Intermediate input tariffs, oligopoly, and free entry:theory and evidence∗

Tomohiro Ara†

Fukushima University

Arpita Chatterjee‡

University of New South Wales

Arghya Ghosh§

University of New South Wales

Hongyong Zhang¶

RIETI

January 8, 2021

Abstract

How does a reduction in tariff on intermediate goods imports affect the number of importers(extensive margin), average volume of imports per firm (intensive margin), and welfare, andwhat, in turn, are the implications for optimal tariffs? Using China’s detailed transaction levelimport data and WTO accession related reductions in import tariffs in post-2001 period, we findthat both extensive margin and average intensive margin increase. Exploring this question the-oretically in a framework with (i) free-entry, homogeneous-product, downstream oligopoly, and(ii) a perfectly competitive upstream sector, we find that the extensive margin always increaseswith reduction in tariffs, while the intensive margin expands – as in the data – if and only ifthe demand function is strictly convex (e.g., a constant elasticity demand function). Concerningoptimal tariff, the classic result – optimal tariff equals the inverse of export supply elasticity– holds if and only if the demand is linear. For all concave demand functions, optimal tariffis strictly positive and larger than the inverse of export supply elasticity. However, negativetariff, i.e., import subsidy, can be optimal for empirically relevant (i.e., strictly convex) demandfunctions. Negative optimal tariff is more likely and in general optimal tariff is lower under (a)product differentiation, (b) oligopolistic upstream sector, and (c) the presence of active competi-tion policy.

Keywords: Oligopoly, free entry, tariff, extensive margin, intensive margin, intermediate in-putJEL Classification Numbers: F12, F13

∗This study was conducted as a part of the Project “Analyses of Trade Costs” undertaken at the Research Instituteof Economy, Trade and Industry (RIETI). Financial support from the RIETI and the JSPS KAKENHI Grant Number16K17106 is gratefully acknowledged. The usual disclaimer applies.

†1 Kanayagawa, Fukushima 960-1296, Japan. Email address: tomohiro.ara@gmail.com‡UNSW Business School, Sydney, NSW, 2052, Australia. Email address: arpita.chatterjee@unsw.edu.au§UNSW Business School, Sydney, NSW, 2052, Australia. Email address: a.ghosh@unsw.edu.au¶1-3-1 Kasumigaseki Chiyoda-ku, Tokyo 100-8901 Japan. Email address: zhang-hong-yong@rieti.go.jp

1 Introduction

Trade in the modern economy is largely in car parts, not cars (Krugman, 2018). A significantfraction of world trade – in fact, nearly two-thirds according to Johnson and Noguera (2012) – isin intermediate goods. A reduction in tariffs across the globe has led to tremendous growth inintermediate goods trade (Hummels et al., 2001; Yeats, 2001; Yi, 2003). Trade in final goods hasalso increased but the magnitude has been significantly larger for intermediate goods.

Figure 1 – China’s imports, 1996-2010

Source: Authors’ own compilation based on the World Integrated Trade Solution (WITS) database.

Note: We divide our full samples into intermediate goods and final goods using the classification of the Broad EconomicCategories (BEC) by the United Nations. Please refer to section 2 for details.

Figure 1, for example, presents China’s import value from the rest of the world during the period1996-2010. It is evident that intermediate goods imports account for a large and growing share ofimports. China’s imports drastically increased after China’s entry to WTO in 2001, and it is moreprominent for intermediate goods than for final goods.

Increased importance of intermediate goods trade has led to development of new models thatembed vertical specialization and rich organizational structure. However, barring a few papers(discussed later in the introduction), little is known about optimal trade policy and the effect oftariff reductions on intermediate goods imports. We take a step in filling this important gap.1

How do tariff reductions on intermediate goods imports affect the number of importers (extensivemargin), average volume of imports per firm (intensive margin), and welfare, and what, in turn,are the implications for optimal tariffs? This is the key question we address in this paper.

1Grossman and Helpman (2020) note that this gap (in knowledge) might be due to the fact that US tariffs onintermediates have been very low, at least until the recent hike in tariffs on China’s imports. For example, average UStariffs applied to imports of intermediate goods were a meagre 0.9% in 2017. Further, tariffs applied to consumptiongoods in the US during 2010-2017 were at least four times as large as those applied to intermediate goods. Bown andCrowley (2016) also report similar findings: MFN tariffs applied by G20 countries to final goods imports are 70-75%higher than those levied on intermediate goods imports. This trend however changed with the introduction of Trumptariffs which are disproportionately on intermediate goods imports.

1

We examine detailed transaction level import data and WTO accession related reductions inimport tariffs in China in the post-2001 period and show that both the number of importers andaverage imports per firm increase. Previous papers in this literature (e.g., Buono and Lalanne,2012; Debaere and Mostashari, 2010) find that tariff reductions do not necessarily lead morefirms to export or import. Instead, the effect is primarily on the intensive margin. Incumbentfirms increase their shipments in response to lower tariffs. We make a clear distinction betweenfinal goods trade and intermediate goods trade, and find that reduction in tariffs due to China’sWTO accession has a significant impact on the extensive margin in intermediate goods trade.

During 2000-2007, intermediate goods imports, extensive, and intensive margins grew at anannual rate of 23.5%, 8%, and 15.5% on average at the country-product level. We find that reduc-tion in average tariff by 1.2% per year accounts for approximately 16% of the import growth, 6%of the extensive margin growth, and 20% of the intensive margin growth. Difference-in-differencespecification attributes even larger role to extensive margin accounting for approximately a quar-ter of the overall response in intermediate goods (in the post-WTO period).

To examine the effect of tariffs on extensive margin, intensive margin, and welfare, we developa free-entry, homogeneous-product oligopoly framework where firms in one country (say Home)specialize in final goods and purchase intermediate goods from perfectly competitive upstreamfirms from the rest of the world. Both the number of Home importers and average volume ofimports increase with a reduction in tariffs – as in the data – if and only if demand functionsare strictly convex but not too convex (e.g., constant elasticity demand, log-linear demand). Forconcave demand functions, the number of importers increases but the average volume of importsdecrease with tariff reductions. The opposite is true for demand functions which are too convex.

As in most trade models, imposing tariff improves terms-of-trade (i.e., reduces input price inthis case) and increases welfare. However, two additional distortions are present in our model:excessive entry and underproduction. Too many firms enter and each firm produces too little out-put in free-entry equilibrium. Tariffs exacerbate welfare loss from underproduction but mitigatethe welfare loss from excessive entry. These two opposing forces exactly offset each other whenthe demand function is linear. Thus, despite oligopolistic downstream sector, in case of lineardemand, optimal tariff equals the inverse of export supply elasticity as it would have been underperfect competition. However, optimal tariffs are strictly lower (higher) than this perfect compe-tition benchmark when the demand functions are strictly convex (concave). In fact, a negativeoptimal tariff can arise for widely used constant elasticity demand function and more generallyfor empirically relevant, strictly convex demand functions.2

Few remarks regarding our modeling choices are in order. First, we focus on market interac-tions (rather than bilateral negotiations) as a large fraction of intermediate goods is internation-ally traded in markets among anonymous final goods producers and intermediate goods suppliers.

2Empirically we find that both extensive and intensive margins expand with a reduction in tariffs. This possibilityarises in our theoretical framework when demand functions are strictly convex. In this sense, strictly convex demandfunctions are empirically relevant.

2

Second, market-based interactions guide us to consider homogeneous (rather than differentiated)intermediate goods as homogeneous intermediate goods tend to be exchanged through marketswhile differentiated intermediate goods tend to be exchanged through non-market mechanisms(Nunn, 2007). Third, we work with a perfectly competitive upstream sector for simplicity, but allour results go through in a vertical oligopoly setting where imperfect competition is present inboth upstream and downstream stages of production (see section 5.1). Finally, we assume thatHome industry is oligopolistic and Home firms produce homogeneous final goods. In doing so, weacknowledge the empirical regularity that a small fraction of firms dominates international tradeand follow the literature in trade theory where differentiated goods go along with monopolisticcompetition and homogeneous goods with oligopolistic competition.3 Our results are robust toproduct differentiation. Nevertheless, a homogeneous product sector provides a nuanced effect oftariffs on entry – different from the one that arises in a differentiated product setting – which inturn influences the sign (and magnitude) of optimal tariffs.

A handful of papers have examined the effect of trade policy on global production linkages.4

Blanchard (2007) shows that in the presence of vertical FDI, governments of source countries havean incentive to improve market access for imported inputs from foreign affiliates of source countryfirms. Blanchard and Matschke (2015) provide empirical support for this argument. Our primaryfocus is, however, on outsourcing rather than FDI. Grossman and Helpman (2020), Antras andStaiger (2012), and Ornelas and Turner (2008, 2012), investigate trade policy in the context ofintermediate goods trade and international outsourcing. Antras and Staiger (2012) study tradeagreements, and Ornelas and Turner (2008, 2012) consider organizational structure. Grossmanand Helpman (2020) look at effects of unanticipated tariffs on intermediate goods in a setting withfirm-to-firm supply relationships. At the heart of all these papers is contractual incompletenessand non-market interactions (e.g., matching, bargaining). We highlight market interactions andexploit the data availability to connect theory and evidence.5

In terms of modeling, our paper is related to the literature on vertical oligopoly and tradepolicy (e.g., Ishikawa and Lee, 1997; Ishikawa and Spencer, 1999; Chen et al., 2004). These papersstudy strategic interaction among firms in vertically related markets and examine effects of exportsubsidies on intermediate goods and welfare, treating market structure as (typically) exogenous.

3See Neary (2003, 2016) for a comprehensive treatment of oligopoly in a general equilibrium framework where firmsare large in their own markets but small in the economy. The number of firms is fixed, and consumers have quadraticpreferences defined over a continuum of goods (which allows for aggregation). Our model is narrower in scope, but wework with free entry and general demand functions, both of which play a crucial role in determining the welfare effectof tariffs.

4Costinot and Rodrıguez-Clare (2014) present a comprehensive survey on the welfare effects of trade liberalizationunder a variety of specifications. They find that the presence of intermediates amplify the welfare effect of tradeliberalization under both perfect competition and monopolistic competition.

5Regarding bilateral negotiations versus market interactions, the suitability of modeling approaches depends on aparticular application (e.g., a particular industry or a particular policy question). Inderst (2010) notes that while somemodels view vertical relations through the lenses of bilateral contracting, others view vertical interactions through thelenses of a “market interface.” He also adds that a framework of bilateral negotiations might be more applicable intight bilateral oligopolies with heterogeneous products, whereas a market interface with uniform contractual terms ismore applicable for a relatively homogeneous upstream sector.

3

In our model, in contrast, market structure is endogenous in the sense that it critically affectsthe responses of entry/exit (extensive margin) to tariff changes and hence optimal trade policy. Insingle-stage oligopoly models, Horstmann and Markusen (1986), Venables (1985), and Etro (2011)have shown that entry/exit considerations can alter optimal trade policy. Closely related to ourpaper is linear Cournot analysis in Bagwell and Staiger (2012a). They find that limiting exportsubsidies – an often used feature of trade agreements – can potentially benefit consumers whenfree entry is invoked in linear oligopoly. Our work complement their analysis by studying multiplestages of production under general demand functions which generates rich set of possibilitiesregarding the effect of input tariffs on extensive and intensive margins as well as welfare.

We also contribute to the literature on optimal tariffs. A classic result in this literature is thatoptimal tariff for a large economy is strictly positive and equal to the inverse of export supplyelasticity. Broda et al. (2008) provide empirical support for this result. They find that, indeed,countries set higher tariffs on goods that are supplied inelastically. In a two-sector setup withproduction linkages, Caliendo et al. (2017) show that the effect of tariffs on entry can reversethe traditional positive optimal tariff argument. In our setup, optimal tariff could be higher orlower than the inverse of export supply elasticity, and in fact be negative when free entry andoligopolistic competition are present in the downstream sector.

There is a large body of work on optimal tariffs under imperfect competition. In monopolisticcompetition models, Gros (1987) and Flam and Helpman (1987) have shown that, even in a smalleconomy, optimal tariff is strictly positive. Consumers underspend on domestic varieties sincethey buy foreign varieties at opportunity costs but domestic varieties come with added markups,and imposing tariffs corrects this distortion. Demidova and Rodrıguez-Clare (2009) and Costinotet al. (2020) study optimal tariffs in models of monopolistic competition and heterogeneous firms.Despite the negative impact of productivity, these papers find that import tariff still improveswelfare. Felbermayr et al. (2013) considers optimal tariffs in a large economy with heterogeneousfirms, extending the results of Gros (1987) and Demidova and Rodrıguez-Clare (2009). Venables(1987) and Ossa (2011) show that import tariffs can induce welfare-inducing entry at Home byprompting consumers to switch from imported varieties to Home varieties.6

The literature cited above primarily examines tariffs on differentiated final goods producedunder monopolistic competition. We complement the literature by developing a framework withtariffs on homogeneous intermediate goods produced under free-entry oligopolistic competition. Akey difference from these papers is the effect of tariffs on entry. Free entry is socially excessivein our homogeneous product oligopoly setup with perfectly competitive upstream firms (Mankiwand Whinston, 1986). Thus, instead of lowering welfare, tariffs improve welfare by curbing entryin our basic framework considered in sections 3 and 4. Nevertheless, a negative tariff, i.e., animport subsidy could arise as optimal policy due to underproduction in oligopoly. Section 5 allows

6See Ossa (2014) for a recent comprehensive discussion of optimal tariffs (cooperative and non-cooperative) in amulti-sector, general equilibrium model that nests traditional (i.e., terms-of-trade), new trade (i.e., profit-shifting), andpolitical-economy motives for protection.

4

Figure 2 – China’s import tariffs, 1996-2010

Source: Authors’ own compilation based on the WITS database. We use China’s simple average of effectively appliedad-valorem tariffs at the HS 6-digit product-year level.

for imperfect competition in the upstream sector, product differentiation in the downstream sec-tor, and active competition policy and shows that optimal tariff is likely to be lower under thesealternative specifications.

2 Evidence

2.1 Background and data

This section reports evidence on the effects of tariff reduction on aggregate volume of intermediateimports, average volume of imports per firm, and the number of importers. We first describeChina’s accession to WTO and present supportive evidence to show that indeed it is a large policychange to Chinese firms. Subsequently, we describe Chinese Customs data used in our analysis.Finally, we conduct a set of robust empirical exercises in section 2.2 and show that both intensiveand extensive margins of intermediate goods imports increase post-WTO in China.

