Trade policies, firm heterogeneity, and variable markups

    Trade Policies, Firm Heterogeneity, and Variable Markups Svetlana Demidova PII: DOI: Reference:

S0022-1996(17)30066-1 doi:10.1016/j.jinteco.2017.05.011 INEC 3052

To appear in:

Journal of International Economics

Received date: Revised date: Accepted date:

2 September 2015 30 May 2017 30 May 2017

Please cite this article as: Demidova, Svetlana, Trade Policies, Firm Heterogeneity, and Variable Markups, Journal of International Economics (2017), doi:10.1016/j.jinteco.2017.05.011

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

T

ACCEPTED MANUSCRIPT

Svetlana Demidova†

SC

McMaster University

RI P

Trade Policies, Firm Heterogeneity, and Variable Markups∗

NU

May 5, 2017

MA

Abstract

We study unilateral trade liberalization in a model of monopolistic competition with heterogeneous firms, endogenous wages, and non-separable and non-homothetic quadratic preferences that generate variable markups. We show that the optimal level of the revenue-generating im-

ED

port tariff is strictly positive so that protection is always desirable, whether the liberalizing economy is large or small. Yet, reductions in cost-shifting trade barriers are welfare-improving,

PT

making free trade optimal. Finally, we show that in both cases, variable markups result in

AC CE

negative pro-competitive effects, reducing gains from trade.

1

Introduction

In the last few decades there has been a general downward trend in import tariffs around the world that resulted in significant welfare gains. For example, by using the new tariff dataset on 189 countries, Caliendo et al. (2015) show that more than 90% of gains from trade liberalization between 1990 and 2010 can be attributed to falling MFN tariffs. Bown and Crowley (2016) demonstrate the similar downward trend in tariffs for their set of 31 major economies between 1993 and 2013. Yet, despite import tariffs being low on average, significant cross-country and cross-sectoral variation remains. Bown and Crowley (2016) report that as of 2013-2014, low-income countries tend to ∗

I would like to thank the editor and two anonymous referees for their helpful and perspective comments. I am

very grateful for comments from Costas Arkolakis, Kyle Bagwell, Seung Hoon Lee, Andrés Rodríguez-Clare, and Ariel Weinberger, and participants at several seminars and conferences for helpful comments and discussions. I also would like to thank McMaster University for financial support. All remaining errors are mine. † Department of Economics, McMaster University, 1280 Main Street West, Hamilton, ON, Canada L8S 4M4. E-mail: [email protected]. Phone: (905) 525 9140, ext. 26095. Fax: (905) 521 8232

1

ACCEPTED MANUSCRIPT

have higher levels of applied import tariffs. Across countries tariffs are higher in agricultural sector, textiles, apparel, and footwear. Moreover, final good producers face higher tariffs than the

T

producers of intermediate inputs. Hence, there is still room for further trade liberalization.

RI P

In this paper we explore the potential gains from reductions in import tariffs in the context of the recent international trade models with imperfect competition and heterogenous firms. In

SC

addition to the expansion in product variety studied by Krugman (1980), these models offer new channels, through which trade affects welfare. A number of papers following on the heels of the seminal work of Melitz (2003) highlight the mechanism of self-selection of more efficient firms into

NU

exporting. In the presence of such a mechanism, trade liberalization leads to reallocation of resources towards more efficient firms, improving average productivity in liberalizing countries and

MA

potentially raising welfare.1 Most of these papers rely on the assumptions of monopolistic competition and constant elasticity of substitution (CES) preferences as in Dixit and Stiglitz (1977), which imply constant markups charged by firms. While being extremely convenient from the analytical

ED

point of view, constant markups are at odds with the empirical evidence. Moreover, models with constant markups ignore so called pro-competitive gains from trade that arise due to the pres-

PT

ence of variable markups.2 This important channel has become the focus in much of the recent literature, which allows for variable markups by deviating from the assumption of monopolistic competition and/or incorporating non-CES preferences.3 The pro-competitive mechanism of trade

AC CE

in these models is twofold. At the firm level, trade liberalization intensifies foreign competition, reducing market power of local producers and forcing them to decrease their markups. This claim is supported by existing empirical literature, with the notable exception of De Loecker, Goldberg, Khandelwal and Pavcnik (2016). At the industry level, trade liberalization has the ability to affect 1

One has to be careful while interpreting productivity growth as a welfare-improving change. See, for example,

Demidova and Rodríguez-Clare (2009), who show that an increase in average productivity (as a result of an export subsidy) does not necessarily mean an increase in welfare. In fact, welfare falls in their model. 2 For empirical evidence on variation in markups across firms see Chen, Imbs and Scott (2009), Feenstra and Weinstein (forthcoming), De Loecker and Warzynski (2012), De Loecker, Goldberg, Khandelwal and Pavcnik (2016) and Hottman, Redding and Weinstein (2016). The importance of pro-competitive gains is emphasized, for example, in Edmond, Midrigan and Xu (2015), who show that the size of these gains is especially large in the presence of significant misallocations and weak cross-country comparative advantage in individual sectors. 3 Some examples of settings with variable markups include, among others, the model of monopolistic competition in Melitz and Ottaviano (2008), the Cournot competition model in Atkeson and Burstein (2008) and Edmond, Midrigan and Xu (2015), and the Bertrand competition setting in Bernard, Eaton, Jensen and Kortum (2003), de Blas and Russ (2015) and Holmes, Hsu and Lee (2014).

2

ACCEPTED MANUSCRIPT

the markup distribution, reducing its dispersion. As shown, for example, by Restuccia and Rogerson (2008) and Hsieh and Klenow (2009), lower markup dispersion is associated with less extensive

T

distortion that arises due to variability of revenue productivity (the product of physical productivity

RI P

and a firm’s output price) across firms. By reducing this misallocation distortion, trade liberalization can potentially raise welfare.4 On the other hand, Edmond, Midrigan and Xu (2015) and

SC

Arkolakis, Costinot, Donaldson and Rodríguez-Clare (2015) point out the possibility of negative pro-competitive effects of trade liberalization due to labor reallocation towards more productive exporting firms, which could internalize the drop in trade costs and charge higher markups. As a

of labor reallocation and markup distribution.

NU

result, whether trade liberalization leads to welfare gains or losses depends on the joint movement

MA

Given new insights from the recent trade literature on variable markups, what can we say about their policy implications? The main goal of this paper is to partially fill the gap in the literature and provide tractable analytical results for the optimal trade policy in the two-country model of

ED

monopolistic competition with firm heterogeneity and variable markups. Our basic environment is a modified version of Melitz and Ottaviano (2008), a well-known extension of Melitz (2003) that

PT

incorporates endogenous markups by using the linear demand system with horizontal product differentiation developed by Ottaviano, Tabuchi and Thisse (2002). We modify Melitz and Ottaviano (2008) along two lines. First, we drop the assumption of the linear outside good so that wages in

AC CE

our setting will be determined endogenously. Hence, we allow for the income effect that is muted in Melitz and Ottaviano (2008). Second, in addition to per-unit barriers that raise costs of exporters5 in Melitz and Ottaviano (2008), we incorporate revenue-generating import tariffs applied to import sales gross of price markups. Our key finding is that the type of trade barriers is crucial for welfare outcomes of unilateral trade liberalization. In particular, from a unilateral perspective of the Home country, a strictly positive revenue tariff maximizes welfare, whereas the complete removal of cost-shifting trade barriers is optimal. These results hold true whether the Home economy is large relative to the Foreign one or Home is modeled as a small country similar to Demidova and Rodríguez-Clare (2009, 2013), who study CES preferences. Our results for cost-shifting trade barriers contradict those in Melitz and Ottaviano (2008). The reason is that in addition to the Melitz (2003) type sector, Melitz and Ottaviano (2008) incorporate 4 5

See Restuccia and Rogerson (2013) for an excellent survey of the literature on misallocations and productivity. The examples of cost-shifting trade barriers include additional costs faced by exporters, e.g., transportation costs

or customs time lags, and cost tariffs without rebates to consumers (see Caliendo et al., 2015, for further discussion).

3

ACCEPTED MANUSCRIPT

a so-called “outside good” sector that produces a freely traded homogenous good under perfect competition. One well-known advantage of this popular assumption is significant simplification

T

of derivations, since wages across all countries become exogenous. Its second advantage is the

RI P

opportunity to study cross-sectoral inefficiencies or perform counterfactuals with one sector in a multi-sector economy. Yet, such an assumption comes with a price. Not only does it exclude

SC

an income effect, but also it brings an extra distortion into the model, since two sectors have different markups. As pointed out by Bhagwati (1971), the presence of distortions can result in the breakdown of Pareto-optimality of laissez-faire. Hence, it is not surprising that the distortion

NU

created by the use of an outside good makes unilateral trade liberalization welfare-reducing in Melitz and Ottaviano (2008).

MA

To gain intuition behind the role of the type of trade barriers, let us follow the approach from Arkolakis, Costinot, Donaldson and Rodríguez-Clare (2015) (hereafter ACDR). As shown in

ED

Appendix A, welfare change due to marginal changes in trade barriers can be written as:   2ρ d ln λ + 1− d ln µ, d ln W = − (1 − η) θ 1−β+θ

(1)

PT

where λ is the share of domestic expenditure on domestic goods, θ is the shape parameter of the 1−β is a non-negative structural parameter that depends, among Pareto cost distribution, η = ρ 1−β+θ

AC CE

other things, on the elasticity of markups with respect to firm productivity, ρ, µ is the ratio of total expenditures and labor income, and β = −1. In the absence of tariff revenues µ = 1 so that as in ACDR, d ln W = − (1 − η) d ln λ/θ, which implies that by reducing the expenditure share λ on locally produced goods, a small reduction in per-unit cost-shifting trade barriers generates welfare gains. However, the use of revenue tariffs with rebates to consumers implies that changes in µ > 1 have to be taken into account, making protection beneficial. In particular, a marginal reduction in prohibitively high import tariffs would reduce λ, but revenues generated by new tariffs would be close to zero, meaning that changes in µ can be ignored. Hence, moving away from autarky is welfare-improving. On the other hand, in the case of revenue tariffs, deviations from the free trade equilibrium, where the market shares of Foreign exporters are high enough, are beneficial as well as the generated tariff revenues together with the terms of trade externality can more than compensate for a fall in welfare due to a rise in λ. Thus, protection becomes a desirable policy. Note that the approch from ACDR cannot be used for calculations of the optimal tariff rate. Hence, our paper is complimentary to ACDR, who consider a large class of demand functions that generate variable markups in the multiple country setting, with the main difference being that 4

ACCEPTED MANUSCRIPT

we consider the non-separable quadratic utility function as well as the case of revenue-generating import tariffs. Moreover, given the complexity in the case of large shifts in trade costs, to evaluate

