Trade, Labor Reallocation Across Firms and Wage Inequality

Board of Governors of the Federal Reserve System International Finance Discussion Papers ISSN 1073-2500 (Print) ISSN 2767-4509 (Online) Number 1348 June 2022 Trade, Labor Reallocation Across Firms and Wage Inequality Mariano Somale Please cite this paper as: Somale, Mariano (2022). “Trade, Labor Reallocation Across Firms and Wage Inequality,” International Finance Discussion Papers 1348. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/IFDP.2022.1348. NOTE: International Finance Discussion Papers (IFDPs) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the International Finance Discussion Papers Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

Trade, Labor Reallocation Across Firms and Wage Inequality Mariano Somale (cid:3) Federal Reserve Board February 2022 Abstract This paper develops a framework for studying the e⁄ects of higher trade openness on the wage distribution that emphasizes within-industry labor reallocation across (cid:133)rms, strong skill-productivity complementarities in production and heterogenous (cid:133)xed export costs across (cid:133)rms. Assuming no entry intheindustry,anautarkiceconomythatopenstotradeexperiencesapervasiveriseinwageinequality; a trade liberalization in a trading economy increases inequality at the lower end of the distribution, but may reduce it elsewhere. Assuming free entry, opening to trade could result in pervasively higher inequality or wage polarization. The analysis highlights the importance of new exporters (extensive margin) in shaping the aggregate relative demand for skills, a channel controlled by the distribution of (cid:133)xed export costs in the model. Keywords: Trade, (cid:133)rms, workers, supermodularity, wage inequality. JEL codes: F10, F12, F16. (cid:3)All the views expressed in this paper are mine and do not necessarily represent those of the Federal Reserve System.

1 Introduction In this paper, I develop a general equilibrium trade model with a large number of skill-groups that emphasizes within-industry labor reallocation across (cid:133)rms as the mechanism through which trade a⁄ects the wage distribution. The impact of international trade on the income distribution has featured prominently in economic analysis since at least Ohlin (1933). This topic has received renewed attention, as economists have tried to elucidate the causes of the signi(cid:133)cant rise in wage inequality in many countries since the late 70s. Three implications following from the empirical research on this phenomenon motivate the analysis. First, as discussed in Goldberg and Pavcnik (2007) and Helpman (2016), the evidence provides little support for trade as an important driver of the rise in inequality through the channels emphasized by the traditional factor-proportions trade theory.1 Second, (cid:133)rms may be an important part of the story behind the changes in the wage distribution. For example, Krishna, Poole, and Senses (2014) (cid:133)nd substantial within-industry labor reallocation across (cid:133)rms following a trade liberalization that cannot be explained by a random assignment of workers to (cid:133)rms.2 Third, divergent trends in inequality in di⁄erent parts of the wage distribution (Autor, Katz, and Kearney 2008) and a rise in within-group (residual) wage inequality (Acemoglu 2002; Attanasio, Goldberg, and Pavcnik 2004), suggests that grouping workers into a few large skill-groups (as typically done in the literature) does not provide enough detail to understand the full distributional consequences of international trade. The framework extends a standard heterogeneous-(cid:133)rms trade model to include heterogenous workers, strong complementarities in production between worker skill and (cid:133)rm productivity and heterogenous (cid:133)xed export costs. As in Melitz (2003), labor is the only factor of production, the labor market is perfectly competitive, and (cid:133)nal goods are produced by monopolistically competitive (cid:133)rms that di⁄er in their productivity. In addition, the presence of (cid:133)xed production and export costs leads to selection into activity and into exporting, i.e. only some (cid:133)rms (cid:133)nd it optimal to produce and only a subset of them export. Departing from Melitz (2003), the labor force comprises heterogeneous workers of a continuum of skill-types, so (cid:133)rms must choose not only the total number of production workers to hire but also the mix of skill-types to employ. With strong production complementarities between worker skill and (cid:133)rm productivity, such a choice implies that more productive (cid:133)rms have workforces of higher average ability in equilibrium. In this setting, a decline in trade costs induces a reallocation of workers across these heterogenous (cid:133)rms that changes the relative demand for skills in the economy. I use the model to study the channels through which this labor reallocation a⁄ects the wage distribution, including the entry and exit of (cid:133)rms into the market, the increased demand of incumbent exporters, and the demand of new exporters. The core of the framework lies in the production and export technology of (cid:133)rms. The output of a (cid:133)rm depends linearly on the number of production workers of each skill-type that it employs. The 1This evidence includes a rise in the skill-premium in developed and developing countries, driven in both cases by higher demandforskilledworkersinallindustries(Autoretal. 1998;GoldbergandPavcnik2007). Inaddition,severalstudies(cid:133)nd little labor reallocation across industries in response to trade liberalizations in developing countries. 2Inaddition,asdiscussedinCardetal. (2016),numerousstudies(cid:133)ndsimilartrendsintheaggregatedispersionofwages and (cid:133)rms(cid:146)productivity. 1

productivity of a production worker at a given (cid:133)rm is a strictly log supermodular function of the worker(cid:146)s skill and the (cid:133)rm(cid:146)s productivity, giving more able workers a comparative advantage in production at more productive (cid:133)rms. As in Costinot and Vogel (2010), these assumptions permit to transform the market equilibrium analysis into the analysis of a matching problem. In particular, the equilibrium allocation of production workers among active (cid:133)rms is characterized by a strictly increasing and continuous matching function that maps the skill-types of the former to the productivity-types of the latter. Moreover, this matching function is a su¢ cient statistic for the dispersion wages in this setting, facilitating the analysis of comparative static predictions about wage inequality. Fixed export costs also play an important role as they determine the shape of the set exporters and their collective demand for skills, a crucial element in the model a⁄ecting the distributional consequences of trade. Therefore, I consider a relatively (cid:135)exible speci(cid:133)cation of (cid:133)xed export costs, so that the analysis can incorporate salient empirical features about this set.3 Speci(cid:133)cally, I posit that (cid:133)xed export costs vary across (cid:133)rms, and model their (cid:133)rm-speci(cid:133)c sizes as independent realizations of a nonnegative random variable with an absolutely continuous and increasing CDF. As a result, exporters are, on average, more productive than nonexporters in equilibrium, but high productivity non-exporters coexists with low productivity exporters. Finally, all (cid:133)xed costs are paid in terms of a "skill bundle" that comprises non-production workers of all skill levels, an assumption that allows me to isolate the impact on the wage distribution of the endogenous assignment of production workers to (cid:133)rms. The cross-section of the model captures several features of the data identi(cid:133)ed by the trade and labor literatures. The dispersion of wages in the model re(cid:135)ects between-(cid:133)rms wage di⁄erences (rather than within-(cid:133)rm di⁄erences), a channel that represents around 60% of the wage dispersion in the United States (Davis and Haltiwanger 1991). In addition, more productive (cid:133)rms tend to be larger (in terms of output), have workforces of higher average ability, and pay higher average wages (Card et al. 2016). Per the stochastic representation of (cid:133)xed export costs, the model features an imperfect positive correlation between size, (cid:133)rm wages and export status (Bernard and Jensen 1995), and between the latter and (cid:133)rm productivity, leading to overlapping productivity distributions for exporters and non-exports (Bernard, Eaton, Jensen, and Kortum 2003). Finally, if workers are classi(cid:133)ed in large skill-groups, possibly re(cid:135)ecting the imperfect observability of workers ability by the econometrician, then the model features wage heterogeneity within each of these skill groups (Acemoglu 2002; Attanasio et al. 2004). I carry out the analysis of the e⁄ects of trade on the wage distribution under two widely-used assumptions about entry, no free entry a-lÆ Chaney (2008) and free entry a-lÆ Melitz (2003). These alternative entry assumptions lead to the no-free-entry and free-entry models analyzed in the paper, whose predictions can be interpreted, respectively, as the short- and long-term e⁄ects of trade.4 These models di⁄er only in the equilibrium condition that pins down the activity cuto⁄, the productivity value below which 3The assumption of common (cid:133)xed export costs, which has been standard in the literature since Melitz (2003), leads to the counterfactual prediction of non-ovelapping productivity distributions for exporters and non-exporters. This unrealistic assumption is not innocuous in this setting. 4Exploring the implications of these two alternative entry assumptions also serves a pedagogical purpose. By delivering sharper results, the no-free-entry model facilitates the analysis of the main forces at play, which in turn simpli(cid:133)es the subsequent discussion of the more nuanced implications of the free-entry model. 2

(cid:133)rms do not (cid:133)nd it pro(cid:133)table to produce. Conditional on the activity cuto⁄, the two models are identical, so they share the cross-sectional implications discussed above. To study the impact of higher trade openness on the wage distribution, I decompose the associated reallocation of production and employment across (cid:133)rms into three channels. The (cid:133)rst channel, the selection-into-activity channel, capturesthereallocationofresourcesdrivenbychangesinthesetofactive (cid:133)rms, i.e. by changes in the activity cuto⁄. The second channel is the intensive margin of trade, and re(cid:135)ects the changes in the production and employment decisions of incumbent exporters that continue serving the foreign market after the decline in trade frictions. Finally, the third channel is the extensive margin of trade, which captures the reallocation of employment associated with changes in the set of exporters. These last two channels are largely determined by the distribution of (cid:133)xed exports costs in the economy, which highlights the importance of properly modeling this element of the model. This decomposition not only highlights the key elements driving the results in the current setting, but also it facilitates the comparison with the implications of other frameworks in the trade literature exploring the connection between international trade, (cid:133)rms and wages. In the no-free-entry model, an initially autarkic economy that opens to trade always experiences an increase in the activity cuto⁄ and a pervasive rise in wage inequality, in the sense that for any pair of workers, the relative wage of the more skilled one rises. In terms of the three channels discussed above, the selection-into-activity channel induces a pervasive rise in wage inequality, as the exit of the least productive (low-skill-intensive) (cid:133)rms leads to a decline in the relative demand of less skilled workers. With no exporters in the initial autarkic equilibrium, the intensive margin channel is not operational in this counterfactual. Finally, the extensive margin channel also leads to a pervasive rise in wage inequality; the (new) exporters in the open economy are, on average, more productive than non-exporters, so their collective labor demand is biased toward more skilled workers. The importance of this channel, which dependsonhowfastthefractionofexporting(cid:133)rmsincreaseswithproductivity, isdeterminedbytheCDF of (cid:133)xed export costs. Regarding the e⁄ect on the level of wages, trade always rises the average real wage, but the least skilled workers in the economy could see their real wage decline. A trade liberalization can lead to additional outcomes.5 Although a decline in variable trade costs in the no-free-entry model always leads to an increase in the activity cuto⁄and a rise in wage inequality at the lower end of the wage distribution, little can be said about its impact elsewhere in the distribution. As the activity cuto⁄rises, the selection-into-activity channel leads to a pervasive rise in wage inequality. The intensive-margin channel also leads to a pervasive rise in wage inequality, re(cid:135)ecting a rise in the more skill-intensive labor demand of incumbent exporters as they expand their production to satisfy a higher foreign demand. In contrast to the previous two channels, the impact of the extensive-margin channel on the wage distribution is ambiguous. Without additional restrictions on the CDF of exports costs, new exporters can be (on average) more or less productive than incumbent (cid:133)rms, so their collective demand may be biased toward more or less skilled workers. Moreover, the ambiguity about the e⁄ects of this third channel extends to the overall impact of a trade liberalization on the wage distribution. This result 5A tradeliberalization isde(cid:133)ned asa declinein thevariabletradecostsfaced by an economy thatalready participatesin international trade. 3

