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Journal of Economic Geography
, Volume 18 (2) – Mar 1, 2018

34 pages

/lp/ou_press/mobile-capital-variable-elasticity-of-substitution-and-trade-03eI32gDPl

- Publisher
- Oxford University Press
- Copyright
- © The Author (2017). Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com
- ISSN
- 1468-2702
- eISSN
- 1468-2710
- D.O.I.
- 10.1093/jeg/lbx022
- Publisher site
- See Article on Publisher Site

Abstract This article investigates the impact of trade liberalization on trade patterns, firm markups, and firm locations in a two-factor monopolistic competition model that features variable elasticity of substitution by a general additively separable utility. We find that, depending on the relative export hurdles, either direction of one-way trade may occur when trade opens up. Its direction determines the responses of firm-level markups and various home market effects to falling trade costs. Our results show that some important findings in the literature are robust only with particular classes of preferences. We provide a possible rationale for some well-known conflicting empirical facts. 1. Introduction Constant elasticity of substitution (CES) preferences are at heart of the Dixit–Stiglitz monopolistic competition model (Dixit and Stiglitz, 1977). Its property—that firms’ profit-maximizing prices display constant markups over marginal costs—considerably simplifies the calculation and allows us to explore markets of imperfect competition and increasing returns in a general equilibrium framework. This overwhelming advantage has caused it to become the workhorse of economists in New Trade Theory/New Economic Geography (NTT/NEG). In their seminal book, The Spatial Economy: Cities, Regions, and International Trade, Fujita et al. (1999, p. 6) even mock their own book, saying that ‘this book sometimes looks as if it should be entitled Games You Can Play with CES Functions’. However, there have been a number of empirical studies on firm-level markups (e.g. De Loecker, 2011; De Loecker and Warzynski, 2012; Feenstra and Weinstein, 2017) reporting that firm markups vary widely across industries and markets. In addition, in the literature of new open economy macroeconomics, a lot of evidence is found against the law of one price, and markup gaps contribute to the variability in price gaps (Krugman, 1987; Lane, 2001; Gopinath et al., 2011). Many authors, presuming that international markets for manufacturing goods are sufficiently segmented, have introduced the so-called ‘pricing-to-market’ into their analyses, calling for models of international trade that allow for variable markups (Atkeson and Burstein, 2008). Although studies following Ottaviano et al. (2002) and Melitz and Ottaviano (2008) work with quasi-linear preferences to introduce endogenous markups, their settings, which feature a freely traded numéraire good produced with linear technology under perfect competition, imply wage equalization across countries. Thus, income effects on prices and agglomeration are diluted in their model. Due to this drawback, their framework fails to account for Simonovska’s (2015) empirical finding that firm markups are higher in countries with higher per capita income. A framework that features variable markups, and that generates income effects is needed, which is the first motivation of our article. Our second motivation lies in explanations for the emergence of zero trade flows, a phenomenon of ‘missing trade’ emphasized by Trefler (1995). In CES preferences, the marginal utility at zero consumption is infinitely large, meaning that it would always be optimal for consumers to consume a (small) positive amount even when prices are very high and/or their budgets are very low. Therefore, international trade models with CES preferences always propose arguments from the supply side. For example, Helpman et al. (2008) suggest that the combination of asymmetric fixed costs of exporting and firm heterogeneity in productivity with a truncated distribution leads to zeros. Eaton et al. (2012) generate zeros by considering an integer number of firms.1 Nevertheless, the supply-side arguments proposed in the literature are definitely not the only possible explanations. Departing from homothetic preferences, how a more general class of variable elasticity of substitution (VES) preferences influences trade patterns should be investigated in a subtler way. To investigate the role of VES in trade patterns, markups, and locations of firms in a globalized world, in which the income effect works as an important force for firms’ pricing-to-market and agglomeration, this article establishes a two-factor trade model of monopolistic competition by relaxing the CES assumption. Specifically, in addition to immobile labor, mobile capital is assumed as another production factor as in the footloose capital model of Takahashi et al. (2013), which is a variant of Martin and Rogers’ (1995) that removes the agricultural sector. By applying a general utility function featuring VES to this footloose capital model, the relationships of trade costs to trade pattern, firm markups, and firm locations are revealed. In particular, we are able to obtain some properties showing how the system reacts when trade costs are intermediate (i.e. between free trade and autarky) in the VES context. This is important because one of the interesting findings of spatial economics is that many economic variables that describe spatial development do not necessarily change in a monotonic way when trade costs fall continuously (Fujita and Thisse, 2013, Chapters 8, 9). It is worth noting that this model does not include firm heterogeneity in productivity. Departing minimally from the standard CES setup used in many NEG/NTT models, we clarify some mechanisms purely from the VES preferences. Our results contribute to the literature in three aspects. First, we develop preference conditions for the one-way trade direction, providing a demand-side explanation of why zero trade flows may occur. Generally, importing prices that are higher than the choke prices in destination markets lead to zero trade flows. Thus, when trade costs decrease from a prohibitive level in autarky, one-way trade—either from the larger to the smaller country or from the smaller to the larger one—may happen. The direction depends on which country can reduce its exporting prices lower than the market choke prices of its trade partner as trade costs decrease. In Figure I of Helpman et al. (2008), we see that two-way, one-way, and no trade account for 30–40, 10–20, and 40–55%, respectively, of country pairs in 158 countries from 1970 to 1997. Their model considers only immobile production factor(s). Subject to the balance of payments, a framework of more than two countries is necessary to derive the result of one-way trade. In contrast, a two-factor model with VES utility allows us to see the other side of the coin—demand side factors can produce one-way trade patterns—even in a two-country space. Thus, our argument does not need to involve any third-country effects, which differentiates our arguments from those of the existing literature. In addition, we reinvestigate bilateral trade flows using the larger dataset, ‘World Trade Flows 1962–2000’, available from Feenstra et al. (2005), to explore the direction of one-way trade, which is not explicitly addressed in Helpman et al. (2008). We observe one-way trade flows both from a larger to a smaller country and in the opposite direction. The upper panel of Figure 1 duplicates the work of Helpman et al. (2008), but in 187 countries from 1962–2000.2 One-way trade still accounts for 10–20% of total country pairs in recent years. The lower panel of Figure 1 partitions the one-way trade country pairs into two groups by their direction of trade flow: from the larger to the smaller country or from the smaller to the larger one.3 It turns out that the average share of one-way trade from a smaller to a larger country in the period between 1962 and 2000 was 34.04%. This share becomes as high as 45.26% in 2000, although trade from a larger to a smaller country still dominates. Figure 1 View largeDownload slide Distribution of country pairs pertaining to trade patterns and one-way trade directions. Figure 1 View largeDownload slide Distribution of country pairs pertaining to trade patterns and one-way trade directions. Second, we study the responses of variable markup patterns to trade liberalization. We show that the domestic markups in the larger country and the smaller country display opposite trends when trade opens. Whether the domestic markups of firms go up or down also depends on the direction of the one-way trade flow. We also examine firm markups in different markets and clarify the directions of these market-specific markups in response to the case of trade opening and the case of being close to free trade. As in Behrens et al. (2014), they are aggregated as the revenue-weighted average markup across all markets for a firm as well as the expenditure-weighted average markup across all origins for a consumer. We show that neither changes monotonically along with falling trade costs. More specifically, although these average markups are indeed higher in autarky than in free trade, they increase when trade costs are falling close to free-trade levels. Third, we study various forms of the home market effect (HME) in this general VES setting. The HME in terms of firm share indicates that the firm share in the larger country is more than its population share, while the HME in terms of wages refers to the fact that the wage rate in the larger country is higher. These two aspects of the HME are known to be equivalent in the CES case. However, we find that they are not equivalent in a general VES framework. In particular, the HME in terms of firm share is reversed in the case of one-way trade from the smaller to the larger country. 