2.1.1 China’s accession to WTO

China joined WTO in 2001. Tariff reductions in 2002, right after the WTO accession, were verydrastic. We measure these tariff reductions endorsed by China’s WTO accession using China’s sim-ple average of effectively applied ad-valorem tariffs at the HS 6-digit product-year level by part-ner countries. The tariff data is obtained from the Trade Analysis Information System (TRAINS)database at the WITS website. It covers more than 5000 products, and for each product at theHS 6-digit level, the tariff database provides detailed information on tariff lines, average, mini-mum and maximum ad-valorem tariff duties. Importantly, our tariff dataset covers 159 countriesincluding both WTO members and non-WTO members.

5

Figure 3 – China’s tariff reductions by initial tariff levels across HS 6-digit products

(a) Intermediate goods (b) Final goods

Source: Authors’ own compilation based on the WITS database.

Figure 2 plots the time trend of China’s simple average applied tariffs during 1996-2010. Thefigure shows that import tariff rates dropped substantially in 1997 but remained largely un-changed during 1997-2001, especially for intermediate goods. However, it started to decreasein 2002 until it stabilized in the late 2000s. Specifically, China’s import tariffs on intermediategoods decreased from 14.2% in 2000 to 6.0% in 2010.7

While Figure 2 shows a general pattern of changes in import tariffs by two production stages(i.e., intermediate goods and final goods), there are large variations across products. Figure 3ashows that the relation between tariffs in 2001 and the tariff reductions for intermediate goodsbetween pre-WTO (2000-2001) and post-WTO (2002-2007) periods across 6-digit level products.Clearly, there is a strong positive correlation between the initial tariff level and the change ofthe tariff because of the WTO accession. That is, intermediate goods with higher initial levelsexperienced larger reductions. As a result, import tariff levels in the post-WTO period are moreuniform across products than the levels in the pre-WTO period. The pattern is quite similarfor final goods (Figure 3b). Thus, a product’s relative protection in 2007 is well explained by itsinitial protection, rather than domestic political economy, pointing towards exogeneity of the tariffchanges.

2.1.2 Trade data

Our data on China’s imports come from the census of annual firm-level import transactions ofChinese firms during 2000-2007, collected by the China Customs. The data contain the tradevalue, quantity, origin and destination countries at the HS 8-digit product classification. Theoriginal data are at the firm-product-country level and we aggregate them at the product-country

7Similar trend is observed for import tariffs on final goods which decreased from 21.3% to 10.8% during 2000-2010.

6

Table 1: Effect of tariffs on intensive and extensive margins of intermediate input import

(1) (2) (3) (3)− (2) (4)

Intermediate goods All Pre-WTO Post-WTO Difference Std Error

Import value 3.765 3.492 3.833 0.342∗∗∗ 0.012(2.754) (2.587) (2.790)

Extensive margin 1.636 1.523 1.663 0.140∗∗∗ 0.005(1.111) (0.991) (1.137)

Intensive margin 2.130 1.968 2.170 0.202∗∗∗ 0.010(2.155) (2.078) (2.171)

Tariff rate 9.177 13.113 8.203 −3.909∗∗∗ 0.028(6.540) (8.541) (5.525)

Final goods

Import value 3.227 3.006 3.280 0.274∗∗∗ 0.016(2.642) (2.523) (2.667)

Extensive margin 1.518 1.415 1.543 0.128∗∗∗ 0.006(1.033) (0.912) (1.058)

Intensive margin 1.709 1.591 1.737 0.146∗∗∗ 0.013(2.094) (2.078) (2.096)

Tariff rate 13.059 19.191 11.594 −7.597∗∗∗ 0.046(8.222) (9.000) (7.300)

Note: Import value, extensive and intensive margins are in logarithm. Standard deviations in parentheses. Thesample size is 175,598 in pre-WTO (2000-2001) period and 309,052 in post-WTO (2002-2007) period, respectively.

level to obtain our key variables, i.e., total imports, the number of importers (extensive margin)and average import value per firm (intensive margin). Then, we use the publicly available 6-digitlevel concordance tables to adjust the product codes consistent over time. The number of countriesis 159 and the number of products is about 5,000, which covers more than 95% of China’s imports.The import value is measured in thousand U.S. dollars and we convert it to thousand ChineseRMB using the annual exchange rate published by China’s National Statistics Bureau.

As mentioned in Figure 1 in Introduction, we divide our full samples into intermediate goodsand final goods using the classification of the Broad Economic Categories (BEC) by the UnitedNations.8 According to this classification, intermediate goods include: (1) industrial supplies notelsewhere specified; (2) fuels and lubricants other than motor spirit; (3) parts and accessories ofcapital goods; (4) parts and accessories of transport equipment; and (5) food and beverages mainlysupplied for the industry. On the other hand, capital goods and consumption goods are classifiedas final goods.9

8For details, please visit: https://unstats.un.org/unsd/tradekb/Knowledgebase/50090/Intermediate-Goods-in-Trade-Statistics.

9In this study, we exclude imports entering China as processing trade and focus only on ordinary imports. Aspointed out by Dai et al. (2016) or Liu and Qiu (2016), processing trade is exempt from import duties on importedinputs and materials, and the effective tariff rate is zero for processing firms in both pre-WTO and post-WTO periods.Also, processing firms are likely to have different production functions and importing behaviors from non-processingfirms, because they are provided with technologies, designs and intermediate goods by their foreign partners.

7

Table 1 reports the summary statistics of the main variables used in this study. The totalimports, extensive and intensive margins are mean values across the HS 6-digit product-countryover the entire sample period (2000-2007). Overall, China imports more intermediate goods thanfinal goods on average. Imports in both intermediate goods and final goods increased significantlyfrom the pre-WTO (2000-2001) period to post-WTO (2002-2007) accession period. Importantly,both extensive and intensive margins of intermediate goods are larger than those of final goodsduring the period 2000-2007, and the growth of the two margins in intermediate goods are largerthan those of final goods from the pre-WTO period to post-WTO accession period. Moreover, bothmean tariff rates and its standard deviations decreased significantly because of China’s WTOaccession.

2.2 Empirical analysis

To examine the effect of import tariffs on extensive and intensive margins of intermediate goodsimports, we conduct a battery of estimations and robustness checks: (i) standard fixed effectsregression with a lagged independent variable (import tariffs), (ii) difference-in-differences (DID)specification, and (iii) Poisson estimation. We show that tariffs significantly affect intermediategoods imports through both extensive and intensive margins.10

We start with the standard fixed effects model. For each product imported from each tradingpartner in each year, the import values are decomposed into the number of importers (extensivemargin) and the average import values (intensive margin). Let Qpct,Mpct and qpct respectivelydenote the import values, the extensive margin, and the intensive margin for product p importedfrom country c in year t, which satisfy Qpct = Mpct ∗ qpct. We estimate the following regression:

ypct = β0 + βln(1 + tariffpct−1) + θpt + γct + αcp + ϵpct, (1)

where ypct is the logarithm of either Qpct, Mpct or qpct, respectively. ln(1 + tariffpct−1) is thelogarithm of (1 + tariffpct−1), where tariffpct−1 is the ad-valorem tariff rates applied on productp imported from country c in year t − 1. We include a set of product-year, country-year, andcountry-product fixed effects. We cluster the standard errors at the product level to deal with thepotential heteroskedasticity and serial autocorrelation. This regression equation is equivalent tothe “within” regressions tested by Buono and Lalanne (2012) and the variation in tariffs come fromtariff changes applied to each foreign country-product pair. This is a comprehensive fixed effectsspecification in which we can control for various patterns of comparative advantage, product andcountry dynamics.

Columns (1) - (3) of Panel A of Table 2 report the results. Compared to Buono and Lalanne(2012), we find a significant extensive as well as intensive margin response of intermediate goodstrade to tariff reductions.

10We have also estimated the effects of tariff on final goods. Detailed results for final goods as well as comparison ofresults between the two cases – final good imports and the intermediate good imports – are available upon request.

8

To further confirm the causal impact of tariff reductions on the margins of imports, we explorethe fact that after China’s accession to WTO, products that had previously been more protected(i.e., products with higher tariffs in 2001) experienced greater tariff reductions under the WTOagreement, whereas previously less protected products (i.e., products with lower tariffs in 2001)observed small changes in tariffs as shown in Figures 3a and 3b. These large variations in extentand the timing of tariff reductions in 2002 allow us to conduct a DID estimation. We compare thechanges in margins of imports in previously more protected products with large tariff reductions(the treatment group) before and after 2001 and the corresponding changes in previously lessprotected products with smaller tariff reductions (the control group) during the same period. Weuse the following specification for our DID estimation:

ypct = β ln(1 + tariffpc2001) ∗ Post2002 + θp + γct + αcp + ϵpct, (2)

where ln(1 + tariffpc2001) is the tariff rate of product p imported from country c in 2001,11 andPost2002 is an indicator of the post-WTO accession period, taking a value of one if t ≥ 2002, andzero otherwise. As before, we include a full set of product, country-year, and country-product fixedeffects in (2). As our main regressor is already at the product-year level, we do not control forproduct-year fixed effects here.

The results are reported in columns (1) - (3) of Panel B of Table 2. Once again the results con-firm that reduction in tariffs raises both margins of intermediate goods trade. While the responseof extensive margin is less than one-fifth of (that of) the intensive margin, in the baseline linearregression, the extensive margin accounts for a quarter of the overall trade response in the DIDspecification. The finding points towards a quantitatively important role that extensive marginplays in the overall response of intermediate goods trade to reduction in import tariffs.

In (2), we use the dummy variable Post2002 to divide the pre-WTO and post-WTO periods.The interaction term ln(1 + tariffpc2001) ∗ Post2002 estimates the average treatment effect whichcompares the difference between the treatment group and control group in the average differencebetween the pre-WTO and post-WTO periods. Since the approach does not consider the year-to-year changes, we also use a more flexible estimation specification to replace the interaction termwith a series of interaction terms as follows:

ypct = βln(1 + tariffpc2001) ∗ dt + θp + γct + αcp + ϵpct, (3)

where dt is a year dummy, taking a value of one if t = 2001, · · · , 2007, and zero otherwise.The regression results are reported in Table A.1 in the Appendix. The estimated coefficients

are statistically significant in all years after 2001 and the magnitudes become larger from 2002onwards. These results imply that the effect of tariff reductions is significantly different betweenthe treatment and control groups. Because we only have two years of data in the pre-WTO (2000-

11Using average tariffs over 2000-2001 or 2000 tariff rates (instead of 2001 tariff rates) yields similar results asimport tariffs did not change much between 2000 and 2001.

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Table 2: Effect of tariffs on intermediate good imports

(1) (2) (3)lnQpct lnMpct ln qpct

Panel A. Linear Regressionln(1 + tariffpct−1) −3.508∗∗∗ −0.468∗∗ −3.040∗∗∗

(1.056) (0.208) (0.936)No. of obs 306, 805 306, 805 306, 805

Adj. R2 0.825 0.925 0.768

Panel B. DID Specificationln(1 + tariffpc2001) ∗ Post2002 −1.105∗∗∗ −0.265∗∗∗ −0.841∗∗∗

(0.305) (0.089) (0.233)No. of obs 408, 971 408, 971 408, 971

Adj. R2 0.776 0.894 0.710

Note: Aggregate Imports, extensive margin, and intensive margin are in logs. Robust standard errors are clustered atthe product level in parentheses. ***, **, and * indicate significance at the 1%, 5% and 10% level, respectively.

2001) period, it is not possible to compare the time trends between our treatment group andcontrol group. This is not a serious problem, however, since, as reported by Lu and Yu (2015) andLiu and Qiu (2016) which use industry-level tariffs during 1998-2007, the high-tariff industriesgroup and low-tariff industries group have quite similar trends in their pre-WTO (1998-2001)period. Nevertheless, the results of the flexible estimations are very similar with those of averagetreatment effect and yearly tariffs.

We also check the robustness of our estimation results by using the fixed effect Poisson modelas an alternative estimation method. Specifically, we use Poisson pseudo maximum likelihood(PPML) model with high-dimension fixed effects, including country-product, product-year andcountry-year fixed effects as (1).12 For simplicity, we focus on the regressions with lagged oneyear tariff rates. The results are reported in Table A.2. We obtain very similar results, as shownin Table 2. In fact, in the Poisson estimations, only the extensive margin of imports registers astatistically significant response to tariff reductions.

In summary, we find that import tariff reductions after China’s WTO accession significantlyincreased China’s ordinary imports of intermediate goods through both extensive and intensivemargins. Reduction in tariffs induced incumbent importers to source more intermediate goodsfrom their trade partners. The effect on the extensive margin is statistically significant in allspecifications. Next, we turn towards building a theoretical framework in section 3 to explore theimpact of trade liberalization on aggregate volume of imports, intensive margin, and extensivemargin of trade. Equipped with the findings in section 3, we characterize and derive the propertiesof optimal tariff in section 4.

12This approach has been used in the literature to account for zeroes in the trade matrix. See, for example, SantosSilva and Tenreyro (2006).

10

3 Model

We develop a model where firms from one country (say Home) produce a final good by using anintermediate good imported from the rest of the world. Home government imposes ad valoremtariffs on the intermediate good. Taking the input price and the tariff rate as given, Home firmscompete in the final good market as Cournot competitors. First, we provide some details of ourmodel (subsection 3.1) and analyze a short-run setting where the number of Home firms is fixed(subsection 3.2). Subsequently, we allow for entry and exit of firms which leads to endogenousdetermination of market structure (subsection 3.3). We focus on the exogenous effect of tariffsthroughout this section. In particular, we examine how reduction in tariffs affect the total volumeof imports, firm-level imports (intensive margin), and the number of importers (extensive margin).Equipped with these findings, in section 4, we turn to characterizing welfare-maximizing optimaltariffs for both settings – with and without free entry.

3.1 Basics

Home comprises unit mass of identical individuals each with L units of labor endowment. Oneunit of labor is required for producing one unit of numeraire good y which is freely traded andproduced under perfect competition. As the price of y is 1 and each individual has L units of labor,the wage rate (i.e., the price of labor) and labor income are unity and L respectively.

Each individual consumes the numeraire good y and a final good Q. Preferences over Q and y isgiven by a quasi-linear utility function V = U(Q)+y where U(0) = 0, U ′(Q) > 0, and U ′′(Q) < 0 forall Q in the relevant range. Assuming L to be high enough, standard utility maximization subjectto budget constraint yields the inverse demand function: P = P (Q)(≡ U ′(Q)), where P denotesthe price of the final good. We assume that P (Q) is twice continuously differentiable, P ′(Q) < 0,and the following inequality holds for all Q > 0:

2P ′(Q) +QP ′′(Q) < 0. (4)

A large class of demand functions – including linear, constant-elasticity demand functions withelasticity greater than unity, and all logconcave demand functions – satisfy (4). This conditionimplies that industry marginal revenue is downward sloping and it rules out demand functionsthat are, loosely speaking, too convex. Occasionally, we will use the notion of slope of elasticity ofdemand function η(Q) = QP ′′(Q)

P ′(Q) which is constant for a class of demand functions including linearand constant-elasticity demand. The condition specified in (4) can be restated as

η(Q) > −2.

