_{Using equations (3), (7), and the definition of m}

i(c) yields mi(c) = ln ˆpi ln pi(c) + 1.

Combining this with equation (6) and the definition of mi(c) yields mi(c) = ln ˆpi ln mi

ln c + 1. By taking the exponential of both sides, we obtain the reported equation.

3_{For the equation d = ze}z_{, the Lambert}

W function solves z as a function of d; that is, z =_{W(d). See Corless et al. (1996) for an overview of this function.}

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si(c) as (10) si(c) =  W ✓ ˆ pi ce ◆ 1

Given ⇡i(c) = ri(c) cri(c) pi(c) and equations (4), (8), and (10), the gross

profits that firms obtain from the domestic market can be calculated as

(11) ⇡i(c) = iwL h W⇣pˆi ce ⌘ 1i2 W⇣pˆi ce ⌘ .

Firms produce for the market if and only if ⇡i 0. Because ⇡i is

monotonically increasing in W and W is higher when the marginal cost is lower, ⇡i is monotonically decreasing in c. Combining it with W(e) = 1,

from equation (11), we can observe that firms produce for the market if and only if c_{ ˆp}i. It is intuitive because the marginal cost must be at most equal

to the highest price a producer can charge to maintain a nonnegative gross profit. This implies that the productivity cuto↵ needed to sell domestically is (12) 'ˆid = aiw ˆ pi .

By following the same logic, we can derive the gross profit obtained abroad. It has the same functional form as that of equation (11) except that ˆpi is

replaced by ˆp⇤_i, wL is replaced by L⇤, and c = ⌧ aiw '. Therefore, the cuto↵

to sell abroad is

(13) 'ˆie =

⌧ aiw

ˆ p⇤_i .

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tivity level for the foreign firms as (14) 'ˆ⇤_id = a⇤i ˆ p⇤ i , and (15) 'ˆ⇤_ie = ⌧ a⇤i ˆ pi .

Free entry implies that the expected gross profit of a potential entrant cannot be greater than the fixed cost; that is

i  Lw Z ₁ ˆ 'id (W 1)2 W ✓b✓ '✓+1d' +L⇤ Z ₁ ˆ 'ie (_W 1)2 W ✓b✓ '✓+1d'  faiw,

where the arguments of_{W are suppressed for notational simplicity. The first} term is associated with the gross profit obtained domestically and the second term is that associated with exports. Using a simple change of variable, v = ˆpi c, and analogously for ˆp⇤i c, the above equation can be rewritten as

(16) i✓b✓1 " Lw ✓ ˆ pi aiw ◆✓ + L⇤ ✓ ˆ p⇤_i ⌧ aiw ◆✓#  faiw, where 1= Z ₁ 1 [_{W (ve)} 1]2 W (ve) v ✓ 1_dv.

By using the same logic, the free entry condition of the foreign firms can be written as (17) i✓b✓1 " L⇤ ✓ ˆ p⇤_i a⇤_i ◆✓ + Lw ✓ ˆ pi ⌧ a⇤_i ◆✓#  fa⇤i.

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Let Li 0 be the labor allocated to sector i in the home country. Then,

labor market clearing implies

Ni Z ₁ ˆ 'id iLw 1 W 1 ✓b ✓ '✓+1d' + Z ₁ ˆ 'ie iL⇤ 1 W 1 ✓b✓ '✓+1d' + f aiw = Liw, (18)

where Ni is the measure of entrants. The first two terms in the square

brackets are related to the demand of labor associated with the variable cost while the last term is that associated with the fixed cost. Using the same change of variable, v = ˆpi c, and analogous to ˆp⇤i c, equation (18) can

be rewritten as Ni ( i✓b✓2 " wL ✓ ˆ pi aiw ◆✓ +L⇤ ✓ ˆ p⇤_i ⌧ aiw ◆✓# + f aiw ) = Liw, (19) where 2 = Z ₁ 1 n 1 [_{W (ve)]} 1ov ✓ 1dv.

Similar logic can be applied to derive the labor market clearing condition of the foreign country:

Ni ( i✓b✓2 " L⇤ ✓ ˆ p⇤_i a⇤_i ◆✓ +wL ✓ ˆ pi ⌧ a⇤_i ◆✓# + f a⇤_i ) = L⇤_i. (20)

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By definition, the measure Mi of surviving entrants is given by Mi =Ni Z ₁ ˆ 'id ✓b✓ '1+✓d✓ + Ni⇤ Z ₁ ˆ '⇤ ie ✓b✓ '1+✓d✓ = b✓hNi( ˆ'id) ✓+ Ni⇤( ˆ'⇤ie) ✓ i . (21)

Similar logic is used to obtain

(22) M_i⇤ = b✓hN_i⇤( ˆ'⇤_id) ✓+ Ni( ˆ'ie) ✓

i .

Using the same change of variable technique we used repeatedly, the fact that the sum of market shares of firms selling in the home country equals one implies (23) ✓b✓3 " Ni ✓ ˆ pi aiw ◆✓ + N_i⇤ ✓ ˆ pi ⌧ a⇤ i ◆✓# = 1, where 3= Z ₁ 1 [_{W (ve)} 1] v ✓ 1dv.

For the firms selling in the foreign country, similar logic is applied to obtain

(24) ✓b✓3 " N_i⇤ ✓ ˆ p⇤_i a⇤ i ◆✓ + Ni ✓ ˆ p⇤_i ⌧ aiw ◆✓# = 1.

Given Li and L⇤i and w, equations (12) to (17) and equations (19) to (24)

can be used to solve the remaining endogenous variables. If Li is greater

than zero, the equality of (16) holds. Combining (16) and equation (19), we obtain the measure of entrants as follows:

(25) Ni = Li aif ⇣ 2 1 + 1 ⌘.