1.1. Related literature There are some important papers on VES preferences in the literature, from which our work gains insight. Behrens and Murata (2007) suggest that the constant absolute risk aversion (CARA) specification may be useful for analyzing market structures that feature the pro-competitive effect. Behrens and Murata (2012) further examine the impact of trade on welfare using this framework. Subsequently, Behrens et al. (2014) apply the CARA utility to disclose how wages, productivity, consumption diversity, and firm markups respond to trade integration. Zhelobodko et al. (2012) propose a model of monopolistic competition based on general additive preferences and use the concept of relative love for variety (RLV) to characterize the pro-competitive and the anti-competitive effects. The application of the setting to international trade is presented in their early working paper version (Zhelobodko et al., 2010), which uncovers dumping and reverse dumping. Parenti et al. (2017) build a general equilibrium model of monopolistic competition to encompass existing models that is capable of mimicking oligopolistic behavior and replicating partial equilibrium results. Instead of starting with utility functions, Mrázová and Neary (2013) introduce a device, called ‘the demand manifold’, which links the elasticity and the convexity of an arbitrary demand function. It relates the demand structure to the behavior of monopoly firms and monopolistically competitive industries. Applying this approach, Mrázová and Neary (2014) combine variable demand elasticities and trade costs to explore their implications on welfare. Bertoletti and Epifani (2014) examine the Melitz (2003) setup with heterogeneous firms and fixed and variable trade costs beyond the CES. They find that extensive margin effects are robust to relaxing the CES assumption but that intensive margin effects and competitive (markup) effects are, instead, fragile. Arkolakis et al. (2015) also adopt a general specification of demand, which is subconvex and features finite choke prices, to clarify the impact of the pro-competitive effect on welfare gains from trade liberalization. Explaining zero trade flows from the demand side, Melitz and Ottaviano (2008) and Okubo et al. (2014) use the quasi-linear preferences of Ottaviano et al. (2002) to model the one-way trade of differentiated goods by choke prices, too. Their findings are limited to the one-way trade of differentiated goods from the larger to the smaller country. In addition, they assume that the numéraire good is transferred from the net importing country of differentiated varieties to equilibrate the trade balance. In their framework, the one-way trade in the sense of Helpman et al. (2008) cannot occur. Finally, as mentioned above, the wage rates in two countries are equalized so that the income effect and the production cost effect are muted in their model. Incorporating non-homothetic preferences into a heterogeneous firm model with one sector of differentiated goods, Behrens et al.’s (2014),CARA and Simonovska’s (2015) Stone-Geary frameworks may also generate zeros in a multi-country setting. However, in this article, we show that these often-used VES preferences are still not general enough to account for one-way trade in both directions. They only predict one-way trade from the smaller to the larger country. Firm markups are also addressed in Behrens et al.’s (2014) paper. By quantifying their model on Canada–U.S. regional trade data, they find that, contrary to pro-competitive effects on consumers’ side, changes in firm-level markups are ambiguous. They suggest that changes in markups result from heterogeneity in firm productivity. In contrast, without firm heterogeneity, we analytically prove how the revenue-weighted average markup of a firm as well as the expenditure-weighted average markup of a consumer respond to the opening of trade differ with each other in the two opposite cases of one-way trade directions. The HMEs in terms of wages and firm share are called the price (of labor) and the quantity (of production) aspects of the HME in Head and Mayer (2004). Redding and Venables (2004) and Hanson (2005) provide strong empirical evidence of the price aspect of the HME. In contrast, after surveying a lot of empirical studies, Head and Mayer (2004, p. 2636) conclude that the evidence on the HME is highly mixed regarding the quantity aspect. The consistent result of the price aspect and the inconsistent empirical facts regarding the quantity aspect all support our theoretical result on the HME using VES preferences. The remainder of this article is organized as follows. The model is specified in Section 2. Then, Section 3 analyzes the equilibrium and one-way trade patterns, while Section 4 discusses markups, and Section 5 considers the HME. Finally, Section 6 concludes the article. 2. The model Consider a world comprising just two countries, n and s. These two countries have the same physical geographical constraints and technologies. However, their populations differ. The total mass of workers (population) is L, and the population in country l∈{n,s} is Ll. The population share in country l is denoted by θl≡Ll/L and, thus, θn+θs=1. We assume that θn∈(1/2,1) so that country n is larger. Each worker owns one unit of labor and one unit of capital,4 which are both supplied inelastically.5 Workers (i.e. capital owners) are immobile; however, they can invest their capital across countries.6 The wage rate (resp., capital return) received from country l∈{n,s} is denoted by wl (resp., rl). Choose the labor in country s as the numéraire, so that ws=1. We further let w denote the ratio of wage rates in two countries: w=wn/ws=wn. A single monopolistically competitive sector produces a continuum of horizontally differentiated consumption varieties.7 Since consumers spend all of their income on goods in this single sector, the economy shows the income effect. Product differentiation ensures a one-to-one relation between firms and varieties. Nl denotes the mass of firms in country l∈{n,s}. 2.1. Preferences and demands We assume that preferences are additively separable over a continuum of horizontally differentiated consumption varieties. Specifically, a consumer in country l,m∈{n,s} solves the following utility maximization problem: maxcll(i),cml(j)Ul≡∫0Nlu(cll(i))di+∫0Nmu(cml(j))dj,s.t. yl=∫0Nlpll(i)cll(i)di+∫0Nmpml(j)cml(j)dj, (1) where m≠l; u(·) is thrice continuously differentiable, strictly increasing, and strictly concave on (0,∞); cml(j) stands for the per capita consumption of variety j∈[0,Nl] produced in country m and consumed in country l; pml(j) is its price; and yl≡wl+rl represents the household income in country l. The varieties are supposed to be symmetric, so we can omit the variety name and simply use cml to refer to the per capita demand of each variety produced in country m and consumed in country l. To guarantee the validity of the analysis, we need to impose some restrictions on u(·). First, the conditions that u(0)=0 and u′(0)<∞ must hold. Thus, we exclude the utility functions that satisfy the Inada conditions, including the CES case. Second, to capture the subconvexity of demand, we assume that the RLV Ru(x)≡-xu″(x)/u′(x) is strictly increasing.8 Third, for the existence and uniqueness of the interior monopolistically competitive equilibrium, we assume that Ru(x)<1 and Ru′(x)≡-xu‴(x)/u″(x)<2.9 In Mrázová and Neary (2014, footnote 3), Ru(x) is called the concavity of the subutility and Ru′(x) measures the convexity of demand because Ru′(x)=-xp″(x)/p′(x). Then, by definition, we further obtain Ru(0)=0 and xRu′(x)=Ru(x)[1+Ru(x)-Ru′(x)]>0. These assumptions are not restrictive. In fact, they are met by the CARA function used in Behrens and Murata (2007) and the hyperbolic absolute risk aversion (HARA) function of Merton (1971, p. 389; see Expression (19) later). Let us summarize all of the restrictions imposed on u(·) as follows: u(0)=0, u′(x)>0, u′(0)<∞, u″(x)<0,Ru′(x)>0, Ru(x)<1, Ru′(x)<2. (2) The first-order conditions associated with utility maximization give the inverse demand functions as follows: u′(cml)=λlpml, l,m∈{n,s}, (3) where Lagrange multiplier λl is the shadow price of the budget constraint of (1). It depends on market aggregates and is taken as given for an individual firm under monopolistic competition. Since the marginal utility at zero consumption is bounded (i.e. u′(0)<∞), the demand for a variety cml=(u′)-1(λlpml), can be zero when price pml is high enough. Consumers have positive demands for a good if and only if the price pml is lower than the choke price, given by plc≡u′(0)/λl. Because varieties are provided by monopolistically competitive firms, an individual firm’s decision has negligible effects on the market. Thus, each firm accurately treats Lagrange multipliers as fixed parameters and chooses its profit-maximizing strategy by anticipating their equilibrium values. Thus, a firm faces a finitely elastic and decreasing demand in its own price: ∂cml∂pml=λlu″(cml)=u′(cml)u″(cml)pml<0, l,m∈{n,s}, (4) where the inequality is from (2). 