T

welfare gains from trade liberalization, ACDR resort to simulations. The advantage of our approach

RI P

is that we can study these gains analytically. Another closely related paper is Bagwell and Lee (2015), who use the Melitz and Ottaviano (2008) setting to study the impact of revenue-generating

SC

import tariffs and export subsidies in the case of two symmetric countries and show, amongst other results, that a marginal increase in the tariff is welfare-improving. Although this result resembles ours, the mechanism behind it is quite different as Bagwell and Lee (2015) maintain the assumption

NU

of the outside good. For instance, their model generates a Metzler Paradox: as Home increases its import tariff, its average price increases, while that abroad falls. As shown in Appendix D, this

MA

is not the case in our model, where both Home and Foreign average prices rise together, proving again that the popular outside good assumption is not innocuous, and one has to be careful while interpreting the results obtained with its help.6

ED

Finally, let us discuss the link between our results and the trade literature on models with constant markups. The comparison with Felbermayr and Jung (2012), Demidova and Rodríguez-

PT

Clare (2009, 2013), and Felbermayr, Jung and Larch (2013) shows that the impact of the type of trade barriers on gains from trade liberalization is similar to the one in our setting with variable markups. Moreover, our analysis of formula (1) resembles the one in Arkolakis, Costinot and

AC CE

Rodríguez-Clare (2012) and Felbermayr, Jung and Larch (2015) for the Melitz (2003) model with the Pareto productivity distribution, where d ln W = −d ln λ/θ + (1 + 1/θ) d ln µ. Given qualitative similarity of our policy results to this literature, what role then do variable markups play, if any? In the last part of the paper we show that in the presence of firm heterogeneity variable markups result in a negative pro-competitive effect due to the misallocation distortion they create.

In

particular, as pointed out by Nocco, Ottaviano and Salto (2014), who study the closed economy case of this model, endogenous markups create an additional within-sector misallocation: more productive firms do not pass on their entire cost advantage to consumers by absorbing part of it in the markup and end up selling quantities below the optimal levels. In contrast, high cost producers’ varieties end up being oversupplied. A similar point is made by Dhingra and Morrow 6

Interestingly, Ludema and Yu (2016), who study incomplete tariff pass-throughs in the extended Melitz and

Ottaviano (2008) model with endogenous quality, show empirically that a quasi-Metzler paradox (quality-unadjusted, tariff-inclusive prices increase) generated by the model may or may not appear in the data depending on the type of product classification. Hence, models with and without an outside good are complementary to each other with their usefulness depending on situation.

5

ACCEPTED MANUSCRIPT

(2012), who study allocation efficiency in the case of separable preferences with a variable elasticity of substitution. Hence, the natural question is what the effects of unilateral trade liberalization are

T

in the presence of such distortion. We show that a small fall in cost-shifting trade barriers faced

RI P

by Foreign exporters causes a reallocation of Home labor towards goods that are oversupplied, i.e., those that have a low markup, which, as pointed out by ACDR, worsens the misallocation distortion

SC

and leads to smaller welfare gains. Moreover, in the case of revenue tariffs, we show that if the level of protection is low enough to begin with, further reductions in the tariff raise the average markup faced by consumers at Home, which, as discussed by Edmond, Midrigan and Xu (2015),

NU

implies smaller welfare gains as well. Thus, variable markups reduce potential gains from trade. The last point worth mentioning is that in the presence of heterogeneous firms, variable markups

MA

provide the social planner an additional incentive to manipulate terms of trade at the firm level. As shown by Costinot, Donaldson, Vogel and Werning (2015) in the setting with a Ricardian economy and perfect competition and Costinot, Rodríguez-Clare and Werning (2016) in the setting with

ED

monopolistic competition and CES preferences, even in the models with constant (positive or zero) markups the social planner has the incentive to discriminate against firms from the same country

PT

based on their productivity. Unfortunately, the approach from Everett (1963) used in these papers is non-applicable to the settings with non-separable preferences, which hinders a more detailed analysis of discriminating trade policies and their use together with other instruments.

AC CE

The rest of this paper proceeds as follows. Section 2 describes the underlying environment. Section 3 solves for the optimal values of trade costs and import tariffs. The role of variable markups is studied in Section 4. Section 5 offers concluding remarks.

2

Model

In this Section we first modify the Melitz and Ottaviano (2008) model of monopolistic competition with firm heterogeneity and the linear demand system over differentiated varieties by dropping the outside good assumption. This brings us back to the case of one-sector economies as in the original Melitz (2003) model. Yet, unlike Melitz (2003), we consider asymmetric countries, which implies endogenously determined wages. Second, in addition to per-unit cost-shifting trade barriers, we introduce revenue tariffs with rebates being distributed equally across consumers in the tariffimposing country. Such tariffs generate discrepancies between wages and individual earnings, which have to be taken into account. The focus of the analysis below will be on the case of two large

6

ACCEPTED MANUSCRIPT

economies. The derivations for the small Home economy can be found in the online Appendix.

Demand

T

2.1

RI P

There are two countries, Home and Foreign, of size Li , i = H, F . Each household in i has the following maximization problem with a non-separable quadratic utility function (α > 0 can be

NU

SC

normalized to 1, but we keep it as it is for ease of comparison with Melitz and Ottaviano, 2008):7 Z 2 Z Z 1 1 q c (ω) dω − γ (q c (ω))2 dω − η q c (ω) dω , max Ui = α (2) 2 Ωi 2 q(ω):ω∈Ωi Ωi Ωi Z p (ω) q c (ω) dω = ei , s.t. Ωi

where Ωi is the set of all available differentiated good varieties in country i, ei is the income (or

MA

expenditure) of a household in country i that in addition to wage may include a portion of tariff revenues, and q c (ω) is the quantity consumed of variety ω ∈ Ωi . η > 0 is the degree of nonseparability across varieties. The degree of product differentiation across varieties is characterized by positive

ED

γ : with lower γ varieties become closer substitutes, and in the limit case of γ = 0, households care

PT

only about the total amount they consume. From the F.O.C., we get Z c q c (ω) dω, λi p (ω) = α − γq (ω) − η

(3)

Ωi

AC CE

where λi is a Lagrangian multiplier. Denote the aggregate quantity of all varieties consumed by an R individual from country i, Ωi q c (ω) dω, by Qi . Then by multiplying both sides of (3) by q i (ω) and integrating over Ωi , we get

λi ei = αQi − γ

Z

(q c (ω))2 dω − η (Qi )2 .

(4)

Ωi

Using the same logic as in Melitz and Ottaviano (2008), it can be shown that the set Ωci of all consumed varieties (q c (ω) > 0) is the largest subset of Ωi that satisfies:   1 1 γα + ηMi p¯i ≡ pmax , p (ω) ≤ i ηMi + γ λi R where Mi is the measure of Ωci , p¯i = (1/Mi ) Ωc p (ω) dω is the average price, and pmax represents i i

the choke price in country i so that only varieties priced below pmax will have non-zero demand.8 i 7 8

In Melitz and Ottaviano (2008), q0c , a consumption of the outside good, is added to the right-hand side of Ui . Note that the absence of the outside good complicates the analysis of the indirect utility function as in Melitz and

Ottaviano (2008). In particular, it can be shown straightforwardly that the relationship between welfare and average

γ −1 α2 − (λ¯ p)2 − 12 N λ2 σ 2p , price p¯, the number of varieties N , and the variance of prices σ 2p becomes U = 21 η + N γ
where λ = (γαN p¯ − eγ (ηN + γ)) / (ηN + γ) N σ 2p + γN p¯2 .

7

ACCEPTED MANUSCRIPT

2.2

Production and Firm Behavior

RI P

T

There is a continuum of domestic firms in country i that derive their unit labor costs from the
θ   , θ > 1. Wage in country i is , c ∈ 0, cM Pareto cost distribution given by Gi (c) = c/cM i i denoted by wi with wF being normalized to unity, wF = 1. Consider a firm from country i with

cost draw c that sells its variety to Lj consumers in country j. Its marginal cost is M Cij (c) .

by choosing the appropriate level of pij (c) . From (3),

1 (α − λj pij (c) − ηQj ) . γ

(5)

NU

c (c) = qij

SC

Hence, it sells qij (c) = Lj qjc (c) and maximizes its profit, π ij (c) = pij (c) qij (c) − M Cij (c) qij (c) ,

Hence, non-separability of the utility function makes the demand for each variety in market j

MA

depend not only on its own price but also on prices of other varieties sold in this market, which, in turn, affect the aggregate quantity consumed, Qj . By using (5) in the expression π ij (c) and maximizing it with respect to pij (c) , we get the demand for a firm as qij (c) =

Lj λj γ

(pij (c) − M Cij (c)) .

ED

In addition, while exporting, each firm faces a per-unit trade barrier that raises its cost, τ ij , where τ ij > 1 and τ ii = 1 for i, j ∈ {H, F } , so that M Cij (c) = wi τ ij c. Moreover, each firm

PT

may face a per-unit import tariff applied to its revenue. In particular, assume that if a firm from country i charges tariff-inclusive price pij (c) in market j, then the country j’ government collects tariff revenues of (tij − 1) pij (c) /tij per unit sold, so that this firm receives only pij (c) /tij . Hence,

AC CE

the revenues earned by firms can differ from their total sales. We assume that only Foreign firms face an import tariff, i.e., the Foreign government is passive, and there are no taxes on local sales: tHH = tF F = tHF = 1 and

tF H ≡ t ≥ 1.

Tariff revenues are distributed equally across consumers so that while in the Foreign country household’s expenditures eF = wF , at Home there are two sources of income, eH = wH + (TH /LH ),   where TH is tariff revenues collected there. Define the cost cutoff c∗ij as a solution of qij c∗ij = 0   (or equivalently, the tariff-inclusive price pij c∗ij equals to wi tij τ ij c∗ij ) so that in country i only

firms with c ≤ c∗ij sell in country j. Then by using qii (c∗ii ) = 0 in (3), we get: wi λi c∗ii = α − ηQi ,

and λj pij (c) = wj λj c∗jj − γqij (c) /Lj . Thus, for each firm with cost c we have pij (c) = rij (c) =



Lj 1 wi tij τ ij c∗ij + c ; qij (c) = wi tij τ ij λj c∗ij − c ; 2 2γ  

2 Lj Lj 2 tij (τ ij wi )2 λj c∗ij − c2 ; π ij (c) = tij (τ ij wi )2 λj c∗ij − c ; 4γ 4γ 8

(6)

ACCEPTED MANUSCRIPT

where the last two equations represent revenues and profits net of tariffs, respectively. Note that while tariff tij and trade cost τ ij have the same effect on tariff inclusive price pij (c) and, in turn,

T

on quantity demanded qij (c) , their effect on the revenues and profits received by firms is different

RI P

as they are net of tariffs.