Mrazova and Neary (2017) use this condition extensively in their work on demand structure andfirm behavior (see footnote 15 for a detailed discussion).

11

On the production side, there is a large number of identical Home firms which can producethe final good Q. Each Home firm first decides whether to incur entry cost K prior to enteringthe downstream sector and engage in production of the final good Q. One unit of the intermediategood X is required to produce one unit of the final good Q. There are no intermediate good sup-pliers at Home. Upon entry, Home firms import the intermediate good from the rest of the worldand transform it into the final good without incurring any additional costs. Subsequently, theycompete in a Cournot fashion in the final goods market.

The upstream sector comprises large number of price-taking firms (in the rest of the world)producing the intermediate good under a strictly convex cost function. This gives rise to upwardsloping supply for intermediate imports:

r = h(X), (5)

where r denote the per-unit world price of X and h′(X) > 0 for all X > 0. We assume 0 ≤ h(0) <

limQ→0 P (Q) which ensures an interior equilibrium.Home government imposes tariffs on intermediate imports. Tariff revenues are rebated back

to Home consumers. Let t denote the ad-valorem tariff rate chosen by the Home government.Throughout this section we assume that t is exogenously given.13

3.2 Short-run analysis

We begin with short-run analysis where the number of Home firms is exogenously fixed. Let M

denote the number of Home firms producing the final good. Given the input price r, ad valoremtariff t, and one-to-one production technology, Home firm’s unit cost is r(1 + t). Each Home firm i

chooses qi to maximize its profits,

πi ≡(P

(qi +

M∑j =i

qj

)− r(1 + t)

)qi,

taking other Home firms’ quantities (qj) as given. Solving the first-order conditions of firms’ profit-maximization problems yields equilibrium quantity for each Home firm:

q = −P (Q)− r(1 + t)

P ′(Q), (6)

where Q solvesMP (Q) +QP ′(Q)−Mr(1 + t) = 0. (7)

Uniqueness of q and Q follows from (4).

13Our results hold for specific tariffs as well. These two forms of tariffs are equivalent in a framework like ours wherethe upstream sector is perfectly competitive.

12

Figure 4 – Equilibrium with fixed number of firms

!, #

$, %

&

&

'

'

(

(

)

)

! = !($)

-(%,., /)

0!

#

(1 + /)#

0$ = 0%

(a)

#

%

'

)

)

#

0%

)′

)′

5

5′

'

(b)

One unit of the final good Q requires one unit of the intermediate good X. Substituting Q withX in (7) and rearranging yields the inverse demand function for intermediate imports:

r =P (X) + XP ′(X)

M

1 + t≡ g(X,M, t), (8)

where

gX =(M + 1)P ′(X) +XP ′′(X)

M(1 + t)< 0, gM = − XP ′(X)

M2(1 + t)> 0, gt = − g(.)

1 + t< 0. (9)

DD in Figure 4 depicts the demand for intermediate imports. From the signs of the partials in(9), it follows immediately that DD is downward-sloping, and furthermore, it shifts outward asthe number of importers M increases or as the ad-valorem tariff rate t decreases. An increasein the number of importers raises the demand for intermediate imports. Similarly, a reductionin ad-valorem tariffs acts as a reduction in unit costs and boosts up the demand for intermediateimports.

Equating demand and supply of intermediate imports – given by (8) and (5) respectively –yields the equilibrium values (X, r) where

r = h(X) = g(X,M, t). (10)

Each Home firm imports x = XM units of the intermediate good to produce q = Q

M units of the finalgood where Q = X and P = P (Q) respectively denote the equilibrium values of aggregate outputand price of the final good.

Figure 4a describes the equilibrium. FF and DD denote the demand function for the final good

13

and the intermediate good respectively. TT plots (1 + t)g(X,M, t) = P (X) + XP ′(X)M against X. SS

denotes the supply function r = h(X). The intersection between DD and SS gives the equilibriumvalues (X, r). The tariff-inclusive input price r(1 + t) and the final good price P are then read offTT and FF respectively at X = X. Proposition 1 describes how these equilibrium outcomes varywith tariffs and the number of importers (see Appendix A.1.1 for proof).

Proposition 1

For a given number of importers (M ), a reduction in ad-valorem tariff rate (t) raises aggregateimports (X) and firm-level import of intermediate input (x). For a given ad-valorem tariff rate,an increase in the number of importers raises aggregate imports but lowers the firm-level importof intermediate input. Input price (r) increases with both reduction in tariffs and increase in thenumber of importers.

Consider Figure 4b which illustrates in isolation the demand and supply for intermediate im-ports. As noted earlier, both a reduction in tariff rate t and an increase in the number of importersM lead to higher aggregate demand for intermediate imports at a given r. Demand curve DD

shifts outward to D′D′ in Figure 4b. The supply curve SS remains unchanged. Consequently,the equilibrium point moves from Z to Z ′. Both aggregate volume of imports and the input priceincrease with tariff reductions. While not reported in Proposition 1, the effect of changes in t or M

on the final good market are immediate from the effect on X. Whenever X increases, aggregateproduction of the final good Q(= X) increases, whereas the final good price P = P (Q) declines.Thus, both lower tariffs and increased competition lead to larger aggregate output and lower pricein the final good market.

What about the effect on the intensive margin, i.e., x = XM ? As aggregate imports increases

with tariff reduction, firm-level imports x increases too, since M is fixed. To examine the effect ofincreased competition on firm-level imports, decompose dx

dM as

dx

dM=

∂x

∂M+

∂x

∂r

dr

dM.

Unless the demand functions are too convex, entry in oligopoly gives rise to business-stealingeffect: an increase in the number of firms reduces each incumbent firm’s output (Mankiw andWhinston, 1986). The lower the production of the final good, the lower the imports of the inter-mediate good and thus ∂x

∂M < 0. With an increase in the input price – induced by an increasein M – increased competition reduces the demand for intermediate good even further and thusdxdM < ∂x

∂M < 0.Recall our empirical findings in section 2. All three – aggregate imports, firm-level imports

(intensive margin), and the number of importers (extensive margin) – increase with a reductionin tariffs. Proposition 1 lends support for the first two but is silent on the third one as we haveassumed that the number of Home firms, M , is fixed. Below, we relax that assumption.

14

3.3 Long-run analysis

The key feature of the long-run analysis is that the number of importers, M , is endogenouslydetermined. Incorporating free entry and exit of importers into our framework, we now explorehow a reduction in ad-valorem tariffs affect aggregate imports, intensive margin and extensivemargin. To avoid notational clutter, we continue to use z to denote equilibrium values of variablez for arbitrary t. For example, X and x respectively denote aggregate imports and firm-levelimports. The only new notation is M – the number of importers which is endogenously determinedvia free entry.

Home firms incur entry cost K(> 0) to enter the downstream sector. Assuming that K is lowenough such that at least one Home firm engages in the final good production, the free entrynumber of Home firms is determined implicitly by the value of M that satisfies the zero-profitscondition:

(P (Q)− r(1 + t))q −K = 0

where q and Q are as defined in (6) and (7) respectively. Rewriting (6) as P (Q)−r(1+ t) = −P ′(Q)q

and replacing q with QM , we can express the zero-profits condition as

− P ′(Q)Q2

M2−K = 0. (11)

It follows from (11) that the free entry level of M and Q has a monotone relationship:

dM

dQ= −q(2P ′(Q) +QP ′′(Q))

2K> 0

Let M(Q) denote the unique value of M that is compatible with the free entry equilibrium levelof Q. Then we can express each Home firm’s Cournot output as

q = −P (Q)− r(1 + t)

P ′(Q)(12)

where Q uniquely solves

M(Q)P (Q) +QP ′(Q)−M(Q)r(1 + t) = 0. (13)

As in the short run, (4) guarantees uniqueness of Q in the long run (see Appendix A.1.2). Notethat (13) is effectively the same as (7) except for one substitution: M(Q) in place of M . Thissubstitution preserves the analogy with the short-run analysis and simplifies the derivation ofthe aggregate demand curve for intermediate imports in the presence of entry.

Substituting Q with X in (13) and rearranging yields the inverse demand function for inter-mediate imports:

r =P (X) + XP ′(X)

M(X)

1 + t≡ g(X,M(X), t) ≡ G(X, t), (14)

15

Figure 5 – Equilibrium with free entry

#

%

#!

%!

-(%,.!, /) -(%,.", /)

6(%, /)

%"

#"

7

8

%#!

(a)

#

%0%

#

'

'

5

5′7

7

7′

7′

(b)

whereGX = gX + gMM ′(X) =

2M(X)P ′(X) +XP ′′(X)

2M(X)(1 + t)< 0, Gt = −G(.)

1 + t< 0.

As in the short run, the demand function for intermediate imports is downward-sloping. Also,aggregate demand increase with reduction in tariff.

Figure 5a illustrates the relationship between the long-run demand curve G(X, t) and theshort-run demand curves g(X,M, t). Suppose the input price is r1 and let X1 and M1 denoteaggregate demand and the number of importers that enter the downstream sector correspondingto r = r1. Point A lies on the short-run demand curve D1D1 as depicted by g(X,M1, t). As theinput price decreases to r2, aggregate demand increases to X1

S if the number of importers is fixedat M1. With free entry, however, as the input price decreases, entry becomes more profitable, andthe number of importers increases to M2(> M1). The relevant short-run demand curve D2D2 isgiven by g(X,M2, t) which lies to the right of D1D1. Entry increases aggregate demand from X1

S

to X2. Thus the long-run demand for intermediate imports G(X, t) passes through points A andB.

Equating (14) and (5) yields the equilibrium values (X, r) where

r = h(X) = G(X, t). (15)

Each Home firm imports x = XM

units of the intermediate good to produce q = Q

Munits of the

final good where Q = X and P = P (Q) respectively denote the equilibrium values of aggregateoutput and the price of the final good. The equilibrium is depicted in Figure 5b where EE denotesthe long run demand for the intermediate good G(X, t), SS denotes the supply function r = h(X),

16

and the intersection point Z denotes the equilibrium: (X, r). Proposition 2 describes how theseequilibrium outcomes vary with tariff reductions under free entry (see Appendix A.1.3 for proof).

Proposition 2

A reduction in ad-valorem tariff rate t raises aggregate imports (X), the number of importers (M )and the input price (r). Firm-level imports (x) increase with tariff reduction if and only if thedemand function is strictly convex.

The effect of tariff reduction on X and r is immediate from Figure 5b. Recalling that Gt <

0, lower tariffs always lead to higher aggregate demand for intermediate imports at a given r.The entry-adjusted demand curve EE shifts outward to E′E′ in Figure 5b. The supply curve SS

remains unchanged. As the equilibrium point moves from Z to Z ′, both aggregate imports X andinput price r increase with a reduction in tariff.

How does a reduction in tariff affect the extensive margin (M ) and intensive margin (x) underfree entry? As X(= Mx) increases with lower tariffs, at least one of the two – M or x – must in-crease. Consider first the extensive margin. Differentiating (11) with respect to t and rearranging,we get:

dM

dt= − x(2P ′(X) + XP ′′(X))

2K

dX

dt= −

P ′(X)x(1 + η(X)2 )

K

dX

dt> 0, (16)

where η(X) ≡ XP ′′(X)

P ′(X)is the elasticity of slope of the inverse demand function for the final good

and the inequality in (16) follows from noting that dX/dt < 0 and 2P ′(.) + XP ′′(.) < 0 (recall 4).If the demand function is not too convex, the number of importers increases with reduction intariffs.

The effect of tariff reduction on the intensive margin depends on demand curvature. When M

is fixed, aggregate imports X and firm-level imports x(= XM ) move in the same direction. That is

not necessarily the case under free entry. Rewriting (11) as

−P ′(X)x2 −K = 0,

differentiating, and rearranging subsequently gives

2xdx

dt=

KP ′′(X)

[P ′(X)]2dX

dt, (17)

which implies that the intensive margin and aggregate imports move in the same direction if andonly if P ′′(.) > 0. Lower tariffs prompt each Home firm to scale up production which in turn raisesthe demand for intermediate good. However, entry (induced by lower tariffs) prompts existingfirms to produce less and import less. These two opposing effects exactly cancel out for lineardemand. For strictly convex demand, x increases with lower tariffs while the opposite is true forstrictly concave demand function.

17

How do our theoretical results tally with the empirical findings? Using the notations in ourmodel, empirical evidence in section 2 can be summarized as follows:

dX

dt< 0,

dM

dt< 0,

dx

dt< 0.

Proposition 2 suggests that the first two inequalities hold under any demand function satisfying2P ′(.) +XP ′′(.) < 0. The third one requires strictly convex demand function, i.e. P ′′(.) > 0. Thus,loosely speaking, we find support for all three if the underlying demand function is convex, butnot too convex. A constant elasticity demand function Q = AP−ed (where ed > 1) and a semi-logdemand function lnQ = A− P satisfy these requirements: P ′′(.) > 0 and 2P ′(.) +XP ′′(.) < 0.

4 Welfare and optimal tariff

Quasilinear preferences together with the assumption that all tariff revenues are rebated back toconsumers allow us to express Home welfare (W ) as

W ≡

[∫ Q

0P (y)dy − P (Q)Q

]︸ ︷︷ ︸

Consumer surplus

+[P (Q)Q− r(1 + t)X − MK

]︸ ︷︷ ︸

Home profits

+ rtX︸︷︷︸Tariff revenues

+ L︸︷︷︸Labor income

,

which can be simplified further to:

W =

∫ X

0P (y)dy − rX − MK + L.

In what follows, we proceed in two stages. First, treating the number of importers, M , as fixed, weexamine how Home welfare varies with tariffs and subsequently determine optimal tariffs for theshort-run setting (subsection 4.1). Subsequently, we derive optimal tariff for the long-run settingin which M is endogenously determined (subsection 4.2).

Terms-of-trade motive for tariffs exists for both short run and long run in our model. Coun-teracting this in the short run is underproduction in oligopoly. Thus optimal tariffs are lowercompared to what it would have been under a perfectly competitive downstream sector: the in-verse of export supply elasticity. In fact, irrespective of the demand function, we find that negativetariff is optimal when M is suitably small. In the long run, however, optimal tariff could be highersince in a homogeneous product setting like ours tariffs can improve welfare by mitigating excessentry in the downstream sector. Optimal tariff is strictly positive for all concave demand func-tions, although for the empirically relevant case – strictly convex demand functions – negativetariff can be optimal for a range of parameter values.