2.2. Prices and profits All firms have access to the same increasing returns to scale technology. Each firm requires one unit of capital as the fixed input to start up. Therefore, the total mass of firms in the world is L, too. Let kl denote the share of all capital located in country l∈{n,s}; thus kn+ks=1. Then, the mass of industry in country l is Nl=klL. To produce one unit of output, a firm requires one unit of labor as its marginal input. Each variety can be traded across countries at iceberg transport cost measured by τ>1. This means that τlm units of a good are needed to be shipped from country l to country m for one unit to arrive, where τlm=τif l≠m1if l=m. Thus, the profit function of a firm in country l is πl=(pll-wl)cllLl+(plm-τwl)clmLm-rl. (5) In what follows, we assume that markets are segmented and firms maximize profits with respect to prices. Combining first order conditions of profit maximization with (4), optimal prices are given by plm=τlmwl1-Ru(clm)>τlmwl, for l,m∈{n,s}. (6) Therefore, it is readily verified that markups in domestic and foreign markets are different and given by μlm=plm-τlmwlplm=Ru(clm), for l,m∈{n,s}. (7) These markups depend on demand, which contrasts with the constant markup in the CES case. In a system of subconvex demands, R′u>0 holds, so markups increase with demand. 3. Equilibrium At first, we claim that there is no corner equilibrium in which all firms agglomerate in one country. Otherwise, the wage rate in the empty country falls to zero, and firms have an incentive to move to the empty country to reduce their labor cost because capital is mobile across countries. Thus, capital returns in two countries are equal in equilibrium, which is denoted by rn=rs=r, subsequently.10 We summarize all conditions for equilibrium as follows: According to (3) and (6), we obtain u′(cll)u′(cml)=pllpml=1τwlwm1-Ru(cml)1-Ru(cll), for l,m∈{n,s}, l≠m. (8) Let ψ(x)≡u′(x)[1-Ru(x)]>0 for all x∈(0,∞). (9) By using inverse demands (3) and optimal prices (6), we know that ψ is the ratio of the marginal cost to the choke price up to a constant u′(0): ψ(cml)=u′(0)pmlplc[1-Ru(cml)]=u′(0)τmlwmplc, for l,m∈{n,s}. (10) Note that Ru(0)=0 implies ψ(0)=u′(0). It means that consumers are choked off from consuming a product once its price is equal to the choke price. As explained later, this ratio ψ plays an important role in determining the direction of one-way trade. Function ψ(x) is strictly decreasing because ψ′(x)=2u″(x)+xu‴(x)=u″(x)[2-Ru′(x)]<0. Then, Equation (8) can be rewritten as ψ(cll)wl=ψ(cml)τwm, for l,m∈{n,s}, l≠m. (11) Also, note that (11) holds if and only if cll and cml are positive. After plugging optimal prices (6) into the budget constraint of each consumer in two countries, we have the following relationship from the differential of two budget constraints between countries: wn-ws=Lknwncnn1-Ru(cnn)-τcns1-Ru(cns)+kswsτcsn1-Ru(csn)-css1-Ru(css). (12) Equality πl=0 holds from the free-entry condition. Thus, profit function (5) leads to LwlθlcllRu(cll)1-Ru(cll)+τθmclmRu(clm)1-Ru(clm)=r, for l,m∈{n,s}, l≠m. (13) All workers in the two countries are fully employed, so that θl=klL[θlcll+θmτclm], for l,m∈{n,s}, l≠m. (14) Since the above labor market clearing condition in each country (14) is satisfied with equality, the equilibrium is interior at which firms are distributed in both countries (i.e. 0<kl<1). To reduce the number of notations, we hereafter let θn=θ (resp., kn=k) so that θs=1-θ (resp., ks=1-k). Subsequently, because we set the unit labor efficiency in country s as the numéraire (i.e. ws=1), for a given trade cost τ, the equilibrium is characterized by the following seven endogenous variables: w, r, k, cnn, cns, css, and csn, which are determined by seven equations (11)–(14). In the following section, we first examine the equilibrium in cases of free trade and autarky. Then we move to the case of general trade costs. 3.1. Free trade When trade is free, the economic interaction between two countries can be regarded as domestic economic activities. We use a superscript f to denote the value of a variable in the case of free trade. The results are as follows: Proposition 1 When trade is free, i.e. τ=1, the following properties hold: All per capita consumptions are equal to 1/L wherever the variety is produced and consumed, i.e. cnnf=csnf=cssf=cnsf=1/L. There is no wage gap across countries, i.e. wf=1. The capital return rf is given by Ru(1/L)/[1-Ru(1/L)]. In each country, the firm share is equal to the population share, i.e. kf=θ. All prices are equal to 1/[1-Ru(1/L)] wherever the variety is produced and consumed, i.e. pnnf=psnf=pssf=pnsf=1/[1-Ru(1/L)]. All markups are equal to Ru(1/L) wherever the variety is produced and consumed, i.e. μnnf=μsnf=μssf=μnsf=Ru(1/L). Proof. See Appendix A. □ To express it in a different way, in an integrated economy, a firm may locate in either country with no difference in shipping its product to markets. Therefore, the location of a firm is determined by production costs only. As a result, the distribution of firms between countries is equal to the share of the market size, and a wage gap between countries does not exist. The HME, either in terms of firm share or wages, vanishes when trade is totally free. 3.2. Autarky ( τ=τa) Since there are no fixed exporting costs and a finite choke price is assumed, we can find a finite threshold value of transport costs τa, above which international trade does not occur. Because of the balance of payments, there is no capital flow in autarky; however, capital starts to flow between the countries at τa. Therefore, we can apply the equalization of capital returns to derive the values of variables at τa. Subsequently, we use a superscript a to denote the value of a variable when τ=τa. We obtain the following results. Proposition 2 In autarky (i.e. τ=τa), we have: The per capita consumptions in the larger (resp., smaller) country are given by cnna=1Lθ (resp., cssa=1L(1-θ)). The wage rate in the larger country is higher than that in the smaller one and is given by wa=Ru(cssa)Ru(cnna)1-Ru(cnna)1-Ru(cssa)>1. The capital return is given by ra=Ru(cssa)1-Ru(cssa). In each country, the firm share is equal to the population share, i.e. ka=θ. The prices in the larger (resp., smaller) country are given by pnna=Ru(cssa)Ru(cnna)[1-Ru(cssa)] (resp., pssa=11-Ru(cssa)). The domestic markups of firms in the larger (resp., smaller) country are μnna=Ru(cnna) (resp., μssa=Ru(cssa)). Proof. See Appendix B. □ When trade is about to start at autarky, capital returns must be identical in both countries. At that moment, the firm share equals the population share ka=θ. Country n hosts more firms and thus offers more varieties. From the labor market clearing condition (14), we also know that individual consumption in country n is smaller, i.e. cssa>cnna. According to the markups (7) and the restriction that Ru′(x)>0, inequality cssa>cnna implies that the markups of firms in country n are smaller than those of firms in country s. Since mobile capital is the fixed input of production in two countries, a firm in country n generates higher revenue to cover the common fixed cost. By the balance of payments in two countries, individual income in country n is higher, implying that wa>1. Note that ka=θ holds so the HME in terms of firm share is not observed, which is exactly the same as the CES result of Takahashi et al. (2013). However, the wage rate in the larger country is higher than that in the smaller country (i.e. w>1), which is in stark contrast with the CES result of Takahashi et al. (2013). The next subsection examines trade patterns when trade costs are between free trade and autarky. 