As in Melitz and Ottaviano (2008), a higher productivity (a lower cost) firm charges a lower

SC

price, makes higher sales, and earns higher profits. Moreover, it charges a higher markups (a markup, mij (c) ≡ pij (c) /M Cij (c) , falls with c). This gives a rise to the misallocation distortion, since more productive firms end up selling too little, while high cost producers tend to oversupply.

NU

Finally, it can be shown directly that the relationships between the cost cutoffs for local sellers in country i and exporters from abroad are (recall that wF = 1):

Equilibrium Conditions

and

c∗F H =

wH c∗HH . tτ F H

(7)

ED

2.3

wj c∗jj c∗F F or c∗HF = wi tij τ ij wH τ HF

MA

c∗ij =

The free entry condition equalizes the expected profits from entering the market to the entry cost. Given the assumption of the Pareto cost distribution and (7), for firms in country i we have cM i


−θ

h
θ+2 i = Const1 , wi Li λi (c∗ii )θ+2 + λj Lj (wj /wi )θ+2 (tij )−θ−1 (τ ij )−θ c∗jj

PT

(F E)i :

(8)

AC CE

where Const1 ≡ 2γfe (θ + 1) (θ + 2) . Note that tariff tij and trade cost τ ij enter this expression differently due to the expected profits being net of tariffs. Next, let us look at the mass of active firms in country i, Mi . Due to free entry, total after-tariff revenues are equal to the labor payment. To see this, note that in addition to the labor payments, the national income in i, Ei , includes tariff revenues Ti : Ei = wi Li + Ti = tRji + Rii , where Rij is the total revenues of firms from country i earned from sales in country j, i.e., Rij is net of tariffs. From Ti = (t − 1) Rji and trade balance Rji = Rij , we get wi Li = Rii + Rij so that Ri = wi Li ,

where



Ri = Mi r¯ii + Gi c∗ij r¯ij /Gi (c∗ii ) ,

(9)

and r¯ij is the firm from country i’ expected revenues from sales in country j conditional on getting a cost draw below the corresponding cutoff, ∗

r¯ij =

Zcij 0



2 1 Lj (wi )2 λj tij τ ij c∗ij / (θ + 2) , rij (c) dGi (c) /Gi c∗ij = 2γ

9

i = H, F.

(10)

ACCEPTED MANUSCRIPT

entrants in country i, Mie , can be calculated as Li , fe (θ + 1)

/fe (θ + 1) . Then the mass of

(11)

RI P

Mie = Mi /Gi (c∗ii ) =


θ

T

By using the (FE)i condition in (9), we get Mi = Li c∗ii /cM i

which means that, as discussed by Caliendo et al. (2015), changes in cost-shifting trade barriers

SC

or revenue tariffs with a rebate do not affect entry margin due to the combined effect of Pareto distribution and the assumption of a single sector in each economy in the model.

NU

Now let us take a closer look at the trade balance condition. It is given by Mij r¯ij = Mji r¯ji ,     where Mij = Gi c∗ij /Gi (c∗ii ) Mi is the mass of exporters from country i to country j. Denote

the relative wage wH /wF by w. Then, by using (7) and (11), we get the trade balance condition
θ

=


θ w2θ+2 λH (τ HF )θ (c∗HH )θ+2 cM . H θ+1 t

MA

(T B) : λF (τ F H )θ (c∗F F )θ+2 cM F

(12)

ED

Finally, we need to derive the equation for Lagrangian multiplier λi . It can be written as   (θ + 2) ei ∗ (λ)i : wi λi cii = α − η . (13) (θ + 1) c∗ii wi

PT

The ratio of the household’s income to its labor earnings in the Foreign country is eF /wF = 1 as there is no tariff revenue collected there. In contrast, as shown in Appendix B,

AC CE

eH (t − 1) RF H =1+ = 1 + λH wH (c∗HH )2 yH , wH wH LH

where yH ≡ (t − 1) MFe (wc∗HH ) / cM F τFH



θ

(14)


/ 2γ (θ + 2) tθ+1 is the term that does not depend

on λH . As a result, as long as there is a positive import tariff t > 1 imposed by Home, yH > 0 and, in turn, eH /wH > 1. We can use (13) and (14) to exclude λH and λF from the set of equilibrium variables so that we end up with 3 unknown variables, w, c∗ii , and c∗jj , and 3 equilibrium conditions (we introduce yF ≡ 0 to write the (FE)i condition in the same format for both countries):9

(F E)i : (T B) : 9

αc∗ii − η (θ+2) (θ+1) 1 + η θ+2 θ+1 yi tθ+1 LH cM H

h i
−θ
−θ (c∗ii )θ Li cM + Lj cM (wi /wj )θ (τ ji )−θ (tij )−θ−1 = Const1 , i j


−θ

+ LF cM F τFH


−θ

wθ = w2θ+1



τ HF cM H τ F H cM F

θ h

LF cM F


−θ

+ LH cM H τ HF


−θ

Note that the (TB) condition implies that we can solve it for the Home wage w. Then all the other variables in

the equilibrium can be calculated as functions of w.

10

i w−θ .

ACCEPTED MANUSCRIPT

2.4

Welfare

SC

so that it rises with Qi .10 From (6) and (13), we get   ei (θ + 2) . Qi = (θ + 1) c∗ii wi

RI P

T

Per capita welfare can be written as a function of aggregate quantity Qi (see Appendix B):   θ+1 α (2θ + 3) Qi − ηQi , Ui = 2 (θ + 2) θ+1

In the absence of tariff rebates, ei /wi = 1 results in the familiar relationship between welfare

NU

and domestic cost cutoff cii , which is common across various extensions of the Melitz (2003) model: the lower is the cost cutoff (or the higher is the productivity cutoff), the higher is welfare in the

MA

economy. It is not surprising then that, as shown in the next Section, higher trade barriers reduce welfare by letting less productive firms survive and unambiguously raising the cost cutoff. The use of revenue tariffs with rebates, however, results in non-monotonic behavior of Qi with respect to a

ED

tariff. While higher protection raises cii , it also brings larger tariff rebates to households, raising the discrepancy between their income and labor earnings. As we show below, this non-monotonicity results in a strictly positive optimal import tariff: while the effect of tariff rebates dominates for

PT

small deviations from free trade, as protection rises further, the loss in productivity outweighs the

AC CE

gains in tariff revenues. The next Section provides a more detailed discussion of these results.

3

Unilateral Trade Liberalization

In this Section we study reductions in the revenue tariff and cost-shifting trade barriers for Foreign firms, i.e., falls in tF H and τ F H , respectively. Let us look at revenue tariffs first.

3.1

Optimal Import Tariff

The derivation of the optimal tariff rate that maximizes QH and, in turn, welfare at Home is complicated by QH being proportional to a product of two components, 1/c∗HH and eH /wH , that move in the opposite directions when tF H changes. Moreover, QH is not concave with respect to tF H . Hence, instead of deriving the optimal level of tF H from the F.O.C. of the maximization problem, we need to look at the behaviour of ∂QH /∂tF H directly. In Appendix C we derive unique topt such that ∂QH /∂tF H R 0 iff tF H ⋚ topt . Hence, we get 10

The restriction α − ηQi > 0 that guarantees non-negative prices in the equilibrium implies that Qi <

α (2θ + 3) /(η (2θ + 2)) so that ∂Ui /∂Qi > 0.

11

ACCEPTED MANUSCRIPT

Proposition 1 There exists a unique import tariff that maximizes welfare at Home. It

1 1 LH + θ θ LF



cM F cM τ H HF

θ

where w is the endogenously determined Home wage.

w−θ ,

RI P

topt = 1 +

T

can be found as a solution of the following equation:

(15)

SC

From (15), the value of the optimal import tariff in the case of two large economies depends on

NU

the shape parameter of the cost distribution and terms of trade externality,11 which is captured by
M θ , openness of the Home economy, the last term and depends on the relative productivity, cM F /cH

(τ HF )−θ , and the relative market size, LH /LF . As LH /LF → 0, Home becomes a small country,

MA

which can be analyzed in line with Demidova and Rodríguez-Clare (2009) by using 3 assumptions: (i) the Foreign demand for Home varieties depends only on their prices, i.e., aggregate variables in the Foreign demand function are not affected by Home; (ii) the cost distribution of Foreign producers

ED

is fixed; and (iii) the mass of available Foreign varieties is fixed. The last two assumptions mean that the mass of firms and wage abroad are not affected by changes at Home. Then we have (see

PT

the online Appendix for more details):

Corollary 1 In the case of a small Home economy, there exists a unique import tariff

AC CE

that maximizes Home welfare:

topt = 1 + 1/θ.

(16)

As shown, protection is higher for the large Home economy, which is not surprising due to the terms of trade externality. Also, Proposition 1 and its Corollary strongly resemble the results derived by Felbermayr, Jung and Larch (2013) for the case of constant markups, and as in their case, the analysis of the properties of topt in the case of two large economies is complicated by the endogenous nature of wage w.12 Yet, in the case of a small Home economy, the level of protection given by (16) is unambiguously higher than that in Felbermayr, Jung and Larch (2013), topt = 1 + (σ − 1) / (θ − σ + 1) , where σ < θ + 1 is the constant elasticity of substitution between 11

Note that non-homotheticity and non-separability of quadratic preferences in our setting do not allow us to follow

the approach from Costinot, Rodríguez-Clare and Werning (2016), who define terms of trade in the generalized Melitz (2003) model as the ratio of aggregate prices of exports and imports, which are obtained with the help of the trade balance condition. Instead, we interpret the terms of trade effect as the relative wage effect: by raising the country’s wage and making its export bundle more expensive, an import tariff improves its terms of trade. 12 See the (TB) condition that implicitly defines w as a function of t.

12

ACCEPTED MANUSCRIPT

varieties. Intuitively, while in both models protection leads to the survival of the least productive firms that would exit without it, compared to the setting with constant markups, in our model

T

these new survivors are the ones with the lowest markups. In other words, protection leads to an

RI P

additional positive effect by shifting resources from high markup firms to low markup ones and reducing misallocation distortion. Thus, variable markups encourage higher levels of protection.13

SC

Moreover, the optimal tariff formula in (16) resembles the one derived by Alvarez and Lucas (2007) for a one-sector Eaton-Kortum (2002) model. The reason is that both settings incorporate the assumption of the unbounded productivity distribution, which provides even a small economy with

NU

market power in some goods and, in turn, allows for terms of trade manipulation, making the positive import tariff optimal with the level of protection being positively related to the dispersion

3.2

MA

of productivities in both cases: lower θ (higher firm heterogeneity) results in higher protection.14

Reduction in Cost-Shifting Trade Barriers

ED

In this Section we ignore import tariffs by assuming that tij = 1 and focus only on a fall in τ F H . By way of illustration, we rely on a graphical analysis. To do this, we first prove:

PT

Lemma 1 The (FE)H condition implies a positive relationship between c∗HH and w, while the (TB) condition implies a negative relationship between c∗HH and w.