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If Li = 0, from equation (19), it is clear that Ni = 0. Ni⇤ can be derived

analogously. If L⇤_i > 0, equality of (17) holds. By using (17) and equation (20), we obtain (26) N_i⇤ = L⇤i a⇤ if ⇣ 2 1 + 1 ⌘.

If L⇤_i = 0, N_i⇤ = 0. Combining equations (12), (15), (21), and (23), we obtain

(27) Mi = ( ✓3) 1.

Similarly, combining equations (13), (14), (22), and (24), we obtain

(28) M_i⇤ = ( ✓3) 1.

Therefore, the mass of available varieties is the same everywhere.

When both Li > 0 and L⇤i > 0, using equations (12), (15), (21), (25),

(26), and (27), we have (29) 'ˆid= b 8 < : ✓3 f⇣2 1 + 1 ⌘ " Li ai +L⇤i a⇤ i ✓ aiw ⌧ a⇤ i ◆✓#9= ; 1 ✓ , and (30) 'ˆ⇤_ie= b 8 < : ✓3 f⇣2 1 + 1 ⌘ " Li ai ✓ ⌧ a⇤_i aiw ◆✓ +L⇤i a⇤ i #9₌ ; 1 ✓ .

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Similarly, using equations (13), (14), (22), (25), (26), and (28), we have (31) 'ˆ⇤_id= b 8 < : ✓3 f⇣2 1 + 1 ⌘ " L⇤_i a⇤_i + Li ai ✓ a⇤_i ⌧ aiw ◆✓#9= ; 1 ✓ , and (32) 'ˆie = b 8 < : ✓3 f⇣2 1 + 1 ⌘ " L⇤_i a⇤_i ✓ ⌧ aiw a⇤_i ◆✓ + Li ai #9₌ ; 1 ✓ .

Using equations (12) and (29), we have

ˆ pi = aiw b 8 < : ✓3 f⇣2 1 + 1 ⌘ " Li ai +L⇤i a⇤ i ✓ aiw ⌧ a⇤ i ◆✓#9= ; 1 ✓ . (33)

Using equations (14) and (31), we have

ˆ p⇤_i = a⇤i b 8 < : ✓3 f⇣2 1 + 1 ⌘ " L⇤_i a⇤_i + Li ai ✓ a⇤_i ⌧ aiw ◆✓#9= ; 1 ✓ . (34)

If Li = 0, ˆ'id, ˆ'⇤ie, ˆ'⇤id, ˆ'ie, ˆpi and ˆp⇤i can be derived by substituting Li= 0

into equations (29) to (34). Similarly, when L⇤_i = 0, those endogenous variables can be derived by substituting L⇤_i = 0 into equations (29) to (34). It should be noted that because there are no entrants in the home country when Li = 0, in this scenario ˆ'id would be the lowest productivity level

needed to maintain zero gross profit at home if there are entrants. Similarly, when L⇤_i = 0, ˆ'⇤_id would be the lowest productivity level needed to maintain zero gross profit if there are entrants.

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Finally, we define the average productivity in a sector as '_i = Z ₁ ˆ 'id 'µ (') d' = ✓ ˆ'id ✓ 1, where µ (') = ✓ ˆ' ✓ id '1+✓

is the distribution of ' conditional on successful entry. ' increases with ˆ'id.

The endogenous variables whose equilibrium level has not been deter-mined are Li, L⇤_i, and w. We analyze these in the next section.

4 Patterns of Specialization and the Equilibrium

Wage

We argue that the patterns of specialization are those described by Figure (1). Only the home country produces in sectors where ai a⇤i  A. Only

the foreign country produces in sectors where ai a⇤i A. Both countries

produce in sectors with ai a⇤_i 2 A, A .

Suppose the equality of (16) holds. Substituting equations (33) and (34) into (16), the free entry condition for the home country can be rewritten as

(35) iL Li ai + L⇤ i a⇤ i (wAi) ✓_⌧ _✓ + (⌧ Ai) ✓ iw 1 ✓L⇤ L⇤ i a⇤ i + Li ai (Ai⌧ w) ✓ = ai, where Ai = ai a⇤_i.

Suppose the equality of (17) holds. Substituting equations (33) and (34) into (17), the free entry condition for the foreign country can be rewritten

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as (36) iL ⇤ L⇤ i a⇤_i + Li ai (Ai⌧ w) ✓ + ⌧ ✓A✓_i iw1+✓L Li ai + L⇤ i a⇤_i (wAi) ✓ ⌧ ✓ = a ⇤ i.

We can obtain (Li, L⇤i) satisfying the above two equations. We use a

two-stage procedure to solve the pair. Treating the denominator as unknown, we can solve the denominator as

8 > > > > < > > > > : Li ai + L⇤ i a⇤ i (wAi) ✓_⌧ ✓₌ iL(1 ⌧ 2✓) ai[1 ⌧ ✓(wAi) 1 ✓] L⇤ i a⇤ i + Li ai (Aiw⌧ ) ✓ ₌ iL⇤(1 ⌧ 2✓) a⇤_i[1 ⌧ ✓_(wA i)1+✓]

The above equations can be used to solve

(37) Li= i L 1 ⌧ ✓_(wA i) 1 ✓ iw 1L⇤ ⌧✓_(wA i) 1 ✓ 1 , (38) L⇤_i = iL ⇤ 1 ⌧ ✓_(wA i)1+✓ iwL ⌧✓_(wA i)1+✓ 1 .

The above two equations can be used to derive the threshold determining the patterns of specialization. It can be observed that Li strictly decreases

with Ai, while L⇤i strictly increases with it. When

Ai= A = ✓ + 1 ⌧ ✓ _{+ ⌧}✓ ◆ 1 1+✓ w 1,

where = wL L⇤, L⇤_i = 0. When Ai < A, if the equality of the free entry

conditions for both countries holds, we must have L⇤_i < 0. However, L⇤_i cannot be less than zero. Therefore, it must be true that the equality of

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the free-entry condition of the foreign country does not hold and the foreign country does not produce in this sector. On the contrary, the equality of the free entry condition of the home country must hold. Because of free entry, we must have aggregate expenditure equal to the aggregate wage bill.