18

4.1 Short run

Differentiating W with respect to t and rearranging yields:

dW

dt= (P − r)

dX

dt− X

dr

dt. (18)

where P = P (X) and M is assumed to be fixed. From Proposition 1 we know that an increase ina tariff rate t lowers aggregate output X as well as the input price r. The first term, (P − r)dXdt ,captures the welfare loss due to tariff-induced output reduction. The second term, −X dr

dt , capturesthe welfare gain due to lower input price. Substituting dr

dt = h′(X)dXdt in (18) and simplifyingsubsequently we get:

dW

dt= 0 ⇔ P = r

(1 +

1

es

), (19)

where es =r

Xh′(X)is the inverse of supply elasticity of intermediate imports. Rearrangement of (6)

yields the markup pricing formula in downstream oligopoly: equilibrium price P equals markup( 11− 1

Med

) times constant marginal cost r(1 + t), i.e.,

P =r(1 + t)

1− 1Med

, (20)

where ed denotes the elasticity of demand for the final good at P = P . Substituting the expressionfor P in (19) and rearranging yields the optimal ad-valorem tariff rate t∗.

Proposition 3

Optimal ad-valorem tariff rate is implicitly given by:

t∗ =1

e∗s− 1

Me∗d

(1 +

1

e∗s

), (21)

where e∗d and e∗s denote demand elasticity for the final good and supply elasticity for the intermedi-ate imports respectively, both of which are evaluated at t = t∗ and (X∗, r∗) = (X, r)|t=t∗ .

(i) limM→∞ t∗ = 1e∗s

> 0, limes→∞ t∗ = − 1Me∗d

< 0.

(ii) Suppose es is constant or strictly decreasing in r. There exists a unique M∗ such that optimaltariff rate is strictly positive if and only if M > M∗.

When the downstream sector is perfectly competitive (i.e., M = ∞) we get the familiar result:

t∗ =1

e∗s> 0.

19

Optimal tariff is strictly positive and its value equals the inverse of export supply elasticity forintermediate imports. Imposing tariff on intermediate imports improves terms-of-trade and in-creases welfare.

Terms-of-trade improvement comes at the cost of exacerbating underproduction in oligopoly.Tariffs reduce output and create a welfare loss as the price-cost margin, P (X)− r(1+ t), is strictlypositive when M is finite. Hence, in the presence of the oligopolistic downstream sector, t∗ < 1

e∗s.

Nevertheless, since limM→∞ t∗ > 0, the case of positive optimal tariffs remains as long as M islarger than a threshold value, M∗ say. Part (ii) of Proposition 3 says that a sharp characterizationexists when either export supply elasticity is constant or decreasing in r. This condition ensuresthat the optimal tariff increases as competition increases in the downstream sector which in turnimplies14

M > M∗ ⇔ t∗ > 0.

To see why optimal tariff could be negative, suppose es is constant and let es → ∞. As thesupply function becomes perfectly elastic, terms-of-trade motive for tariffs disappears. In thiscase, an import subsidy raises aggregate output which increases Home welfare. This setting issimilar to a single-stage, Cournot oligopoly with M domestic firms and constant marginal cost.In such settings, an import subsidy increases welfare by mitigating underproduction in oligopoly.Optimality of an import subsidy is not restricted to the case when es is large. Even for arbitrarilysmall es there exists suitable demand functions and market structure such that t∗ < 0.

4.2 Long run

We continue to use z to denote equilibrium values of variable z for arbitrary t, and z∗ to denotez at t = t∗. However, here, equilibrium values refer to the ones obtained in section 3.3 where thenumber of Home firms, M , is endogenously determined. Differentiating W with respect to t yields

dW

dt= (P − r)

dX

dt− X

dr

dt− dM

dtK. (22)

As before, the first two terms respectively capture the welfare loss from reduction in aggregateoutput and the welfare gains from reduction in input price. The new term, −dM

dt K, captures thewelfare gain from fixed cost savings due to tariff-induced exit of Home firms. We can simplify theexpression for dM

dt in (16) as:

dM

dt= −

P ′(X)x(1 + η(X)2 )

K

dX

dt=

(P − r(1 + t))(1 + η(X)2 )

K

dX

dt, (23)

14See Appendix A.1.4 for a formal proof of the statement.

20

where the last equality follows from rewriting (6) as P − r(1+t) = −P ′(X)x. Using (23) and notingthat X dr

dt = Xh′(X)dXdt , we obtain the following implication from (22):

dW

dt= 0 ⇔ P = r

(1 +

1

es

)+ (P − r(1 + t))

(1 +

η

2

), (24)

where η = η(X). Substituting the markup pricing formula from (20) into (24) and simplifyingfurther yields the long-run optimal tariff rate t∗.

Proposition 4

Optimal ad-valorem tariff rate t∗ is implicitly given by

t∗ =1

e∗s+

( η∗

2M∗e∗d

1− 1M∗e∗d

− η∗

2M∗e∗d

)(1 +

1

e∗s

), (25)

where e∗d, e∗s, and η∗ denote demand elasticity for the final good, export supply elasticity for theintermediate imports, and elasticity of slope of the inverse demand function respectively, all ofwhich are evaluated at t = t∗ and (X∗, r∗,M∗) = (X, r, M)|t=t∗ . Furthermore,

(i) t∗ ⋛ 1e∗s

⇔ P ′′(.) ⋛ 0,

(ii) limK→0 t∗ = 1

e∗s> 0, limes→∞ t∗ ⋛ 0 ⇔ P ′′(.) ⋛ 0,

(iii) Optimal tariff is negative only if the demand function is strictly convex. For all concavedemand functions, optimal tariff is strictly positive.

As the entry cost becomes arbitrarily small, the downstream sector becomes almost perfectlycompetitive and the value of the optimal tariff t∗ approaches 1

e∗s– the inverse of the export supply

elasticity for intermediate imports. Apart from this terms-of-trade route that exists irrespec-tive of market structure, tariffs also affect welfare through two additional channels in free-entryoligopoly. As in the fixed M case, an increase in tariff worsens welfare by lowering aggregate out-put. However, tariffs improve welfare by lowering M and saving entry costs. The welfare gainsfrom entry cost savings exactly offset the welfare losses from output reduction when demand islinear. Thus, for linear demand, t∗ = 1

e∗sholds irrespectively of whether there is perfect competi-

tion or (free-entry) oligopoly in the downstream sector. However, in general, t∗ could be larger orsmaller than 1

e∗s, depending on the demand curvature. From the expression of t∗ in (25), it follows

thatt∗ ⋛ 1

e∗s⇔ η∗ ⋛ 0 ⇔ P ′′(.) ⋛ 0. (26)

where η∗ = XP ′′(X)

P ′(X)

∣∣t=t∗

. When e∗s → ∞, (26) translates to part (ii): a strictly positive tariff isoptimal if and only if the demand function is strictly concave.

21

To understand why demand curvature plays an important role, let e∗s → ∞, and consider aninfinitesimally small increase in t from t = 0. Decomposing dX

dt as

dX

dt= M

dx

dt+ x

dM

dt,

and noting that(P − r)x−K = 0

holds in free-entry, free-trade equilibrium, we obtain the following implication from (22):

dW

dt

∣∣∣∣t=0

⋛ 0 ⇔ M(P − r)dx

dt− X

dr

dt⋛ 0.

When e∗s → ∞, drdt → 0 and thus

dW

dt

∣∣∣∣t=0

⋛ 0 ⇔ dx

dt⋛ 0 ⇔ P ′′(.) ⋚ 0.

where the last inequality follows from Proposition 2.Homogenous product oligopoly exhibits underproduction and excessive entry, i.e., more firms

enter in equilibrium than socially optimal (Mankiw and Whinston, 1986; Suzumura and Kiyono,1987). An increase in tariffs reduces X which exacerbates underproduction and lowers welfarebut it also reduces M which mitigates excessive entry and increases welfare. Loosely speaking,which effect dominates depends on the effect of tariffs on x = X

M, or equivalently the effect of

tariffs on firm-level output q. In the presence of entry costs K, higher firm-level output impliesgreater economies of scale and consequently higher welfare. As we have shown in Proposition 2,firm-level output increases (decreases) with tariffs when the demand function is strictly concave(convex). Thus, starting from free trade, a small ad-valorem import tariff (subsidy) improveswelfare irrespective of entry costs and market structure if and only if the demand function isstrictly concave (convex). With es < ∞, however, the terms-of-trade motive for tariff resurfaces.Thus, a positive tariff becomes optimal not only for strictly concave demand functions but for alarger range of parameterizations as indicated in part (iii) of Proposition 4.

Empirically we found that reduction in tariff raises firm-level imports of intermediate goods.This arises in the long-run setting for strictly convex demand functions. For such demand func-tions, an import subsidy or an import tariff could be optimal. Consider for example a constantelasticity demand function Q = AP−ed where ed is constant and greater than unity. Also, sup-pose that the supply function of intermediate imports has a constant elasticity es > 0. Using theexpression for the optimal tariff from (25), it is easy to show that

t∗ < 0 ⇔ M <1

ed

(1 +

es2

(1 +

1

ed

)).

22

An import subsidy is optimal when es is large and K is large (implying that M is small). Whatto make of t∗ < 0? The implication of negative optimal tariff is that starting from a positivelevel, a reduction in tariff, i.e., moving towards free trade can improve welfare even when tradeliberalization is pursued unilaterally.

Note that the sign of optimal tariff depends on demand curvature and entry considerations.In terms of dependence of optimal policy on demand curvature, our results have similar flavor tosome of the existing results in the trade literature. For example, the classic result that the signof optimal tariff in the presence of a Foreign monopoly depends on whether there is incompletepass-through, which in turn depends on whether the demand curve is flatter than the marginalrevenue curve (e.g., Brander and Spencer, 1984a, b; Helpman and Krugman, 1989, Ch. 4). As forthe difference in optimal policy between the endogenous and exogenous market structures, ourresults are reminiscent of Horstmann and Markusen (1986) and Venables (1985), who show thatfree entry can alter optimal trade policy due to the firm-delocation effect in oligopoly models. Ina single-stage Cournot oligopoly with linear demand, Bagwell and Staiger (2012a, b) analyze therole of the firm-delocation effect in context of trade agreements.

5 Discussion

We have shown that both aggregate imports and the number of importers increase with a reduc-tion in tariffs while firm-level imports can increase or decrease depending on the elasticity of slopeparameter η ≡ QP ′′(Q)

P ′(Q) . In particular, firm-level imports increase with tariff reduction if and onlyif η < 0. Irrespective of the sign of η, the optimal tariff is strictly lower than the inverse of exportsupply elasticity when M is fixed. However, the sign of η plays a key role in determining optimaltariffs when M is endogenously determined via free entry. For η > 0, the optimal tariff is alwayshigher than the inverse of export supply elasticity. For η < 0, the optimal tariff can be negative.Diagrammatic summary of our findings is presented in Figure 6.15

In demonstrating the results, we assumed that (i) the upstream sector is perfectly competitive,(ii) downstream firms produce homogeneous goods, and (iii) import tariff is the only policy instru-ment. Due to lack of data on exporters of intermediate inputs, we chose the simplest possiblemarket structure in the upstream sector: perfect competition. Simplicity notwithstanding, thischoice highlights the key role of oligopoly and free entry in the downstream sector in determiningthe effect of tariffs on market structure and welfare. Our choice of homogeneous goods is in line

15The condition η > −2, assumed in Mrazova and Neary (2017), is a succinct way of saying that industry marginalrevenue is decreasing in output. Mrazova and Neary (2017) primarily focus on monopoly, i.e. M = 1. For M > 1,η > −2 is not necessary for existence or uniqueness of Cournot equilibrium. We could work with η > −(M + 1).Allowing for η < −2 opens up the possibility that extensive margin decreases with trade. Thus, a more comprehensivesummary (than the one offered in the sections 3 and 4) of our findings is as follows: both extensive and intensive marginincreases with reduction in tariffs if and only if −2 < η < 0; else, if η > 0, extensive margin increases but intensivemargin declines with trade liberalization, while the opposite is true when η < −2. Despite richer set of possibilitiesarising under η > −(M + 1), we chose to work with η > −2 in the main text. This is partly due to our focus onendogenous market structure. More specifically, we want η > −(M + 1) to hold for all M including M = 1 which leadsto the requirement: η > −2.

23

Figure 6 – Summary of the results

# $%#& > 0# $%

#& < 0

* = 0

* = 0

* = −2

* = −2

# ./#& < 0# ./

#& > 0

&∗ > 11"

&∗ < 11"

Note: The shaded area is empirically relevant ( dxdt

< 0, dMdt

< 0). The demand functions satisfying −2 < η < 0 includeconstant elasticity Q = AP−ed (where ed > −1) and semi-log lnQ = A− P .

with the literature on imperfect competition in international trade where, typically, oligopoly ispaired with homogeneous goods, and monopolistic competition is paired with differentiated goods.Below, in sections 5.1 and 5.2, we show how our model can be extended to incorporate imperfectcompetition in the upstream sector as well as product differentiation in the downstream sector.Following the standard practice in the trade policy literature (see, for example, a recent surveyby Ossa, 2016), we have focused exclusively on tariffs as the only policy instrument. In section5.3, we consider an extension where Home government chooses both tariffs and the number ofimporters (or equivalently entry taxes). We show that the presence of active competition policylowers optimal tariff. In fact, while the issues raised in (i)-(iii) might seem disparate, the analysesin subsections 5.1-5.3 suggest a common theme: optimal tariffs are lower and negative optimaltariffs are more likely compared to the baseline case (with free entry).

5.1 Vertical oligopoly

Consider a vertical oligopoly setting, where firms in both upstream and downstream sectors actas Cournot competitors. We continue to assume that there are no intermediate goods suppliersat Home. All final goods producers at Home import intermediate goods from the rest of the world(which we refer to as Foreign in the discussion below). There is free entry of firms in both sectors.A self-contained, detailed analysis of this subsection is presented in Appendix B.

24

Recall the inverse demand function for the intermediate good faced by Foreign firms:

r =P (X) + XP ′(X)

M

1 + t≡ g(X,M, t),

where g(X,M, t) satisfies (9). Suppose N(≥ 2) Foreign firms enter the upstream sector paying theentry cost KF . For simplicity, we assume that Foreign firm faces constant marginal cost c. EachForeign firm i ∈ {1, 2, ..., N} chooses xi to maximize

πFi ≡(g

(xi +

N∑j =i

xj ,M, t

)− c

)xi,

taking other Foreign firms’ quantities (xj) as given. Furthermore, we assume that

(N + 1)gX(X,M, t) +XgXX(X,M, t) < 0,

which ensures uniqueness of the upstream Cournot equilibrium. Solving the first-order conditionsyields the equilibrium quantity for each Foreign firm:

x = −g(X,M, t)− c

gX(X,M, t),

where X uniquely solvesNg(X,M, t) +XgX(X,M, t)−Nc = 0.