3.3. One-way trade Once τ becomes smaller than τa, trade between countries starts. Because there are two factors of production in this model, the trade imbalance in goods can be balanced by mobile capital. When country sizes are asymmetric, the trade flow from country n to s and that from country s to n may emerge at different transport cost levels.11 To put it another way, different directions of trade are choked off at different levels of τ. As a result, there is a threshold value τ˜∈(1,τa) such that one-way trade prevails when τ∈[τ˜,τa), and two-way trade occurs when τ∈[1,τ˜). Our general framework captures both the VES and the income effect. In this subsection, it is used to disclose one-way trade in two directions. We show that the direction of one-way trade is determined by a comparison of importing prices and market choke prices when trade opens up. According to Equation (11) for the two countries, we define τ1a≡waψ(0)ψ(cnna), τ2a≡ψ(0)waψ(cssa), where wa, cssa, and cnna are given in Proposition 2, and ψ(·) is defined by (9). Thus, τ1a>τ2a holds iff (wa)2ψ(cssa)>ψ(cnna). (15) The following proposition gives the value of τa and the direction of one-way trade. Proposition 3 Patterns of one-way trade are determined by Condition (15) as follows: If Condition (15) holds, then τa=τ1a and one-way trade from the smaller country to the larger one occurs, i.e. csn>cns=0 when τ∈[τ˜,τa). If the reverse inequality of Condition (15) holds, then τa=τ2a, and one-way trade from the larger country to the smaller one occurs, i.e. cns>csn=0 when τ∈[τ˜,τa). Proof. See Appendix C. □ The economic intuition of Condition (15) is clear if we recast it as follows by using expressions of ψ in (10) and the values of wa, cssa, and cnna given in Proposition 2: (wa)2τaψ(cssa)>τaψ(cnna) ⇔ pnsapsc=waτapsc>τapnc=psnapnc, where pnsa≡wnaτa/[1-Ru(0)]=waτa and psna≡wsaτa/[1-Ru(0)]=τa. Because pnsa is the import price (in autarky) for a variety of country n to be sold to country s, and psc is the highest price allowed in country s, pnsa/psc measures the export hurdle from n to s. Similarly, psna/pnc measures the export hurdle from s to n. Therefore, the inequality in Condition (15) implies that trade from n to s is more difficult than trade from s to n, which results in one-way trade from s to n. In other words, under Condition (15), the opening of trade leads to capital outflow from the larger country to the smaller country and imports from the smaller country to the larger country to pay for the capital service. Note that the choke prices in a specific country are determined only by its own market aggregates in autarky. Now consider a multi-country space in which countries are identical except for size (population). In such a case, the countries have different levels of choke prices in autarky. When trade opens up, whether an exporter in country i can enter a destination market j depends on whether its exporting prices are lower than the choke prices in the destination market. Thus, different import prices and market choke prices may result in bilateral trade zeros of a specific country pair in a multi-country world. Using Proposition 2, Condition (15) can be written as Δ≡Φ1L(1-θ)-Φ1Lθ>0, (16) where Φ(·) is defined by Φ(x)≡u′(x)[1-Ru(x)]Ru(x)1-Ru(x)2. Expression (16) gives the condition of one-way trade by the primitives of the model. It shows that trade direction depends on solely on preferences. Now we examine the implications of Condition (16) for various preference systems. Since θ>1/2 > 1-θ, whether Condition (16) holds or not depends on the properties of function Φ(·). Function Φ(·) may have four different shapes: (i) strictly decreasing, (ii) strictly increasing, (iii) constant, and (iv) non-constant and non-monotonic over R+. In case (i), the reverse inequality of (16) holds, meaning that a larger country starts exporting first. In case (ii), Condition (16) always holds, which implies that one-way trade goes from the smaller country to the larger one. In case (iii) (which is a zero-measure event but still seems interesting as a borderline), one-way trade never occurs. Finally, in case (iv), the direction of one-way trade is determined by L and θ. Condition Φ′(x)>0 can be equivalently reformulated as follows: Φ′x>0⇔εxRu>Rux1-Rux2-Rux, where εxRu≡xRu′x/Rux is the elasticity of the RLV. Thus, corresponding to the above four cases, the implications given by this inequality are listed as follows: Condition (16) never holds when the RLV is decreasing, i.e. using Mrázová and Neary’s (2013) terminology, when demand is superconvex. Indeed, in this case we have εxRu<0<Ru1-Ru/(2-Ru), meaning that Φ(·) is decreasing. Since Ru<1 is assumed, it can be shown that Ru(1-Ru)/(2-Ru)≤(2-1)2. Hence, a sufficient (but not necessary) condition for Condition (16) to hold is εxRu>(2-1)2≈0.1716, which explains why Condition (16) always holds for the CARA utility function u(x)=1-e-βx, where β>0 (see Behrens and Murata, 2007). In this case, εxRu=1 holds. Condition (16) is also satisfied by the Stone–Geary utility function u(x)=ln(1+βx), where β>0 (see Murata, 2009; Simonovska, 2015), because inequalities 1>εxRu>Ru1-Ru/(2-Ru)>0 hold. The borderline case (when τ<τa implies bilateral trade) occurs iff Ru(·) solves the following first-order ordinary differential equation: Ru′(x)[Ru(x)]22-Ru(x)1-Ru(x)=1x. (17) Thus, Ru(x) is implicitly given by Ru(x)1-Ru(x)e-2Ru(x)=ax, (18) where a>0 is a constant. Equation (17) implies that Ru′(x)>0 so Ru(x) of (18) is an increasing function with range in (0,1). Finally, for the preferences with increasing RLV and non-monotonic Φ(·), the direction of one-way trade will essentially depend on the world population L and the cross-country population distribution (θ,1-θ). We can easily find examples in this case under the HARA preferences of Merton (1971, p. 389): u(x)≡1-γγβx1-γ+1γ-1, (19) where parameter β>0 gives the degree of love for variety, while parameter γ∈(-∞,1) represents the extent of differentiation across varieties. When γ is very close to -∞, varieties are extremely differentiated. If γ approaches 1, varieties become less differentiated, so that the substitutability between varieties becomes higher.12 Then, the RLV under the HARA is increasing since Ru′(x)=β(1-γ)2/(1-γ+βx)2>0, and function Φ(·) is given by Φ(x)=β3(1-γ)3x2(1-γ+βx)2[1-γ(1-βx)]βx1-γ+1γ. Given the parameter set, β=12 and γ=0.5, the left panel in Figure 2 displays the non-monotonicity of Φ(·). Furthermore, based on the same parameter set and a fixed value of the world population (e.g. L=10), the right panel in Figure 2 shows that Δ can be either positive (see 0.5<θ<0.763263) or negative (see 0.763263<θ<1), depending on the cross-country population distribution (θ,1-θ). Figure 2 View largeDownload slide The function Φ(·) and Δ under HARA preferences. Figure 2 View largeDownload slide The function Φ(·) and Δ under HARA preferences. 4. Markups This section analyzes how markups respond to trade liberalization. By differentiating both sides of the seven equations (11)–(14) with respect to τ and applying Proposition 1, we find that dk(τ)/dτ and dw(τ)/dτ are positive at τ=1 while dcsn(τ)/dτ, dμsn(τ)/dτ, dcns(τ)/dτ, and dμns(τ)/dτ are negative at τ=1 (see Appendix D for their expressions). This demonstrates that once τ rises slightly from 1, the larger market attracts firms more than proportionately, and the wage rate in the larger country is increased by stronger labor demand. Meanwhile, imported goods become more expensive than domestic goods, so consumers reduce their consumption of foreign goods, which lowers firms’ markups in their export markets. However, the signs of dcnn(τ)/dτ, dμnn(τ)/dτ, dcss(τ)/dτ, and dμss(τ)/dτ at τ=1 are ambiguous. In other words, when τ rises slightly from 1, an individual consumer may either increase or decrease her consumption of domestic goods. According to (7), firms’ markups in their domestic markets may either increase or decrease. Let us consider a consumer in country l to see how consumers’ preferences and the cross-country population distribution determine outcomes. From Proposition 1, we know that consumers’ per capita consumption of each domestic and imported variety are all 1/L at τ=1. When τ moves upward from 1, if the consumer in country l chooses to increase her consumption of a local good from 1/L to cll⋆, there is an associated decline in her consumption of a foreign good from 1/L to cml⋆. Consequently, the consumer has a subutility increase of u(cll⋆)-u(1/L) for each domestic variety and a subutility decrease of u(1/L)-u(cml⋆) for each imported good. By taking into consideration the change in the mass of varieties provided locally, from Nl to Nl⋆, and the change in the mass of varieties from abroad, from Nm to Nm⋆, we know that the total change in her utility is then given by [u(cll⋆)Nl⋆-u(1/L)Nl]+[u(cml⋆)Nm⋆-u(1/L)Nm]. Otherwise, if she chooses to reduce both her consumption from 1/L to cll† and cml†, this results in subutility losses for each foreign variety, u(1/L)-u(cml†), and for each local good, u(1/L)-u(cll†). In this case, the mass of domestic varieties changes from Nl to Nl†, while the mass of foreign goods shifts from Nm to Nm†. As a result, her total change in utility becomes [u(cll†)Nl†-u(1/L)Nl]+[u(cml†)Nm†-u(1/L)Nm]. Even though we know that the mass of domestic varieties in the larger country will increase (i.e. dk(τ)/dτ is positive at τ=1), which decision is optimal for the consumer depends on the tradeoff between the change in the subutilities and the change in the cross-country distribution of varieties. In this model, these two channels are affected by the concavity of subutilities and the cross-country population distribution. The first choice is more reasonable when the concavity Ru(·) is large, while the second is more reasonable when Ru(·) is small. In particular, if the sizes of two countries are close (i.e. θ is close to 1/2) and the concavity of subutilities is very large (i.e. Ru(·) is close to 1), then the second option is chosen in both countries, resulting in both negative dμnn(τ)/dτ and dμss(τ)/dτ at τ=1. This theoretical indeterminacy