4 _{Therefore, we have L}

i = i

⇣

L + w 1_L⇤⌘_{. Following the same logic, we}

obtain A = ✓ ⌧ ✓_{+ ⌧}✓ + 1 ◆ 1 1+✓ w 1. When Ai A, Li = 0 and L⇤i = i ⇣

wL + L⇤⌘. To summarize, the demand for labor is represented by the following equation

Li = 8 > > > > < > > > > : i ⇣ L + w 1L⇤⌘ if Ai  A iL 1 ⌧ ✓_(wA_i₎ 1 ✓ i w 1_L⇤ ⌧✓_(wA_i₎ 1 ✓ ₁ if Ai 2 A, A 0 otherwise

for the home country. It can be shown that the values of the first and the second expressions are equalized when Ai = A. For the foreign country, the

demand for labor is represented by

L⇤_i = 8 > > > > < > > > > : 0 if Ai A iL⇤ 1 ⌧ ✓_(wA i)1+✓ iwL ⌧✓_(wA i)1+✓ 1 if Ai2 A, A i ⇣ wL + L⇤⌘ otherwise

It can be shown that the values of the second and the third expressions are equalized when Ai = A.

The patterns of specialization are determined by Ricardo’s comparative

4_{Suppose the equality of (16) holds. Multiplying both sides of (16) by N}

i, we have

aggregate gross profit equal to aggregate fixed cost. Therefore, aggregate expenditure must be distributed to labor to form the variable and the fixed costs, that is, the total

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advantage and the home market e↵ect. From equation (37), for a given w, it can be observed that Li strictly decreases in Ai. This e↵ect captures

Ricardo’s comparative advantage. Equation (37) also implies that Li L

strictly increases in . This e↵ect captures the home market e↵ect. There-fore, A increases with , so that the range of sectors in which production takes place wholly in the home country increases. Analogously, equation (38) implies that both e↵ects also determine the pattern of specialization in the foreign country. L⇤_i increases with Ai and L⇤i L

⇤

decreases with . Furthermore, the home market e↵ect is also reflected in the result that A decreases when decreases.

w is determined by the labor market clearing condition: Z i2[0,i] i 1 + 1 di + Z i2[i,i] lidi = 1, (39)

where i corresponds to A, i corresponds to A, and li = Li L. It is obvious

that both A and A go to zero and goes to infinity when w goes to infinity; hence, the hand side of equation (39) goes to zero. Conversely, the left-hand side of equation (39) must be greater than one when w goes to zero because in this scenario, A and A go to infinity and goes to zero. Taking the derivative of the left-hand side of equation (39) with respect to w, we have Z i2[0,i] i 2 L L⇤di + Z i2[i,i] @li @wdi < 0.

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equation (37) with respect to w. Therefore, w can be uniquely determined.

5 Impact of Trade Liberalization

In this section, we focus on the home country. The result of the foreign country can be obtained analogously.

5.1 A Partial Equilibrium Analysis

Similar to Fan et al. (2013), we fix the wage in this section. We first analyze the impact on patterns of specialization. Then we analyze the impact on cuto↵ productivity level. The average sectoral productivity level increases with the cuto↵ productivity level.

Taking the derivative of A and A with respect to ⌧ , we can observe that depending on , trade liberalization has di↵erent impacts on patterns of specialization. The result is summarized by Figure (2). It is caused by the interaction between the home market e↵ect and Ricardo’s comparative advantage.

When the di↵erence in size is significant, the decrease in ⌧ induces pro-ducers to leave sectors of the large country where that country has a weak comparative advantage. Therefore, if > ⌧2✓_{, the range of sectors in which}

production occurs in only the home country is reduced, and the range of sectors in which only the foreign country produces increases. By contrast, when < ⌧ 2✓, the range of sectors in which production occurs in only the home country increases, and the range of sectors in which only the foreign country produces is reduced. When the di↵erence in size is not so signif-icant, that is, when 2 ⌧ 2✓_{, ⌧}2✓ _{, the impact is similar to the ordinary}

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Labor reallocation accompanies the change of specialization patterns. It also depends on the relative size and is summarized by Figures (4) to (6). The solid line shows the allocation of labor before trade liberalization. The dotted line shows the result after trade liberalization. As will be explained latter, the dash-dot lines in Figure (4) are used to analyze the impact of trade liberalization. The relationship is not necessarily linear. We draw a linear relationship to illustrate the result clearly without a↵ecting the generality of the qualitative result. In general, because the home market e↵ect is weakened when trade costs fall, when is higher, more sectors experience a decrease in labor demand while fewer sectors experience an increase in demand.

We briefly describe how the figures are derived. Without loss of gener-ality, the figures are drawn under the assumption that i is the same for all

i. Therefore, the maximum labor demand, i

⇣

L + w 1L⇤⌘, does not vary with i. This level of demand is reached when the production occurs in only the home country. It is represented by the horizontal line in those figures. The line with negative slope shows the labor demand when both countries produce. As we mentioned, in this scenario, labor demand starts decreasing with Ai; when Ai is high enough, labor demand reaches zero.

To show how labor demand changes with ⌧ , by taking the derivative of the right-hand side of equation (37) with respect to ⌧ , it can be shown that the derivative is positive if Ai is greater than

(40) Aˆl= ⌧✓+ 21 1 + ⌧✓ 21 ! 1 1+✓ w 1,

where the subscript l means the threshold is related to the labor reallocation. The derivative is negative if Ai< ˆAl. It is zero when Ai = ˆAl.