highlights the importance of paying close attention to the modeling of the extensive-margin channel in any study emphasizing the role of heterogenous (cid:133)rms in the distributional consequences of higher trade openness. I present su¢ cient conditions on the CDF of export costs under which wage inequality rises pervasively after a trade liberalization. Finally, the average real wage always increases following a trade liberalization, but the real wage of the least skilled workers in the economy could decline. Moving to the predictions of the free-entry model, opening to trade has an ambiguous e⁄ect on the wage distribution, as the impact on the activity cuto⁄cannot be determined without imposing additional restrictions on primitives. However, despite the generality of the assumptions, trade can have only two broad e⁄ects on the wage distribution. If the activity cuto⁄ increases, then inequality is pervasively higher in the open economy for the same reasons discussed in the case of the no-free-entry model. If the cuto⁄ decreases, then trade leads to wage polarization, i.e. wage inequality decreases among the least skilled workers, but increases among the most skilled ones. In this case, the selection-into-activity and extensive-margin channels lead to a pervasive decline and a pervasive rise in wage inequality, respectively, with the former channel dominating at lower end of the wage distribution and the latter at the upper end. As in the no-free-entry model, the average real wage is always higher after the economy opens to trade. Perhaps surprisingly, trade raises the real wage of all workers in the economy if and only if it leads to pervasively higher inequality. When trade leads to wage polarization, the real wage of the least skilled workers in the economy necessarily declines. Thecaseofatradeliberalizationinthefree-entrymodelcombinesallthesourcesofambiguitydiscussed above. If the activity cuto⁄ increases, wage inequality necessarily increases at the lower end of the distribution but may decline elsewhere, a possibility that is eliminated by the same su¢ cient conditions presented for the no-free-entry model. If the activity cuto⁄ decreases, then wage inequality necessarily decreases at the lower end of the distribution and increases somewhere else in the distribution, but additional outcomes beyond a wage polarization are possible. That said, the analysis shows that higher tradeopennessneverleadstoapervasivedeclineinwageinequality. Theaveragerealwageintheeconomy always increases with a trade liberalization, while the real wages of the least skilled workers increase if and only if the activity cuto⁄increases. Finally, in a methodological contribution, I establish the existence and uniqueness of the equilibrium in this setting, a prerequisite for a theoretical analysis of comparative statics. Conditional on the activity cuto⁄, the market equilibrium is characterized by a system of nonlinear di⁄erential equations involving the matching, price and revenue functions, which together with a set of boundary conditions, de(cid:133)nes a nonlinear two-point boundary value problem (BVP). In contrast to the cases of initial value problems (IVP) and linear BVPs, where the standard mathematical theory can handle a wide array of problems, establishing existence and uniqueness of solutions is not trivial in the case of non-linear BVPs. Because of the complexity of the subject, the mathematical literature has typically focused on particular cases of the problem, leading to a multitude of theoretical approaches tailored to each of these cases.6 In addition, most results in the literature are based on restrictive and not-easily-veri(cid:133)able assumptions, while those 6Bernfeld and Lakshmikantham (1974) survey some of the most common theoretical approaches used in the literature. See Kiguradze (1988) for some results for the general, (cid:133)rst-order, two-point BVP. 4

based on less restrictive assumptions (resembling those used in the standard theory of IVPs) have a local (cid:135)avor.7 Despitethesedi¢ culties,severalstudiesinthetradeliteraturethatuseassignmentmodelsleading tosimilarBVPssimplyassumeorstatewithoutprooftheexistenceanduniquenessofthesolution. Inthis paper, I (cid:133)ll this gap in the trade literature by presenting existence and uniqueness results for a nonlinear two-point boundary BVP that encompasses those in this paper and others in the literature.8 In addition, I derive a set of results which characterize the dependence of the solution to this BVP on its parameters. This paper is related to a now large literature proposing heterogeneous-(cid:133)rms models in which international trade can a⁄ect wage inequality through within-industry mechanisms. Motivated by developments in within-group wage inequality, one line of research introduces labor market frictions so that ex-ante identical workers can earn di⁄erent wages at di⁄erent (cid:133)rms. Davis and Harrigan (2011) develop a model of e¢ ciency wages in which the wage that induces worker e⁄ort varies across (cid:133)rms due to di⁄erences in monitoring technology. Egger and Kreickemeier (2009, 2012) and Amiti and Davis (2012) develop fairwage models in which the perceived fair wage at which workers supply e⁄ort depends on (cid:133)rms(cid:146)revenue. Finally, bargaining over production surplus resulting from search and matching frictions in labor markets induce wages to vary across (cid:133)rms in Helpman, Itskhoki, and Redding (2010) and Helpman et al. (2016). Another strand of the literature uses models with competitive labor markets and ex-ante heterogeneous workers (from the perspective of (cid:133)rms) to study the e⁄ect of trade on wage inequality through its impact on (cid:133)rms(cid:146)technological choices or on the endogenous assignment of workers to (cid:133)rms; see, for example, Yeaple (2005), Bustos (2011), Monte (2011), Sampson (2014) and Somale (2015). This paper is also related to a growing number of studies using assignment models to study the distributional consequences of international trade and o⁄shoring, such as Grossman and Maggi (2000), Ohnsorge and Tre(cid:135)er (2007), Costinot (2009), Antr(cid:224)s, Garicano, and Rossi-Hansberg (2006) and Nocke and Yeaple (2008). Methodologically, this paper is closer to a branch of this literature that, building on Costinot (2009), develops two-sided heterogeneity models by embedding in di⁄erent general equilibrium frameworksaproductiontechnologysimilartooneconsiderinthispaper,givingrisetosimilarassignment problems. In models with neoclassical roots, Costinot and Vogel (2010) study the assignment of workers to tasks while Grossman, Helpman, and Kircher (2017) study the matching of managers and workers and their sorting into di⁄erent industries. In monopollistically competitive settings, Sampson (2014) and Somale (2015) extend standard frameworks in the trade literature to study the matching of workers to (cid:133)rms. Sampson(2014)presentsageneralequilibriumanalysisofamodelofendogenoustechnologychoice that extends Yeaple (2005) to a continuum of skill-types, and presents some partial equilibrium results for a model that extends Melitz (2003) to allow for worker heterogeneity. Somale (2015) presents a general equilibrium analysis of the e⁄ects of opening to trade on wage inequality by extending Chaney (2008). Relative to the last two studies, this paper present a general equilibrium analysis of the e⁄ects of opening to trade and of a trade liberalization in models extending both Chaney (2008) and Melitz (2003). In addition, this paper allows (cid:133)xed export costs to vary across (cid:133)rms, a generalization that makes the model 7The existence of a solution is guaranteed only over su¢ ciently small intervals. 8The general BVP considered in this paper encompasses those in Costinot and Vogel (2010), Sampson (2014), Somale (2015), Grossman, Helpman, and Kircher (2017). 5

better suited to study the e⁄ects of trade openness on wage inequality as discussed above. Moreover, this generalization actually simpli(cid:133)es some parts of the analysis, as the matching function does not exhibit kinks as in Sampson (2014) and Somale (2015). The rest of the paper is organized as follows. Section 2 describes the basic setup of the framework. Sections3and4characterizetheequilibriumintheno-free-entrymodelandpresentexistenceanduniqueness results. Section 5 studies the e⁄ects of higher trade openness on wage inequality in the no-free-entry model. Finally, section 6 extends the analysis to the free-entry model. 2 The Model 2.1 Demand The preferences of the representative consumer are given by a C.E.S utility function over a continuum of goods indexed by ! : (cid:27) (cid:27) 1 (cid:27) 1 U = u(!) (cid:0)(cid:27) d! (cid:0) ; (cid:20)Z ! 2 (cid:10) (cid:21) where u(!) is the quantity consumed of good !, the measure of the set (cid:10) represents the mass of available goods and (cid:27) > 1 is the elasticity of substitution between goods. The demand and expenditure for individual varieties generated by this utility function are u(!) = EP(cid:27) 1p(!) (cid:27); E(!) = EP(cid:27) 1p(!)1 (cid:27); (1) (cid:0) (cid:0) (cid:0) (cid:0) where P is the aggregate price level and E is aggregate expenditure, 1 P = p(!)1 (cid:27)d! 1 (cid:0) (cid:27) ; E = E(!)d!: (2) (cid:0) (cid:20)Z ! 2 (cid:10) (cid:21) Z ! 2 (cid:10) 2.2 Production There is a continuum of active, monopolistically competitive (cid:133)rms in the market, each producing a di⁄erent variety !.9 As in Melitz (2003), (cid:133)rms di⁄er in their productivity level (cid:30); which they obtain as an independent draw from a distribution G((cid:30)) with density function g((cid:30)): I assume that the support of G, (cid:8) (cid:30) : g((cid:30)) > 0 ; is equal to some bounded interval of non-negative real numbers, (cid:30);(cid:30) R+ . In (cid:17) f g (cid:18) contrast to Melitz (2003), the labor force is heterogenous, consisting of a continuum of workers of mass (cid:2) (cid:3) L that di⁄er in their skill level s. The distribution of worker(cid:146)s skills is represented by a nonnegative density V (s), so LV (s) 0 represents the inelastic supply of workers with skill s. I only consider skill (cid:21) distributions such that the support of V, denoted by S, is equal to some bounded interval of non-negative real numbers, i.e. S s : V (s) > 0 = [s;s] R+ . In addition, I assume that the density V (s) is (cid:17) f g (cid:18) continuously di⁄erentiable on S. 9A (cid:133)rm is active in the market if it produces positive output. 6