AC CE

Proof. Using the (FE)H and (FE)F , we can re-write the (TB) condition as   h i M θ
−θ θ

θ θ+2 θ τ HF cH M −θ −θ ∗ ∗ w + L τ − ) = Const1 cM . η (c wθ+1 LF cM αc c H F H HH HH F HF M θ+1 τ F H cF

This equation implies that w and c∗HH are negatively related. Next, by using (λ)H and (λ)F , we can re-write the (FE)H condition as " #  
θ+2 Const1 θ M −θ ∗ ∗ η (cHH ) + LF LH αcHH − = Const1 , cF
−θ
M −θ θ+1 τ θHF wθ+1 LF cM + wL c H F H

which implies a positive relationship between w and c∗HH . 13

Note that this conclusion relies strongly on the positive relationship between a firm’s productivity and its markup.

Hence, it might not hold in general. 14 The relationship between θ and topt in the case of two large economies is quite complicated because of endogeneity of w in (15). Felbermayr, Jung and Larch (2013) show that ∂w/∂θ = 0 in the neighborhood of the symmetric equilibrium. Given that we study unilateral policies, two countries are not symmetric a priori, which does not allow us to follow this approach.

13

ACCEPTED MANUSCRIPT

wage 6

RI P

FE curve

T

Figure 1: Unilateral Trade Liberalization by Home

SC

?

TB curve

0

-

Cost cutoff for domestic sellers

MA



NU

?

We depict both relationships in Figure 1, where the FE and TB curves represent the (FE)H and

ED

(TB) conditions, respectively.15 Intuitively, the FE curve is upward sloping, since high wages deter entry, letting less efficient firms survive. The TB curve is downward sloping, since, to maintain the

PT

trade balance, high wages at Home must be compensated by higher efficiency of Home firms. The intersection of two curves gives the unique equilibrium values of w and c∗HH .

AC CE

Next, it is straightforward to show that a reduction in τ F H affects only the TB curve by shifting it down as shown in Figure 1, which immediately proves that both w and c∗HH fall as τ F H falls. Also, from the (F E)F condition, c∗F F falls with falling w. Finally, recall that in the absence of tariff rebates welfare in both economies falls with the cost cutoff there. This leads to: Proposition 2 Unilateral trade liberalization (UTL) by Home in the form of falling cost-shifting trade barriers raises welfare there. Moreover, in the case of two large economies, welfare of the Foreign country rises as well. Proposition 2 shows that the use of the outside good assumption in Melitz and Ottaviano (2008) and its absence in our model are crucial and lead to the opposite outcomes in two settings. Intuitively, as argued, for example, by Ossa (2011), the effect of UTL depends on the interplay between production relocation and terms-of-trade effects. In particular, higher trade barriers for 15

Note that this Figure resembles the one in Demidova and Rodríguez-Clare (2013) for CES preferences. This is

not a coincidence, since they depict the relationships between w and the productivity cutoff for exporters, ϕ∗HF ,.which is an inverse of the cost cutoff.

14

ACCEPTED MANUSCRIPT

Foreign exporters generate positive profits to local firms at Home. They can be competed away either through entry (i.e., a production relocation effect) or through an increase in wages (i.e., a

T

terms-of-trade effect). The result of the combination of these two effects depends on the elasticity

RI P

of the labor supply in the differentiated good sector. In Melitz and Ottaviano (2008), the outside good can release labor at a constant wage, making labor supply perfectly elastic, so that the term-of

SC

trade effect is muted.16 In contrast, in our setting labor supply in the differentiated good sector is perfectly inelastic as it is fixed by a country endowment so that the production relocation effect is isolated. Hence, the impact of UTL is the opposite in two models.17

NU

Finally, this result is in stark contrast to Proposition 1 and its corollary: in the case of costshifting trade barriers, full trade liberalization, and not protection, is preferable for Home. Hence,

MA

the choice of the optimal policy depends crucially on the type of barriers the government deals with.

Role of Variable Markups

ED

4

Our results for UTL in the setting with variable markups seem to be quite similar to those obtained

PT

for the CES models with constant markups. Given such similarity, the natural question is what role variable markups play. We answer it in this Section. To begin, let us discuss a reduction in cost-

AC CE

shifting trade barriers. Even though the mean and dispersion of local markups remain unchanged as τ F H falls, this does not mean that variable markups play no role as countries liberalize. In particular, as Edmond, Midrigan and Xu (2015) and ACDR point out, tracking down the changes in the local markup distribution alone is not enough. First, Edmond, Midrigan and Xu (2015), who study the revenue-weighted harmonic average of markups, argue that higher markups on imported goods can outweigh the reduction in local markups and make the misallocation distortion worse. In our case, however, a unilateral fall in trade barriers does not affect the average markup due to the assumption of a Pareto cost distribution. Second, ACDR show that variable markups can create a new source of gains or losses from trade liberalization, depending on whether low cost firms, which 16

Note that this statement seem to contradict to Costinot, Rodríguez-Clare and Werning (2016), who still have

terms of trade externality in the presence of the outside good. This confusion can be resolved by noticing that they use a new definition of terms-of-trade manipulation at the macro-level, the manipulation of the relative price of sector-level aggregate prices, while we follow a more traditional approach based on the manipulation of relative wages, based on which TOT externality in Melitz and Ottaviano (2008) is indeed muted. 17 See Haaland and Venables (2016), who allow the elasticity of labor supply take intermediate values between two polar cases, while analyzing trade policies in the CES model of the small open economy.

15

ACCEPTED MANUSCRIPT

charge high markups and under-supply their varieties, end up growing in size. In this case the effect of trade liberalization on welfare of country j depends on the sign of the covariance of the

T

markup, m (ω, i) charged by a firm in country j that produces variety ω for market i and a change

RI P

in its labor share that is needed to produce this variety for this market:  XZ    dl (ω, i) dl (ω, i) = dω, m (ω, i) cov m (ω, i) , Lj Lj ω∈Ωji

(17)

SC

i

where l (ω, i) is the total employment associated with a production of variety ω in country j for sales in country i, 18 and Ωji is the set of all varieties produced in country j for country i. In other

NU

words, trade liberalization has a positive (negative) effect on welfare in country j, if this covariance is positive (negative). The important property of (17) is that it depends not only on the firms’

MA

decisions in their local market, but also on their exporting decisions. In our model,   Z c∗ Z c∗ jj ji Mjj dGj (c) dGj (c) Mji dl (ω, i)   , = [p (c) dqjj (c)] [pji (c) dqji (c)] + cov m (ω, i) , Lj wj Lj 0 w L G ∗ j j 0 j (cji ) G c j

jj

ED

where dx denotes a marginal change in x. As we show in Appendix E, for a small fall in τ F H , the expression above is negative. Hence, as in the case of separable preferences in ACDR that generate

PT

variable markups, in our setting with non-separable preferences, although UTL is beneficial for the Home economy as a whole, its gains are mitigated by the misallocation distortion that gets worse

AC CE

as τ F H starts to fall.19 In other words, while the UTL results seem to be qualitatively similar to the ones in the Melitz (2003) model with CES preferences, variable markups reduce welfare. The next point worth emphasizing is that unlike movements in cost-shifting trade barriers that leave the aggregate markup unchanged, per-unit revenue tariffs have the ability to affect it. To avoid analytical difficulties, let us consider two symmetric countries with zero variable trade costs, M i.e., assume that LH = LF , cM H = cF , τ F H = τ HF = 1. Then the average markup becomes

m ¯H =

X

i=H,F

MiH MHH + MF H

Z

c∗iH

miH (c)

0

dG (c) 1 2θ − 1 MHH + tMF H
= . ∗ 2 θ − 1 MHH + MF H G ciH

As we show in Appendix E, when the import tariff t starts to rise from 1, the average markup falls. In other words, when the initial level of protection is already small, further reductions in t raise the average markup faced by consumers at Home. Similarly, a small decrease from t = topt raises the average markup, which, according to Edmond, Midrigan and Xu (2015), implies a negative 18

Note that (17) follows from the fact that for any labor re-allocations, the size of the economy remains the same P R so that i ω∈Ω [dl (ω, i) /Lj ] dω = 0. ji 19 Due to complexity of the analysis of large falls in τ F H , we leave it to future work.

16

ACCEPTED MANUSCRIPT

pro-competitive effect. To summarize, one has to be careful when using the results derived for

Conclusion

RI P

5

T

trade costs to analyze changes in import tariffs.

In this paper we have studied the implications of reductions in cost-shifting trade barriers and

SC

revenue tariffs with rebates to consumers in the extended version of the Melitz and Ottaviano (2008) model without the outside good. Our conclusions can be broadly summarized as follows. First, we

NU

find that in contrast to Melitz and Ottaviano (2008), a reduction in per-unit trade barriers raises, and does not reduce, welfare of the liberalizing country as well as welfare of its trading partner. Thus, the breakdown of optimality of laissez-faire in Melitz and Ottaviano (2008) can be explained

MA

by the distortion created by the presence of the outside good sector. Second, we derive the optimal values of import tariffs for the large and small Home economies and show that as in the models

ED

with monopolistic competition and CES preferences, protection is always a desirable policy for the Home government. The main difference between the policy implications from the CES models and our setting is the negative pro-competitive effect created by variable markups.

PT

Given our results, there are several potential avenues for future research. First, the absence of pro-competitive gains in our model might be partially explained by the assumption made about

AC CE

the cost distribution specified as Pareto. The Pareto distribution assumption can be modified: Feenstra (2014), who studies the model with non-CES preferences, shows that once the support of the distribution becomes bounded, other channels of pro-competitive gains from trade begin to work, affecting the average markup as well as the markup dispersion, which otherwise remain unchanged. Alternatively, the Pareto distribution can be replaced by another one. For example, as argued by Fernandes et al. (2015), in the Melitz-Pareto models conditional on the fixed costs of exporting, falling trade costs affect exports only through the number of exporters, i.e., the extensive margin, while in their empirical analysis half of the variation in exports occurs on the intensive margin. The authors show that one way to reconcile the theory with the data is to use the log-normal distribution with firm-destination fixed trade costs. Another limitation of our analysis is that as Arkolakis, Costinot and Rodríguez-Clare (2012) and ACDR, it focuses on single-sector economies only, resulting in entry in each economy being not affected by trade policies.20 This might not be the case in a setting with multiple sectors, where the 20

In the case of CES preferences, Caliendo et al. (2015) show that entry becomes endogenous if tariff revenues are

17

ACCEPTED MANUSCRIPT

mass of entrants in a particular sector depends on the relative expenses on goods produced there. See, for example, Spearot (2016), who studies changes in revenue-generating import tariffs in the

T

Melitz and Ottaviano (2008) model without the outside good assumption. The author modifies the

RI P

model further by incorporating multiple countries and multiple industries with heterogeneity in the country-by-industry shape parameters of the Pareto cost distributions21 and estimates the model

SC

empirically. Although his focus is different from ours, Spearot (2016) provides some interesting results for the case of unilateral trade liberalization by showing that the US gains both from an increase in all its tariffs by 10% and a removal of the observed tariffs. Note that in addition

NU

to producing potentially interesting third-country effects, having multiple countries as in Spearot (2016) can reduce the negative effect of variable markups by affecting the level of absorption of the

MA

drop in trade costs by exporters.