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. It is summarized by Figure (3). If _{2 ⌧} 2✓_{, ⌧}2✓ _{, ˆ}_A

l 2 A, A . Hence,

when trade is liberalized, as is shown by Figure (5), the line is rotated clock-wise, as ˆAl is the pivot point. The rotated line intersects with the upper

bound of labor demand and the horizontal axis at A0 and A0, respectively. Those are the new thresholds used to determine the patterns of specializa-tion. In Figure (4), if we also draw the graph of equation (37) when it is defined over Ai  A or Ai  A0, we obtain the dash-dot lines. The dash-dot

line extending from the solid line represents the scenario before trade liber-alization. The other represents the scenario afterwards. As shown by Figure (3), when > ⌧2✓, ˆAl < A; therefore two dash-dot lines must intersect at

ˆ

Al. As shown by Figure (4), when trade is liberalized, the solid line and its

extended dash-dot line rotate clockwise, as ˆAl is the pivot point. The new

thresholds used to determine patterns of specialization are also obtained. Hence, for sectors with Ai2 A0, A , their labor demand decreases. Similar

logic can be applied to derive Figure (6).

Now we analyze the impact of trade liberalization on ˆ'id. The average

sectoral productivity strictly increases in it. By substituting Li and L⇤_i

obtained in the previous section into equation (29), we obtain

ˆ 'id= 8 > > > > > > < > > > > > > : ⇣ L+w 1_L⇤ ai⇠ ⌘1 ✓ if Ai< A, ⇢ L(1 ⌧ 2✓₎ ⇠ai[1 ⌧ ✓(Aiw) 1 ✓] 1 ✓ if Ai2 ⇥ A, A⇤, ⇣ wL+L⇤ a⇤ i⇠ ⌘1 ✓ _wA_i ⌧ if Ai> A. (41)

where ⇠ = f ( i✓b✓1) 1. If we substitute Ai in the second expression with

the definition of A, after some calculation it can be shown that the values of the first and second expressions are equalized. A similar calculation shows

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Ai = A.

As we have mentioned, because there are no entrants in sectors with Ai A, in this scenario ˆ'id would be the lowest productivity level needed

to maintain zero gross profit at home if there are entrants. Because the home country does not produce, in this case the change of ˆ'id should not

be viewed as an impact on productivity distribution. It is the change of ˆ

'⇤_id that reflects this impact because production takes place wholly in the foreign country .5 The change in ˆ'⇤_id can be derived when we analyze the impact on productivity distribution in the foreign country. We still analyze how ˆ'idchanges although there are no production activities because welfare

is a function of it as we show in appendix.

The relationship between ˆ'id and i before trade liberalization is shown

by the solid lines in Figures (7) to (9). The dotted lines show how the relationship is changed by the trade liberalization. i0 and i0 correspond to A0 and A0, respectively. The impact is a function of . Without loss of generality, those figures are drawn under the assumption that ai increases

when i increases while a⇤_i is the same for all i. Under these assumptions, ˆ'id

decreases with i and it starts increasing in sectors where the home country does not produce. The relationship is not necessarily linear. We draw a linear relationship to illustrate the result clearly without a↵ecting the generality of the qualitative result.

We use Figure (10) as an example to demonstrate how we derive the result of trade liberalization. The figure shows the case when 2 1, ⌧2✓ _.

The solid lines are the graph before trade liberalization. The dotted lines

5_{The change of ˆ}_'⇤

ie may be even more appropriate to reflect the impact because all

products come from the foreign country via exports. However, it can be immediately confirmed that ˆ'⇤ie= ˆ'⇤idby substituting Li= 0 into equations (30) and (31). Therefore,

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show the result of trade liberalization. Line AA is the graph of the first expression of (41). The graph also shows the value of the expression when i > i. Line BB is the graph of the second expression. The value when i < i or i > i is also shown. Line CC is the graph of the last expression. We also draw the graph of this expression when i < i. If we take the derivative of the second expression of (41) with respect to ⌧ and after some rearrangement, then we obtain @ ˆ'id @⌧ > 0 if Ai > ˆA' = ✓ ⌧✓_{+ ⌧} ✓ 2 ◆ 1 1+✓ w 1,

where the subscript ' means that the threshold is associated with produc-tivity. The derivative is negative when Ai < ˆA'. The derivative is zero

when Ai = ˆA'. Simple calculation shows that the relative position of ˆA',

A, and A on the real line depends on the value of . Figure (11) summarizes this information. When _{2 1, ⌧}2✓ , ˆA' 2 A, A . Therefore, after trade

liberalization, line CC rotates clockwise with the pivot point lying between i and i. Because the value of the last expression increases when ⌧ decreases, line CC moves up. The new thresholds, i0 and i0 determining the patterns of specialization are also obtained. The impact of trade liberalization when

> ⌧2✓ or when < 1 can be derived analogously.

As these figures illustrate, because of the home market e↵ect the impact of trade liberalization is a function of . When > 1, sectors where the large country has a weak—but not too weak—comparative advantage experience a decrease in ˆ'id. However, it is impossible to observe a decrease in ˆ'idwhen

< 1. This result is in contrast to the one-sector models in Melitz (2003) and Melitz and Ottaviano (2008) where trade liberalization always increases

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is because high-productivity firms are more capable of overcoming the fixed exporting costs not because of higher markup since elasticity is exogenous. In our model, it results from the decrease of the markup; therefore, trade liberalization does not necessarily reduce markup.

5.2 Why Productivity Cuto↵ Decreases with Trade Liberal-ization

Now we analyze why, in some sectors, trade liberalization decreases ˆ'idand

therefore it also decreases the average sectoral productivity. Taking the derivative of equation (29) with respect to ⌧ , we have

@ ˆ'id @⌧ = b✓ ✓ ✓3 f⇣2 1 + 1 ⌘ ˆ'1 ✓_id  ✓⌧ ✓ 1(Aiw)✓ L ⇤ i a⇤_i +1 ai @Li @⌧ + 1 a⇤ i ✓ Aiw ⌧ ◆✓ @L⇤_i @⌧ # .