The solution denotes the aggregate volume of imports in the short run, i.e., given market structure(M,N). In addition, the following pair of equations hold in the long run:

(P (Q)− r(1 + t))Q

M−KH = 0,

(r − c)X

N−KF = 0,

which describe the zero-profits conditions in the downstream and upstream sector respectively.Substituting Q = X, using r = g(X,M, t), and solving the above aggregate first-order conditionand free entry conditions simultaneously yields implicit solutions of X, M , and N in a free-entry,vertical oligopoly equilibrium.

As in section 3, a reduction in tariffs increases the aggregate volume of imports (exports)and the number of importers (exporters). The effect on the intensive margin continues to beambiguous. Firm-level imports increase with tariff reductions if and only if the demand function isstrictly convex. While the properties of optimal tariffs are similar as before, imperfect competitionin the upstream sector means that there is no explicit export supply curve. When M and N arefixed in the short run, an import subsidy is optimal when the number of Home importers (M )is relatively small. When these numbers are endogenous in the long run, an import subsidy

25

is optimal for a larger range of parameterizations. This is because tariffs on imported inputsdiscourage entry not only of Home importers (M ) but also Foreign exporters (N ). An increasedconcentration in the upstream sector puts upward pressure on the input price. This dampens theterms-of-trade motive of tariffs and increases the likelihood of an import subsidy.

Terms-of-trade can worsen, i.e., the input price (r) can increase, with an increase in tariff. Tosee why, rewrite the zero-profits condition in the upstream sector as

r = c+KF

x,

where r and x are the input price and quantity of upstream imports respectively. An increase in r

is equivalent to a reduction in x. We have that

dx

dt=

∂x

∂t+

∂x

∂N

dN

dt+

∂x

∂M

dM

dt.

When M and N are exogenously fixed, each upstream firm cuts back its production in responseto increased t. Thus ∂x

∂t < 0. There is a further cutback as an increase in t reduces the numberof Home firms which in turn reduces the demand for the intermediate good. Thus ∂x

∂MdMdt < 0.

Counteracting these negative effects (on x) is a positive effect arising from weaker competitionamong upstream firms. An increase in t reduces the number of Foreign firms which in turnincreases the scale of production for surviving Foreign firms. Thus ∂x

∂NdNdt > 0. Combining these

effects, we find that dxdt < 0 for η(≡ QP ′′(Q)

P ′(Q)) ) > 1. However, for all demand functions satisfying η < 1

– a class of demand that includes both constant elasticity demand and linear demand – there existKH and KF such that dx

dt < 0, and consequently drdt > 0.

Our finding drdt > 0 is akin to Lerner paradox which occurs when import tariffs worsen country’s

terms-of-trade. While rare in perfectly competitive settings, Venables (1987) and more recentlyBagwell and Lee (2020a, b) have shown that the paradox can arise in single-stage, monopolisticcompetition models.16 As tariffs raise the price of Foreign varieties, Home consumers substituteForeign varieties with Home varieties, which induces exit of Foreign firms and entry of Homefirms. An increased concentration of Foreign firms raises the Foreign price index which in turncan worsen the terms-of-trade. One-to-one production technology implies that the final good pro-duced by Home firms and the intermediate good produced by Foreign firms are complements inproduction. Thus, an increase in tariffs not only prompts exit of Foreign firms but also exit ofHome firms. Exit of Home firms shifts the input demand inward but has no direct effect on theinput price for constant elasticity demand or linear demand as the elasticity of input demand isunaffected by the shift. For those demand functions, and more generally for demand functionsthat are not too concave, the Lerner paradox can arise in our model when the indirect upwardpressure (of tariffs) on the input price induced by exit of Foreign firms outweighs the direct down-ward pressure (of tariffs) on the input price.

16See also Bagwell and Staigher (2012a) for a similar result in linear Cournot framework with free entry.

26

5.2 Product differentiation

Incorporating differentiated products in our framework is straightforward. Suppose that thereare M Home firms each producing a distinct variety. Let qi and pi denote the quantity and priceof variety i. Assume that the representative consumer at Home maximizes U(q1, q2, ..., qM ) + y,where y is the numeraire good and

U(q1, q2, ..., qM ) = a

M∑i=1

qi −M∑i=1

q2i2

− 1

2

∑j =i

qiqj .

This quadratic utility specification gives rise to a linear demand system:

pi = a− qi − b∑j =i

qj .

The parameter b ∈ [0, 1] captures the degree of product differentiation. As b increases, varietiesbecome less differentiated. When b = 1 we get linear demand for homogeneous goods. As in ourbaseline case, we assume one-to-one production technology and a perfectly competitive upstreamsector.

Proceeding as in section 3, we find that the short-run demand for the intermediate good for M

firms is

r =a− 2+(M−1)b

M X

1 + t= g(X,M, t).

Using the zero-profits condition (pi − r(1 + t))qi −K = 0 where K denotes entry costs, we obtainthe long-run demand curve for intermediate imports:

r =a− (2− b)

√K − bX

1 + t= G(X, t).

The equilibrium value of aggregate imports is given by X where G(X, t) = h(X). The number ofimporters and volume of imports per firm are given by M = X√

Kand x = X

M=

√K respectively.

Both aggregate volume of imports (X) and the number of importers (M ) increase with reductionsin tariffs. However, as in section 3, firm-level imports (x) do not vary with tariffs when demand islinear.

Substituting qi = x = XM

=√K in U(.) and simplifying, we can express Home welfare as

W = X

(a− (1 + b(M − 1)

2

√K

)− rX − MK + L.

Setting dWdt = 0 and rearranging yields

t∗ =1

e∗s− (1− b)

√K

2r∗, (27)

27

where r∗ = r at t = t∗. Observe that, as in section 4, optimal tariff equals the inverse of exportsupply elasticity when products are homogeneous (b = 1) or entry costs are vanishingly small(K → 0). Comparing (27) with (25) shows that the optimal tariff is lower in the presence ofproduct differentiation (b < 1). Imposing tariff worsens welfare by exacerbating underproductionand increases welfare by improving terms-of-trade and saving entry costs (by prompting exit).These effects are present irrespective of whether the final good is homogeneous or differentiated.However, when products are differentiated, fewer firms (resulting from higher tariffs) also implyfewer varieties which lowers welfare. This additional source of welfare losses lowers the optimaltariff. In fact, it follows immediately from (27) that when the export supply elasticity is large,an import subsidy could be optimal – a possibility that does not arise for linear demand in ahomogeneous product setting.

5.3 Competition policy

We have assumed that market structure in the downstream sector is either exogenously givenor endogenously determined via free entry. Here, we consider another alternative: Home gov-ernment directly chooses the number of firms (M ) to maximize welfare. In reality, a governmentindirectly affects market structure by entry taxes/subsidies, merger approvals, and ownership re-strictions. However, choosing the number of firms directly captures the essence of competitionpolicy in a simple fashion (Horn and Levinsohn, 2001).

To highlight the role of competition policy, consider first an environment with free trade (i.e.,t = 0). Differentiating welfare with respect to M yields:

dW

dM= (P − r)

dX

dM− X

dr

dM−K.

Let Mf denote the number of importers in the free-entry equilibrium. Decomposing dXdM as x +

M ∂x∂M and noting that (P − r)x−K = 0 holds in free-entry equilibrium, we get:

dW

dM

∣∣∣∣M=Mf

= (P − r)∂x

∂M− X

dr

dM.

From Proposition 1 we know that an increase in the number of importers decreases firm-levelimports x and increases the input price r. Thus ∂x

∂M < 0 and drdM > 0, which in turn implies that

dW

dM

∣∣∣∣M=Mf

< 0.

Restricting the number of importers at margin always improves Home welfare, since there is anexcessive number of importers in free-entry equilibrium. Marginal entrant/importer ignores thebusiness stolen from existing firms, leading to excessive entry in homogeneous product oligopoly

28

(Mankiw and Whinston, 1986; Suzumura and Kiyono, 1987).17 In our trade setting, marginalentrants also ignore the negative impact on terms-of-trade which exacerbates excess entry. Thus,for any tariff rate t, the following relationship holds:

Mf (t) > M0(t)

where Mf (t) and M0(t) are free-entry and welfare maximizing number of firms respectively.Now let us re-introduce tariffs. Suppose a welfare-maximizing Home government chooses both

M and t, so that it can use both trade and competition policies at the same time. How does thataffect optimal tariff? Observe that

dW

dt=

∂W

∂t+

∂W

∂M

dM0(t)

dt.

As the number of importers adjusts optimally in equilibrium, solving for optimal tariff, t∗, boilsdown to choosing the value of t such that ∂W

∂t = 0 or equivalently:

(P − r)∂X

∂t− X

∂r

∂t= 0.

Using P − r(1 + t) = −P ′(X)x and X ∂r∂t = Xh′(X)∂X∂t in the equation above and rearranging,

optimal tariff in this case is expressed as

t∗ =1

e∗s− 1

M∗e∗d

(1 +

1

e∗s

),

where all the expressions are evaluated at t = t∗,M = M∗ = M0(t∗).Note that the above expression for optimal tariff in the presence of active competition policy

is the same as t∗ in (21) – short-run optimal tariff – with one difference: the value of M(= M∗)

is endogenously determined by a welfare-maximizing Home government. As in the short run,optimal tariff is always less than the inverse of export supply elasticity, and in fact be negative ifentry costs are large (and M∗ is small). The possibility of t∗ exceeding 1

esarises with free entry as

higher tariffs can partially mitigate the welfare loss from excessive entry. With competition policyin place, that welfare enhancing effect of tariffs goes away. However, the welfare worsening effectof tariffs (through exacerbating underproduction) still remains which in turn lowers t∗ below 1

es

– the optimal tariff under perfect competition (where only the terms-of-trade motive of tariffs ispresent).

17In a closed economy setting, maintaining the assumption of a homogeneous product, Ghosh and Morita (2007)show that entry might be insufficient in a vertical oligopoly model like the one seen in section 5.1. The trade literaturetypically focuses on differentiated goods and CES preferences where insufficient entry is the norm. An exception isBagwell and Lee (2020a, b). Considering quadratic preferences used by Melitz and Ottaviano (2008), they show thatentry can be insufficient or excessive in equilibrium depending on demand and cost parameters.

29

6 Conclusion

Recent years have witnessed a significant increase in vertical specialization and a concomitantgrowth in trade in intermediate goods. While there is a burgeoning literature exploring firms’participation in global supply chains, relatively scant attention has been paid to the positive andnormative aspects of trade policy in this context. This lacuna in the literature is more glaringnow in the context of a large fraction of Trump tariffs against China being on intermediate goods(Bown 2019a, 2019b). A recent example in the literature to examine normative aspects of interme-diate input tariffs is Grossman and Helpman (2020) who investigate the welfare effects of tariffsin the context of firm-to-firm supply relationships. We examine the effect of intermediate inputtariff on output, market structure, and welfare in a model with market based interactions be-tween upstream and downstream producers. The number of Home importers (extensive margin)increases with a reduction in tariffs. At the same time, however, the average volume of importsper firm (intensive margin) increases – as we find in the data – if and only if the demand functionis strictly convex. Using the effect of tariffs on these two margins, we characterize optimal tariffsin our framework.

One of the key findings is that optimal tariff in our model can differ from the inverse of exportsupply elasticity – the perfect competition benchmark. It can be higher as tariffs improve welfareby mitigating excess entry in homogeneous product oligopoly. Optimal tariff can also be lower thanthe inverse of export supply elasticity as tariffs worsen welfare by exacerbating underproductionin oligopoly. In fact, for a widely used constant elasticity demand function – or more generally,strictly convex demand functions – we find that a negative tariff (i.e., an import subsidy) might beoptimal.

While we have used homogeneous product oligopoly in the downstream sector and a perfectlycompetitive upstream sector in sections 3 and 4, our findings are robust to alternative specifi-cations. For example, as we show in section 5, our results are robust with respect to productdifferentiation in the downstream sector as well as imperfect competition in the upstream sector.The possibility of a negative optimal tariff is higher under both alternatives.

An important assumption maintained throughout the paper is a non-strategic rest of the worldwhich does not engage actively in trade policy. Theoretically, a natural next step will be to considera vertical specialization setting with two countries (Home and Foreign say) where consumersreside in both countries that set tariffs strategically – Home on intermediate goods and Foreign onfinal goods. As in all trade frameworks, the terms-of-trade motive for tariffs will be present hereas well. However, complementarity between intermediate goods and final goods might prompteach government to set lower tariffs than it would have in the absence of vertical specialization.In future research, we plan to explore this issue and related implications for trade agreementswhen countries are specialized in different layers of production.

30

Appendix A: Proofs and additional tables

A.1 Proofs

A.1.1 Proof of Proposition 1

Differentiating (10) and rearranging, we get

dX

dt=

gth′(X)− gX

< 0,dX

dM=

gMh′(X)− gX

> 0, (A.1)

where the inequalities follow from noting that gt < 0, gM > 0, gX < 0 from (9) and h′(X) > 0.Applying (A.1), we get

dr

dt= h′(X)

dX

dt< 0,

dr

dM= h′(X)

dX

dM> 0,

anddx

dt=

1

M

dX

dt< 0.

Finally, rearranging (7), we getP (X) + P ′(X)x = r(1 + t).

Differentiating the above equality yields:

P ′(X)dx

dM= (1 + t)

dr

dM− (P ′(X) + xP ′′(X))

dX

dM. (A.2)

The right-hand side of (A.2) is positive since drdM > 0, dX

dM > 0 and P ′(X) + xP ′′(X) < 0. Thus,dxdM < 0 since P ′(X) < 0.

A.1.2 Proof of uniqueness of Q

Consider first the short-run equilibrium. Define the left-hand side of (7) as

f(Q) = MP (Q) +QP ′(Q)−Mr(1 + t).

Differentiating f(Q) with respect to Q and rearranging we get

f ′(Q) = (M + 1)P ′(Q) +QP ′′(Q) < 0,

where the inequality follows from applying (4). The inequality implies that the left-hand sideof (7) is strictly decreasing in Q which in turn ensures that q and Q specified in (6) and (7) areunique.

Consider next the long-run equilibrium. Define the left-hand side of (13) as

F (Q) = M(Q)P (Q) +QP ′(Q)−M(Q)r(1 + t).

31

Differentiating F (Q) with respect to Q and rearranging we get

F ′(Q) =2M(Q)P ′(Q) +QP ′′(Q)

2< 0,

where the inequality follows from applying (4). The inequality implies that free entry level of qand Q specified in (12) and (13) are unique.