The production technology of (cid:133)rms is represented by a cost function that exhibits constant marginal cost and (cid:133)xed overhead costs. After paying the (cid:133)xed costs described below, a (cid:133)rm must decide the mix of workers to use in production. The total output of a (cid:133)rm with productivity (cid:30), q((cid:30)), is given by q((cid:30)) = A(s;(cid:30))l(s;(cid:30))ds; (3) s S Z 2 where A(s;(cid:30)) is the marginal productivity of a worker of skill s, and l(s;(cid:30)) is the total number of production workers of that skill level employed by the (cid:133)rm.10 More skilled workers are more productive than less skilled workers, regardless of the productivity of the (cid:133)rm that employs them. Also, more productive (cid:133)rms have lower labor input requirements than less productive (cid:133)rms no matter the type of worker considered. In terms of the production function (3), I formally assume that the productivity function A(s;(cid:30)) is strictly positive, strictly increasing and continuously di⁄erentiable, i.e. A(s;(cid:30)) > 0, A (s;(cid:30)) > 0 andA (s;(cid:30)) > 0.11 s (cid:30) In addition to the absolute productivity advantage described above, more skilled workers have a comparative advantage in production at more productive (cid:133)rms. Speci(cid:133)cally, I follow Costinot and Vogel (2010) and assume that the function A(:;:) is strictly log-supermodular, i.e. A s;(cid:30) A(s;(cid:30)) > A s;(cid:30) A s;(cid:30) for all s > s and (cid:30) > (cid:30): (4) 0 0 0 0 0 0 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Since A(s;(cid:30)) > 0, condition (4) can be rearranged as A s;(cid:30) =A(s;(cid:30)) > A s;(cid:30) =A(s;(cid:30)), showing 0 0 0 0 that the productivity gains from switching to a more productive (cid:133)rm are higher for more skilled workers. (cid:0) (cid:1) (cid:0) (cid:1) Alternatively, the gains from hiring a more skilled worker are higher for more productive (cid:133)rms. Following a standard practice in the international trade literature, I assume that (cid:133)xed costs are paid in terms of labor. Speci(cid:133)cally, I assume that (cid:133)rms pay a (cid:133)xed cost of fV (s) units of each skill s S, 2 implying that the total (cid:133)xed cost of a (cid:133)rm is s f w(s)V (s)ds = fw; s Z where w(s) is the wage of a worker with skill level s; and w is the average wage in the economy and the numeraire,w = 1. Thisspeci(cid:133)cationof(cid:133)xedcostsguaranteesthatthedistributionofskillsintheeconomy is still given by V (s) after all (cid:133)xed costs have been paid, implying that the demand of labor induced by (cid:133)xed-costs requirements has no e⁄ect on the wage schedule w(s) . As I discuss later, the wage schedule f g is completely determined by the interactions between the exogenous relative supply of skills, captured by the distribution V(s), and the endogenous relative demand of skills derived from the (cid:133)rm(cid:146)s demand of production workers. 10In addition to production workers, a (cid:133)rm employs non-production workers to satisfy the (cid:133)xed costs requirements. 11For any function F(x ;:::;x ), F denotes the partial derivative of F with respect to variable x . 1 n xi i 7

2.3 Variable Costs and Prices Perthelinearproductiontechnology(3),workersareperfectsubstitutesinproduction. Accordingly, (cid:133)rms employ only those worker-types that entail the lowest cost per unit of output, implying that the marginal cost of a (cid:133)rm with productivity (cid:30), c((cid:30)), is given by w(s) c((cid:30)) = min : (5) s S A(s;(cid:30)) 2 (cid:26) (cid:27) For any wage schedule, the marginal cost c((cid:30)) is strictly decreasing in the productivity level (cid:30), as a (cid:133)rm can always hire the same type of workers employed by a less productive competitor and obtain a strictly lower marginal cost due its absolute productivity advantage, A (s;(cid:30)) > 0, i.e. (cid:30) (cid:30) > (cid:30) c((cid:30)) < c((cid:30)): (6) 0 0 , Faced with the iso-elastic demands in (1), (cid:133)rms optimally set their price equal to a constant markup over their marginal costs, p((cid:30)) = (cid:27) c((cid:30)). This pricing rule and the cost minimization condition (5) (cid:27) 1 (cid:0) imply (cid:27) w(s) (cid:27) w(s) p((cid:30)) for all s S; p((cid:30)) = if l(s;(cid:30)) > 0: (7) (cid:20) (cid:27) 1A(s;(cid:30)) 2 (cid:27) 1A(s;(cid:30)) (cid:0) (cid:0) 2.4 Entry I carry out the analysis under two widely-used assumptions regarding entry; no free entry a-lÆ Chaney (2008) and free entry a-lÆ Melitz (2003). In the (cid:133)rst case, there is a (cid:133)xed mass of (cid:133)rms in the industry. In the second case, there is unbounded pool of prospective (cid:133)rms that must pay a (cid:133)xed entry-cost to develop a new product variety and enter the industry. The results obtained under the no-free-entry assumption can be interpreted as the short-term consequences of trade, before investment in the development of new varieties leads new (cid:133)rms to enter the industry. In contrast, the results obtained under the free-entry assumption can be viewed as the long-term e⁄ects of trade. 3 No-Free-Entry Equilibrium in the Closed Economy As in Chaney (2008), there is a (cid:133)xed mass M of (cid:133)rms in the industry. A (cid:133)rm is active in the market if and only if it (cid:133)nds it pro(cid:133)table to produce. The pricing rule (7), the consumer(cid:146)s demand and expenditure functions in (1), and the goods-market clearing condition (u(!) = q(!)), imply that a (cid:133)rm(cid:146)s output, revenue and pro(cid:133)t from serving the domestic market are given by12 qd((cid:30)) = EP(cid:27) 1 (cid:27) c((cid:30)) (cid:0) (cid:27) ; rd((cid:30)) = EP(cid:27) 1 (cid:27) c((cid:30)) 1 (cid:0) (cid:27) ; (cid:25)d((cid:30)) = rd((cid:30)) f; (8) (cid:0) (cid:0) (cid:27) 1 (cid:27) 1 (cid:27) (cid:0) (cid:20) (cid:0) (cid:21) (cid:20) (cid:0) (cid:21) 12I make explicit reference to the domestic market here and use the superscript d to denote relevant domestic variables because it facilitates the comparison with the open-economy case analyzed in the next section. 8

where aggregate expenditure, E, equals aggregate income.13 The last expression, together with a decreasing marginal cost function c((cid:30)), implies that a (cid:133)rm(cid:146)s pro(cid:133)t is an increasing function of the (cid:133)rm(cid:146)s productivity. There are combinations of parameters such that all (cid:133)rms are active in equilibrium, (cid:25)d((cid:30)) 0. How- (cid:21) ever, since this case is not theoretically interesting nor empirically relevant, I focus on the conditions that characterize an equilibrium featuring selection into activity, i.e. the least productive (cid:133)rms (cid:133)nd it unprofitable to produce and remain inactive, (cid:25)d((cid:30)) < 0.14 In such an equilibrium, there is a cuto⁄productivity value (cid:30) ((cid:30);(cid:30)) such that only (cid:133)rms with productivity above this value are active in the market. The (cid:3) 2 value of this activity cuto⁄corresponds to the level of productivity at which (cid:133)rms make zero pro(cid:133)ts,15 (cid:25)d((cid:30) ) = 0: (9) (cid:3) In turn, the activity cuto⁄(cid:30) determines the total mass of active (cid:133)rms in the industry, (cid:3) M = [1 G((cid:30) )]M: (10) (cid:3) (cid:0) Finally, the labor market of each type of worker must clear, (cid:30) g((cid:30)) LV (s) = ld(s;(cid:30)) d(cid:30)M +MfV (s) for all s S: (11) [1 G((cid:30) )] 2 Z (cid:30)(cid:3) (cid:0) (cid:3) The left- and right-hand sides of the last expression capture, respectively, the total supply and demand of workersofskills,withthetotaldemandcomprisingthedemandofproduction workers((cid:133)rstterm),andthe demand of non-production workers derived form the presence of (cid:133)xed costs of production (second term). Having described all the components of the economy, I state the formal de(cid:133)nition of the equilibrium. De(cid:133)nition 1 A no-free-entry equilibrium of the closed economy is a mass of active (cid:133)rms M > 0, a productivity activity-cuto⁄ , (cid:30) (cid:3) ((cid:30);(cid:30)), an output function qd : [(cid:30) (cid:3) ;(cid:30)] R+ , a labor allocation function 2 ! ld : S [(cid:30) (cid:3) ;(cid:30)] R+ , a price function p : [(cid:30) (cid:3) ;(cid:30)] R+ and a wage schedule w : S R+ such that the (cid:2) ! ! ! following conditions hold,16 (i) consumers behave optimally, equations (1) and (2); (ii) (cid:133)rms behave optimally given their technology, equations (3), (7), (9) and (10); (iii) goods and labor markets clear, equations (8) and (11), respectively; (iv) the numeraire assumption holds, w = 1. 13Aggregate income is given by the sum of total labor income and (cid:133)rms(cid:146)pro(cid:133)ts. 14Intuitively, if the size of the market, captured by the mass of workers L, is su¢ ciently large relative to the mass of potential(cid:133)rms,thenall(cid:133)rms(cid:133)nditpro(cid:133)tabletoproduce. Theconditionsonprimitivesthatruleoutthispossibility,sothat the equilibrium features selection into activity, are formally stated in proposition 1. 15In an equilibrium in which all (cid:133)rms are active in the market, we have (cid:30)(cid:3) =(cid:30). In addition, in such an equilibrium it is posible that even the least productive (cid:133)rms make strictly positive pro(cid:133)ts, so condition (9) does not need to hold. 16Technically,thisde(cid:133)nitioncorrespondstoanequilibriumfeaturingselectionintoactivity. However,sinceIonlyconsider equilibria of this type, there is no risk of confusion. 9