The terms in the brackets capture all e↵ects that influence ˆ'id. This occurs

through two channels. The first term captures the anticompetitive e↵ect. The increase of ⌧ reduces the competitiveness of the foreign producers when they sell in the home country. Therefore, it reduces the competition level and increases markup. This reduces ˆ'id. The second channel comes from the

reallocation of labor across sectors. It is captured by the last two terms. As the measure of entrants in a sector strictly increases with the labor allocated to the sector, the competition level, markup, and therefore ˆ'id, change. The

impact of this channel can be positive, negative, or zero depending on how the labor is reallocated across sectors.

It is now clear why ˆ'iddoes not change in those sectors in which

produc-tion occurs in only the home country before and after trade liberalizaproduc-tion. Because there are no foreign competitors, there is no e↵ect operating through

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the first channel. When w is fixed, Li does not change and L⇤i remains zero;

thus, the impact of the second channel is also missing. On the contrary, for sectors in which only the foreign country produces before and after trade liberalization, ˆ'id increases. Because trade liberalization increases the

com-petitiveness of foreign producers and Li remains equal to zero and L⇤i does

not change because the wage is fixed, trade liberalization increases the cuto↵ productivity level mainly through the first channel.

When the e↵ects of both channels are not zero, the result is ambiguous. If the e↵ect of the second channel is negative and it dominates that of the first channel, the cuto↵ productivity level decreases when trade is more liberalized. We will use the home country as an example to show that the necessary condition for the second channel to be negative is dLi < 0. For

the foreign country, the same logic can be applied to show that the condition is dL⇤_i < 0. It helps to explain why trade liberalization tends to decrease

ˆ

'id when Ai is higher than a threshold level but lower than an upper bound

and when > 1.

Because the wage is fixed, total expenditure in sector i is unchanged. This implies that wdLi + dL⇤i = 0. Therefore, dL⇤i dLi = w. Using

this information, it can be shown that if the e↵ect of the second channel is negative, we have

(42) dLi

⇣

1 A1+✓_i w1+✓⌧ ✓⌘< 0.

Therefore, when dLi is negative, the e↵ect of the second channel is negative

if and only if

(43) Ai < w 1⌧

✓ 1+✓.

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The result is intuitive. As the decrease in Li must accompany the

pro-portional increase in L⇤_i, the home producers are replaced by the foreign producers. The reallocation decreases the competition level if and only if the efficiency of the home producers is higher than a threshold level. Fur-thermore, the home producers have to be more efficient to satisfy inequality (42) when the wage is higher or when ⌧ is lower, because both imply that foreign producers are more competitive in the home market. On the con-trary if dL > 0, the e↵ect of the second channel is negative if and only if

(44) Ai > w 1⌧

✓ 1+✓.

In this case, the reallocation decreases the competition level if and only if the labor requirement of the home producers relative to that of the foreign producers is large enough.

It can be shown that ˆAl is smaller than the right-hand side of the

in-equalities (43) and (44). When Ai< ˆAl, the inequality (44) will never hold.

Therefore, if dLi > 0 (this happens when Ai < ˆAl), it must be true that the

e↵ect of the second channel is positive. On the contrary, if dLi < 0 (this

happens when Ai > ˆAl) and inequality (43) is satisfied, the e↵ect of the

second channel is negative.

As dLi < 0 is necessary, it is clear why ˆ'id decreases when is large

enough and when Ai belongs to a certain interval, that is, when Ai is high

enough but not too high. When two countries are symmetric in size, trade liberalization tends to reduce labor demand in sectors with weak compara-tive advantage. However, when two countries are asymmetric, the reduction is more significant when is higher because the home market e↵ect is weak-ened. Conversely, it is impossible to observe a reduction in labor demand

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when is sufficiently small. Furthermore, the reduction of labor demand, if it occurs, is small when the comparative disadvantage is very strong. This is because most of the production activities of those sectors have been concen-trated in the foreign country. If the disadvantage is sufficiently strong, all production occurs in the foreign country before trade liberalization. Then the fall in trade costs has no impact on Li.

5.3 A General Equilibrium Analysis

The change of w a↵ects Li and L⇤i. It also a↵ects the competitiveness of

home relative to foreign producers. Therefore, ˆ'id must change with it.

Without endogenizing w, the results obtained in the partial equilibrium analysis might not be robust. Furthermore, because labor demand from the sectors in which production occurs in only one country changes with w, the cuto↵ productivity level of those sectors also changes. This e↵ect is not captured by the partial equilibrium analysis.

Taking the derivative of the left-hand side of equation (39) with respect to ⌧ while holding w constant, we have

Z

i2[i,i]

@li

@⌧ _wdi.

The sign of the derivative is ambiguous and depends on the distributions of A and , the size of , and the level of ⌧ . Therefore, the endogenous level of w might change nonmonotonically with ⌧ . Because of the ambiguity, we rely on numerical examples to show the impact under general equilibrium. We also analyze how the change in the cuto↵ productivity level a↵ects welfare. In the appendix, we show that after a monotonic transformation, the utility

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level of the representative consumer in the home country can be written as (45) ln eV = Z i i ✓ ln " Li ai +L⇤i a⇤ i ✓ Aiw ⌧ ◆✓# di.

The utility level of the representative consumer in the foreign country can be obtained analogously. In the next section, we provide numerical examples to show the impact of trade liberalization on cuto↵ productivity level, on patterns of specialization, and on welfare. The result is in contrast to what we obtained in the partial equilibrium analysis.

6 Numerical Examples

We provide two numerical examples. In the first example, we set L = 2_{⇥ 10}10, L⇤ = 1010, ⌧ _{2 [1.1, 2], and ✓ = 3. There are three sectors:}

1 = 0.5, 2 = 1/3, and 3 = 0.5/3. a1 = (1.1) 1, a2 = 1, and a3 = 1.1.

a⇤₁ = a⇤₂ = a⇤₃ = 1. f is chosen so that ( ✓b✓1) 1f = 1. The increment

of ⌧ is 0.01. In all figures illustrating the results, the horizontal axis is the inverse of ⌧ .