Next, it would be interesting to see whether our trade policy results change, if one considers alternative demand structures that generate variable markups. For instance, Jung, Simonovska

ED

and Weinberger (2015) show that the use of the generalized CES utility function allows to match qualitatively and quantitatively the stylized facts about exporters and their markups that other

PT

frameworks including ours fail to do.

Finally, we have left Nash trade policies out of the scope of this paper. The question is whether protection remains the optimal policy when all trading countries, and not just the Home economy,

AC CE

have the ability to choose their tariffs.22 We leave all of these questions to future work.

References

[1] Alvarez, F., Lucas, R.E., 2007. General Equilibrium Analysis of the Eaton—Kortum Model of International Trade. Journal of Monetary Economics 54, 1726—1768. [2] Arkolakis, C., Costinot, A., Donaldson, D., Rodríguez-Clare, A., 2015. The Elusive ProCompetitive Effects of Trade. Mimeo. [3] Arkolakis, C., Costinot, A., Rodríguez-Clare, A., 2012. New Trade Models, Same Old Gains?” American Economic Review 102(1), 94-130. not rebated to consumers or if one replaces revenue tariffs with the revenue ones. 21 Note that preferences in Spearot (2016) differ from those in Melitz and Ottaviano (2008) and in our setting, since he excludes the total quantity squared from the industry-level sub-utility, making the preferences separable. 22 Grossman (2016) offers an excellent review of trade literature on cooperative tariffs. See also Bagwell and Lee (2015) for the discussion of Nash trade policies in the Melitz and Ottaviano (2008) model with the outside good.

18

ACCEPTED MANUSCRIPT

[4] Atkeson, A., Burstein, A., 2008. Pricing-to-Market, Trade Costs, and International Relative

T

Prices. American Economic Review 98(5), 1998-2031.

RI P

[5] Bagwell, K., Lee, S.H., 2015. Optimal Trade Policies with Delocation and Selection Effects. Mimeo.

[6] Bernard, A., Eaton, J., Jensen, J.B., Kortum, S.S., 2003. Plants and Productivity in Interna-

SC

tional Trade. American Economic Review 93(4), 1268-1290.

NU

[7] Bhagwati, J., 1971. The Generalized Theory of Distortions and Welfare, in: Bhagwati, J., Jones, R., Mundell, R., Vanek, J. (Eds), Trade, Balance of Payments and Growth. North-

MA

Holland, Amsterdam.

[8] Bown, C. P., Crowley, M.A., 2016. The Empirical Landscape of Trade Policy, in: Bagwell, K., Staiger, R.W. (Eds.), The Handbook of Commercial Policy, Vol. 1A. North-Holland, Amster-

ED

dam.

[9] Caliendo, L., Feenstra, R.C., Romalis, J., Taylor, A.M., 2015. Tariff Reductions, Entry, and

PT

Welfare: Theory and Evidence for the Last Two Decades. NBER Working Paper 21768. [10] Chen, N., Imbs, J., Scott, A., 2009. The Dynamics of Trade and Competition. Journal of

AC CE

International Economics 77(1), 50—62. [11] Costinot, A., Donaldson, D., Vogel, J., Werning, I., 2015. Comparative Advantage and Optimal Trade Policy. Quarterly Journal of Economics 130(2), 659-702. [12] Costinot, A., Rodríguez-Clare, A., Werning, I., 2016. Micro to Macro: Optimal Trade Policy with Firm Heterogeneity. NBER Working Paper 21989. [13] de Blas, B., Russ, K.N., 2015. Understanding Markups in the Global Economy. American Economic Journal: Macroeconomics 7(2), 157-180. [14] De Loecker, J., Goldberg, P.K., Khandelwal, A.K., Pavcnik, N., 2016. Prices, markups and Trade Reform. Econometrica 84(2), 445-510. [15] De Loecker, J., Warzynski, F., 2012. Markups and Firm-level Export Status. American Economic Review 102(6), 2437-2471.

19

ACCEPTED MANUSCRIPT

[16] Demidova, S., 2008. Productivity Improvements and Falling Trade Costs: Boon or Bane?

T

International Economic Review 49(4), 1437-1462.

RI P

[17] Demidova, S., Rodrígues-Clare, A., 2009. Trade Policy under Firm-Level Heterogeneity in a Small Economy. Journal of International Economics 78(1), 100-112.

[18] Dhingra, S., Morrow, J., 2012. Monopolistic Competition and Optimum Product Diversity

SC

Under Firm Heterogeneity. Mimeo.

American Economic Review 67, 297-308.

NU

[19] Dixit, A. K., Stiglitz, J.E., 1977. Monopolistic Competition and Optimum Product Diversity.

MA

[20] Eaton, J., Kortum, S.S., 2002. Technology, Geography, and Trade. Econometrica 70, 1741— 1779.

[21] Edmond, C., Midrigan, V., Xu, D., 2015. Competition, Markups, the Gains from International

ED

Trade. American Economic Review 105(10), 3183-3221. [22] Everett , H., 1963. Generalized Lagrange Multiplier Methods for Solving Problems of Optimal

PT

Allocation of Resources. Operations Research 11, 399—417.

AC CE

[23] Feenstra, R.C., 2014. Restoring the Product Variety and Pro-competitive Gains from Trade with Heterogeneous Firms. NBER Working Paper 19833. [24] Feenstra, R.C., Weinstein, D.E. (forthcoming). “Globalization, Markups and U.S. Welfare. Journal of Political Economy. [25] Felbermayr, G., Jung, B., 2012. Unilateral Trade Liberalization in the Melitz Model: A Note. Economics Bulletin 32(2), 1724-1730. [26] Felbermayr, G., Jung, B., Larch, M., 2013. Optimal tariffs, retaliation, and the welfare loss from tariff wars in the Melitz model. Journal of International Economics 89(1), 13-25. [27] Felbermayr, G., Jung, B., Larch, M., 2015. The welfare consequences of import tariffs: A quantitative perspective. Journal of International Economics 97(2), 295-309. [28] Fernandes, A., Klenow, P.J., Meleshchuk, S., Pierola, M.D., Rodríguez-Clare, A., 2015. The Intensive Margin in Trade: Moving Beyond Pareto. Mimeo.

20

ACCEPTED MANUSCRIPT

[29] Grossman, G.M., 2016. The Purpose of Trade Agreements, in Bagwell, K., Staiger, R.W.

T

(Eds.), The Handbook of Commercial Policy, Vol. 1A. North Holland, Amsterdam.

RI P

[30] Haaland, J.I., Venables, A., 2016. Optimal trade policy with monopolistic competition and heterogeneous firms. Journal of International Economics 102, 85-95.

[31] Hsieh, C.-T., Klenow, P.J., 2009. Misallocation and Manufacturing TFP in China and India.

SC

The Quarterly Journal of Economics 124(4), 1403-1448.

NU

[32] Holmes, T., Hsu, W., Lee, S., 2014. Allocative Efficiency, Mark-ups, and the Welfare Gains from Trade. Journal of International Economics 94(2), 195-206.

MA

[33] Hottman, K., Redding, S.J., Weinstein, D.E., 2016. Quantifying the Sources of Firm Heterogeneity. Quarterly Journal of Economics 131(3), 1291-1364. [34] Jung, J.W., Simonovska, I., Weinberger, A., 2015. Exporter Heterogeneity and Price Discrim-

ED

ination: A Quantitative View. NBER Working Paper 21408. [35] Krugman, P.R., 1980. Scale Economies, Product Differentiation, and the Pattern of Trade.

PT

American Economic Review 70, 950-959.

AC CE

[36] Ludema, R.D., Yu, Z., 2016. Tariff pass-through, firm heterogeneity and product quality. Journal of International Economics 103, 234-249. [37] Melitz, M.J., 2003. The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity. Econometrica 71, 1695-1725. [38] Melitz, M.J., Ottaviano, G.I.P., 2008. Market Size, Trade, and Productivity. Review of Economic Studies 75, 295-316. [39] Nocco, A., Ottaviano, G.I.P., Salto, M., 2014. Monopolistic Competition and Optimum Product Selection. American Economic Review 104(5), 304-09. [40] Ossa, R., 2011. A “New Trade” Theory of GATT/WTO Negotiations. Journal of Political Economy 119(1), 122-152. [41] Ottaviano, G.I.P., Tabuchi, T., Thisse, J.F., 2002. Agglomeration and trade revisited. International Economic Review 43, 409-435.

21

ACCEPTED MANUSCRIPT

[42] Restuccia, D., Rogerson, R., 2008. Policy Distortions and Aggregate Productivity with Het-

T

erogeneous Plants. Review of Economic Dynamics 11, 707-720.

RI P

[43] Restuccia D., and Rogerson, R., 2013. Misallocation and Productivity. Review of Economic Dynamics 16(1), 1—10.

[44] Spearot, A. C., 2016. Unpacking the Long Run Effects of Tariff Shocks: New Structural

SC

Implications from Firm Heterogeneity Models. American Economic Journal: Microeconomics

NU

8(2), 128-167.

Appendix A

MA

In this Appendix we derive formula (1) that describes the welfare effect of a small shock in import tariffs or trade costs in the Melitz and Ottaviano (2008) model without the outside good. To deal with non-separability of the utility function as well as with the complications caused by accounting

ED

for changes in tariff revenues, we modify the approach used in the 2012 version of Arkolakis, Costinot, Donaldson and Rodríguez-Clare (2015) (hereafter ACDR 2012). Before we start, note

PT

that to simplify the comparison with ACDR 2012, here we use their notation instead of that used in the main body of the paper.