3.1 Characterization of the Equilibrium The log-supermodularity of the productivity function, A, implies that the equilibrium labor allocation is characterized by positive assortative matching, i.e. more productive (cid:133)rms employ production workers of higher ability. Speci(cid:133)cally, there exists a continuous and strictly increasing matching function N : S ! [(cid:30) ;(cid:30)] such that, all (cid:133)rms of productivity N (s) employ production workers of skill s, and all production (cid:3) workers of skill s are employed at (cid:133)rms with the productivity N (s).17 Behind this result, formally stated inlemma1,liesasimpleintuition. Thecost-minimizationcondition(5)impliesthata(cid:133)rmofproductivity (cid:30) employing a worker of skill s cannot reduce its marginal cost of production by employing a worker 0 0 of a di⁄erent skill, i.e. w(s)=A s;(cid:30) w(s)=A s;(cid:30) for all s S. This observation and the strict 0 0 0 0 (cid:20) 2 log-supermodularity of A imply that, for any skill level s > s and any productivity level (cid:30) < (cid:30), the (cid:0) (cid:1) (cid:0) (cid:1) 0 0 following inequalities hold, A(s;(cid:30)) < A(s;(cid:30)0) w(s) : Accordingly, a (cid:133)rm with productivity (cid:30) < (cid:30) does A(s 0 ;(cid:30)) A(s 0 ;(cid:30)0) (cid:20) w(s 0 ) 0 not employ workers of skill s > s, as it can obtain a strictly lower marginal cost by hiring a worker of 0 skill s. Although this argument only proves that the matching function is weakly increasing, it highlights 0 the connection between the log-supermodularity of A and positive assortative matching in equilibrium. Armedwiththepreviousresult,theequilibriumcanbecharacterizedintermsofthematchingfunction N,revealingatightconnectionbetweenthelatterandwageinequalityinthecurrentframework. Aworker of skill s is matched to a (cid:133)rm with productivity N (s) in equilibrium if and only if the skill level s solves the cost minimization problem (5) for any (cid:133)rm with productivity (cid:30) = N (s). The (cid:133)rst order condition for an interior solution of this problem yields the following equilibrium condition,18 dlnw(s) @lnA(s;N (s)) = : (12) ds @s The last expression is central in the analysis of wage inequality. It implies that the matching function N is a su¢ cient statistic for the dispersion of wages in the economy, as it is the only endogenous variable a⁄ecting the slope of the wage schedule. The connection between N and wage inequality can be seen more clearly by integrating (12) between s and s > s to get w(s )=w(s) = exp s 00 @lnA(t;N(t)) dt : 0 00 0 00 0 f s 0 @s g Thelastexpression, togetherwiththestrictlog-supermodularityofA, impliesthattheratiow(s )=w(s) R 00 0 is increasing in the values that the matching function takes on the interval [s;s ]. Then, any change in 0 00 the environment leading to an upward shift of the matching function on a given interval also leads to higher relative wages for more skilled workers in that interval. Moreover, the new distribution of wages in the interval is second-order stochastically dominated by the old one, i.e. inequality is pervasively higher after the change.19 Letting H : [(cid:30) ;(cid:30)] S denote the inverse function of the matching function N, the optimal pricing (cid:3) ! rule (7) and the expression for revenues in (8) can be used to express (cid:133)rm(cid:146)s prices and revenues as 17Insomewhatdi⁄erentsettings,CostinotandVogel(2010),Sampson(2014)andGrossman,Helpman,andKircher(2017) alsoobtainpositiveassortativematchinginequilibriumasaresultofassumingstriclog-supermodularityofrelevantfunctions. 18The assumptions we made on the primitives of the model imply that all the endgogenous functions considered in this section are di⁄erentiable, a result that is formally stated in lemma 1 and proved in the appendix. 19In appendix A.1.2, I show that the new distribution is Lorenz dominated by the previous one. The equivalence between Lorenz dominance and normalized second-order stochastic dominance was (cid:133)rst shown in Atkinson (1970). 10

functions of the productivity level (cid:30) and the value of the function H at that productivity level. Totally di⁄erentiatingthesefunctionswithrespectto(cid:30)andusingequation(12)intheresultingexpressionsyields @lnA(H((cid:30));(cid:30)) p ((cid:30)) = p((cid:30)) ; (13) (cid:30) (cid:0) @(cid:30) @lnA(H((cid:30));(cid:30)) rd((cid:30)) = ((cid:27) 1)rd((cid:30)) : (14) (cid:30) (cid:0) @(cid:30) The last two equations imply that the equilibrium matching of workers and (cid:133)rms is also a su¢ cient statistic for the dispersion of (cid:133)rms(cid:146)prices and revenues. In particular, integrating equation (14) between (cid:30) and (cid:30) > (cid:30) yields rd (cid:30) =rd (cid:30) = exp ((cid:27) 1) (cid:30)00 @lnA(H(t);t) dt , so the ratio of the revenues of any 0 00 0 00 0 f (cid:0) (cid:30)0 @(cid:30) g two (cid:133)rms depends only on the productivity levels of these (cid:133)rms and the values of the function H between (cid:0) (cid:1) (cid:0) (cid:1) R thesetwolevels. Notethatthelog-supermodularityofAimpliesthattheratioofrevenuesrd (cid:30) =rd (cid:30) 00 0 is increasing in the values that the inverse of the matching function takes on (cid:30);(cid:30) , so a shift in the 0 00 (cid:0) (cid:1) (cid:0) (cid:1) matching function will have opposite e⁄ects on the dispersion of wages and revenues. (cid:2) (cid:3) The equilibrium labor allocation implied by the matching function must be consistent with market clearing in the labor and goods markets, i.e. N (or H) must be consistent with conditions (1), (3), (8) and (11). This consistency requirement yields the following equilibrium condition, rd((cid:30))g((cid:30))M H ((cid:30)) = ; (15) (cid:30) A(H((cid:30));(cid:30)) L f[1 G((cid:30) )]M V (H((cid:30)))p((cid:30)) (cid:3) (cid:0) (cid:0) (cid:2) (cid:3) which, after some re-arrangement, states that consumers(cid:146)expenditure accruing to (cid:133)rms with productivity (cid:30), rd((cid:30))g((cid:30))M, must equal the total value of the output that those (cid:133)rms can produce with the workers they employ.20 Note that the last expression implies that the output of a (cid:133)rm depends positively on the slope of the function H. Intuitively, (cid:133)rms on the interval [(cid:30) d(cid:30);(cid:30)+d(cid:30)] employ workers on the interval (cid:0) [H((cid:30)) H ((cid:30))d(cid:30);H((cid:30))+H ((cid:30))d(cid:30)] in equilibrium. Then, for a given value of H((cid:30)), a higher value of (cid:30) (cid:30) (cid:0) H ((cid:30)) implies that the same (cid:133)rms employ more workers, so their output is higher. (cid:30) Given the equilibrium activity cuto⁄, (cid:30) , equations (13)-(15) form a system of nonlinear di⁄erential (cid:3) equations that the price function, p, the revenue function, rd, and the inverse of the matching function, H, must satisfy in equilibrium. As is well-known, there is an uncountable family of functions that satisfy a system like (13)-(15), so a set of boundary conditions is needed to pin down a particular solution. Two of these boundary conditions are provided by the labor market clearing condition, as all workers must be assigned to some (cid:133)rm in equilibrium, H((cid:30) ) = s, H (cid:30) = s. A third boundary condition is provided by (cid:3) the zero-pro(cid:133)t condition for (cid:133)rms with productivity (cid:30) , rd((cid:30) ) = (cid:27)f. Finally, the activity cuto⁄(cid:30) can (cid:0) (cid:3)(cid:1) (cid:3) (cid:3) be determined from the the following equilibrium condition, (cid:30) (cid:27) 1 rd((cid:30))g((cid:30))d(cid:30)M +f[1 G((cid:30) )]M = L; (16) (cid:0)(cid:27) (cid:0) (cid:3) Z (cid:30)(cid:3) which states that the total wages paid by (cid:133)rms to production and non-production workers (left) equals 20The total value of the output of (cid:133)rms with productivity (cid:30) is A(H((cid:30));(cid:30)) L f[1 G((cid:30)(cid:3))]M V (H((cid:30)))p((cid:30))H (cid:30) ((cid:30)): (cid:0) (cid:0) (cid:2) (cid:3) 11