Figure (12) shows how patterns of specialization change with the degree of trade liberalization. The results are consistent with intuition. Initially, both countries produce in all sectors. When trade cost is sufficiently low, only the home country produces in the first sector, and only the foreign country produces in the third sector. The second sector is always produced by both countries in the range of ⌧ we consider. Figure (15) shows the relative wage changes nonmonotonically. Initially, it decreases with trade liberalization. It then increases and decreases again.

Figures (16) to (18) show the change of the cuto↵ productivity level ( ˆ'id) for each sector in the home country under general equilibrium. For

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the first sector, the cuto↵ productivity level increases with trade liberaliza-tion. For the second sector, there is a range over which it decreases when trade is more liberalized. The third sector experiences decreases, although not apparent, in the cuto↵ productivity level when ⌧ 1 is between 0.68 and 0.705.

However, the results are di↵erent under partial equilibrium. Figure (14) shows the results when w _{⌘ 1. Because A}2 is smaller than ˆA' it is

im-possible for the second sector to experience cuto↵ productivity reduction. Furthermore, because A < A3 < ˆA' < A when ⌧ 1 is between 0.68 and

0.705, ˆ'id must increase when ⌧ 1 falls within this interval.

Figures (19) to (21) show the change of the cuto↵ productivity level for each sector in the foreign country under general equilibrium. For the first two sectors, the cuto↵ productivity level increases with trade liberalization. For the third sector, there is a range over which the cuto↵ productivity level decreases. This happens because production in this sector is completely located in the foreign country, and because the relative wage decreases. The relatively expensive labor of the foreign country reduces demand of the varieties produced in the sector. It decreases labor allocated to the sector, which decreases the measure of entrants and hence the competition level. Therefore, the cuto↵ productivity level decreases. The result cannot be generated under partial equilibrium.

Although some sectors in both countries experience a decrease in the cuto↵ productivity level, Figure (22) shows that welfare increases under trade liberalization when the wage is endogenized. The loss from decreasing the cuto↵ productivity level is o↵set by the increase in efficiency caused by the reallocation of labor toward comparatively advantaged sectors.

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are unchanged. Figure (13) shows how patterns of specialization change with the degree of trade liberalization. Initially, both countries produce in all sectors. When ⌧ 1 _{is higher than about 0.7, only the home country}

produces in the first sector. Only the home country produces in the second sector and only the foreign country produces in the third sector when ⌧ 1 _is

higher than about 0.8. Figure (23) shows the relative wage is monotonically decreasing.

Figures (24) to (29) show the change of the cuto↵ productivity level as a function of ⌧ 1 under general equilibrium. For the home country, the cuto↵ productivity level of all sectors is nondecreasing with the degree of trade liberalization. For the foreign country, the cuto↵ productivity level of the first sector increases with the degree of trade liberalization. The cuto↵ productivity level of the second sector decreases with the degree of trade liberalization when ⌧ 1 is in the interval between 0.76 and 0.79. The cuto↵ productivity level of the third sector decreases with the degree of trade liberalization when ⌧ 1 = 0.8; otherwise it is nondecreasing. It is because when ⌧ 1= 0.8, the sector is produced by only the foreign country. However, the relative wage slightly decreases when ⌧ 1 _{increases from 0.8}

to 0.8065. Figure (30) shows that welfare increases with the degree of trade liberalization in both countries under general equilibrium.

The results are di↵erent under partial equilibrium. For instance, un-der partial equilibrium, according to Figure (14), ˆ'3d should decrease with

the degree of trade liberalization when A > A3 > ˆA'.6 However, once we

endogenize the wage, because the wage is decreasing, the amount of labor leaving the third sector is not large enough to decrease the cuto↵ productiv-ity level. Furthermore, under partial equilibrium, trade liberalization never

6_{Notice that A, A and ˆ}_A

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decreases cuto↵ productivity level of sectors in the small country. However, it is the small country instead of the large one that experiences the decrease in the cuto↵ productivity level when trade is more liberalized under general equilibrium.

7 Conclusion

We established a full-fledged general equilibrium model incorporating firm heterogeneity, endogenous markup, homothetic demand, and patterns of trade that emerge because of Ricardian comparative advantage and the home market e↵ect. Trade liberalization induces reallocation of labor within and across sectors and changes the relative competitiveness of foreign and domestic producers, which contributes to the change in the lowest productiv-ity level required to survive and therefore the average sectoral productivproductiv-ity. Contrary to one-sector models such as those of Melitz (2003) and Melitz and Ottaviano (2008), trade liberalization may reduce sectoral productivity. The competitiveness of foreign producers is enhanced by trade liberalization. However, more labor may be reallocated toward other sectors, which reduces the measure of entrants and the competition level there. When the latter e↵ect dominates the former, we observe a decrease in sectoral productivity. The problem we considered is similar to those studied by Fan et al. (2013) and Demidova (2008), but is di↵erent in results. We consider how the change in the wage a↵ects labor reallocation. This minor modification generates results di↵erent from those found in Fan et al. (2013). It is not necessarily true that only the comparative disadvantage sectors of the large country can experience reductions in average sectoral productivity. The sec-tors in the small country may also experience reductions and the reductions

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tion is completely located in only one country, no matter what the size of that country, may experience decreases or increases in sectoral productivity, depending on how wage moves. Our result is also in contrast to that of Demidova (2008) who argues that trade liberalization reduces average sec-toral productivity in technologically inferior countries. For instance, in our model, the average sectoral productivity of sectors in which production is located in only one country may decrease with trade liberalization. How-ever, whether production in a particular sector will be concentrated in only one country depends on comparative instead of absolute technological di↵er-ences. Finally, the reason for the exit of low-productivity firms is di↵erent. In Fan et al. (2013) and Demidova (2008), it is because of the asymmet-ric ability that only high-productivity firms are capable of overcoming the fixed cost to export. In our model, it is because of the markup decreasing with the level of competition. Therefore, we also found that the markup in-creases in sectors whose average sectoral productivity dein-creases with trade liberalization.