AC CE

Consider a world economy comprising j = 1, ..., n countries, one factor of production, labor, and a continuum of differentiated goods ω ∈ Ω. All individuals are endowed with one unit of labor, are perfectly mobile across the production of different goods, and are immobile across countries. Lj denotes the total endowment of labor and wj denotes the wage in country j. All consumers have the same preferences. Similar to the case of the separable non-CES and translog utility functions explored in ACDR 2012, if a consumer in country j with expenditure (or income) ej faces a schedule of prices p ≡ {pω }ω∈Ω , her Marshallian demand for any differentiated good ω is ln qw (p, ej ) = −β ln pω + γ ln ej + d (ln pω − ln p∗ (p, ej )) , where p∗ (p, ej ) is symmetric in all prices. Note that in the absence of tariffs, income ej =wage wj , while with import tariffs, when tariff revenue, Tj , is distributed equally across Lj workers, ej = wj + (Tj /Lj ). Let us derive a similar formula for our case of a quadratic, non-separable utility function. From the F.O.C. described in Section 2, λp (ω) = α − γ˜ q c (ω) − η˜

Z

Ω

22

q c (ω) dω,

ACCEPTED MANUSCRIPT

h i R where λ is a Lagrangian multiplier, λ = αQ − γ˜ Ω (q c (ω))2 dω − η˜Q2 /ej . Using the same logic

as in Melitz and Ottaviano (2008), one can show that the set Ωc of all varieties that are consumed

RI P

T

(qωc > 0) is the largest subset of Ω that satisfies:   1 1 γ˜ α + η˜M p¯ ≡ p∗ , pω ≤ ηM + γ˜ λ

SC

R where M is the measure of consumed varieties in Ωc and p¯ = (1/M ) ω∈Ωc pω dω. p∗ represents the R choke price. By definition, λp∗ = α − η˜ Ω q c (ω) dω. Let us define the income multiplier µI as (we use µI instead of µ here to make the comparison with ACDR 2012, who use µ for markups, more

NU

straightforward. In the final formula for welfare we will drop superscript “I” to relate this formula to the one in Felbermayr, Jung and Larch, 2015):

MA

µI ≡ Ej /wj Lj = expenditure/wage = ej /wj , so that R   ∗ Z αQ − γ˜ Ω (q c (ω))2 dω − η˜Q2 p (ω) p c c q (ω) = α − η˜ q (ω) dω − λp (ω) = . −1 w p (ω) γ˜ µI Ω

ED

We can find p∗ and λ by solving the following system of equations: λ= p∗ =

PT



αQ−˜ γ

R

1 ηM +˜ γ

Ω (q

c (ω))2 dω−˜ η Q2

e 1 γ ˜ λ α

Hence, we can re-write the Marshallian demand as c



−(ln p(ω)−ln p∗ )

AC CE

ln q (ω) = ln p (ω) − ln w + ln e



,
+ η˜M p¯ .

− 1 + ln

"

αQ − γ˜

R

# c (ω))2 dω − η 2 (q ˜ Q Ω . (A.1) γ˜ µI

The last term in the expression above appears due to non-separability of the utility function and the presence of tariff revenues (it is absent in ACDR 2012). It absorbs the characteristics of the destination market only and is the same for all varieties sold there. Thus, if a consumer with income e = µI w faces a schedule of prices p ≡ {pω }ω∈Ω , her Marshallian demand for any differentiated good ω takes the form of ln qw (p, e) = −β ln pω + γ ln w + d (ln pω − ln p∗ (p, e)) + ln ∆. (18) i h R γ µI is the aggrewhere p∗ (p, e) is symmetric in all prices, ∆ ≡ αQ − γ˜ Ω ( q c ( ω) )2 dω − η˜Q2 /˜

gate term, ∂∆/∂ ln pω = 0. Finally, β = γ = −1 and d (x) = ln (e−x − 1) . Note that in ACDR

2012 for the non-CES separable utility function β = γ = 0 and for the translog utility function β = γ = 1. Hence, in all cases, theirs and ours, β = γ ≤ 1. “A” Assumptions from ACDR 2012. Now let us verify that the “A” assumptions from ACDR 2012 hold in our case. 23

ACCEPTED MANUSCRIPT

A1. [Existing Demand Systems]. As we have shown already, β = γ = −1 ≤ 1. A2. [Choke Price] It is straightforward to show that for all x ≥ 0, d (x) = ln (e−x − 1) = −∞.

1 e−x = − −x − 1, −x e −1 e −1

−e−x < 0. (e−x − 1)2

RI P

d′ (x) = −

T

A3. [Log-concavity] For all x ≤ 0, d′′ (x) ≤ 0. In our case d′′ (x) =

SC

A4. [Pareto] Firm environment is exactly the same as in ACDR 2012 including the assumption of the Pareto productivity distribution, where the per unit labor cost of a firm with productivity z

NU

is c = 1/z. For more details see Section 2.

Firms in Trade Equilibrium. The main difference between ACDR 2012 and our setting is that in addition to trade costs, we allow for import tariffs. In particular, while exporting, each

MA

firm faces an iceberg transportation costs τ ij , where τ ij > 1 and τ ii = 1, and if a firm from country i charges price pij in market j, then the country j’ government collects tariff revenues of

ED

(tij − 1) pij /tij per unit sold, so that this firm receives only pij /tij , where tij ≥ 1. To deal with the complications caused by tariffs, we are going to deviate from ACDR 2012 by looking at firm characteristics inclusive of tariffs. In particular, if M Cij = τ ij wi /z is the marginal

PT

cost of a firm from country i that serves market j, then c ≡ tij M Cij = cij /z is its marginal cost

AC CE

inclusive of a tariff and the firm’s profit decision is max (pij /tij − M Cij ) qij (pij ) pij

⇐⇒

max π ij = max (pij − c) qij (pij ) , pij

pij

where pij and π ij are the firm’s price and profit inclusive of a tariff. Hence, we get expressions similar to ACDR 2012 for profits and sales, but now they are inclusive of tariffs: π (c, p∗ , w) = (p − c) q (p, p∗ , w)

and

x (c, p∗ , w) = pq (p, p∗ , w) .

Define the firm-level markups as m = ln (p/c) and ν = ln (p∗ /c) . Then from F.O.C., p−c 1 =− , ∗ p ∂ ln q (p, p , w) /∂ ln p so that we get equation (3) from ACDR 2012:   β − d′ (m − ν) m − ln = 0. β − 1 − d′ (m − ν)

(A.3)

(A.3) is the same as in ACDR 2012, since ∂∆/∂ ln pω = 0. Thus, the derivative ∂ ln q (p, p∗ , w) /∂ ln p is also the same as in ACDR 2012. 24

ACCEPTED MANUSCRIPT

T

Remark. The exact form of d (·) and β = γ = −1 allow us to solve equation (A.3) explicitly:       ′ 1 1 p∗ p 1 d +1 m ν = (1 + e = ln 1 + , , or e = ) = m = ln d′ + 2 c 2 2 c 2 − e−(m−ν)

RI P

so that p = (c + p∗ ) /2, which is exactly the formula for the price derived in Section 2. Define p (c, ν) = ceµ(v) , where µ (ν) is the optimal markup defined as a solution of (A.3). Then

SC

the sales and profits of a firm with marginal cost c are

NU

 1−β ed(µ(v)−ν) ∆, x (c, ν, w, L) = Lwγ ceµ(ν) ! eµ(ν) − 1 x (c, ν, w, L) . π (c, ν, w, L) = eµ(ν)

(A.4) (A.5)

MA

Note that the only difference between these formulas from those in ACDR 2012 is term ∆, which is specific to the destination market. This allows us to use the same logic as in ACDR 2012 to get tariff-inclusive total sales and profits earned by firms from country i in country j : = χNi bθi (wi τ ij tij )−θ Lj wjγ p∗j

Πij

= πNi bθi (wi τ ij tij )−θ Lj wjγ p∗j

ED

Xij


1−β+θ


1−β+θ

∆j ,

(A.6)

∆j .

(A.7)

PT

In the case of our preferences, χ = (2 (θ + 2))−1 and π = (2 (θ + 2) (θ + 1))−1 . An important feature of (A.6) and (A.7) is that even though we study non-separable preferences, the term that

AC CE

“absorbs” non-separability, ∆j , enters both Xij and Πij multiplicatively so that Πij = πXij /χ.

(A.8)

Before making the next step let us point out another difference from ACDR 2012. As explained earlier, we look at the tariff-inclusive sales and profits. This means that the total expenditure in country j, Ej = Σj Xij , is no longer equal to the total (after-tariff) revenue earned by firms in this country, Σij Xij /tij . In turn, we need to make adjustments in the trade balance, free entry, and market clearing conditions. As a result, we have Σi Πji /tji = wj Nj Fj , Σi Xji /tji = wj Lj .

(A.9) (A.10)

Nevertheless, even with the adjustments made in the case of import tariffs, we still get Nj = πLj /χFj .

25

(A.11)

ACCEPTED MANUSCRIPT

This is an important result, since it shows that the mass of entrants in each country is not affected by changes in trade costs and/or import tariffs.

T

Remark. As shown in Section 2, in our case, Nj = Lj / [(θ + 1) Fj ] .

RI P

Finally, let us derive the gravity equation. Using the expression for the price inclusive of tariffs, pij (z) = (wi τ ij tij /z) eµ(ln(pj (pj ,ej )z/wi τ ij )) , if z ≥ wi τ ij tij /p∗j (pj , ej ) , ∗

Xij =

SC

we can re-write (A.6) as

Ni bθi (wi τ ij tij )−θ Ej

Σk Nk bθk (wk τ kj tkj )−θ

,

(A.12)

(A.13)

NU

where Ej = Σk Xkj . This equation is similar to equation (13) in ACDR 2012, since to derive it, one needs to use ratios of Xij , i = 1, .., n, so that term ∆j gets cancelled.

MA

Compensating Variation. Consider a small change from t = {tij } to t′ = {tij + dtij } . In order to look at compensating variation, we need to study expenditure function of a representative   consumer in country j denoted by exj , for which we have dexj /dpω,j = q pω,j , p∗j , ej ≡ qω,j for

ED

all ω ∈ Ω. We consider only infinitesimal changes in tariffs, which allows us to ignore creation of

“new” goods and destruction of “old” ones. The reason is that, first, these changes do not affect

PT

entry (as shown above, Nj does not depend on trade costs and import tariffs). Second, there are no fixed costs of production in our setting, which means that the price of firms with the productivity

AC CE

close to the cutoff is almost equal to the choke price, i.e., in the absence of fixed costs of production their quantities, revenues, and profits are almost zero. Hence, as in ACDR 2012, the change in Z expenditures in country j is dexj = Σi [qω,j dpω,j ] dω. Then, using the same logic as in ACDR ω∈Ωij

2012, we can rewrite it as:

d ln ej = Σi

Z

[λω,j d ln pω,j ] dω,

(A.14)

ω∈Ωij

where λω,j ≡ pω,j qω,j /ej . Using (A.12) in (A.4), we get XZ ∞ d ln exj = λij (z) [d ln cij + dmij (z)] dGi (z) , where i

(A.15)

∗ zij

∗ ∗ ∗ ∗ Xij Ni e−(1−β)(ln(z/zij )−µ(ln(z/zij ))) ed(µ(ln(z/zij ))−ln(z/zij )) λij = = R∞ ∗ ∗ ∗ ∗ ′ ′ ′ Ej Σk Nk z ∗ −(1−β)(ln(z /zkj )−µ(ln(z /zkj ))) ed(µ(ln(z /zkj ))−ln(z/zkj ))

(A.16)

kj

is the expenditure share in country j on goods from country i.