total labor income in the economy, where the expression for the latter uses the numeraire assumption. Theconditionsderivedabovearenotonlynecessarybutalsosu¢ cientforanequilibrium. Inparticular, in the appendix I show that if a number (cid:30) ((cid:30);(cid:30)) and a triplet of functions p;rd;H satisfy those (cid:3) 2 conditions, then they can be used to construct a wage schedule w(s), an output function qd((cid:30)), and a (cid:8) (cid:9) labor allocation function ld(s;(cid:30)) such that all the conditions in the de(cid:133)nition of equilibrium are satis(cid:133)ed. I summarize the results in this section in the following lemma that I formally prove in the appendix. Lemma 1 In a no-free-entry equilibrium of the closed economy there exists a continuous and strictly increasing matching function N : S [(cid:30) ;(cid:30)] (with inverse function H) such that (a) ld(s;(cid:30)) > 0 if and (cid:3) ! only if N (s) = (cid:30), (b) N (s) = (cid:30) , and N (s) = (cid:30). In addition, the following conditions hold (cid:3) (i) The wage schedule w is continuously di⁄erentiable and satis(cid:133)es (12). (ii) The price, revenue and matching functions, p;rd;N(and H) ; are continuously di⁄erentiable. Given (cid:30) , the triplet p;rd;H solves the boundary value problem (BVP) comprising the system of di⁄erential (cid:3) (cid:8) (cid:9) equations (13)-(15) and the boundary conditions rd((cid:30) ) = (cid:27)f, H((cid:30) ) = s, H (cid:30) = s. (cid:8) (cid:9) (cid:3) (cid:3) (iii) The activity cuto⁄ (cid:30) and the revenue function rd satisfy (16). (cid:3) (cid:0) (cid:1) Moreover, if a number (cid:30) (cid:3) ((cid:30);(cid:30)), and functions p;rd : [(cid:30) (cid:3) ;(cid:30)] R+ and H : [(cid:30) (cid:3) ;(cid:30)] S satisfy 2 ! ! conditions (ii)-(iii), then they are, respectively, the productivity activity-cuto⁄, the price function, the revenue function, and the inverse of the matching function of a no-free-entry equilibrium of the closed economy. As discussed earlier, one of the contributions of this paper is to formally show the existence and uniqueness of the equilibrium characterized above. However, I defer the formal discussion of this issue to section 4.2, where a more general setting is considered. 4 No-Free-Entry Equilibrium in the Open Economy Balanced trade takes place between n+1 symmetric (identical) economies of the type described above, so the description presented in section 2, including equations (1)-(7), holds for each of these economies. Given that the symmetry assumption ensures that all countries share the same equilibrium variables, I restrict the analysis to the home country. Firms face (cid:133)xed and variable trade costs. Per-unit trade costs are common to all (cid:133)rms and are modeled in the standard iceberg formulation, whereby (cid:28) > 1 units of a good must be shipped in order for 1 unit to arrive in a foreign destination. In contrast, (cid:133)xed export costs vary across (cid:133)rms. A (cid:133)rm that wishes to export to country i must incur an idiosyncratic (cid:133)xed cost of y units of a "bundle of skills" comprising fxV (s) workers of each skill s S. Setting the average wage as 2 the numeraire, w = 1, the unit-cost of this bundle of skills is fx, so the total (cid:133)xed export cost of the (cid:133)rm is fxy per foreign market.21 I model the (cid:133)rm-speci(cid:133)c size of (cid:133)xed export costs, y, as the realization of a nonnegative random variable Y with distribution F, which I assume is independent of the productivity distribution, absolutely continuous, and satis(cid:133)es F(y) = 0 for y y, dF(y) > 0 for y y, where y is the (cid:20) (cid:21) 21The unit-cost of the bundle of skills is fx sw(s)V (s)ds =fxw=fx: s R 12

lower bound of the support of Y. In addition, I assume that f y(cid:28)(cid:27) 1 > f, which guarantees that a (cid:133)rm(cid:146)s x (cid:0) pro(cid:133)t in the domestic market is always higher than in any individual foreign market.22 These assumptions about (cid:133)xed export costs have three important implications. First, formulating export costs in terms of the bundle of skills described above guarantees that the demand of labor induced by (cid:133)xed-export-costs requirements does not a⁄ect the wage schedule.23 Second, in the presence of heterogeneous (cid:133)xed export costs, a highly productive (cid:133)rm may not (cid:133)nd it pro(cid:133)table to export if it faces high (cid:133)xed export costs, while a less productive competitor may choose to serve the foreign market if its (cid:133)xed export costs are su¢ ciently low. As a result, the productivity distributions of exporters and nonexporters overlap in equilibrium, consistent with the evidence in Bernard, Eaton, Jensen, and Kortum (2003). Third, an implication of the restriction f y(cid:28)(cid:27) 1 > f is that, as in Melitz (2003), the activity x (cid:0) status of a (cid:133)rm in the open economy continues to be determined by its domestic pro(cid:133)t. Although not essential for the qualitative results in the paper, this implication simpli(cid:133)es the exposition as the condition determining the activity cuto⁄is unchanged relative to the closed economy.24 The determination of the set of active (cid:133)rms and their operations in the domestic market are little changed relative to the closed economy. There is a (cid:133)xed mass M of potential (cid:133)rms in the industry. A (cid:133)rm is active if and only if it makes non-negative pro(cid:133)ts in the domestic market. The pricing rule (7) and the expenditure functions in (1) imply that the potential domestic output, qd, revenue, rd, and pro(cid:133)t, (cid:25)d, of a (cid:133)rm with productivity (cid:30) are still given by (8). As before, domestic pro(cid:133)ts are strictly increasing in (cid:30), so the equilibrium is characterized by a cuto⁄productivity level, (cid:30) ((cid:30);(cid:30)), such that a (cid:133)rm is active (cid:3) 2 in the market if and only if its productivity is above this level.25 Firms with productivity (cid:30) make zero (cid:3) domestic pro(cid:133)t, condition (9), while the mass of active (cid:133)rms, M, is given by (10). The equilibrium in the open economy features selection into trade, i.e. only a subset of active (cid:133)rms export. An active (cid:133)rm serves a foreign market if and only if it can make non-negative pro(cid:133)ts there. In the presence of variable trade costs, consumers in each country face higher prices for imported goods, px((cid:30)) = (cid:28)p((cid:30)), so conditions (7) and (1) and the symmetry assumption imply that the potential export output, revenue and pro(cid:133)t of a (cid:133)rm with productivity (cid:30) and (cid:133)xed export costs fxy are given by (cid:28)1 (cid:27)rd((cid:30)) qx((cid:30)) = (cid:28)1 (cid:27)qd((cid:30)); rx((cid:30)) = (cid:28)1 (cid:27)rd((cid:30)); (cid:25)x((cid:30)) = (cid:0) fxy: (17) (cid:0) (cid:0) (cid:27) (cid:0) Then, such a (cid:133)rm exports if and only if y (cid:28)1 (cid:27)r ((cid:30))=(cid:27)fx, which, together with the assumptions (cid:0) d (cid:20) about y; implies that only a fraction F (cid:28)1 (cid:27)r ((cid:30))=(cid:27)f of (cid:133)rms with productivity (cid:30) (cid:30) export. Note (cid:0) d x (cid:3) (cid:21) that this fraction is a continuous and increasing function of the productivity level (cid:30), so exporters are, (cid:0) (cid:1) on average, more productive than non-exporters.26 These observations imply that the mass of exporters 22A similar relationship between domestic and foreing pro(cid:133)ts is featured in Melitz (2003). 23See the discussion following the speci(cid:133)cation of (cid:133)xed production costs. 24Alternatively, I could have just assumed that a (cid:133)rm is active if and only if it makes positive pro(cid:133)ts in the domestic market, regardless of its potential export pro(cid:133)ts. 25As before, we focus on the conditions that characterize an equilibrium featuring selection into activity, i.e. (cid:25)d((cid:30))<0: 26The productivity distribution of exporters (cid:133)rst-order stochastically dominates the distribution of non-exporters. 13

with productivity (cid:30) is Mx((cid:30)) = g((cid:30))F (cid:28)1 (cid:27)rd((cid:30))=(cid:27)fx M: (18) (cid:0) (cid:16) (cid:17) Finally, the labor market of each type of worker must clear, (cid:30) (cid:30) (cid:28)1 (cid:0) (cid:27)rd((cid:30)) LV (s) = [ld(s;(cid:30))g((cid:30))M+lx(s;(cid:30))Mx((cid:30))]d(cid:30)+fMV (s)+ nfx (cid:27)fx ydF (y)g((cid:30))Md(cid:30)V (s): Z (cid:30)(cid:3) Z (cid:30)(cid:3) Z0 (19) The left- and right-hand sides of the last expression capture, respectively, the total supply and demand for workers of skill s. Total demand comprises the demand of production workers to supply the domestic and foreign markets, (cid:133)rst term, and the demand of non-production workers derived form the presence of (cid:133)xed costs of production and (cid:133)xed export costs, the second and third terms. Conditions (1)-(3), (7)- (10), (17)-(19) and the numeraire assumption completely describe the equilibrium, prompting the formal de(cid:133)nition of equilibrium in the appendix, analogous to that for the closed economy. 4.1 Characterization of the Equilibrium The equilibrium of the open economy shares several features with its closed-economy counterpart. Cost minimization by (cid:133)rms and the strict log-supermodularity of A imply that the equilibrium labor allocation in the open economy is characterized by a strictly increasing matching function, N, that maps the set of skills, S; to the set of productivity levels of active (cid:133)rms, [(cid:30) ;(cid:30)]. In addition, equation (12), connecting the (cid:3) wage schedule to the matching function, and equations (13) and (14), connecting the price and domesticrevenue functions to the inverse of the matching function, H, continue to hold, as the arguments used in their derivation do not depend on the trade regime of the economy. As before, these equilibrium conditions imply that the labor allocation of workers to (cid:133)rms(cid:151)as captured by the matching function, N, and its inverse, H(cid:151)is a su¢ cient statistic for the dispersion of wages, prices and domestic revenues. The equilibrium labor allocation must be consistent with labor and goods markets clearing, i.e. N (or H) must be consistent with conditions (3), (8), (17) and (19). This observation and the expression for the mass of exporters, equation (18), yield the following equilibrium condition, rd((cid:30))(cid:28)1 (cid:27) rd((cid:30)) 1+F (cid:0) n(cid:28)1 (cid:27) g((cid:30))M (cid:27)fx (cid:0) H (cid:30) ((cid:30)) = (cid:20) (cid:18) (cid:19) rd((cid:30)0) (cid:21) (cid:28)1 (cid:0) (cid:27) : (20) A(H((cid:30));(cid:30))V(H((cid:30)))p((cid:30)) 2 L (cid:0) fM (cid:0) (cid:30) (cid:30) (cid:3) nfx 0 (cid:27)fx ydF(y)g((cid:30)0)Md(cid:30)03 6 R R 7 4 5 After some re-arrangement, the last expression states that the total revenue that (cid:133)rms with productivity (cid:30) make from their sales in the domestic and foreign markets, the numerator on the right-hand side of (20), must equal the total value of the output that those (cid:133)rms can produce with the workers they employ. Given the equilibrium activity cuto⁄, (cid:30) , equations (13), (14) and (20) form a system of nonlinear (cid:3) di⁄erential equations that the price function, p, the domestic revenue function, rd, and the inverse of the matching function, H, must satisfy in equilibrium. Two boundary conditions for this system are provided by the labor market clearing condition, as all workers must be assigned to some (cid:133)rm in equilibrium, 14