The model generates the policy implication that trade liberalization is not always productivity improving by forcing low-productivity firms to exit. Trade liberalization also does not necessarily reduce markup. Besides, in the numerical examples, we show that the qualitative impact on productivity distribution can be sensitive to the parameters of the model. Although the examples show that some sectors experience decreases in productivity and increases in markup, the welfare is accreted with the degree of trade liberalization.

This model lends itself to testable hypotheses. In particular, it sug-gests that how trade liberalization a↵ects the average sectoral productivity and markup in each sector is a↵ected by how the factors are reallocated

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across sectors. Sectors that expand after trade liberalization are more likely to experience markup reduction and productivity improvement by weeding out low-productivity firms. Testing these hypotheses can be a direction for future research.

Appendices

A

Welfare Function

We derive the welfare of the home country. That of the foreign country can be obtained analogously. In equation (2), ln Qi can be written as

ln Qi = ln iy ln Pi, where ln Pi= 1 2 Mi + 1 Mi Z !2⌦i ln pi(!)d! + 2Mi Z Z !,!0_2⌦_i ln pi(!)⇥ln pi(!0) ln pi(!)⇤d!0d!,

which can be rearranged as

ln Pi = 1 2 Mi + 1 Mi Z !2⌦i ln pi(!)d! + 2Mi Z !2⌦i ln pi(!)d! 2 2 Z !2⌦i [ln pi(!)]2d!. (46)

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Arkolakis et al. (2010) derive the closed form of equation (46). We briefly describe it. It can be shown that

(47) Z !2⌦i ln pi(!)d! = Mi  1 ln ✓ ˆ 'id aiw ◆ 4 ,

where 4 = ✓R₁1W(ve)v 1 ✓dv and

Z !2⌦i [ln pi(!)]2d! = Mi ( 1 ln ✓ ˆ 'id aiw ◆ 2 + 5 24  1 ln ✓ ˆ 'id aiw ◆ , (48)

where 5 = ✓R₁1[W(ve)]2v 1 ✓dv. According to equations (8), (12), and

the definition of the Lambert_{W function, we have}

(49) ln pi(c) = 1 ln ˆ 'id aiw W ✓ ' ˆ 'id e ◆ ,

if the good is produced by the home country. Analogously, if the good is produced by the foreign country, we have

(50) ln pi(c) = 1 ln ˆ '⇤_ie ⌧ a⇤ i W ✓ ' ˆ '⇤ ie e ◆ .

By using the Pareto distribution we assume, we obtain Z !2⌦i ln pi(!)d! = Ni Z ₁ ˆ 'id  1 ln ✓ ˆ 'id aiw ◆ ✓b✓ '1+✓d' + N_i⇤ Z ₁ ˆ '⇤ ie  1 ln ✓ ˆ '⇤_ie ⌧ a⇤ i ◆ ✓b✓ '1+✓d' Ni Z ₁ ˆ 'id W ✓ ' ˆ 'id e ◆ ✓b✓ '1+✓d' N_i⇤ Z ₁ ˆ '⇤ ie W ✓ ' ˆ '⇤_iee ◆ ✓b✓ '1+✓d'. (51)

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According to equations (12) and (15), we have ˆ'id (aiw) = ˆ'⇤ie ⌧ a⇤i. Fur-thermore, by definition, Mi = Ni b ˆ'id ✓ + Ni⇤ b ˆ'⇤ie ✓ . Therefore, we have Ni Z ₁ ˆ 'id  1 ln ✓ ˆ 'id aiw ◆ ✓b✓ '1+✓d' + N_i⇤ Z ₁ ˆ '⇤ ie  1 ln ✓ ˆ '⇤_ie ⌧ a⇤_i ◆ ✓b✓ '1+✓d' = Mi  1 ln ✓ ˆ 'id aiw ◆ . (52) By using Mi = Ni b ˆ'id ✓+ Ni⇤ b ˆ'⇤ie ✓

and a simple change of variable we use repeatedly, we have

Ni Z ₁ ˆ 'id W ✓ ' ˆ 'id e ◆ ✓b✓ '1+✓d' + N_i⇤ Z ₁ ˆ '⇤ ie W ✓ ' ˆ '⇤_iee ◆ ✓b✓ '1+✓d' = ✓Mi Z ₁ 1 [_{W(ve)] v} 1 ✓dv. (53)

Equation (47) can be derived from equations (51) to (53).

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the Pareto distribution we assume, we can derive Z !2⌦i [ln pi(!)]2 = Ni Z ₁ ˆ 'id  1 ln ✓ ˆ 'id aiw ◆ 2 ✓b✓ '1+✓d' + N_i⇤ Z ₁ ˆ '⇤ ie  1 ln ✓ ˆ '⇤_ie ⌧ a⇤ i ◆ 2 ✓b✓ '1+✓d' + Ni Z ₁ ˆ 'id  W ✓ ' ˆ 'id e ◆ 2 _✓b✓ '1+✓d' + N_i⇤ Z ₁ ˆ '⇤ ie  W ✓ ' ˆ '⇤_iee ◆ 2 _✓b✓ '1+✓d' 2Ni Z ₁ ˆ 'id  1 ln ✓ ˆ 'id aiw ◆ W ✓ ' ˆ 'id e ◆ ✓b✓ '1+✓d' 2N_i⇤ Z ₁ ˆ '⇤ ie  1 ln ✓ ˆ '⇤_ie ⌧ a⇤_i ◆ W ✓ ' ˆ '⇤_iee ◆ ✓b✓ '1+✓d'. (54)