A.1.3 Proof of Proposition 2

Differentiating (15) and rearranging, we get

dX

dt=

Gt

h′(X)−GX< 0, (A.3)

where the inequality follows from noting that Gt < 0, GX < 0 and h′(X) > 0. Applying (A.3),

dr

dt= h′(X)

dX

dt< 0.

From (16) we know thatdM

dt= −

P ′(X)x(1 + η(X)2 )

K

dX

dt> 0.

From (17) we havedx

dt=

1

2x

KP ′′(X)

[P ′(X)]2dX

dt.

Since dXdt < 0, it follows that

dx

dt< 0 ⇐⇒ P ′′(.) > 0.

A.1.4 Proof of Proposition 3

The expression for t∗ in (21) as well as the proof part (i) follows from the text preceding Proposition3. To prove part (ii) it suffices to show that t∗ is strictly increasing in M . Rewrite (19) as

P

r−(1 +

1

es

)= 0 (A.4)

where P , r, and es respectively are the price of the final good, input price, and export supplyelasticity respectively evaluated at t = t∗. Suppose t remains at t∗. From Proposition 1, weknow that as M increases, P declines and r increases. Thus, P

r decreases with an increase inM . Furthermore, as r increases, es decreases. Thus, the left-hand side of (A.4) decreases with anincrease in M and falls below zero. Increase in t restores the equality in (A.4), since, Proposition1 implies that effect of an increase on t on P , r, and consequently on es is exactly the opposite tothat of an increase in M . Thus t∗ is strictly decreasing in M .

32

A.2 Additional tables

Table A.1 — DID estimations in (3)

(1) (2) (3)

lnQpct lnMpct ln qpct

ln(1 + tariffpc2001) ∗ d2001 −0.576∗∗∗ −0.204∗∗∗ −0.372∗

(0.240) (0.047) (0.203)

ln(1 + tariffpc2001) ∗ d2002 −0.778∗∗∗ −0.220∗∗∗ −0.558∗∗∗

(0.259) (0.062) (0.214)

ln(1 + tariffpc2001) ∗ d2003 −1.335∗∗∗ −0.430∗∗∗ −0.905∗∗∗

(0.318) (0.084) (0.256)

ln(1 + tariffpc2001) ∗ d2004 −1.376∗∗∗ −0.394∗∗∗ −0.982∗∗∗

(0.398) (0.105) (0.313)

ln(1 + tariffpc2001) ∗ d2005 −1.517∗∗∗ −0.372∗∗∗ −1.145∗∗∗

(0.467) (0.126) (0.362)

ln(1 + tariffpc2001) ∗ d2006 −1.593∗∗∗ −0.370∗∗∗ −1.223∗∗∗

(0.498) (0.136) (0.384)

ln(1 + tariffpc2001) ∗ d2007 −1.849∗∗∗ −0.434∗∗∗ −1.415∗∗∗

(0.431) (0.1227) (0.335)

No. of obs 408, 971 408, 971 408, 971

Adj. R2 0.776 0.894 0.710

Note: Aggregate volume of imports (Qpct), extensive margin (Mpct), and intensive margin (qpct) are in logarithm.

Robust standard errors are clustered at the product level in parentheses. ***, **, and * indicate significance at the

1%, 5% and 10% level, respectively.

33

Table A.2 — Poisson estimations in (1)

(1) (2) (3)

Qpct Mpct qpct

ln(1 + tariffpct−1) −4.675∗∗∗ −1.107∗∗ −4.207

(1.673) (0.436) (2.677)

No. of obs 283, 763 283, 763 283, 763

Note: Aggregate volume of imports (Qpct), extensive margin (Mpct), and intensive margin (qpct) are in real numbers.

Robust standard errors are clustered at the product level in parentheses. ***, **, and * indicate significance at the

1%, 5%, and 10% level, respectively.

34

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38

Appendix B: Supplementary material to section 5.1 (Not for publi-cation)

B.1 Model

This Appendix contains the detailed analyses for the vertical oligopoly model outlined in section5.1. As the analysis is self-contained, some parts read very similar to the main text.

B.1.1 Basics

Two countries, say Home and Foreign, respectively produce a final good and an intermediategood. Home comprises a unit mass of identical consumers L with a quasi-linear utility function,V = U(Q)+ y, where Q is total output produced under oligopoly and y is a competitively producednumeraire good. Assuming income to be high enough, standard utility maximization subject tobudget constraint gives the inverse demand function P = P (Q). We assume that P (Q) is twicecontinuously differentiable P ′(Q) < 0, and furthermore,

P ′(Q) +QP ′′(Q) ≤ 0 (B.1)

for all Q ≥ 0.1 This implies that industry marginal revenue (QP (Q))′ decreases with Q. We canrestate (B.1) as ϵ(Q) ≡ QP ′′(Q)

P ′(Q) ≥ −1 where ϵ(Q) is the elasticity of slope of demand. For analyticalsimplicity, we focus on a class of demand functions such that ϵ(Q) is constant, i.e., ϵ(Q) = ϵ for allQ ≥ 0. When demand is P (Q) = a − Qb, ϵ = b − 1 (note ϵ = 0 if demand is linear b = 1). Thusconstant elasticity of slope holds for a range of demand functions.

There are M Home firms producing the final good. One unit of final good requires one unitof intermediate good. We assume that Home firms transform the intermediate good costlesslyinto the final good. Furthermore, there are N Foreign firms producing the intermediate good ata constant marginal cost c > 0. Let r and t respectively denote per-unit price of the intermediategood and ad valorem tariff rate on imports. Home firms pay a tariff-inclusive input price r(1 + t)

per unit. Define τ ≡ 1 + t. Home firms’ per unit cost can then be expressed as rτ .The timing is as follows. A large number of identical potential entrants exist in both Home

and Foreign, each of whom must decide whether to enter. Should a Foreign firm decide to enterthe upstream sector, it must incur entry cost KF . Similarly, a Home firm incurs entry cost KH if itdecides to enter the downstream sector. Stage 2 involves Cournot competition among N Foreignfirms in the upstream sector. Similarly, M Home firms compete in quantities in the downstreamsector.

1(B.1) holds if and only if the demand function Q(P ) is logconcave (see Appendix).

B-1

B.1.2 Short-run analysis

Consider first an environment where M and N are exogenously given. Home firms compete inquantities in the final-good market. Each Home firm i chooses qi to maximize its profits,

πHi ≡(P

(qi +

M∑j =i

qj

)− rτ

)qi,

taking other Home firms’ quantities (qj) as given. Solving the first-order conditions of the firms’profit-maximization problem yields the equilibrium quantity for each Home firm:

q = −P (Q)− rτ

P ′(Q),

where Q solvesMP (Q) +QP ′(Q)−Mrτ = 0. (B.2)

Uniqueness of q and Q follows from (B.1).Let X denote the aggregate input demanded at any given input price r, which satisfies X = Q.

Substituting Q with X in (B.2) and rearranging yields

r =P (X) + XP ′(X)

M

τ≡ g(X,M, τ). (B.3)

Applying (B.1), we get

gX(X,M, τ) =(M + 1)P ′(X) +XP ′′(X)

Mτ=

P ′(X)(M + 1 + ϵ)

Mτ< 0. (B.4)

Moreover,XgXX(X,M, τ)

gX(X,M, τ)=

XP ′′(X)(M+1+ϵ)M

P ′(X)(M+1+ϵ)M

= ϵ, (B.5)

which says that the elasticity of slope of input demand is constant and the same as that of final-good demand. Since ϵ ≥ −1, we have that

gX(X,M, τ) +XgXX(X,M, τ) ≤ 0, (B.6)

for all X ≥ 0. This condition - analogous to (B.1) implies that upstream industry’s marginalrevenue (Xg(X,M, τ))

′ in the upstream sector decreases with X.As with Home firms, Foreign firms compete in quantities in the intermediate-good market.

Each Foreign firm i chooses xi to maximize its profits,

πFi ≡(g

(xi +

N∑j =i

xj ,M, τ

)− c

)xi,

B-2

taking other Foreign firms’ quantity (xj) as given. Solving the first-order conditions of the firms’profit-maximization problem yields the equilibrium quantity for each Foreign firm:

x = −g(X,M, τ)− c

gX(X,M, τ),

and X solvesNg(X,M, τ) +XgX(X,M, τ)−Nc = 0. (B.7)

Uniqueness of x and X follows from (B.6).To summarize, for a given M , N and τ , an equilibrium consists of an output vector (Q, X, q, x)

and a price vector (P , r) are such that:

• X solves (B.7), Q = X;

• q = QM , x = X

N ;

• P ≡ P (Q), r ≡ g(X,M, τ).

We are now ready to state and prove the comparative statics results with respect to M , N andτ(≡ 1 + t).

Proposition B.1

(i) For a given market structure (M,N ), a reduction in ad valorem tariff rate (τ ) raises aggregateoutput (Q, X) and firm-level output (q, x).

(ii) For a given ad valorem tariff rate, an increase in the number of firms in any sector raisesaggregate output but lowers firm-level output in that sector. However, firm-level output invertically related sector increases.

(iii) Input price received by each Foreign firm (r) increases with a reduction in tariff and with anincrease in the number of exporters (N ) but is independent of importers (M ).

Proposition B.1 says that aggregate output increases and the price of final good declines withan increase in tariff rate. The effects are exactly opposite for an increase in the number of firms.An increase in the tariff rate acts as increase in cost while an increase in the number of firmsacts as increased competition. The effects of increased cost and competition on aggregate outputsand prices are well understood from standard oligopoly theory. Output per firm also declines inboth sectors as the tariff rate increases. The effect of increased competition on firm level outputhowever is nuanced. Entry in any sector steals business from existing firms in that sector butcreates more business for these firms in the vertically related sector. As a consequence, outputper Home (Foreign) firm declines with entry of Home (Foreign) firms but increases with Foreignentry.

B-3

Finally, we turn to input price. An increase in tariff rate increases elasticity of input demandwhich in turn leads to lower input price. A reduction in input price acts as a terms-of-trade im-provement for Home. Later, we show that if the upstream sector is relatively thin, the welfaregains from improved terms of trade can outweigh the welfare losses from reduced aggregate out-put. Notwithstanding a reduction in r, Home firms that import inputs from Foreign firms face thehigher input price rτ with an increase in τ as the decline in r is proportionately less than an in-crease in τ . Regarding the effect of increased competition on the input price, while the input pricedecreases with the number of Foreign firms, a change in the number of Home firms has no effecton the input price. We can understand this invariance result ( ∂r

∂M = 0) in terms of the elasticity ofinput demand. Consider, for example, linear demand Q = a − P which gives rise to the followinginput demand function:

X =M

M + 1(a− r).

The elasticity of input demand, − rX

∂X∂r = r

a−r , is independent of M which in turn implies that theequilibrium input price is invariant to M . This property holds for all demand functions with theconstant elasticity of slope.2

B.1.3 Long-run analysis

Now we examine the effect of tariffs in an environment where the market structure is endoge-nously determined via free entry.

Cournot competition works exactly the same way for a given market structure (M,N). Con-sider the entry stage in which to determine M and N . Each Home firm pays KH to enter thedownstream sector which produces final goods. Each Foreign firm pays KF to enter the upstreamsector which produces intermediate goods. We assume that the entry costs are low enough so thatat least one Home firm and one Foreign firm enters the respective sector. Let

πH ≡ (P ∗ − r∗τ)q∗,

πF ≡ (r∗ − c)x∗,

denote the post-entry profit of a Home firm and a Foreign firm respectively where

Q∗ = X∗, q∗ =Q∗

M, x∗ =

X∗

N, r∗ ≡ g(X∗,M, τ).

Entry occurs until the post-entry profit of firms equals the entry cost, i.e., the net profit becomeszero. Using q∗ = Q∗

M and x∗ = X∗

N , the zero-profits conditions for the downstream and upstream

2For general demand functions, it can be shown that ∂r∂M

⋛ 0 ⇔ ϵd ⋛ ϵu where ϵd ≡ QP ′′(Q)P ′(Q)

and ϵu ≡ XgXX (X,M,τ)gX (X,M,τ)

.For demand functions with constant elasticity of slope, we have ϵd = ϵu and hence ∂r

∂M= 0.

B-4

sectors of production respectively can be expressed as:

(P ∗ − r∗τ)Q∗

M= KH , (B.8)

(r∗ − c)X∗

N= KF , (B.9)

where P ∗ = P (Q∗) and Q∗, X∗ and r∗ are functions of M and N . Solving (B.8) and (B.9) gives thefree entry number of Home and Foreign firms in each sector.

To summarize, the equilibrium outcomes (of the overall game) are:

• endogenously determined market structure (M, N ) - solution to (B.8) and (B.9),

• (Q, X) = (Q∗, X∗)|(M,N)=(M,N)

• (q, x) = (q∗, x∗)|(M,N)=(M,N),

• (P , r) = (P ∗, r∗)|(M,N)=(M,N).

Now we are ready to report the effect of a change in τ .

Proposition B.2

(i) A reduction in ad valorem tariff rate τ raises aggregate output (Q, X) and number of firms ineach sector (M, N ).

(ii) Each Home firm’s output (q) is strictly increasing in τ if and only if ϵ > 0, while each Foreignfirm’s output (x) is strictly increasing in τ if and only if

r >c

1 + ϵN

− 2+ϵN(2M+ϵ)

. (B.10)

(iii) The input price received by each Foreign firm (r) is decreasing in τ if and only if x is strictlyincreasing in τ , i.e., dx

dτ > 0 for all ϵ ≥ 1. If ϵ < 1, drdτ > 0 as long as (B.10) holds.

Recall from section B.1.2 that an increase in the tariff rate lowers aggregate output and raisesthe price of the final good. Here, an increase in the tariff rate discourages entry and reduces thenumber of firms in each sector, which in turn lowers aggregate output and raises the price of thefinal good even further. Note, as Home output and Foreign input are complements in our verticaloligopoly model, an increase in tariff rate on input import not only discourages entry of Foreignfirms but it also discourages entry of Home firms by raising the production costs in both sectors(dMdτ < 0, dNdτ < 0). This stands in sharp contrast to horizontal oligopoly models in which Homefirms’ and Foreign firms’ outputs are substitutes and tariffs create a firm-delocation effect (e.g.,Bagwell and Staiger 2012a, b): tariffs on Foreign final goods discourages entry of Foreign firmsand encourage entry of Home firms.

B-5

To understand part (ii), decompose dqdτ as:

dq

dτ=

∂q

∂τ+

∂q

∂M

dM

dτ+

∂q

∂N

dN

dτ.