H((cid:30) ) = s, H (cid:30) = s. A third boundary condition is provided by the zero-domestic-pro(cid:133)t condition for (cid:3) (cid:133)rms with productivity (cid:30) , rd((cid:30) ) = (cid:27)f. Finally, the open-economy counterpart of equation (16) can be (cid:0) (cid:1) (cid:3) (cid:3) used to determine the activity cuto⁄(cid:30) , (cid:3) (cid:27) 1 (cid:30) rd((cid:30))[1+F rd((cid:30))(cid:28)1 (cid:0) (cid:27) n(cid:28)1 (cid:27)]g((cid:30))d(cid:30)M+ (cid:0)(cid:27) (cid:30)(cid:3) (cid:27)fx (cid:0) R rd(cid:16)((cid:30)0)(cid:28)1 (cid:0) (cid:27) (cid:17) = L; (21) fM + (cid:30) nfx (cid:27)fx ydF (y)g (cid:30) Md(cid:30) (cid:30)(cid:3) 0 0 0 R R (cid:0) (cid:1) which states that the total value of wages paid by (cid:133)rms to production and non-production workers (left) equals total labor income in the economy, where the expression for the latter uses the numeraire assumption. As in the closed economy case, the conditions derived in this section are not only necessary, but also su¢ cient for an equilibrium. This characterization of the equilibrium is summarized in lemma 3 in the appendix, which can be easily proved adapting the arguments in the proof of lemma 1. I conclude this section with a summary of the qualitative properties of the equilibrium in the open economy. In equilibrium more productive (cid:133)rms employ production workers of higher ability and pay them higher wages. The stochastic speci(cid:133)cation of (cid:133)xed export costs, though, imply an imperfect positive correlation between (cid:133)rms(cid:146)productivity, average workforce ability, size and export status, which is consistent with the empirical evidence documented in Bernard and Jensen (1995) and Bernard, Eaton, Jensen, and Kortum (2003). In addition, as each (cid:133)rm employs workers of all types ((cid:133)xed costs as skill-bundles), the dispersion of wages in the model has a between- and within-(cid:133)rm component. 4.2 Existence and Uniqueness of the Equilibrium Istartthissectionbystudyingtheexistenceanduniquenessofsolutionstothenonlinear, two-pointBVPs characterizing the equilibrium in the closed and open economies. In contrast to the cases of initial value problems (IVPs) and linear BVPs, for which there is a standard theory that provides fairly general results under relatively mild restrictions on the data of the problem, such a study is not trivial in the case of nonlinear BVPs for two reasons.27 First, there is no uni(cid:133)ed theory that can be applied to study these issues for an arbitrary problem. Because of the complexity of the subject, the mathematical literature has typically focus on particular cases of the problem, leading to a multitude of theoretical approaches tailored to these cases.28 Second, most results in the literature are based on restrictive and not-easilyveri(cid:133)able assumptions, while those results based on less restrictive assumptions, resembling those used in the standard theory of IVPs, have a local (cid:135)avor.29 Despite these di¢ culties, several studies in the trade 27Most textbools on ordinary di⁄erential equations cover the standard existence and uniqueness theory for IVPs. Some examples include Coddington and Levinson (1987) and Agarwal and O(cid:146)Regan (2008a), with the latter also covering basic resutlsforlinearBVPs. Fora more comprehensivetreatmentoflinearBVPssee Stakgold (1998)and Agarwaland O(cid:146)Regan (2008b). 28Bernfeld and Lakshmikantham (1974) present a survey of some of the most common theoretical approaches used in the literature,togetherwith theparticularproblemstowhich they havebeen applied. SeeKiguradze(1988)forsomeresultsfor the general, (cid:133)rst-order, two-point BVP. 29Followingawidely-usedapproachinthetheoryofIVPs,Bailey,Shampine,andWaltman(1968)presentseveralexistence anduniquenessresultsfornonlinearBVPsusingPiccard(cid:146)sIterationmethod(Banach(cid:133)xedpointtheorem)whenthefunctions involved satisfy certain Lipschitzian conditions. In all cases, the resuls are local in nature, i.e. the interval over which the 15

literature that use assignment models and arrive to characterizations of the equilibrium involving a BVP similar to those above, simply assume or state without proof the existence and uniqueness of the solution. In this section I (cid:133)ll this gap in the trade literature by presenting existence and uniqueness results for a nonlinear BVP that encompasses the two BVPs considered above and others in the literature.30 For any (cid:30) ;(cid:30) [(cid:30);(cid:30)] and s ;s [s;s], with (cid:30) < (cid:30) and s < s , I consider the nonlinear, two-point 0 1 0 1 0 1 0 1 2 2 BVP (22), comprising the system of di⁄erential equations (22a)-(22c) and the boundary conditions (22d), @lnA((cid:0)((cid:30));(cid:30)) z ((cid:30)) = z((cid:30)) ; (22a) (cid:30) (cid:0) @(cid:30) @lnA((cid:0)((cid:30));(cid:30)) x ((cid:30)) = ((cid:27) 1)x((cid:30)) ; (22b) (cid:30) (cid:0) @(cid:30) x((cid:30))[1+F (K x((cid:30)))K ](cid:11)((cid:30))g((cid:30)) 0 1 (cid:0) ((cid:30)) = ; (22c) (cid:30) A((cid:0)((cid:30));(cid:30))V ((cid:0)((cid:30)))z((cid:30)) x((cid:30)) = 1; (cid:0)((cid:30) ) = s , (cid:0)((cid:30) ) = s , (22d) 0 0 1 1 where (cid:11)((cid:30)) is a strictly positive continuous function, (cid:11) : [(cid:30);(cid:30)] R++ , K 0 and K 1 are nonnegative ! constants and A;g;V;F are the functions de(cid:133)ned earlier. f g The general BVP de(cid:133)ned above nests the BVPs corresponding to the closed and open economies, as the latter can be obtained as particular parametrizations of the former. If we set K = (f=f )(cid:28)1 (cid:27), 0 x (cid:0) K = n(cid:28)1 (cid:27), (cid:30) = (cid:30) , (cid:30) = (cid:30) and (cid:11)((cid:30)) = 1 for all (cid:30) [(cid:30);(cid:30)], the resulting BVP is equivalent to the BVP 1 (cid:0) 0 (cid:3) 1 2 of the open economy, in the sense that any solution to one of these two BVPs can be used to construct a solution to the other. To see this, let z;x;(cid:0) be a solution to the BVP (22) parametrized as above. f g If we de(cid:133)ne rd((cid:30)) (cid:27)fx((cid:30)), p((cid:30)) z((cid:30))(cid:27)fM=[L fM (cid:30) nf fx((cid:30)0)(cid:28)1 (cid:0) (cid:27)=fxydF (y)g (cid:30) Md(cid:30)] (cid:17) (cid:17) (cid:0) (cid:0) (cid:30)(cid:3) x 0 0 0 and H = (cid:0), then p;rd;H is a solution to the BVP of the open economy. A similar argument shows R R (cid:0) (cid:1) that any solution to the BVP of the open economy can be used to construct a solution to this particular (cid:8) (cid:9) parametrization of BVP (22). Finally, if we set K = 0 in the parametrization above, the resulting BVP 1 is equivalent to the BVP of the closed economy de(cid:133)ned in lemma 1.ii. Lemma 2 states the main results about the general BVP considered in this section. Lemma 2 Under the assumptions on the data of the problem, A;g;V;F;(cid:11);K ;K , there is a unique 0 1 f g continuously di⁄erentiable solution to the BVP (22) for any (cid:30) ;(cid:30) [(cid:30);(cid:30)] and s ;s [s;s], with (cid:30) < (cid:30) 0 1 0 1 0 1 2 2 and s < s . As a function of ((cid:30) ;s ), the solution to the BVP, z(:;(cid:30) ;s );x(:;(cid:30) ;s );(cid:0)(:;(cid:30) ;s ) , 0 1 0 0 0 0 0 0 0 0 f g satis(cid:133)es the following conditions, (i) (no crossing) If K = 0 and (cid:0) 1 denotes the inverse of (cid:0), then sa < sb implies (cid:0)((cid:30);(cid:30) ;sa) < 1 (cid:0) 0 0 0 0 (cid:0) (cid:30);(cid:30) ;sb on [(cid:30) ;(cid:30) ); while (cid:30)a > (cid:30)b implies (cid:0) 1(s;(cid:30)a;s ) > (cid:0) 1 s;(cid:30)b;s on [s ;s ): 0 0 0 1 0 0 (cid:0) 0 0 (cid:0) 0 0 0 1 (ii) (cid:30)a > (cid:30)b implies x((cid:30);(cid:30)a;s ) < x (cid:30);(cid:30)b;s on [(cid:30)a;(cid:30) ]. (cid:0) 0 (cid:1)0 0 0 0 0 0 1 (cid:0) (cid:1) (cid:0) (cid:1) The formal proof of the last lemma can be found in the appendix, so here I present a brief outline of solution is de(cid:133)ned has to be su¢ ciently small. 30The general BVP considered in this section encompasses those in Costinot and Vogel (2010), Sampson (2014), Somale (2015), Grossman, Helpman, and Kircher (2017). 16