may account for the conflicting empirical results in the literature. For example, Feenstra and Weinstein (2017) document the reduced markups in the U.S. market between 1992 and 2005, whereas De Loecker et al. (2016) find increasing markups of Indian firms between 1989 and 1997. Nevertheless, markups in the export market always decrease with trade costs, regardless of whether the country is large or small. In other words, when trade costs fall to a low level, firms are always able to benefit from their rising markups in the export markets, and it is not necessarily at the expense of reducing their markups in domestic markets. As in Behrens et al. (2014), the average markup from the viewpoint of a firm in country n (resp., s) is given by the revenue-weighted average of markups across the two markets as follows: Ωl¯≡θlcll1-Ru(cll)μll+τθmclm1-Ru(clm)μlmθlcll1-Ru(cll)+τθmclm1-Ru(clm),for\l,m∈n,s, l≠m. Then, by using (D.1)—(D.8) and Proposition 1, we know that the signs of dΩ¯n(τ)/dτ and dΩ¯s(τ)/dτ are both negative at τ=1 (see Appendix D for their expressions). As the central statement of welfare impact, the average markup that consumers face is the expenditure-weighted average of markups across all origins of their consumed goods: ωl¯≡klwlcll1-Ru(cll)μll+τkmwmcml1-Ru(cml)μmlklwlcll1-Ru(cll)+τkmwmcml1-Ru(cml),forl,m∈n,s, l≠m. Similarly, by using (D.1)—(D.8) and Proposition 1, we find that the signs of dω¯n(τ)/dτ and dω¯s(τ)/dτ are also negative at τ=1 (see Appendix D for their expressions). We find that when τ is very close to 1, both average markups for firms and consumers increase along with trade integration. For a firm, even though its markup in the domestic market might decrease when trade gets freer, the rising trend of its markup in the export market is dominant. By the same token, the markup of a consumer’s consumption of import goods dominates the markup of her consumption of domestic goods. The results are summarized as follows. Proposition 4 In free trade, we have: All of the average markups for firms, Ω¯nf and Ω¯sf, as well as for consumers, ω¯nf and ω¯sf, are equal, given by Ru(1/L). When τ is very close to 1, average markups for firms in both countries increase along with trade integration, and the gradients are the same, i.e. dΩ¯n(τ)dττ=1=dΩ¯s(τ)dττ=1<0. When τ is very close to 1, average markups for consumers in both countries increase along with trade integration; however, the gradients are different from each other, i.e., dω¯n(τ)dτ|τ=1<0, dω¯s(τ)dτ|τ=1<0, and dω¯n(τ)dτ|τ=1≠dω¯s(τ)dτ|τ=1. Both firms’ and consumers’ average markups are equal to the domestic markups in autarky since there is no export, i.e. Ω¯na=ω¯na=μnna and Ω¯sa=ω¯sa=μssa. Thus, we know that Ω¯na=ω¯na<Ω¯sa=ω¯sa. In comparing average markups between free trade and autarky, we find that average firm-level markups as well as average consumer-faced markups in free trade are always lower than those in autarky, as summarized in the following proposition. Proposition 5 In each country l∈{n,s}, we have μllf<μlla and Ω¯lf=ω¯lf<Ω¯la=ω¯la. Proof. The results are obtained by the following equations. μnnf-μnna=Ru(1/L)-Ru(1/(Lθ))<0,Ω¯nf-Ω¯na=ω¯nf-ω¯na=Ru(1/L)-Ru(1/(Lθ))<0,μssf-μssa=Ru(1/L)-Ru(1/[L(1-θ)])<0,Ω¯sf-Ω¯sa=ω¯sf-ω¯sa=Ru(1/L)-Ru(1/[L(1-θ)])<0. □ Given the one-way trade direction, other important variables can be explored. We are able to calculate their derivatives with respect to τ at τ=τa to show how they change when trade opens up. The results are given below. Proposition 6 Under(15), one-way trade from s to n starts at τa with zero consumption cns(τ) when τ is close to τa and dcns(τ)/dτ is zero at τa. The signs of dk(τ)/dτ, dw(τ)/dτ, dcss(τ)/dτ, dμss(τ)/dτ, dΩ¯s(τ)/dτ, and dω¯s(τ)/dτ are positive at τa. The signs of dcnn(τ)/dτ, dcsn(τ)/dτ, dμnn(τ)/dτ, dμsn(τ)/dτ, and dΩ¯n(τ)/dτ are negative at τa. The sign of dω¯n(τ)/dτ is ambiguous at τa. If the reverse inequality of (15) holds, one-way trade from n to s starts at τa with zero consumption csn(τ) when τ is close to τa and dcsn(τ)/dτ is zero at τa. The signs of dk(τ)/dτ, dw(τ)/dτ, dcss(τ)/dτ, dcns(τ)/dτ, dμss(τ)/dτ, dμns(τ)/dτ and dΩ¯s(τ)/dτ are negative at τa. The signs of dcnn(τ)/dτ, dμnn(τ)/dτ, dΩ¯n(τ)/dτ, dω¯n(τ)/dτ, and dω¯s(τ)/dτ are positive at τa. Proof. See Appendix E. □ The above results reveal that opening trade has different effects on markups μnn, μss, Ω¯n, Ω¯s, ω¯n, and ω¯s across these two cases of one-way trade. When τ∈[τ˜,τa) in the case of one-way trade from s to n, only psn gets lower than choke price pnc while pns is still higher than choke price psc. Therefore, decreasing τ lowers psn further to increase per capita consumption csn, whereas cns remains zero. Consequently, firm markup μsn, which depends on csn, also increases with decreasing τ. In addition, the mass of firms in country n decreases when τ drops from τa. The reduced competition increases domestic markup μnn in country n when τ falls. Then, for a firm in country n, its average markup Ω¯n rises because all of its sales revenue comes from its domestic market. Yet the average markup of a firm in country s (i.e. Ω¯s) drops, since the revenue weight of increasing μsn is too small to cover the impact of decreasing μss in its more competitive domestic market when τ decreases from τa. To a consumer in country s, her average markup ω¯s goes down when τ falls because her total expenditure is spent on domestic consumption. However, whether the consumer-side average markup in country n (i.e. ω¯n) increases or decreases is ambiguous. It depends on if the convexity of demand in the importing country n at τa (i.e. Ru′(cnna)) is larger than 1 (see Appendix E). Conversely, in the case of one-way trade from n to s, only pns gets lower than choke price psc, while psn is still higher than choke price pnc when τ∈[τ˜,τa). Decreasing τ further drags down pns to raise cns, whereas csn remains zero. Consequently, firm markup μns also increases with decreasing τ. Meanwhile, the decreasing (resp., increasing) mass of firms in country s (resp., n) raises (resp., lowers) its domestic markups μss (resp., μnn) when τ drops from τa. Therefore, average markups from a firms’ viewpoint behave in the opposite way against their counterparts in the case of one-way trade from s to n. The average markup for consumers in country n, i.e. ω¯n, only depends on μnn, since there is no consumption on imported goods from country s. However, the average markup for consumers in country s, i.e. ω¯s, drops without ambiguity when τ falls from τa. Note that markup μns of imported goods is significantly lower than that of domestic varieties μss. Furthermore, although trade is just opening from autarky, the expenditure weight of these imported goods also rises quickly since the mass of domestic firms shrinks. Therefore, for a consumer in country s, the average markup goes down when τ falls from τa. Next, we employ HARA preferences of (19) again to illustrate how these firm markups respond to changes in trade costs τ in the two different cases of one-way trade. The first set of parameters L=10, β=7, γ=0.5, and θ=0.6, which satisfies Condition (15), is used to display the case of one-way trade from s to n. The second set of parameters L=10, β=50, γ=0.5, and θ=0.6, which does not meet Condition (15), illustrates the opposite case. Figure 3 compares firms’ market-specific markups, while Figure 4 contrasts firms’ revenue-weighted average markups in the two cases of one-way trade. Figure 3 View largeDownload slide Firm markups in various markets. Left: one-way trade from s to n; right: one-way trade from n to s. Figure 3 View largeDownload slide Firm markups in various markets. Left: one-way trade from s to n; right: one-way trade from n to s. Figure 4 View largeDownload slide Firms’ revenue-weighted average markups. Left: one-way trade from s to n; right: one-way trade from n to s. Figure 4 View largeDownload slide Firms’ revenue-weighted average markups. Left: one-way trade from s to n; right: one-way trade from n to s. With respect to the consumers’ side, we find that the average markups for consumers react in a different manner from the manner in which average markups for firms react. In the case of one-way trade from s to n, when τ falls from τa, ω¯n may rise or fall, while ω¯s becomes lower. The result depends on the convexity of demand, as shown analytically in Appendix E(a). Figures 5 and 6 contrast two opposite examples about how ω¯n changes based on different parameter sets of HARA preferences. In contrast, in the case of one-way trade from n to s, both ω¯s and ω¯n always drop when trade opens from autarky, as shown in Figure 7. Figure 5 View largeDownload slide Consumers’ expenditure-weighted average markups (one-way trade from s to n) { L=10, β=3, γ=0.5, and θ=0.6}. Left: the original diagram; right: the enlarged view of ω¯n. Figure 5 View largeDownload slide Consumers’ expenditure-weighted average markups (one-way trade from s to n) { L=10, β=3, γ=0.5, and θ=0.6}. Left: the original diagram; right: the enlarged view of ω¯n. Figure 6 View largeDownload slide Consumers’ expenditure-weighted average markups (one-way trade from s to n). { L=10, β=8, γ=0.5, and θ=0.6}. Left: the original diagram; right: the enlarged view of