From Proposition B.1, we know that ∂q∂τ < 0, i.e. an increase in tariff rate τ reduces Home firms’

output q for a given market structure. Here, additional effects arise from tariff-induced exit ofHome and Foreign firms. Fewer Foreign firms imply higher input price which prompts each Homefirm to cut back their output. Thus ∂q

∂NdNdτ < 0. Fewer Home firms also encourages surviving

Home firms to increase their scale of production. Thus ∂q∂M

dMdτ > 0. This positive impact of tariffs

on Home firms’ output dominates if and only if the inverse demand function is strictly concave.For linear demand (i.e. ϵ = 0), the positive and negative effects exactly offset each other thusdqdτ = 0.

Concerning the effect of import tariffs on Foreign firms’ output x we have:

dx

dτ=

∂x

∂τ+

∂x

∂N

dN

dτ+

∂x

∂M

dM

dτ.

From Proposition B.1, we know that ∂x∂τ < 0. When M and N are exogenously fixed, each Foreign

firm cuts back its production in response to increased τ . There is a further cutback as an increasein τ reduces the number of Home firms which in turn reduces the demand for intermediate input.Thus ∂x

∂MdMdτ < 0. Counteracting these negative effects (on x) is a positive effect arising from

weaker competition among Foreign firms. An increase in τ reduces the number of Foreign firmswhich in turn increases the scale of production for surviving Foreign firms. Thus ∂x

∂NdNdτ > 0. This

positive effect can dominate the negative effect for a range of parameterizations. For example,dxdτ > 0 for all strictly concave demand functions satisfying ϵ > 1. For linear demand (i.e. ϵ = 0), itfollows immediately from (B.10) that

dx

dτ> 0 =⇒ r >

c

1− 1MN

.

Clearly, this is more likely to hold if the number of Home firms is large enough.To summarize, an increase in the tariff rate reduces aggregate output as well as the number

of firms in both upstream and downstream sectors. Firm-level outputs - q and x - can increase ordecrease by an increase of tariffs, however, as the reduction in aggregate output might be moreor less than proportionate with reduction in the number of firms. The ambiguity on firm-leveloutputs is in line with the recent literature of heterogeneous-firm trade models where countriestrade differentiated final goods in monopolistic competition, even though most work focuses onthe impact of distance (rather than tariffs like ours). We show that the result similarly arises in afree-entry, vertical oligopoly model where countries specialize in different sectors of production.

Finally, we turn to the effect of tariffs on terms-of-trade. Recall that, in the absence of entry-exit considerations, an increase in tariff rate τ improve terms-of-trade (i.e. reduces r) for all

B-6

log-concave demand functions. Here, terms-of-trade improves irrespective of market structureif inverse demand is suitably concave in (B.1) or else if downstream sector is sufficiently thick.As in section B.1.2, higher tariff rates put downward pressure on input price r. However, byprompting exit of Foreign firms, higher tariff rates also put upward pressure on r which actstowards worsening of the terms-of-trade. As a consequence, the terms-of-trade improvement isless likely in the presence of entry-exit considerations.

B.2 Welfare and optimal tariff

Quasilinear preferences together with the assumption that all tariff revenues are rebated back tothe consumers allow us to express Home welfare (W ) as

W ≡

[∫ Q

0P (y)dy − P (Q)Q

]︸ ︷︷ ︸

Consumer surplus

+[P (Q)Q− rτ X − MKH

]︸ ︷︷ ︸

Home profits

+ (τ − 1)rX︸ ︷︷ ︸Tariff revenues

+ L︸︷︷︸Labor income

,

which, upon cancellation of common terms, could be further simplified to:

W =

∫ Q

0P (y)dy − rX − MKH + L.

Tariffs on intermediate input raises the price of the final good which hurts Home consumers.It also raises the tariff-inclusive price for the intermediate input which hurts Home producers.Despite both consumers and producers are worse off, tariffs on intermediate input can improvewelfare by improving the terms-of-trade (i.e., by lowering the input price r). Below we derive theconditions for welfare-improving tariffs and optimal tariff on intermediate inputs for the two sce-narios - first, when M and N are exogenously given and subsequently when they are endogenouslydetermined by free entry.

B.2.1 Short run

Differentiating W with respect to τ and rearranging we get:

∂W

∂τ= (P (Q)− r)

∂Q

∂τ− ∂r

∂τX, (B.11)

where M is assumed to be fixed. From Proposition B.1, we know that an increase in ad valoremtariff rate lowers aggregate output as well as the input price. The first term captures the welfareloss due to the tariff-induced output reduction and the second term captures the welfare gainsarising from the terms-of-trade improvement. Rewriting (B.11) as P (Q) − r = −P ′(Q)Q

M + (τ − 1)r

B-7

and using that and ∂Q∂τ = M ∂q

∂τ yields:

∂W

∂τ= −X

(∂r

∂τ+ P ′(Q)

∂q

∂τ

)+ (τ − 1)r

∂X

∂τ. (B.12)

Now consider an infinitesimally small increase in t from t = 0 or equivalently a small increasein τ from τ = 1. Starting from free trade (i.e. τ = 1) a small tariff on intermediate input importsimprove the terms-of-trade (i.e. ∂r

∂τ < 0) which increases welfare. However the firm-level outputof final goods decreases as well (i.e. ∂q

∂τ < 0) which reduces welfare. Proposition B.3a provides thenecessary and sufficient condition when an infinitesimally small positive tariff improves welfare.It also highlights how market structure affects the likelihood of welfare improvement.

Proposition B.3a

(i) Starting from free trade, a small increase in ad-valorem tariffs improves Home welfare, i.e.∂W∂τ |τ=1 > 0, if and only if the following condition holds:

r −(1− 1 + ϵ

N + 1 + ϵ

)(1 +

1

M + 1 + ϵ

)c > 0. (B.13)

(ii) For all M ≥ 1, there exists NM such that ∂W∂τ |τ=1 < 0 holds if and only if N > NM . For all

N ≥ 1, there exists MN such that ∂W∂τ |τ=1 > 0 holds if and only if M > MN .

To understand Proposition B.3a, fix M and let N → ∞. As the upstream sector becomesclose to perfectly competitive, the terms-of-trade motive (for tariffs) disappears. Only the harmfuleffect of the tariff – an increase in the final good price – remains. In this case, an import subsidyraises Home welfare by increasing aggregate output and lowering the price of the final good.More generally, when N is sufficiently large, upstream profits are negligible which makes thesetting akin to a single-stage, Cournot oligopoly with M domestic firms. In such settings, subsidyincreases welfare by narrowing the wedge between final good price and marginal cost. The samehappens here. Standard continuity argument then suggests that a positive subsidy continues tobe welfare-improving if N is larger than a threshold NM .3

Consider the other extreme: M → ∞. As the downstream sector becomes almost perfectlycompetitive, P − r → 0, and the welfare loss from tariff-induced output reduction, −(P (X)− r)∂X∂τ ,becomes vanishingly small whereas the welfare gain from terms-of-trade improvement, − ∂r

∂τ X,still remains. This setting is effectively equivalent to Home importing the final good from Foreignand its welfare is composed of consumer surplus and tariff revenues. From single-stage oligopolymodels we know that tariffs improve welfare in such settings as long as pass through from tariffto prices is not complete. Logconcave demand or equivalently 1 + ϵ ≥ 0 ensures incomplete pass-through in our setup. As in the above paragraph, standard continuity argument suggests that a

3Uniqueness of NM follows from noting that r − (1− 1+ϵN+1+ϵ

)(1 + 1M+1+ϵ

)c is strictly decreasing in N .

B-8

tariff continues to be welfare improving if M is larger than a threshold MN .That Home welfare could increase or decrease with small tariffs (in the neighborhood of free

trade) suggests that the optimal ad-valorem tariff rate could be positive or negative. Let τ = τ∗

denote the optimal ad-valorem tariff rate, i.e. the value of τ that maximizes Home welfare W .Proposition B.3b below characterizes τ∗ and describes how it varies with market structure.

Proposition B.3b

The optimal ad-valorem tariff rate is given by

τ∗ =r − Nc

N+1+ϵ(1 +1

M+1+ϵ)c

N+1+ϵ(r

r−c +N

M+1+ϵ), (B.14)

where r denote the equilibrium input price at τ = τ∗. Furthermore,

(i) τ∗ is increasing in M and decreasing in N .

(ii) For all N ≥ 1, there exists MN ≥ 1 such that τ∗ > 0 if and only if M ≥ MN . Similarly, for allM ≥ 1, there exists NM ≥ 1 such that τ∗ < 0 if and only if N > NM .

To understand the properties in (i) and (ii), note that finding optimal tariff is equivalent tofinding a solution to the fixed point problem f(τ) = τ where

f(τ) =r − Nc

N+1+ϵ(1 +1

M+1+ϵ)c

N+1+ϵ(r

r−c +1

M+1+ϵ).

We have f ′(τ) < 0 because f(.) is strictly increasing in r where r is strictly decreasing in τ (fromProposition B.1(iii)). Figure B.1 depicts a 45-degree line plus two f(τ) - one with f(0) > 0 and theother one with f(0) < 0.

Optimal tariff τ∗ is given by the intersection of 45-degree line and f(.). Observe that

f(0) ⋛ 0 =⇒ τ∗ ⋛ 0.

This observation, together with the following -

f(0) ⋛ 0 ⇐⇒ r − Nc

N + 1 + ϵ

(1 +

1

M + 1 + ϵ

)⋛ 0 ⇐⇒ dW

dτ

∣∣∣∣τ=1

⋛ 0

implies that optimal tariff is strictly positive (negative) if and only if starting from free trade aninfinitesimally small tariff increases (decreases) welfare. As the conditions under which τ∗ ⋛ 0

are the same as the ones under which dWdτ |τ=1 ⋛ 0, Proposition B.3b(ii) follows immediately from

Proposition B.3a(ii).To understand Proposition B.3b(i), note that f(τ) shifts up when M increases while it shifts

down when N increases. From Figure B.2 then it immediately follows that τ∗ increases when M

B-9

!(#)

##∗ < 0 45°

!(#)

! 0 < 0

(a) τ∗ < 0

!(#)

##∗ > 0

45°

!(#)

! 0 > 0

(b) τ∗ > 0

Figure B.1 – Optimal tariff in the short run

increases and τ∗ decreases when N increases. (While Figure B.2 focuses on the case for f(0) > 0,the same applies to the other case for f(0) < 0.)

The upshot of the above discussion applies when the market structure is exogenously given,optimal tariff is strictly positive if the number of Home firms is large enough while it is strictlynegative if the number of Foreign firms is large enough. Negative optimal tariff does not arisein single-stage oligopoly models with logconcave demand functions (unless there is free entry ofForeign firms). Negative tariff, i.e. import subsidy, is rarely seen in reality. One way to interpretnegative optimal tariff is that for parameterizations - which entail τ∗ < 0 - a unilateral reductionin tariffs from any positive level improves welfare.

B.2.2 Long run

Differentiating W with respect to τ and rearranging we get

dW

dτ= (P (Q)− r)

dQ

dτ− X

dr

dτ−KH

dM

dτ. (B.15)

The first two terms are familiar from those in the exogenous market structure. However, unlikethat setting, the terms-of-trade might improve or worsen depending on ϵ and market structure(see Proposition B.2(iii)). In addition, the new term, −KH

dMdτ , captures the welfare gains from a

reduction in overall entry costs as fewer Home firms enter when tariffs increase.Rewriting the zero-profits condition in the downstream sector as (P (Q) − rτ)X − MKH = 0,

B-10

!(#)

##∗

45°

!(#)

(a) An increase in M

!(#)

##∗

45°

!(#)

(b) An increase in N

Figure B.2 – Impact of M and N

totally differentiating it with respect to τ yields

(P (Q)− rτ)dQ

dτ+

(P ′(Q)

dQ

dτ− r − τ

dr

dτ

)X −KH

dM

dτ= 0,

which enables us to write (B.15) as

dW

dτ= −X

(P ′(Q)

dQ

dτ− r

)+ (τ − 1)

d(rX)

dτ. (B.16)

Let us consider again the welfare effect of tariff in the neighborhood of free trade (i.e. τ = 1).(B.16) indicates that, as τ rises slightly from τ = 1, consumer surplus decreases by −P ′(Q)dQdτ X

while tariff revenues increase by rX. Home welfare W increases with imposition of tariffs if andonly if increase in tariff revenues outweigh the loss in consumer surplus. Proposition B.4a statesthe precise condition for welfare improvement and how it relates to demand curvature and marketstructure.

Proposition B.4a

(i) Starting from free trade, a small increase in ad-valorem tariffs improves Home welfare, i.e.∂W∂τ |τ=1 > 0, if and only if the following condition holds:

r >c

1 + ϵ( 1M

+ 1N) + ϵ2−ϵ−2

2MN

(B.17)

where r, M , and N denote the input price, and the number of Home and Foreign firms in

B-11

free-entry equilibrium.

(ii) For all demand functions satisfying (B.1), a small ad-valorem tariff improves welfare irre-spective of market structure. Furthermore,

(a) limKH→0dWdτ

∣∣τ=1

> 0 if g(.) is logconcave in X,

(b) limKF→0dWdτ

∣∣τ=1

⋛ 0 if and only if ϵ ⋛ 0.

Proposition B.4a says that Home welfare improves with imposition of tariffs if equilibriuminput price is larger than a threshold. The input price and threshold depend on endogenouslydetermined market structure (M, N). To gain insight, it is useful to consider some specific cases.First consider KH → 0. As Home entry costs become arbitrarily small M → ∞ and equation (B.17)reduces to r > c

1+ ϵN

. Since r > c the condition always holds if g(.) is concave (i.e. ϵ ≥ 0). With somealgebra, it can also be shown that the condition holds for logconcave g(.) too (see Appendix). Thus,as in the case with exogenous (M,N), tariffs improve welfare if the number of downstream firmsis large enough.

Now consider KF → 0 in which case N → ∞ and the input price r approaches the marginalcost c. Equation (B.17) reduces to r = c > c

1+ ϵM

which holds if and only if ϵ > 0. In section B.2.1we argued that as N becomes arbitrarily large, the environment becomes like the domestic single-stage oligopoly setting in which subsidy unambiguously improves welfare. However, even in suchsettings, subsidy is not necessarily optimal in the presence of free entry. Subsidy prompts entrywhich reduces price and improves allocative efficiency. At the same time, production efficiencyworsens (i.e., average cost per firm increases) as entry prompts each firm to scale down theiroutput. The welfare losses from reduced production efficiency exactly offsets the welfare gainsfrom increased allocative efficiency if and only if ϵ = 0 (i.e., linear demand). For ϵ > 0, the welfarelosses from reduced production efficiency dominate the welfare gains, which implies that a positiveimport tariff (rather than an import subsidy) can improve Home welfare. The opposite is true forϵ < 0.