the argument. To prove existence, I follow O(cid:146)Regan (2013) and recast the BVP as a (cid:133)xed point problem. In particular, I show that a function (cid:0) is part of a solution, z;x;(cid:0) , to the BVP (22) if and only if f g it is a (cid:133)xed point of some compact functional, (cid:9); de(cid:133)ned over a convex and closed set K, (cid:9)((cid:0)) = (cid:0). Then, a direct application of the Schauder (cid:133)xed point theorem yields the existence result. The uniqueness of the solution is established as a consequence of the particular structure of the problem and the strict log-supermodularity of A. The claim in (i) is obtained as a corollary of the uniqueness result. For the case K = 0 (closed economy), the claim in (ii) immediately follows from the no-crossing result in (i), 1 (22b) and the log-supermodularity of A. However, this argument cannot be extended to the case K > 0 1 (open economy), as the no-crossing property no longer holds. In the appendix, I present a slightly longer argument that is valid for the general case K 0, which also establishes the result as a consequence of 1 (cid:21) the strict log-supermodularity of A. An important corollary of the discussion so far is that, for a given activity cuto⁄(cid:30) , the functions rd (cid:3) and H that solve the BVPs of the closed and open economies do not depend on the mass of (cid:133)rms, M, nor the mass of production workers.31 This feature of the solution follows from the uniqueness result in theorem 7, equation (22c) and the correspondence between said BVPs and BVP (22) described above. In fact, the mass of (cid:133)rms and the mass of production workers a⁄ect only the level of the solution function p. This result will prove useful in the analysis of the free-entry model in section 6. Armed with lemma 2, the existence and uniqueness of the equilibrium in the open economy can be easily derived. The structure of the model, together with the fact that the BVP of the open economy has a unique solution, implies that there is exists a unique equilibrium if and only if there is a unique value of the activity cuto⁄, (cid:30) , that solves equation (21). In turn, this last result can be establish by analyzing the (cid:3) properties of the left- and right-hand sides of said equation as functions of (cid:30) . As discussed above, if rd is (cid:3) part of the solution to the open-economy BVP, then rd((cid:30)) = (cid:27)fx((cid:30)), where x is part of the solution to a particular parametrization of the BVP (22). Then, lemma 2.ii implies that rd((cid:30)) is strictly decreasing in the activity cuto⁄(cid:30) , making the left-hand side of (21) strictly decreasing in the value of (cid:30) . In addition, (cid:3) (cid:3) it is readily seen that the right-hand side of (21) does not depend on the value of (cid:30) , so there is a unique (cid:3) solution to (21) if the size of the market, as captured by L, is not too large. The intuition behind this restriction on the market size is simple. The de(cid:133)nition and characterization of the equilibrium in the open economy (and the closed economy) correspond to an equilibrium featuring selection into activity. However, such an equilibrium does not exist if the size of the market, as captured by L, is su¢ ciently large relative to the mass of (cid:133)rms, M, as in this case even the least productive (cid:133)rms (cid:133)nd it pro(cid:133)table to produce. The same argument can be used to show that there exists a unique equilibrium in the closed economy. I summarize this discussion in the next proposition, which also establishes the (constrained) e¢ ciency of the equilibrium in the closed and open economies. Proposition 1 Let p;rd;H and pa;rd;a;Ha be, respectively, the solution to the BVPs characterizing the open and closed economies with (cid:30) =(cid:30). In addition, let (cid:12)(rd;(cid:30) ) and (cid:12)a(rd;(cid:30) ) denote the functions (cid:8) (cid:9) (cid:8)(cid:3) (cid:9) (cid:3) (cid:3) de(cid:133)ned by left-hand sides of equations (21) and (16), respectively, in terms of (cid:30) and rd. Then, (cid:3) 31The mass of production workers in the closed and open economies are given by the term in brackets in the denominator of the right-hand side of equations (15) and (20), respectively. 17

(i) If (cid:12)(rd;(cid:30)) > L, then there is a unique no-free-entry equilibrium of the open economy. (ii) If (cid:12)a(rd;(cid:30)) > L, then there is a unique no-free-entry equilibrium of the closed economy. In addition, the equilibrium of the closed economy is e¢ cient, while that of the open economy is e¢ cient when f f (cid:28)1 (cid:27), and constrained e¢ cient when f > f (cid:28)1 (cid:27). x (cid:0) x (cid:0) (cid:20) Being a su¢ cient statistic for the dispersion of wages in the model, the matching function takes center stage in the subsequent analysis, as any result about wage inequality in this model is essentially a statement about the impact on the matching function of the shock under consideration. Lemma 4 in the appendix collects several results related to the BVP (22) that are instrumental to the analysis, which characterize the dependence of the solution function (cid:0) (and some functionals of (cid:0)) on the parameters of the problem. 5 No-Free-Entry, Trade and Wage Inequality In this section, I study the e⁄ects of higher trade openness on wage inequality in the no-free-entry model described above. In the model, a decline in trade frictions induces a reallocation of production and employment across (cid:133)rms with heterogenous skill demand, a⁄ecting the aggregate relative demand for skills and the relative wages in the economy. In the analysis, I distinguish three channels through which trade induces this reallocation, and study the impact of each of these channels on wage dispersion. The (cid:133)rst channel is the intensive margin of trade, and re(cid:135)ects the changes in the production and employment decisions of those (cid:133)rms that were exporters before the shock and that remain exporters afterwards. The secondchannelistheextensive margin oftrade, whichcapturesthereallocationofemploymentassociated withchangesinthesetofexporters. Finally, thethirdchannelistheselection-into-activity e⁄ectoftrade, capturing the reallocation of resources driven by changes in the set of active (cid:133)rms. 5.1 Autarky vs. Trade The (cid:133)rst instance of higher trade openness that I consider is the case of an initially autarkic economy that opens up to trade. I start this section with one of the main results of the paper, Proposition 2, which states that opening to trade leads to a pervasive increase in wage inequality. Proposition 2 Let (cid:30) ;Na and (cid:30) ;N(cid:28) be the activity cuto⁄ and matching function corresponding to (cid:3)a (cid:3)(cid:28) f g f g the no-free-entry equilibrium of the closed and open economies, respectively. Then the following conditions hold: (i) (cid:30) > (cid:30) and N(cid:28) (s) > Na(s) for all s [s;s), so inequality is pervasively higher in the open economy. (cid:3)(cid:28) (cid:3)a 2 (ii)Theselection-into-activityandextensive-marginchannelsleadtopervasivelyhigherinequality(intensivemargin channel not operational). The (cid:133)rst result in the last proposition, (cid:30) > (cid:30) , states that the selection-into-activity e⁄ects of trade (cid:3)(cid:28) (cid:3)a highlighted in Melitz (2003) always hold in the no-free-entry model of this paper, i.e. trade induces the least productive (cid:133)rms to exit the market. Although somewhat trivial in homogenous-workers models a-lÆ 18

Melitz/Channey,thisresultisnotimmediateinthecurrentframework.32 Forexample,inanhomogenousworkers version of the no-free-entry model above, assuming that (cid:133)rms with productivity (cid:30) are still active (cid:3)a after the economy starts trading results in unchanged domestic revenues and labor costs. With aggregate laborcostspinneddownbyanequilibriumcondition, thisobservation, togetherwithpositiveexportlabor costs, implies that a higher activity cuto⁄is required in the open economy. In contrast, making the same assumption in the heterogeneous-worker framework above leads to lower domestic revenues and labor costs, so establishing the result requires proving that the decline in the latter is more than o⁄set by the new labor costs of exporting (variable and (cid:133)xed). I do so in the appendix by showing that total wages paid to production workers necessarily increase if the activity cuto⁄remains unchanged, which together with the presence of (cid:133)xed export labor costs, leads to a rise in the the total wages paid by (cid:133)rms. With total wages pinned down by the numeraire assumption, condition (21), a higher activity cuto⁄is required in the open economy.33 To gain more insight into the e⁄ect of trade on wage inequality, I decompose the overall e⁄ect into the three channels mentioned above. First of all, note that the intensive-margin channel is not operational in this case, as there were no exporters before the economy started to trade. The selection-into-activity channel captures the impact on wage inequality of the trade-induced increase in the activity cuto⁄, excluding the impact of changes in the set of exporters. To isolate the e⁄ect of this channel, I contrast the matching function of the closed economy with that of an ancillary autarkic economy that di⁄ers from the former only in that its activity cuto⁄ is given by that of the open economy. That is, the equilibria of the closed and ancillary economies are characterized by the BVP in lemma 1.ii with (cid:30) = (cid:30) and (cid:3) (cid:3)a (cid:30) = (cid:30) , respectively. The typical situation is depicted in (cid:133)gure 1, where the solid and dashed red lines (cid:3) (cid:3)(cid:28) are,respectively,thematchingfunctionsoftheclosed(Na)andancillary(N0)economies. Theno-crossing result in lemma 2.i. implies that the latter lies strictly above the former on [s;s) as shown in the (cid:133)gure. Intuitively, as the (cid:133)rms with productivity in the range [(cid:30) ;(cid:30) ) become inactive, the aggregate demand (cid:3)a (cid:3)(cid:28) for workers with skills in the range [s;Na((cid:30) )) drops to zero barring any change in the wage schedule. (cid:3)(cid:28) Per the labor market clearing condition, these workers must be reallocated among the (cid:133)rms that remain active, requiring an increase in the relative wages of more skilled workers. The extensive-margin channel re(cid:135)ects the impact on wage inequality of the increased labor demand by new exporters as they expand their production to serve the foreign market, excluding the e⁄ects of changes in the activity cuto⁄. Put another way, this channel captures the e⁄ects of replacing [1 + F rd((cid:30))(cid:28)1 (cid:27)=(cid:27)fx n(cid:28)1 (cid:27)]with1intheBVPoftheopeneconomy, preciselywhatthedi⁄erencebetween (cid:0) (cid:0) thematchingfunctionsoftheancillary(N0)andopen(N(cid:28))economiesin(cid:133)gure1captures, withthelatter (cid:0) (cid:1) shown in blue. To see why N(cid:28) necessarily lies above N0 as depicted in the (cid:133)gure, suppose for a moment that the wages of the ancillary economy also prevail in the open economy. In this case, (cid:133)rms of a given productivity level demand the same skill-type of workers in both economies, with exporters in the open economy demanding more labor due to the foreign demand they face. With a constant fraction of exporters across productivity levels, this additional export-driven labor demand would a⁄ect all skill 32Chaney (2008) develops a parametrized version of Melitz (2003) featuring no free entry. 33As explained earlier, the left-hand side of (21) is strictly decreasing in the activity cuto⁄. 19