Next, using (A.13) together with the assumption of the Pareto productivity distribution, we get d ln exj =

X

λij d ln cij − ρ

i

X i

26

∗ λij d ln zij ,

ACCEPTED MANUSCRIPT

where ρ is the weighted average of the markup elasticities µ′ (v) across all firms. In our case, Z

0

∞

µ′ (v) R ∞ 0

e−(1−β)(v−µ(v)) ed(µ(v)−v) e−θv dv θ+2 . = ′ −µ(v ′ )) d(µ(v ′ )−v ′ ) −θv ′ −(1−β)(v ′ 2 (θ + 1) e e e dv

T

ρ=

d ln exj

RI P

As shown in ACDR 2012, a useful decomposition of the last expression for d ln exj is = Σi λij d ln cij + (−ρ) Σi λij d ln cij + ρ ln p∗j

SC

= (Change in MC) + (Direct Mark-up effect) + (GE markup effect).

(A.17)

NU

Given that in our case ρ > 0, if trade liberalization reduces the choke price p∗j , then markups fall and gains from trade liberalization are higher.

Up to this point, most of our derivations strongly resemble those in ACDR 2012. However, the

MA

next step introduces one of the crucial differences for our case of import tariffs. In particular, given the use of new term ∆j and the fact that by Xij we denote the sales inclusive of tariffs, the labor market clearing condition becomes

ED

Σi Xij /tij = wj Lj ,

PT

which is derived by using the (TB) condition, Σi6=j Xij /tij = Σj6=i Xji /tji , in (A.10). Then we have

AC CE

Lemma 2 For the individual expenditure in country j we have   d ln λjj 1−β 2ρ + d ln wj + d ln µIj . d ln exj = 1 − ρ 1−β+θ θ 1−β+θ

(A.18)

Proof. We can re-write Σi Xij /tij = wj Lj as Σi Xij /tij wj Lj = 1 or Σi ψ ij = 1, where ψ ij ≡ (Xij /tij ) /wj Lj is the after-tariff revenue share of firms from country i in the after-tariff revenues earned by all firms in country j. Note that λij =

Xij Ej

and

ψ ij =

Xij /tij , wj Lj

so that without tariffs, ψ ij = λij . Using (A.6) and totally differentiating Σi Xij /tij wj Lj = 1, we get X i

  ψ ij −θd ln cij + (1 − β + θ) d ln p∗j + (γ − 1) wj − d ln tij − d ln ∆j = 0.

Then by using Σi ψ ij = 1 and the fact that for infinitesimal changes in t, changes in aggregate quantities can be ignored so that d ln ∆j = −d ln µIj , we get: d ln p∗j =


θ 1−γ 1 Σi ψ ij d ln cij + d ln wj + Σi ψ ij d ln t + d ln µIj . 1−β+θ 1−β+θ 1−β+θ 27

(A.19)

ACCEPTED MANUSCRIPT

Next, from λij /λjj = (cij /cjj )−θ , where cjj = wj , we have 1 (d ln λjj − d ln λij ) + d ln wj , θ

Σi ψ ij d ln cij =

d ln λjj 1 + d ln wj − Σi ψ ij d ln λij . θ θ (A.20)

T

and

RI P

d ln cij =

SC

Then by using the expression above and noting that β = γ, we get     d ln λjj 1 ∗ I d ln pj = θ + d ln wj + (1 − β) d ln wj + Σi ψ ij (d ln tij − d ln λij ) + d ln µj 1−β+θ θ
d ln λjj θ 1 = d ln wj + + Σi ψ ij d ln tij µIj /λij . 1−β+θ θ 1−β+θ

NU

Moreover, from (A.20)

ED

MA

d ln λjj d ln λjj 1 Σi λij d ln cij = + d ln wj − Σi λij d ln λij = + d ln wj , and θ  θ θ 
d ln λjj 1−β ρ + d ln wj + Σi ψ ij d ln tij µIj /λij , where d ln exj = 1 − ρ 1−β+θ θ 1−β+θ  2
2 wj Lj Ej tij Ej Ej tij µIj /λij = ∗ = = µIj /ψ ij , so that wj Lj Xij wj Lj Xij /tij

2 Σi ψ ij d ln tij µIj /λij = d ln µIj Σi ψ ij − Σi ψ ij d ln ψ ij = 2d ln µIj ,

PT

resulting in (A.18), which looks the same as equation (18) in ACDR 2012 except for the last term that appears due to tariff revenues being a part of the national income.

AC CE

Finally, note that the compensating variation in the case of import tariffs implies that d ln Wj


= d ln incomej − d ln exj = d ln µIj wj − d ln ej = d ln µIj + d ln wj − d ln exj     d ln λjj 1−β 2ρ = − 1−ρ + 1− d ln µIj , 1−β+θ θ 1−β+θ

which leads to formula (1). In our special case of non-separable quadratic preferences, ρ = 1−β and β = γ = −1 gives η = ρ 1−β+θ =

1 θ+1 ,

θ+2 2(θ+1)

so that we can re-write (1) as:

    2ρ θ d ln λ d ln λ + 1− + d ln µ . d ln µ = − d ln Wj = − (1 − η) θ 1−β+θ 1+θ θ

(A.21)

Appendix B: Equilibrium Conditions Derivations for Lagrangian Multipliers. From (6), λi pq c = (α − ηQi − γq c ) q c = (λi wi c∗ii − γq c ) q c . By integrating both parts over all varieties sold in country i, we get Z Z λi p (ω) q c (ω) dω = ei λi = λi wi c∗ii Qi − γ (q c (ω))2 dω, Ωi

Ωi

28

so that

ACCEPTED MANUSCRIPT

λi =

γ

R

Ωi

(q c (ω))2 dω

wi c∗ii Qi − ei

,

(B.1)

Z

(q c (ω))2 dω =

i 1

1 wi (λi )2 (c∗ii )θ+2 h e 1 −θ θ+1 M −θ e M −θ w + M (t τ ) (w ) = wλi c∗ii c c Qi . M i ji ji i i i j j 2γ 2 (θ + 1) (θ + 2) γ θ+2

SC

Ωi

h i
−θ
1 1 −θ θ+1 e M −θ + M (t τ ) (w ) λi (c∗ii )θ+1 Mie wi cM , c ji ji i i j j 2γ θ + 1

RI P

Qi =

T

where from (7),

Thus, (B.1) can be re-written as λi (wi c∗ii Qi − ii ) = wi λi c∗ii Qi / (θ + 2) , so that

θ + 2 ei , and (B.2) θ + 1 wi c∗ii 2  Z θ + 2 ei θ + 2 ei 1 2 2 c (q (ω)) dω − η (Qi ) = α = αQi − γ − ei λi , −η θ + 1 wi c∗ii θ+1 θ + 1 wi c∗ii Ωi

MA

ei λi

NU

Qi =

which we can solve for ei λi :

θ + 2 ei η , θ + 1 wi c∗ii

ED

wi cii λi = α −

where

(tji − 1) (tji − 1) ei =1+ Mji r¯ji = 1 + tji Mje wi wi Li 2γ (θ + 2) tθ+2 ji 

θ

PT

with yi ≡

tji −1 2γ(θ+2)(tji )θ+1

wi c∗ii w j cM j τ ji

(B.3)

wi c∗ii wj cM j τ ji

!θ

c∗ii (wi λi c∗ii ) ,

Mje . Then by using (B.2) in the equation above, we get:

  θ+2 = (1 + αc∗ii yi ) / 1 + η yi , so that θ+1     θ+2 θ + 2 ei θ+2 1 / 1+η η ∗ yi . = α−η wi cii λi = α − θ + 1 wi c∗ii θ + 1 c∗ii θ+1 R 1 1 Welfare. Since γ Ωi (q c (ω))2 dω = wλi c∗ii θ+2 Qi = (α − ηQi ) θ+2 Qi , per capita welfare is   Z α (2θ + 3) γ η θ+1 2 2 c (q (ω)) dω − (Qi ) = Q − ηQ . Ui = αQi − 2 Ωi 2 2 (θ + 2) θ+1

AC CE

ei wi

Appendix C: Optimal Import Tariff As discussed in Section 2.4, we need to study the behavior of z ≡ wλH c∗HH . Using the new notation and replacing λF (c∗F F )θ+2 from the (TB) condition, we get: "  M θ θ #
θ cH w (F E)H : z (c∗HH )θ+1 LH + LF = Const1 cM , H M θ+1 t cF τ F H (F E)F :

z (c∗HH )θ+1

"

LF



cM H τ HF cM F τFH

θ

θ+1 w2θ+1 −θ w + L (τ ) H F H tθ+1 tθ+1

29

#

= Const1 cM F

(C.1)
θ

,

(C.2)

ACCEPTED MANUSCRIPT

(z) : z +

η θ+2 [1 + zyH c∗HH ] = α. c∗HH θ + 1

(C.3)



cM H M cF τ F H

θ

wθ = LF



cM cM H τ HF ∗ H M cF τ F H cM H

θ

w2θ+1 + LH



cM H M cF τ F H

RI P

LH tθ+1 + LF

T

Combining (C.1) and (C.2) gives θ

wθ+1 .