ω¯n. Figure 6 View largeDownload slide Consumers’ expenditure-weighted average markups (one-way trade from s to n). { L=10, β=8, γ=0.5, and θ=0.6}. Left: the original diagram; right: the enlarged view of ω¯n. Figure 7 View largeDownload slide Consumers’ expenditure-weighted average markups (one-way trade from n to s). { L=10, β=50, γ=0.5, and θ=0.6}. Left: the original diagram; right: the enlarged view of ω¯n. Figure 7 View largeDownload slide Consumers’ expenditure-weighted average markups (one-way trade from n to s). { L=10, β=50, γ=0.5, and θ=0.6}. Left: the original diagram; right: the enlarged view of ω¯n. In the literature, firm productivity is often used to explain the various responses of firm-level markups (i.e. the average markups for firms) to the opening of trade. For instance, using Canada–U.S. regional trade data, Behrens et al. (2014) find that responding to trade integration, average markups on the consumers’ side show pro-competitive effects (the markup decreases as the mass of firms increases) while firm-level markups are ambiguous. They attribute the indeterminacy of firm-level markups to the heterogeneity of firm productivity. More specifically, the decrease in trade costs induces less productive firms to export and, in turn, allows existing more-productive exporters to charge higher markups. As a result, with respect to a single firm, whether the opening of trade leads to a lower or higher average markup depends on its relative position in the firm productivity distribution. Our two-factor VES setup without firm heterogeneity provides another insight on markups. It does not display the different responses of firms’ average markups to trade opening within a country due to the assumption of homogeneous firm productivity. Instead, we focus on characteristics in the preference side and the distribution of market size. It is shown that firm markups may display opposite responses in the two opposite cases of one-way trade. In this way, we provide another theoretical base for empirical studies that compare the responses of firm markups to the opening of trade across countries. Another prevailing view seems to be that globalization and trade widen the variety available to consumers and enforce competition between firms. Because of this pro-competitive effect, firms’ markups should show a decreasing trend with increasing levels of integration (Krugman, 1979; Feenstra and Weinstein, 2017). By just comparing firm markups between free trade and autarky, Proposition 5 provides results consistent with this perception. However, Propositions 4–6 tell us that average markups, both for firms and for consumers, do not change monotonically when trade costs fall from autarky to free trade. These features have been illustrated in Figures 4–7. Our analysis, based on a setup with mobile production factor (capital) and without firm selection, offers insights that complement to those works that consider firm heterogeneity without mobile factors, such as Arkolakis et al. (2015). They find that a decrease in trade costs reduces the markups of domestic producers but increases the markups of foreign producers. Thus, they point out that the overall pro-competitive effects of trade may be very different from the effects on domestic producers, and that focusing solely on domestic producers may provide a misleading picture of the so-called pro-competitive effects of trade. In contrast, in our model with symmetric firms, the change in competing firm mass in a market is driven by the capital movement rather than the selection of firms. We find that a decrease in trade costs may increase the markups when trade costs are relatively small. A key message from our analysis is that the overall pro-competitive effects of trade may be more complicated in a world with mobile production factors. The non-monotonicity of firms’ markup changes also offers a possible rationale for the conflicting empirical findings. For instance, based on the dataset of price-cost-margins (PCMs) in 17 OECD countries between 1970 and 2003, Boulhol (2010) finds a negative correlation between trade liberalization and PCMs only in Japan and Spain. De Loecker et al. (2016) also observe increasing markups when trade gets freer. Although both Boulhol (2010) and De Loecker et al. (2016) indeed find strong and robust evidence of expected pro-competitive effect (lowering markups), they conclude that other factors, such as the input tariff liberalization and incomplete pass-through of this cost saving, have compensated for the pro-competitive effect and, furthermore, increased the markups. Since we choose the labor in the smaller country as the numéraire, the absolute changes in marginal cost are less meaningful in our model. However, we prove that even when other things remain constant, just lowering trade costs alone may result in rising markups in the framework of VES. Besides, since the average markups for consumers lie at the center of welfare analysis, its non-monotonicity also implies that the welfare change with trade liberalization may not be monotonic. 5. Home market effects As an important issue in the literature of agglomeration economics, the HME predicts the following conclusions for an industry characterized by monopolistic competition, increasing returns to scale, and trade costs. In a world of two countries, (i) wages are higher in the larger country; (ii) a more-than-proportionate relationship exists between the larger country’s share of world production and its share of world demand. Two results are theoretically derived from different models in Krugman (1980), and they are shown to be equivalent in the CES framework of Takahashi et al. (2013). Furthermore, Zeng and Uchikawa (2014) find that these two HMEs are quite robust in general frameworks of multiple countries in which the CES utility function is assumed. The HMEs in terms of wages and firm share are called the price (of labor) and the quantity (of production) aspects, respectively, by Head and Mayer (2004). Empirical studies examine the two aspects separately. Redding and Venables (2004) and Hanson (2005) provide strong empirical evidence of the price aspect of the HME. Head and Mayer (2011, p. 288) also conclude that ‘Larger and/or more centrally located countries are much richer than countries characterized by a small local market and few or small neighbors’. A survey paper of Redding (2011) documents evidence clarifying that the close relationship between market access and wages is causal. In contrast, regarding the quantity aspect, after surveying many empirical studies, Head and Mayer (2004, p. 2636) conclude that ‘the evidence on HMEs accumulated by these papers is highly mixed’. Indeed, Davis and Weinstein (1999) find positive support in only 8 of 19 manufacturing sectors. Davis and Weinstein (2003) examine HMEs in a majority of industries, with significantly positive support in four industries and significantly negative support in two. The pooled estimate results of Brülhart and Trionfetti (2009) ‘paint an inconsistent picture’, while only 7 of 17 manufacturing industries exhibit a supportive response to home-biased demand in their industry-by-industry estimates. Therefore, Head and Mayer (2004, p. 2642) conclude that HMEs ‘generally take the form of higher factor incomes in large demand areas rather than magnified production shares of IRS industries’. This section shows that the general VES framework fills the gap between theoretical and empirical studies of the HME. In fact, Proposition 6 reveals that different cases of one-way trade are associated with different characteristics of the HME. Once one-way trade from s to n starts when τ drops from τa, some firms in country n move to country s to escape higher wages in country n and extend their markets by serving both countries. In contrast, in the case of one-way trade from n to s, firms in country s have an incentive to move to the larger market, n. Based on the same parameter sets employed in the previous section for the respective patterns of one-way trade, Figure 8 displays the firm share curves, while Figure 9 plots the labor wage rates of the two examples. We find that the HME in terms of firm share occurs (i.e. k>θ) in the one-way trade pattern from n to s, while the inverse happens (i.e. k<θ) in the one-way trade pattern from s to n. Inequalities remain, even when two-way trade begins. Nevertheless, the wage rate in the larger country is higher in both cases. Therefore, the equivalence of the two aspects of the HME is not valid in the appearance of VES. Proposition 7 summarizes the results. Proposition 7 In the case of one-way trade from s to n, the HME in terms of firm share is reversed; however, the HME in terms of wages is observed. In the case of one-way trade from n to s, both HMEs in terms of firm share and wages are observed. Figure 8 View largeDownload slide The HME in terms of firm share. Left: one-way trade from s to n; right: one-way trade from n to s. Figure 8 View largeDownload slide The HME in terms of firm share. Left: one-way trade from s to n; right: one-way trade from n to s. Figure 9 View largeDownload slide The HME in terms of wages. Left: one-way trade from s to n; right: one-way trade from n to s. Figure 9 View largeDownload slide The HME in terms of wages. Left: one-way trade from s to n; right: one-way trade from n to s. Proof. See Appendix F. □ 6. Concluding remarks To investigate the impact of trade liberalization on trade patterns, firm markups, and firm locations under the general VES framework, this article establishes a two-factor model of monopolistic competition by a general additively separable utility. Based on this framework, we find one-way trade and two possible directions when trade opens up, which are observed in empirical data. Furthermore, different one-way trade patterns lead to different markup responses to the opening of trade, and a higher wage rate may not be accompanied by a more-than-proportionate share of firms. As expected, our general setting allows us to identify which findings are robust against alternative preference specifications, and those that depend on particular classes of preferences. These results have three implications. First, for theoretical studies, a more general functional form of utility and a setup of mobile capital is necessary to allow for the two possible cases of one-way trade. Otherwise, the impact of VES on the direction of one-way trade is opaque. Second, for empirical studies, we provide a testable prediction regarding the relationship between the direction of one-way trade and the existence of the HME in terms of firm share. They should be highly correlated, as our model predicts. Third, our results raise a question regarding the measurement of industrial agglomeration. One of the great findings of NTT/NEG is the mechanism revealing how increasing returns of technology lead to industrial agglomeration. This is the so-called HME in terms of firm share—that the larger country has a more-than-proportionate firm share. However, we find that this result may not be true in the case of one-way trade from the smaller to the larger country. Funding This work was supported by JSPS KAKENHI (Grant Numbers 24330072, 17H02514, and 26380282) of Japan, and the Economic System Science Research Project for Long Sustainable Growth in East Asia at Tohoku University. Ching-mu Chen also appreciates the postdoctoral fellowship in Academia Sinica and the financial support from Shin-Kun Peng’s Academia Sinica Investigator Award (2316-105-105-1100) in Taiwan. Footnotes 1 See Head and Mayer (2014) for a more detailed survey. 2 We exclude titles of country groups and count 187 countries. We construct a matrix of trade flows, measured in thousands of US dollars, with 187×186=34,782 observations. A list of the countries is available from the authors. Since we cover more countries which are disintegrated or new born during this period and their trade flows are zero before they were established or after they disappeared, the shares of no trade is much higher than that in Helpman et al. (2008). 3 Country size is measured by population, which is available from the Penn World Tables 8.1 (Feenstra et al., 2013, 2015). 4 As in Baldwin et al. (2003, p. 69), capital is viewed as knowledge capital or physical capital. The former resembles patents, blue prints, and know-how for producing a variety, while the latter includes buildings, factories, equipments, and other assets. 5 We impose this assumption to remove comparative advantages in technology and resource endowments (first nature features). Of course, such factors play an important role in determining economic structure. However, this article aims to clarify the role of increasing returns and monopolistic competition; therefore, we need to remove these first nature features. This is analogized to peeling an orange to taste the flesh by Zeng and Uchikawa (2014). 6 As in the Heckscher–Ohlin model, some papers (e.g. Kichko et al., 2014) in the literature consider multiple production factors, in which all factors are immobile. They are useful in examining the role of factor endowments. We assume the mobility of capital because the mobility of factors can balance the trade in goods even if we do not add an outside good. 7 According to Krugman (1980, Section II), a one-industry setting is enough to justify the HME, by which firms concentrate production in the larger market to realize economies of scale and minimize transport costs. The specialization of countries is beyond the scope of the discussion in this article, although we can expect that the HME also plays an important role in countries’ specialization in a multi-industry setting. 8 Krugman (1979, p. 476) argues that the subconvexity of demand is plausible. Mrázová and Neary (2012, p. 17) document that subconvexity is sometimes called Marshall’s Second Law of Demand. 9 Less restrictively, we assume that Ru(x)<1 and Ru′(x)<2 hold over some interval (x̲,x¯)⊆R+, while L and θn satisfy x̲<1/(Lθn)<1/[L(1-θn)]<x¯. 10 In autarky, there is no capital flow due to zero international trade in good. Thus, the capital returns in both countries are not necessarily equalized. However, the capital rents in the two countries are equal as long as trade occurs. This equalization of capital returns lasts until trade between these two countries just stops at a delimiting point of trade costs. Therefore, in the subsequent analysis of autarky, we maintain the equality of capital rents to find the properties of variables at that delimiting point of trade costs. 11 Without mobile capital, trade in goods is balanced. 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Additionally, the budget constraints in the two countries are as follows: wf+rf=kfLpnnfcnnf+(1−kf)Lpsnfcsnf, (A.1) 1+rf=kfLpnsfcnsf+(1−kf)Lpssfcssf. (A.2) Subtracting (A.2) from (A.1) yields wf-1 = 0 because pnnf=pnsf, pssf=psnf, cnnf=cnsf, and cssf=csnf. Thus, we obtain wf=1. Plugging τ=1 and wf=1 into (11) for the two countries yields cnnf=csnf and cssf=cnsf. Thus, we know that cnnf=cnsf=cssf=csnf=1/L, rf=Ru(1/L)/[1-Ru(1/L)], and pnnf=pnsf=pssf=psnf=1/[1-Ru(1/L)] from the budget constraints and the free-entry condition. Then, μnnf=μsnf=μssf=μnsf=Ru(1/L) is derived directly. Lastly, taking these results into a condition of labor market clearing (14) in country n gives kf=θ. Appendix B: Proof of Proposition 2 In autarky, two countries do not trade with each other, i.e. cnsa=csna=0. Then, the labor market clearing condition (14) for each country derives cnna=1Lka, cssa=1L(1-ka). (B.1) Taking (B.1) into the free-entry conditions (i.e. πn=0 and πs=0) yields ra=(1-θ)Ru1L(1-ka)(1-ka)1-Ru1L(1-ka), wa=ka(1-θ)Ru1L(1-ka)1-Ru1Lka(1-ka)θRu1Lka1-Ru1L(1-ka). (B.2) Next, we obtain ka=θ by plugging (B.1) and (B.2) into (12). This simplifies (B.1) and (B.2), deriving the corresponding results of Proposition 2. Finally, the prices are given by (6) while μnna and μssa are given by (7). Appendix C: Proof of Proposition 3 In this two-country setup, there are only two possible one-way trade patterns, either from s to n or the opposite. We examine them one by one. (a) Unilateral trade from s to n This case is described by csn>cns=0 for τ∈[τ˜,τa). On one hand, the fact of csn=0 at τ=τa gives τa=τ1a, according to Proposition 2 and Equation (11) for country n. On the other hand, the fact of cns=0 for τ∈(τ˜,τa] implies that consumers in country s have no incentive to buy any foreign goods because the prices of imported goods are higher than the choke prices, i.e. pns>psc=u′(0)/λs. Accordingly, Equation (11) for country s is replaced by wτψ(css)>ψ(0). (C.1) Thus, by using Proposition 2, we know that the inequality of (C.1) at τ=τa=τ1a is equivalent to τ1a=τa>ψ(0)waψ1L(1-θ)=τ2a, which holds iff (wa)2ψ(cssa)>ψ(cnna) is true. In other words, the one-way trade from s to n occurs iff the inequality of (15) is true. (b) Unilateral trade from n to s In contrast, the case of one-way trade from n to s is expressed by cns>csn=0 when τ∈[τ˜,τa). The fact that cns=0 holds at τa derives τa=τ2a, according to Proposition 2 and Equation (11) for country s. Meanwhile, the fact that csn=0 holds for τ∈(τ˜,τa] gives psn>pnc=u′(0)/λn. Consequently, Equation (11) for country n becomes τψ(cnn)>wψ(0). (C.2) Similar to the previous case, by using Proposition 2, we know that the inequality of (C.2) at τ=τa=τ2a violates τ1a>τ2a. Therefore, this case does not happen under (15). Appendix D: The Derivatives at τ=1 By differentiating both sides of the seven equations (11)–(14) with respect to τ and applying Proposition 1, we obtain the slopes of important variables at τ=1 as follows: dk(τ)dτ|τ=1=(2θ−1)θ(1−θ)>0,dw(τ)dτ|τ=1=2θ−1>0, dcnn(τ)dτ|τ=1=L2(1−θ)u'(1L)L{Ru′(1L)u'(1L)−u″(1L)[1−Ru(1L)]}{Ru′(1L)+L[1−Ru(1L)]}×{2(1−θ)[1−Ru(1L)]3+A0[Ru(1L)]2([1−Ru(1L)]−2θA0)}, (D.1) dcsn(τ)dτ|τ=1=−L2(1−θ)u'(1L)L{Ru′(1L)u'(1L)−u″(1L)[1−Ru(1L)]}{Ru′(1L)+L[1−Ru(1L)]} ×{2θ([1−Ru(1L)]3+[A0Ru(1L)]2) +A0Ru(1L)[2−Ru(1L)][1−Ru(1L)]} <0,(D.2) dcss(τ)dτ|τ=1=L2θu'(1L)L{Ru′(1L)u'(1L)−u″(1L)[1−Ru(1L)]}{Ru′(1L)+L[1−Ru(1L)]} ×{2θ[1−Ru(1L)]3+A0[Ru(1L)]2([1−Ru(1L)]−2(1−θ)A0)}, (D.3) dcns(τ)dτ|τ=1=−L2θu'(1L)L{Ru′(1L)u'(1L)−u″(1L)[1−Ru(1L)]}{Ru′(1L)+L[1−Ru(1L)]} ×{2(1−θ)([1−Ru(1L)]3+[A0Ru(1L)]2) +A0Ru(1L)[2−Ru(1L)][1−Ru(1L)]} <0,(D.4) dμnn(τ)dτ|τ=1=dcnn(τ)dτ|τ=1Ru'(1L), (D.5) dμsn(τ)dτ|τ=1=dcsn(τ)dτ|τ=1Ru'(1L)<0, (D.6) dμss(τ)dτ|τ=1=dcss(τ)dτ|τ=1Ru'(1L), (D.7) dμns(τ)dτ|τ=1=dcns(τ)dτ|τ=1Ru'(1L)<0, (D.8) where