By following the same logic we have described, it can be verified that

Ni Z ₁ ˆ 'id  1 ln ✓ ˆ 'id aiw ◆ 2 _✓b✓ '1+✓d' + N_i⇤ Z ₁ ˆ '⇤_ie  1 ln ✓ ˆ '⇤_ie ⌧ a⇤_i ◆ 2 _✓b✓ '1+✓d' = Mi  1 ln ✓ ˆ 'id aiw ◆ 2 , (55) Ni Z ₁ ˆ 'id  W ✓ ' ˆ 'id e ◆ 2 _✓b✓ '1+✓d' N_i⇤ Z ₁ ˆ '⇤ ie  W ✓ ' ˆ '⇤ ie e ◆ 2 ✓b✓ '1+✓d' = ✓Mi Z ₁ 1 [W(ve)]2v 1 ✓dv, (56)

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and 2Ni Z ₁ ˆ 'id  1 ln ✓ ˆ 'id aiw ◆ W ✓ ' ˆ 'id e ◆ ✓b✓ '1+✓d' 2N_i⇤ Z ₁ ˆ '⇤ ie  1 ln ✓ ˆ '⇤_ie ⌧ a⇤_i ◆ W ✓ ' ˆ '⇤_iee ◆ ✓b✓ '1+✓d' = 2✓Mi  1 ln ✓ ˆ 'id aiw ◆ Z ₁ 1 W(ve)v ✓ 1_dv. (57)

Equation (48) can be obtained from equations (54) to (57). By using equations (46), (47), and (48), we have

(58) ln Pi= ln ✓ ˆ 'id aiw ◆ + 1 4+ 1 2 Mi + Mi 2  2 4 5 .

Using equations (1), (2), (46), and (58), after applying a monotonic trans-formation to remove the constant, we obtain equation (45).

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References

Arkolakis, C., A. Costinot and A. Rodriguez-Clare, “Gains From Trade under Monopolistic Competition : A Simple Example with Translog Expenditure Functions and Pareto Distributions of Firm-Level Productiv-ity,” Yale University, MIT, and Penn State University, 2010.

Bernard, A. B., S. Redding and P. K. Schott, “Comparative Advan-tage and Heterogenous Firms,” Review of Economic Studies 74 (2007), 31–66.

Corless, R., G. Gonnet, D. Hare, D. Jeffrey and D. Knuth, “On the Lambert W Function,” Advances in Computational Mathematics 5 (1996), 329–359.

Demidova, S., “Productivity Improvement and Falling Trade Costs: Boon or Bane?,” International Economic Review 49 (2008), 1437–1462.

Dornbusch, R., S. Fischer and P. A. Samuelson, “Comparative Ad-vantage, Trade, and Payments in a Ricardian Model with a Continuum of Goods,” The American Economic Review 67 (1977), 823–839.

Fan, H., E. L.-C. Lai and H. Qi, “A Model of Trade with Ricardian Com-parative Advantage and Intra-sectoral Firm Heterogeneity,” Department of Economics, Hong Kong University of Science and Technology, 2013. Feenstra, R. C., “A Homothetic Utility Function for Monopolistic

Com-petition Models, Without Constant Price Elasticity,” Economics Letters 78 (2003), 79–86.

Huang, Y.-Y., C.-T. Lee and D.-S. Huang, “Home Market E↵ects in the Chamberlinian-Ricardian World,” Bulletin of Economic Research

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Krugman, P., “Scale Economies, Product Di↵erentiation, and the Pattern of Trade,” American Economic Review 70 (1980), 950–959.

Melitz, M. J., “The Impact of Trade on Intra-industry Reallocations and Aggregate Industry Productivity,” Econometrica 71 (2003), 1695–1725. Melitz, M. J. and G. I. P. Ottaviano, “Market Size, Trade, and

Pro-ductivity,” Review of Economic Studies 75 (2008), 295–316.

Okubo, T., “Firm Heterogeneity and Ricardian Comparative Advantage Within and Across Sectors,” Economic Theory 38 (2009), 533–559. Rodriguez-Lopez, J. A., “Competition and O↵shoring,” Department of

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(1) Patterns of Specialization

(2) Impact on Patterns of Specialization

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(4) Impact on Li, > ⌧2✓

(44)

(6) Impact on Li, < ⌧ 2✓

(45)

(8) Impact on ˆ'id, ⌧2✓> > 1

(46)

(10) Impact on ˆ'id, ⌧2✓> > 1

(47)

(48)

(14) Partial Equilibrium 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 (15) Equilibrium Wage 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 1 2 3 4 5 6 7 8 9 (16) 'ˆ1d

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0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 −1 0 1 2 3 4 5 6 (17) 'ˆ2d 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 −2 0 2 4 6 8 10 12 14 (18) 'ˆ3d 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 5 10 15 20 25 30 (19) 'ˆ⇤1d 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 2 4 6 8 10 12 14 (20) 'ˆ⇤2d (21) 'ˆ⇤3d (22) Welfare

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0.5 0.6 0.7 0.8 0.9 1 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 (23) Equilibrium Wage 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 1 2 3 4 5 6 7 8 (24) 'ˆ1d 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 2 4 6 8 10 12 14 (25) 'ˆ2d 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 2 4 6 8 10 12 14 16 18 20 (26) 'ˆ3d 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 5 10 15 20 25 (27) 'ˆ⇤ 1d 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 −2 0 2 4 6 8 10 12 14 16 18 (28) 'ˆ⇤ 2d 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 −2 0 2 4 6 8 10 12 14 16 (29) 'ˆ⇤ _{(30) Welfare}

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科技部補助計畫衍生研發成果推廣資料表

日期:2016/01/31

科技部補助計畫

計畫名稱: 本國市場效果、比較利益以及生產力與加價分配計畫主持人: 徐則謙計畫編號: 103-2410-H-004-011- 學門領域: 國際經濟學

無研發成果推廣資料

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103年度專題研究計畫研究成果彙整表

計畫主持人：

徐則謙

計畫編號：

103-2410-H-004-011-計畫名稱：