(C.4)

From the implicit function theorem applied to (C.4) and (C.2), we get:

w



 = 1 +

θLF



cM H τ HF cM F τFH

∗

cM H cM H

θ

w2θ+1 + LF

(θ + 1) LH tθ+1



θ

SC

t

w′

cM H cM F τFH

wθ

−1  

,

and

(C.5)

MA

 h  i−1 
η θ+2 M M θ θ 1 + (t − 1) L + L τ /c w = α. w L c F H F H HF F c∗HH θ + 1

ED

(z) : z +

NU


M θ w 2θ+1 + (θ + 1) L w θ+1 (2θ + 1) LF cM 1 z′ w′ H H τ HF /cF = (C.6) . − −
θ M M ∗ 2θ+1 θ+1 t (θ + 1) z (θ + 1) w L cH τ HF /cF w + LH w i h 
M θ wθ Finally, from (F E)F , zyH c∗HH = (t − 1) LF / w LH + LF cM , so that τ /c HF H F (c∗HH )′ c∗HH

PT

After applying the implicit function theorem to the equation above, we get    ′ ∗ η θ + 2  (cHH )  (t − 1) LF z′ − ∗ 1+ 
θ   −  cHH θ + 1 c∗HH w L + L cM τ /cM wθ w L +L

AC CE

+

H

LF


M θw w LH + LF cM H τ HF /cF

F

H

HF

F

w′  LH + (θ + 1) LF 2 w θ

Using the expression for (c∗HH )′ /c∗HH , we can re-write it as z′

LF

M M H F cH τ HF /cF  
M θ θ  cM w  = 0. H τ HF /cF


θ

wθ



  −1 (t − 1) LF 1 θ+2 1  θ+2   1+  1+ η =
tc∗HH θ + 1 (θ + 1) z θ + 1 c∗HH M θ wθ w LH + LF cM τ /c HF H F   (t − 1) LF ∗ 1 + 
θ   M M w LH + LF cH τ HF /cF wθ     M cH τ HF θ 2θ+1 θ+1 + (2θ + 1) L (θ + 1) L w w ′ H F tLF tw cM   F  ∗ 1 − −     M  θ M (θ + 1) w c τ cH τ HF θ θ LH wθ+1 + LF HcMHF w2θ+1 w LH + LF w cM F F 
M θ wθ (t − 1) LF tw′ LH + (θ + 1) LF cM H τ HF /cF . + 
θ
θ  LH + LF cM τ HF /cM wθ w L + L cM τ /cM wθ w η

H

F

H

HF

H

F

30

F

ACCEPTED MANUSCRIPT

(t − 1) tw′ 2 θ (θ + 1) w

1−

LH

RI P

in squared brackets that can be re-written as 
LF  sign z ′ = sign  
M θ wθ w LH + LF cM τ /c H HF F

T

The first multipliers are positive so that to know the sign of z ′ , we need to look at the last term

SC

M LH + LF cM H τ HF /cF #
M θ w 2θ+1 τ /c (θ + 1) LH wθ+1 + (2θ + 1) LF cM tw′ HF H F . +1 −
M θ w 2θ+1 (θ + 1) w LH wθ+1 + LF cM τ /c H HF F


θ

wθ

Note that the (FE)H condition can be written as r˜HH + r˜HF = wfe (θ + 1) , so that from (C.1),

Moreover, from (C.4),

w LH + LF Finally,


M θ wθ cM H τ HF /cF

=

ED



LF

MA

NU


M θ θ r˜HF /˜ rHH = LF cM w /LH tθ+1 , and from (C.5) H /τ F H cF   −1 ˜ 
tw′ HF M θ θ+1 r = (θ + 1) + 1 + θ M τ /c w . HF H F (θ + 1) w r˜HH M LF cM H /τ F H cF

tθ+1 LH + LF


θ

(C.7)

wθ

r˜HF = .
θ M r˜HH + r˜HF wθ cM H τ HF /cF

PT


M θ w 2θ+1
(θ + 1) LH wθ+1 + (2θ + 1) LF cM r˜HF H τ HF /cF M θ = (θ + 1) + θwθ+1 cM .
H τ HF /cF θ M M θ+1 2θ+1 r ˜ ˜HF HH + r LH w + LF cH τ HF /cF w

AC CE

Using all the relationships together, we get ! " ′
L r˜HF tw r ˜ HF H (t − 1) θ2 1 − + sign z ′ = sign
θ M M θ r˜HH + r˜HF (θ + 1) w r ˜ ˜HF HH + r LH + LF cH τ HF /cF w  
r˜HF tw′ M θ − (θ + 1) + θwθ+1 cM H τ HF /cF (θ + 1) w r˜HH + r˜HF "
M θ wθ LF cM tw′ r˜HH r˜HF H τ HF /cF 2 (t − 1) θ − (θ + 1) + = sign
θ M θ r˜HH + r˜HF (θ + 1) w r˜HF LH + LF cM τ /c w HF F   H

r˜HH M M θ θ+1 θ+1 M M θ . cH τ HF /cF − (θ + 1) + 1 − θw +1 + θ cH τ HF /cF w r˜HF By noting that w′ > 0 and rearranging the term in brackets, we have " !#
M θ wθ
LF cM H τ HF /cF ′ 2 (t − 1) θ −θ . sign z = sign
M θ wθ LH + LF cM τ /c HF H F Thus, dz/dt R 0 for t R topt ≡ 1 + 1θ + 1θ LLH F



cM F M cH τ HF

θ

w−θ . Given (C.5), w rises with t, meaning

that topt is unique. Finally, recall that as z falls, welfare rises. Hence, we proved Proposition 1.

31

!

−1

!

ACCEPTED MANUSCRIPT

Appendix D: No Metzler Paradox

T

Given the Pareto cost distribution assumption, the average price in country i can be written as

RI P

p¯i = (2θ + 1) wi c∗ii /2 (θ + 1) . First, let us look at wH c∗HH = wc∗HH . From (C.5) and (C.6),   
M /cM θ w 2θ+1 ′ ′ τ c θL z 1 w F HF H F  t  ,  (wc∗HH )′ = wc∗HH − + 1 −
θ (θ + 1) z t M M (θ + 1) LF τ HF cH /cF w2θ+1 + LH wθ+1 w

SC

  ′ /w is positive where for t ∈ 1, topt , z ′ /z < 0, 0 < tw′ /w < 1, and the multiplier in front of twH

NU

and less than 1. Hence, we get

d¯ pH 2θ + 1 d (wc∗HH ) = > 0, dt 2 (θ + 1) dt

MA

i.e., the average price at Home rises with t. Next, let us look at c∗F F . From the (TB) condition,    θ τ HF cM w2θ+1 θ+2 H η = θ+1 wλH (c∗HH )θ+2 λF (c∗F F )θ+2 = (c∗F F )θ αc∗F F − , θ+1 t τ F H cM F

PT

ED

where the left-hand side is monotonically increasing in c∗F F . From the (F E)F condition, " #   −θ M θ
θ τ HF cM w2θ+1 θ+2 τ HF cH −θ 1 ∗ H wλH (cHH ) (τ F H ) . LF + LF = Const1 cM F M M θ+1 θ t w τ F H cF τ F H cF Given that w rises with t, the term in the squared brackets above falls, so that the multiplier

AC CE

in front of these brackets has to rise, meaning that c∗D rises too. Thus, we have d¯ pF 2θ + 1 d (c∗F F ) = > 0, dt 2 (θ + 1) dt

so that the average prices everywhere rise with an increase in t, implying no Metzler paradox.

Appendix E: Role of Variable Markups A rise in misallocation distortion in the case of falling trade costs. Here we show that the Home country’s covariance introduced in Section 4 is negative for a marginal fall in τ F H . First, note that dqHi (c) = d (Li wH λi (c∗Hi − c) /2γ) = Li [d (wH λi cHi ) − cd (wH λi )] /2γ, i = H, F, so that using the Pareto cost distribution assumption, we get (here we use dτ HF = 0), Z

c∗Hi

1 dGH (c) MHi LF
= ∗ 4γwH LH (θ + 1) (θ + 2) GH cHi 0 h i
∗ (2θ + 1) (θ + 2) (τ Hi )2 cHi d (wH λi cHi ) − 2θ2 + 3θ (τ Hi c∗Hi )2 d (wH λi ) . MHi wH LH

pHi (c) d [qHi (c)]

32

ACCEPTED MANUSCRIPT

Z

c∗HH

d [cqHH (c)]

0

dGH (c)
+ MHF GH c∗HH

Z

c∗HF

d [τ HF cqHF (c)]

NU

MHH

SC

RI P

T

Hence, the expression for the covariance of markups and labor share changes is  X  Z c∗ Hi MHi dl (ω, i) dGH (c)
= cov m (ω, i) , pHi (c) d [qHi (c)] LH wH LH 0 GH c∗Hi i h i (2θ + 1) MHH LH c∗HH d (wH λH c∗HH ) + MHF LF (τ HF )2 c∗HF d (wH λF c∗HF ) = 4γwH LH (θ + 1)
i h 2θ2 + 3θ − MHH LH (c∗HH )2 d (wH λH ) + MHF LF (τ HF c∗HF )2 d (wH λF ) . 4γwH LH (θ + 1) (θ + 2) P R The total labor size at Home does not change. Hence, i ω∈ΩHi [dl (ω, i) /LH ] dω = 0, or 0

dGH (c)
= 0, GH c∗HF

or

i h X θ MHi Li (θ + 2) (τ Hi )2 c∗Hi d (wH λi c∗Hi ) − (θ + 1) (τ Hi c∗Hi )2 d (wH λi ) = 0. 2γ (θ + 1) (θ + 2)

MA

i

Multiplying the expression above by 2γ (2θ + 3) and rearranging the terms result in

ED

=

i 2θ + 3 h MHH LH c∗HH d (wH λH ) + MHF LF (τ HF c∗HF )2 d (wH λF ) θ+2 i 2θ + 3 h MHH LH c∗HH d (wH λH c∗HH ) + MHF LF (τ HF )2 c∗HF d (wH λF c∗HF ) . θ+1

AC CE

PT

We can use this equation in the expression for the covariance to get:   MHH LH c∗HH d (wH λH c∗HH ) + MHF LF (τ HF )2 c∗HF d (wH λF c∗HF ) dl (ω, i) cov m (ω, i) , . = LH 4γwH LH (θ + 1)2 Finally, note that from (7)and (11), we get d (wH λi c∗Hi ) =   dl (ω, i) cov m (ω, i) , = LH

 1 θ+2  ∗ η dcii / (c∗ii )2 , and τ Hi θ + 1   dc∗HH dc∗F F (θ + 2) η MHH LH ∗ . + MHF LF cHH wH c∗F F 4γwH LH (θ + 1)3

From Proposition 1, c∗HH and c∗F F fall as τ F H falls, so the covariance is negative for dτ HF < 0. A fall in the average markup in the case of a rising import tariff. By definition, 1 2θ − 1 M + tMx∗ 1 2θ − 1 tθ+1 + wθ = , so that 2 θ − 1 M + Mx∗ 2 θ − 1 tθ+1 + twθ    2θ − 1 1 θ−1 θ+1 dw 2θ θ θ = − w − θt w 1 + − t . t 2 (1 − t) θw dt θ 2 (θ − 1) (tθ+1 + twθ ) m ¯H =

dm ¯H dt

When t starts to rise from 1 (consider t between 1 and 1 + 1/θ), w rises too (see (C.5)), so that for small increases in t we get dm ¯ H /dt|t=1 < 0. In addition, it can be shown straightforwardly that we get dm ¯ H /dt|t=topt < 0 so that a small deviation from topt by reducing t raises the average markup at Home. 33