To better understand the effect of a small tariff for arbitrary (KH ,KF ), we rewrite the zeroprofit condition in the downstream sector as

P = rτ +KH

q.

Totally differentiating the equation with respect to τ , substituting the resultant expression in(B.16) and evaluating at τ = 1 we get:

dW

dτ

∣∣∣∣τ=1

> 0 ⇐⇒ −dr

dτ+

KH

q2dq

dτ> 0.

A small import tariff reduces r for a given (M, N). This direct effect is present here as well.However, the number of Foreign firms decreases with tariff which in turn puts upward pressure

B-12

on r. Thus, the terms-of-trade may not necessarily improve with imposition of tariffs, as shownin Proposition B.2. Since the terms-of-trade improvement becomes less likely with free entry, thepossibility that tariffs can improve welfare becomes less likely too. Counteracting this effect ispossible gains in production efficiency arising from tariff-induced rationalization of industries -an effect that is absent under endogenous market structure. Free entry often leads to sociallyexcessive number of firms in homogeneous product oligopolies - too many firms enter and eachfirm produces too little output (Mankiw and Whinston, 1986; Suzumura and Kiyono, 1987). Inour model, an imposition of tariffs implies fewer firms, allowing for a larger scale of productionand a lower average cost, which work towards the welfare improvement. Under condition (B.1),the input price declines ( drdτ > 0) and each Home firm’s scale of production increases ( dqdτ > 0). Forsuch demand functions welfare improves with an imposition of a small import tariff irrespectiveof market structures.

Both proof and intuition are similar to the ones outlined above for Proposition B.4a. Althoughwe omit them here to avoid repetition, it is worth noting what to make of the facts that (a) demandcurvature matters for the sign of optimal tariffs and (b) the sign of optimal tariff can differ be-tween two environments - endogenous and exogenous market structures. In terms of dependenceof optimal policy on demand curvature, our results have similar flavor to some of the existingresults in the trade literature. For example, the classic result that the sign of optimal tariffs inthe presence of a Foreign monopoly depends on whether there is incomplete pass-through, whichin turn depends on whether the demand curve is flatter than the marginal revenue curve (e.g.,Brander and Spencer, 1984a, b). In terms of the difference in optimal trade policy between the en-dogenous and exogenous market structures, our result is reminiscent of the recent re-examinationof Cournot oligopoly models in which free entry can alter the conventional implication of optimaltrade policy due to a firm-relocation effect (e.g., Bagwell and Staiger, 2012a, b).

B-13

Proofs for Appendix B

(B.1) ⇔ Q(P ) is logconcave

The assumption Q(P ) is logconcave implies that

d

dP

[d lnQ(P )

dP

]=

d

dP

[Q′(P )

Q(P )

]=

Q(P ) ·Q′′(P )− [Q′(P )]2

[Q(P )]2≤ 0,

which can be expressed asQ(P )Q′′(P )

[Q′(P )]2≤ 1. (B.18)

Differentiating P = P (Q(P )) with respect to P , we get

1 = P ′(Q(P ))Q′(P ).

Differentiating this once again with respect to P gives

0 = P ′′[Q′(P )]2 + P ′Q′′(P ).

Rewriting this equation, we getQ′′(P )

[Q′(P )]2= −P ′′

P ′ .

Substituting this relationship into (B.18), we find that

−QP ′′(Q)

P ′(Q)≤ 1,

which implies P ′(Q) +QP ′′(Q) ≤ 0. □

Proof of Proposition B.1

(i) Differentiating (B.7) with respect to τ and rearranging, we get

∂X

∂τ=

Nc

(N + 1)gX(X,M, τ) + XgXX(X,M, τ)=

MNc

P ′(Q)(M + 1 + ϵ)(N + 1 + ϵ). (B.19)

B-14

Using (B.19), we get

∂Q

∂τ=

∂X

∂τ=

MNc

P ′(Q)(M + 1 + ϵ)(N + 1 + ϵ)< 0,

∂P

∂τ= P ′(Q)

∂Q

∂τ=

MNc

(M + 1 + ϵ)(N + 1 + ϵ)> 0,

∂q

∂τ=

1

M

∂Q

∂τ=

Nc

P ′(Q)(M + 1 + ϵ)(N + 1 + ϵ)< 0,

∂x

∂τ=

1

N

∂X

∂τ=

Mc

P ′(Q)(M + 1 + ϵ)(N + 1 + ϵ)< 0,

(B.20)

where the signs follow from noting that P ′(.) < 0, M + 1 + ϵ > 0 and N + 1 + ϵ > 0. Finally,

∂r

∂τ= −g(X,M, τ) + gX(X,M, τ)

∂X

∂τ= −1

τ

(r − Nc

N + 1 + ϵ

)< 0. (B.21)

That ∂r∂τ < 0 follows from noting that r > c and 1 + ϵ ≥ 0.

(ii) Differentiating (B.7) with respect to N and rearranging, we get

∂X

∂N= − g(X,M, τ)− c

(N + 1)gX(X,M, τ) + XgXX(X,M, τ)

= −g(X,M,τ)−c

gX(X,M,τ)

(N + 1) + XgXX(X,M,τ)

gX(X,M,τ)

=x

N + 1 + ϵ.

Using the expression for ∂X∂N , we get

∂Q

∂N=

∂X

∂N=

x

N + 1 + ϵ,

∂P

∂N= P ′(Q)

∂Q

∂N=

xP ′(Q)

N + 1 + ϵ.

Using the expressions for ∂Q∂N , ∂X

∂N and noting that Q = Mq and X = Nx, we get

∂q

∂N=

q

N(N + 1 + ϵ),

∂x

∂N= − (N + ϵ)x

N(N + 1 + ϵ).

B-15

Finally, noting that r = g(X,M, τ), we have

∂r

∂N= gX(X,M, t)

∂X

∂N

=xP ′(X)(M + 1 + ϵ)

M(N + 1 + ϵ).

Signs of ∂X∂N , ∂Q

∂N , ∂P∂N , ∂q

∂N , ∂x∂N and ∂r

∂N follow from noting that P ′(.) < 0, M+1+ϵ > 0 and N+1+ϵ > 0.

(iii) Differentiating (B.7) with respect to M and rearranging, we get

∂X

∂M= − NgM (X,M, τ) + XgXM (X,M, τ)

(N + 1)gX(X,M, τ) + XgXX(X,M, τ).

Recall that g(X,M, τ) = P (X) + XP ′(X)M , which implies that

gM (X,M, τ) = −XP ′(X)

M2,

gXM (X,M, τ) = −P ′(X)(1 + ϵ)

M2.

Substituting these into the expression for ∂X∂M and simplifying, we get

∂X

∂M=

q

M + 1 + ϵ.

Then we have that

∂Q

∂M=

∂X

∂M=

q

M + 1 + ϵ,

∂P

∂M= P ′(Q)

∂Q

∂M=

P ′(Q)q

M + 1 + ϵ.

Using Q = Mq, X = Nx and the expressions for ∂Q∂M and ∂X

∂M , we get

∂q

∂M= − (M + ϵ)q

M(M + 1 + ϵ),

∂x

∂M=

x

M(M + 1 + ϵ).

B-16

Finally, note that

∂r

∂M=

[gX(X,M, τ)

∂X

∂M+ gM (X,M, τ)

]

=

[P ′(X)(M + 1 + ϵ)

M· q

M + 1 + ϵ− XP ′(X)

M2

]= 0

since P ′(X)qM = P ′(X)X

M2 . Signs of ∂X∂M , ∂Q

∂M , ∂P∂M , ∂q

∂M , ∂x∂M and ∂r

∂M follow from noting that P ′(.) < 0,M + 1 + ϵ > 0 and N + 1 + ϵ > 0.

Proof of Proposition B.2

Let a dot represent proportional rates of change (e.g. Q ≡ dQQ ) and totally differentiating (B.7),

(B.8) and (B.9) respectively gives

(N + 1 + ϵ)X =

(N + 1 + ϵ

M + 1 + ϵ

)M + N +

(MNc

XP ′(X)(M + 1 + ϵ)

)τ , (B.22)

(2 + ϵ)X = 2M, (B.23)

(2 + ϵ)X =

(1 + ϵ

M + 1 + ϵ

)M + 2N + τ , (B.24)

where KH and KF hold constant. Note that (B.22), (B.23) and (B.24) are the three equations thathave the three unknowns X(= Q), M and N . Solving the three equations for the three unknownsand rearranging, we obtain

τ

X

dX

dτ=

−2(M + 1 + ϵ) + 4MNcXP ′(X)

(2M + ϵ)(2N + ϵ)− (2 + ϵ), (B.25)

τ

M

dM

dτ=

(2 + ϵ

2

)τ

X

dX

dτ, (B.26)

τ

N

dN

dτ= −1

2+

((2 + ϵ)(2M + 1 + ϵ)

4(M + 1 + ϵ)

)τ

X

dX

dτ, (B.27)

That all these expressions are negative follows from noting that (2M + ϵ)(2N + ϵ) − (2 + ϵ) > 0.Observe that dQ

dτ = dXdτ < 0 while dP

dτ = P ′(X)dXdτ > 0.

(ii) Differentiating (B.8) and using (B.26), we get

τ

q

dq

dτ= − ϵ

2

τ

X

dX

dτ,

B-17

which implies that dqdτ ⋛ 0 ⇔ ϵ ⋛ 0. Similarly, differentiating (B.9) and using (B.27), we get

τ

x

dx

dτ=

1

2+

((2 + ϵ)− ϵ(2M + ϵ)

4(M + 1 + ϵ)

)τ

X

dX

dτ.

Substituting (B.26) and rearranging, dxdτ > 0 if

1− MNτc

XP ′(X)(M + 1 + ϵ)<

N(2M + ϵ)

2 + ϵ− ϵ(2M + ϵ).

Note from (B.4) and (B.7) that

MNτc

XP ′(X)(M + 1 + ϵ)=

Nτc

XgX(X, M , τ)= 1 +

Ng(X, M , τ)

XgX(X, M , τ).

Substituting the last equality and noting r = g(X, M , τ) establishes the result.

(iii) Rewrite (B.9) asr = c+

KF

x

and observe that drdτ < 0 ⇔ dx

dτ > 0.

Proof of Proposition B.3a

(i) From (B.20) and (B.21) in the proof of Proposition B.1, we know that

∂P

∂τ=

MNc

(M + 1 + ϵ)(N + 1 + ϵ),

∂r

∂τ= −1

τ

(r − Nc

N + 1 + ϵ

).

Substituting these expressions in (B.12) and then rearranging and evaluating the resultant ex-pression at τ = 1 leads immediately to (B.13).

(ii) Define h(r;M,N) ≡ r −(1− 1+ϵ

N+1+ϵ

)(1 + 1

M+1+ϵ

)c. We have

hM (.) =Nc

(N + 1 + ϵ)(M + 1 + ϵ)2> 0, lim

M→∞h(.) = r −

(Nc

N + 1 + ϵ

)>

(1 + ϵ)c

N + 1 + ϵ> 0,

which implies existence of a unique MN ∈ [1,∞) such that dWdτ |τ=1 > 0 ⇔ M > MN . Similarly,

hN (.) =∂r

∂N− (1 + ϵ)(M + 2 + ϵ)c

(M + 1 + ϵ)(N + 1 + ϵ)2< 0, lim

N→∞h(.) = − c

M + 1 + ϵ< 0,

which implies existence of a unique NM ∈ [1,∞) such that dWdτ |τ=1 < 0 ⇔ N > NM .

B-18

Proof of Proposition B.3b

Note first that the expressions for ∂X∂τ , ∂P

∂τ and ∂r∂τ in (B.16) are already derived in (B.19), (B.20)

and (B.21) respectively. Substituting them into (B.16) and rearranging, we get

∂W

∂τ= −X

(Nc

(M + 1 + ϵ)(N + 1 + ϵ)− 1

τ

(r − Nc

N + 1 + ϵ

))+

rτMNc

P ′(X)(M + 1 + ϵ)(N + 1 + ϵ). (B.28)

Noting that gX(X,M, τ) = P ′(X)(M+1+ϵ)M and XgX(X,M,τ)

τN = −(r − c), we can rewrite (B.28) as

∂W

∂τ= −X

τ

(Nτc

(M + 1 + ϵ)(N + 1 + ϵ)− r +

Nc

N + 1 + ϵ

)− rτ

(Xc

τ(r − c)(N + 1 + ϵ)

).

Setting ∂W∂τ = 0 and solving for τ yields the expression in (B.14).

Proof of Proposition B.4a

(i) From (B.16), we know thatdW

dτ

∣∣∣∣τ=1

> 0 ⇐⇒ r >dP

dτ, (B.29)

where

dP

dτ= P ′(X)

dX

dτ

=

XP ′(X)τ

(−2(M + 1 + ϵ) + 4MNτc

XP ′(X)

)(2M + ϵ)(2N + ϵ)− (2 + ϵ)

(using (B.25))

=−2XP ′(X)(M+1+ϵ)

τ + 4MNc

(2M + ϵ)(2N + ϵ)− (2 + ϵ)

=2MN

(−xgX(X, M , τ) + 2c

)(2M + ϵ)(2N + ϵ)− (2 + ϵ)

(using (B.4))

=2MN(r − c+ 2c)

(2M + ϵ)(2N + ϵ)− (2 + ϵ)(using (B.3) and (B.7))

=2MN(r + c)

(2M + ϵ)(2N + ϵ)− (2 + ϵ). (B.30)

Substituting (B.30) into (B.29) gives us (B.17).

(ii) Let us consider (a) which states that limKH→0 τ∗ > 0 if g(X,M, τ) is logconcave in X. As Home

entry costs become arbitrarily small, the number of Home firms becomes arbitrarily large, andhence (B.17) reduces to

limKH→0

dW

dτ

∣∣∣∣τ=1

> 0 ⇐⇒ r >c

1 + ϵN

. (B.31)

B-19

Further, using (B.3), let us rewrite (B.7) as

c = r

(1 +

1

N

X

r

dr

dX

).

Substituting this into (B.31) and rearranging, (B.31) further reduces to

1 >g(X, M , τ) · gXX(X, M , τ)

[gX(X, M , τ)]2. (B.32)

As in section A.1, the assumption g(X,M, τ) is logconcave in X implies that

d

dX

[d ln g(X, M , τ)

dX

]=

d

dX

[gX(X, M , τ)

g(X, M , τ)

]

=g(X, M , τ) · gXX(X, M , τ)− [gX(X, M , τ)]2

[gX(X, M , τ)]2≤ 0,

which can be expressed as (B.32). Thus, (B.31) is always satisfied under the condition. As for(b) which states that limKF→0

dWdτ

∣∣τ=1

⋛ 0 if and only if ϵ ⋛ 0, it follows immediately from theexplanation in the main text.

B-20