A0≡2-Ru′(1/L)>0. Then, we can calculate the derivatives of average markups at τ=1 by using (D.1)—(D.8) and Proposition 1: dΩ¯n(τ)dττ=1=Ru′1L(1-θ)dcns(τ)dττ=1+θdcnn(τ)dττ=1=-2θ(1-θ)LRu′1L<0,dΩ¯s(τ)dττ=1=Ru′1L(1-θ)dcss(τ)dττ=1+θdcsn(τ)dττ=1=-2θ(1-θ)LRu′1L<0,dω¯n(τ)dττ=1=-(1-θ)Ru′1L2θRu′1L+L1-Ru1L2(1-θ)+(2θ-1)Ru1LLRu′1L+L1-Ru1L<0,dω¯s(τ)dττ=1=-θRu′1L2(1-θ)Ru′1L+L1-Ru1LRu1L+2θ1-Ru1LLRu′1L+L1-Ru1L<0. Appendix E: The Derivatives at τ=τa (a) Unilateral trade from s to n Condition (15) implies τ1a>τ2a. Therefore, cns=0 holds when τ is close to τa, which implies dcns(τ)/dτ=0 at τ=τa. We then plug cns=0 into Equations (11)–(14), and differentiate both sides with respect to τ. Propositions 2 and 3 (i) give dk(τ)dτ|τ=τa=Lθ2(1−θ)2u'(0)u'(cnna)[Ru(cnna)]2[1−Ru(cssa)]3A1+(1−θ)[u'(0)]2Ru(cssa)[1−Ru(cssa)]B1>0,dw(τ)dτ|τ=τa={(1−θ)2Ru'(cnna)[1−Ru(cssa)]2Ru(cssa) +θ[1−Ru(cnna)]Ru(cnna) {L(1−θ)[1−Ru(cssa)]Ru(cssa)[1−(1−θ)Ru(cssa)]+θRu'(cssa)}} ×u'(0)u'(cnna)A1+(1−θ)[u'(0)]2Ru(cssa)[1−Ru(cssa)]B1 >0,dcnn(τ)dτ|τ=τa=−(1−θ)2u'(0)u'(cnna)[Ru(cnna)]2[1−Ru(cssa)]3A1+(1−θ)[u'(0)]2Ru(cssa)[1−Ru(cssa)]B1<0,dcsn(τ)dτ|τ=τa=−θ(1−θ)[u'(cnna)]2[Ru(cnna)]3[1−Ru(cssa)]3A1+(1−θ)[u'(0)]2Ru(cssa)[1−Ru(cssa)]B1<0,dcss(τ)dτ|τ=τa=θ2u'(0)u'(cnna)[Ru(cnna)]2[1−Ru(cssa)]2A1+(1−θ)[u'(0)]2Ru(cssa)[1−Ru(cssa)]B1>0,dμnn(τ)dτ|τ=τa=dcnn(τ)dτ|τ=τaRu'(cnna)<0,dμsn(τ)dτ|τ=τa=dcsn(τ)dτ|τ=τaRu'(0)<0,dμss(τ)dτ|τ=τa=dcss(τ)dτ|τ=τaRu'(cssa)>0,dΩ¯n(τ)dτ|τ=τa=−(1−θ)2u'(0)u'(cnna)[Ru(cnna)]2[1−Ru(cssa)]3A1+(1−θ)[u'(0)]2Ru(cssa)[1−Ru(cssa)]B1Ru'(cnna)<0,dΩ¯s(τ)dτ|τ=τa=θ2u'(0)u'(cnna)[Ru(cnna)]2[1−Ru(cssa)]2A1+(1−θ)[u'(0)]2Ru(cssa)[1−Ru(cssa)]B1 ×{Ru'(cssa)+L(1−θ)[Ru(cssa)]2[1−Ru(cssa)]}>0,dω¯n(τ)dτ|τ=τa=(1−θ)2u'(0)u'(cnna)[Ru(cnna)]2[1−Ru(cssa)]3A1+(1−θ)[u'(0)]2Ru(cssa)[1−Ru(cssa)]B1 ×LθRu(cnna)[Ru'(cnna)−1],dω¯s(τ)dτ|τ=τa=θ2u'(0)u'(cnna)[Ru(cnna)]2[1−Ru(cssa)]2A1+(1−θ)[u'(0)]2Ru(cssa)[1−Ru(cssa)]B1Ru'(cssa)>0, where A1≡θ2[u′(0)]2Ru(cnna)Ru′(cssa)+θ(1-θ)[u′(0)Ru′(0)-u″(0)]u′(cnna)[Ru(cnna)]2Ru(cssa)[1-Ru(cssa)]2>0,B1≡LRu(cnna)[(1-θ)2Ru(cssa)+2θ-1]+(1-θ)[1-Ru(cssa)]Ru′(cnna)+Ru(cnna)θcnna[1-θRu(cnna)]>0. Note that the sign of dω¯n(τ)/dτ at τ=τa is ambiguous and depends on whether the convexity of demand in the importing country n at τ=τa is larger than 1. For example, in the case of CARA, where u(x)=1-e-βx, dω¯n(τ)/dτ at τ=τa is negative since Ru′(x)=Ru(x)=βx<1 always holds. However, if the utility is as general as the HARA in (19), Ru′(cnna) may be larger or smaller than 1 when the one-way trade is from country s to n. (b) Unilateral trade from n to s The reverse inequality of (15) implies τ1a<τ2a. Thus, csn=0 holds when τ is close to τa, which implies dcsn(τ)/dτ=0 at τ=τa. We plug csn=0 into Equations (11)–(14), and differentiate both sides with respect to τ. According to Propositions 2 and 3 (ii), we have dk(τ)dτ|τ=τa=−Lθ2(1−θ)2u'(0)u'(cssa)[Ru(cssa)]2[1−Ru(cnna)]3A2+θ[u'(0)]2Ru(cnna)[1−Ru(cnna)]2B2<0,(E.1)dw(τ)dτ|τ=τa=−u'(0)u'(cssa)[Ru(cssa)]2[1−Ru(cnna)]2[Ru(cnna)]2[1−Ru(cssa)]2{A2+θ[u'(0)]2Ru(cnna)[1−Ru(cnna)]2B2} ×{θ2Ru′(cssa)Ru(cnna)[1−Ru(cnna)]2 +(1−θ)Ru(cssa)[1−Ru(cssa)] ×[(1−θ)Ru′(cnna)+LθRu(cnna)[1−Ru(cnna)][1−θRu(cnna)]]}<0,dcnn(τ)dτ|τ=τa=(1−θ)2u'(0)u'(cssa)[Ru(cssa)]2[1−Ru(cnna)]2A2+θ[u'(0)]2Ru(cnna)[1−Ru(cnna)]2B2>0,dcss(τ)dτ|τ=τa=−θ2u'(0)u'(cssa)[Ru(cssa)]2[1−Ru(cnna)]3A2+θ[u'(0)]2Ru(cnna)[1−Ru(cnna)]2B2<0,dcns(τ)dτ|τ=τa=−θ(1−θ)[u'(cssa)]2[Ru(cssa)]3[1−Ru(cnna)]3A2+θ[u'(0)]2Ru(cnna)[1−Ru(cnna)]2B2<0,dμnn(τ)dτ|τ=τa=dcnn(τ)dτ|τ=τaRu′(cnna)>0, dμss(τ)dτ|τ=τa=dcss(τ)dτ|τ=τaRu′(cssa)<0,dμns(τ)dτ|τ=τa=dcns(τ)dτ|τ=τaRu′(cnsa)<0,dΩ¯n(τ)dτ|τ=τa=(1−θ)2u'(0)u'(cssa)[Ru(cssa)]2[1−Ru(cnna)]2A2+θ[u'(0)]2Ru(cnna)[1−Ru(cnna)]2B2 ×{Ru′(cnna)+Lθ[Ru(cnna)]2[1−Ru(cnna)]}>0,dΩ¯s(τ)dτ|τ=τa=−θ2u'(0)u'(cssa)[Ru(cssa)]2[1−Ru(cnna)]3A2+θ[u'(0)]2Ru(cnna)[1−Ru(cnna)]2B2Ru′(cssa)<0,dω¯n(τ)dτ|τ=τa=(1−θ)2u'(0)u'(cssa)[Ru(cssa)]2[1−Ru(cnna)]2A2+θ[u'(0)]2Ru(cnna)[1−Ru(cnna)]2B2Ru′(cnna)>0,dω¯s(τ)dτ|τ=τa=θ2u'(0)u'(cssa)[Ru(cssa)]2[1−Ru(cnna)]3A2+θ[u'(0)]2Ru(cnna)[1−Ru(cnna)]2B2 ×L(1−θ)Ru(cssa)[Ru'(cssa)−1]>0, where A2≡(1-θ)2[u′(0)]2Ru(cssa)Ru′(cnna)+θ(1-θ)[u′(0)Ru′(0)-u″(0)]u′(cssa)[Ru(cssa)]2Ru(cnna)[1-Ru(cnna)]2>0,B2≡θRu′(cssa)+Ru(cssa)cssa1-θRu(cnna)1-Ru(cnna)-θRu(cssa)>0. Note that dω¯s(τ)/dτ>0 at τ=τa results from Ru′(cssa)-1 > 0, which is supported by Lemma E.1 below. Lemma E.1 If the reverse inequality of (15) holds, then Ru′(x)-1 > 0.Proof. By using Proposition 2 and the definition of Ru(x), the reverse inequality of (15) can be re-written as [cssau″(cssa)]2u′(cssa)+cssau″(cssa)<[cnnau″(cnna)]2u′(cnna)+cnnau″(cnna). (E.2) Let g(x)≡[xu″(x)]2/[u′(x)+xu″(x)]. Then, the inequality (E.2) means g(cssa)<g(cnna). For all cssa>cnna, the inequality g(cssa)<g(cnna) must be always satisfied. It requires that function g(x) is strictly decreasing, which yields g′(x)=xu″(x)[u′(x)+xu″(x)]2C<0, where C≡[2u′(x)+xu″(x)][u″(x)+xu‴(x)]-x[u″(x)]2>0 must be true. In turn, dividing each term of expression C by u′(x)[-u″(x)] gives [2-Ru(x)][Ru′(x)-1]>Ru(x)>0. Thus, Ru′(x)-1 > 0 must hold. □ Appendix F: Proof of Proposition 7 (a) Unilateral trade from s to n Lemma F.1 For all τ∈[τ˜,τa), k<θ always holds.Proof: According to Proposition 3, csn≥cns=0 for all τ∈[τ˜,τa] iff (15) holds. Then, taking k=θ and cns=0 into the equations and applying (12)–(14) to the two countries yield cnn=1Lθ, css=1L(1-θ), csn=0, w=Ru1L(1-θ)1-Ru1LθRu1Lθ1-Ru1L(1-θ), which are equivalent to all solutions when τ=τa. This means that k=θ only occurs when τ=τa for τ∈[τ˜,τa]. Therefore, k≠θ when τ∈[τ˜,τa). Moreover, because dk(τ)dττ=τa>0, inequality k<θ holds for all τ∈[τ˜,τa). □ Lemma F.2 For all τ∈[τ˜,τa), inequality w<1 holds if cnn=css.Proof: Plugging cns=0 and css=cnn into the labor market clearing condition (14) for each country, we obtain cnn=1/(Lk) and csn=(2k-1)(1-θ)/[L(1-k)kθτ]. Since csn>0, k>1/2 must hold. Then, by using cns=0, css=cnn=1/(Lk), and csn=(2k-1)(1-θ)/[L(1-k)kθτ] in (12) and (13), we obtain w=kθ1-2k-1k1Ru(cnn)<1, as a result of k>1/2 and Lemma F.1. Thus, for all τ∈[τ˜,τa), inequality w<1 holds if cnn=css. □ Lemma F.3 For all τ∈[τ˜,τa), inequalities css>cnn>csn>cns=0 hold.Proof: Proposition 2 gives cssa>cnna and wa>1 when autarky. If w > 1 fails in [τ˜,τa), we let τw = 1 be the largest τ∈[τ˜,τa) at which w = 1. In the case of one-way trade from country s to n, we have τψ(cnn)τ=τw=1=ψ(csn)<ψ(0)=ψ(cns)<τψ(css)τ=τw=1, (F.1) where the first equality is from (11), the first inequality holds because ψ is decreasing, and the second inequality is from (C.1). The monotonicity of ψ and (F.1) give the result of cnn>css at τw=1. Since w, cnn, and css are all continuous functions of τ, the above results imply that the relationship between cnn and css changes from css>cnn to cnn>css when τ falls from τa to τw=1. Accordingly, there is a τ†∈(τw=1,τa) at which cnn=css. Since τw=1 is the largest τ∈[τ˜,τa) where w=1, the value of w at τ† should satisfy w†>1, which contradicts Lemma F.2. Therefore, w>1 holds for all τ∈[τ˜,τa). Accordingly, we also obtain the result that css and cnn do not intersect according to Lemma F.2. In other words, inequalities css>cnn>csn>cns=0 hold when τ∈[τ˜,τa). □ Let css=tcnn, where t>1 according to Lemma F.3. Taking cns=0 and css=tcnn into labor market clearing conditions (14) for the two countries yields cnn=1/(Lk) and csn=[(t+1)k-t](1-θ)/[L(1-k)kθτ]. Then, k>t/(t+1) must hold due to csn>0. As a result, k>1/2 holds for t>1. Thus, 1/2<k<θ for all τ∈[τ˜,τa). (b) Unilateral trade from n to s Lemma F.4 For all τ∈[τ˜,τa), k > θ always holds.Proof: According to Proposition 3, cns≥csn=0 for all τ∈[τ˜,τa] iff the reverse of (15) holds. In the same way as the above, taking k = θ and csn=0 into (12)–(14) for the two countries yields cnn=1Lθ, css=1L(1-θ), csn=0, w=Ru1L(1-θ)1-Ru1LθRu1Lθ1-Ru1L(1-θ), which are also equivalent to all solutions when τ=τa. This implies that k=θ only occurs when τ=τa for τ∈[τ˜,τa]. Therefore, k≠θ when τ∈[τ˜,τa). Moreover, as a result of (E.1) inequality k>θ>1/2 holds for all τ∈[τ˜,τa). □ Lemma F.5 For all τ∈[τ˜,τa), css>cnn always holds.Proof: Plugging csn=0 and css=cnn into labor market clearing conditions (14) for the two countries, we obtain cnn=1/[L(1-k)] and cns=-(2k-1)θ/[L(1-k)k(1-θ)τ]<0. Because cns<0 contradicts Proposition 3 (ii), css and cnn do not intersect when τ∈[τ˜,τa). Moreover, Proposition 2 gives cssa>cnna. Thus, we can conclude that for all τ∈[τ˜,τa), css>cnn always holds. □ Lemma F.6 For all τ∈[τ˜,τa), w > 1 always holds.Proof: Since Proposition 2 gives wa>1 when autarky, there must exist w = 1 if w < 1 occurs when τ∈[τ˜,τa). Taking csn=0 and w = 1 into (12)–(14) for the two countries yields k=θRu(cnn)[1-θRu(css)]-(1-θ)Ru(cns)Ru(css)θRu(cnn)[1-Ru(css)]+(1-θ)Ru(css)[1-Ru(cns)]. Due to k > θ from Lemma F.4, we know that Ru(cnn)[1-θRu(css)]-(1-θ)Ru(cns)Ru(css)θRu(cnn)[1-Ru(css)]+(1-θ)Ru(css)[1-Ru(cns)]>1. (F.2) Then, (F.2) can be simplified as (1-θ)[Ru(cnn)-Ru(css))]>0. It means cnn>css, which contradicts Lemma F.5. Therefore, w=1 does not exist. In other words, for all τ∈[τ˜,τa), w>1 always holds. □ © The Author (2017). Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

Journal of Economic Geography – Oxford University Press

**Published: ** Mar 1, 2018

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