Figure 1: The E⁄ects of Trade on the Matching Function Note: The solid red and blue lines represent, respectively, the matching functions of the closed (Na) and open (N(cid:28)) economies. The dashed red line depicts the matching function of an ancillary autarkic economy (N0)whichisobtainedbysolvingtheBVPinlemma1.iiwith(cid:30)(cid:3)=(cid:30)(cid:3)(cid:28) . Thedi⁄erencebetweenNa andN0 capturestheimpactoftheselection-into-activitychannel,whilethedi⁄erencebetweenN0 andN(cid:28) captures the impact ofthe extensive-margin channel. levels proportionally, leaving unchanged the overall relative demand for skills in the economy. However, as the fraction of exporters in the model increases with (cid:133)rms(cid:146)productivity, this additional export-driven labor demand is tilted towards more able workers. The resulting rise in the overall relative demand for more skilled workers is inconsistent with labor market clearing, so the relative wages of these workers must be higher in the open economy.34 I conclude this section with a discussion of the impact of trade on the level of real wages. Although trade always rises the average real wage, the least skilled workers in the economy may see their real wage decline. The pricing rule (7) and the zero pro(cid:133)t condition (9) imply that the aggregate price indices of the closed (Pa) and open (P(cid:28)) economies satisfy Pi (cid:27) = (cid:27)f (cid:27) wi(s) (cid:27) (cid:0) 1 for i = a;(cid:28); (23) Ui ((cid:27) 1)A(s;(cid:30) ) (cid:20) (cid:0) (cid:3)i (cid:21) (cid:0) (cid:1) whereUi istheaggregaterealexpenditure/incomeintheeconomy. Perthee¢ ciencyresultinproposition 1, real income is higher in the open economy, U(cid:28) > Ua.35 In addition, proposition 2.i, together with the numeraire assumption (wi = 1), implies that the open economy exhibits a higher activity cuto⁄, (cid:30) > (cid:30) ; (cid:3)(cid:28) (cid:3)a and a lower wage for the least able workers, w(cid:28) (s) < wa(s). Accordingly, P(cid:28) < Pa, so the average real 34Formally, in the appendix I show that the BVPs of the ancillary and open economies can be conceived as particular parameterizations of the general BVP (22) with K =0 that di⁄er only in the parameter function (cid:11)((cid:30)), which is constant 1 in the former and increasing in the latter. The result then follows from a direct application of lemma 4.i in the appendix. 35Notethattheclosed economy allocation isavailableto theplanneroftheopen economy,soasimplerevealed-preference argument yields U(cid:28) >Ua. See the proof of proposition 1 in the appendix for more details. 20

wage, w=P, is higher in the open economy. Finally, recalling that Ui = Ei=Pi, equation (23) can be rearranged to get the an expression for the real wage of workers with the lowest skill level, wi(s)=Pi = ((cid:27) (cid:0) (cid:27) 1) A(s;(cid:30) (cid:3)i ) Ei=(cid:27)f (cid:27) (cid:0) 1 1. This expression implies that trade improves the real wage of even the least able workers in the economy when it induces a (cid:2) (cid:3) riseinaggregateexpenditure/income. However, insomeparameterizationsofthemodel, tradecaninduce a decline in the real wage of these workers, as the drop in aggregate expenditure more than o⁄sets the boost coming from a higher activity cuto⁄. 5.2 Trade Liberalization Although the preceding analysis sheds light into the e⁄ects of higher trade openness on wage inequality, very few, if any, of the countries in the world operate in autarky. For this reason, in this section I study the e⁄ects on wage inequality of a trade liberalization, de(cid:133)ned as a decline in the variable trade costs faced by an economy that participates in international trade. I (cid:133)nd that these e⁄ects may di⁄er from those described in the previous section. In particular, although a trade liberalization necessarily rises wage inequality among the least skilled workers in the economy, wage inequality may decline elsewhere in the wage the distribution. Proposition 3 presents the main results of this section. Proposition 3 Consider a trade liberalization consisting in a decline in variable trade costs from (cid:28) to h (cid:28) , and let (cid:30) ;Nh and (cid:30) ;Nl represent, respectively, the pre- and post-liberalization activity cuto⁄s l (cid:3)h (cid:3)l and matching functions. Then, the following conditions hold: (cid:8) (cid:9) (cid:8) (cid:9) (i) (cid:30) > (cid:30) , so a trade liberalization raises wage inequality among the least skilled workers in the economy. (cid:3)l (cid:3)h (ii) The selection-into-activity and intensive-margin channels lead to pervasively higher inequality, while the e⁄ect of the extensive-margin channel is ambiguous. (iii) If the functions (cid:17)F (t;(cid:21)) Fy(t(cid:21))(cid:21) and (cid:17)F (t;(cid:21)) Fy(t(cid:21))(cid:21)2 are, respectively, strictly decreasing 0 (cid:17) [1+F(t(cid:21))k] 1 (cid:17) [1+F(t(cid:21))k] and strictly increasing in (cid:21) for all t;k R++ , then a trade liberalization rises wage inequality pervasively. 2 The (cid:133)rst result of the proposition states that, as in the Melitz/Channey models, a trade liberalization always leads to the exit of the least productive of (cid:133)rms from the market, (cid:30) > (cid:30) . The general line (cid:3)l (cid:3)h of argument used in the proof of proposition 2.i. can be applied here as well. If the activity cuto⁄ remainsunchangedafterthedeclineintradecosts,thentotalwagespaidtoproductionandnonproduction workers necessarily increase. With total wages pinned down by condition (21), the activity cuto⁄ must be higher after the liberalization. This result and the continuity of the matching functions imply that Nl(s) > Nh(s) on some interval of the form [s;s), which is equivalent to the second part of the claim in 0 proposition 3.i.36 As before, the overall impact of a trade liberalization on wage inequality can be decomposed into the three channels mentioned above. The selection-into-activity channel captures the changes in wage dispersion associated with the rise in the activity cuto⁄, excluding the impact of changes in the labor 36Of note, establishing the consequences of an unchanged activity cuto⁄ is more complicated in the case of a trade liberalization, as multiple crossings of relevant matching functions cannot be ruled out. In this case, the formal argument is based on the results in lemma 4.iv-v. 21

Figure 2: The E⁄ects of a Trade Liberalization on the Matching Function Note: Thesolidredandbluelinesrepresent,respectively,thepre-(Nh)andpost-liberalization(Nl)matching functionsdescribedinProposition3. StartingatNh,increasingtheactivitycuto⁄to(cid:30)(cid:3)l >(cid:30)(cid:3)h ,whilekeeping the set of exporters and the level of variable trade costs unchanged yields the matching function depicted by thedashed red line,N0. Allowing variablestradecoststo increaseto theirpost-liberalization levelwhile keeping the set of exporters unchanged yields N1, the dashed blue line. Adjusting the set of exporters to their post-liberalization composition yields, Nl. Accordingly, the e⁄ects of the selection-into-activity, intensive-margin,andextensive-marginchannelsonthematchingfunctionarecaptured,respectively,bythe di⁄erences between the pairs Nh;N0 , N0;N1 ,and N1;Nl . f g f g f g demand of incumbent exporters and of changes in the set of exporters. To isolate the e⁄ect of this channel, I contrast the matching function of the open economy before the liberalization, Nh, with that of an ancillary open economy, N0, that di⁄ers from the former only in that its activity cuto⁄ is given by that prevailing after the liberalization, (cid:30) . That is, as I explain in more detail in the appendix, the (cid:3)l BVPs associated with Nh and N0 can be conceived as parameterization of the general BVP (22), with K = 0 and (cid:11)h((cid:30)) [1+F rd;h((cid:30))(cid:28)1 (cid:27)=(cid:27)fx n(cid:28)1 (cid:27)], that di⁄er only in their boundary conditions.37 1 (cid:17) h(cid:0) h(cid:0) Accordingly, the no crossing result in lemma 2.i. implies that N0 lies strictly above Nh on [s;s) as (cid:0) (cid:1) depicted by the dashed and solid red lines in (cid:133)gure 2. The intuition for the e⁄ects of this channel are the same as before, i.e. the exit of the least productive (cid:133)rms from the market reduces the relative demand of less skilled workers, pushing down their relative wages. Theintensive-marginchannelcapturestheimpactonwageinequalityoftheliberalization-inducedrise in the labor demand of incumbent exporters. I isolate this channel by contrasting the matching function N0 with that of a second ancillary open economy, N1, with the same set of exporters and active (cid:133)rms, but with variable trade costs given by (cid:28) . That is, N1 is obtained by replacing the parameter function l (cid:11)h((cid:30)) with (cid:11)1((cid:30)) [1+F rd;h((cid:30))(cid:28)1 (cid:27)=(cid:27)fx n(cid:28)1 (cid:27)] in the BVP associated with N0. As shown by the (cid:17) h(cid:0) l(cid:0) dashed blue and red lines in (cid:133)gure 2, N1 necessarily lies above N0 on (s;s) for the same reasons laid out (cid:0) (cid:1) inthediscussionoftheextensive-marginchannelinproposition2. Supposethattheseancillaryeconomies 37rd;h is the domestic revenue function of the open economy with variable trade costs (cid:28) . h 22

share the same wage schedule. Then, (cid:133)rms of a given productivity level demand the same skill-type of workers in both economies, with exporters in the N1-economy (lower trade costs) demanding more labor due to the larger foreign demand they face. As the (common) fraction of exporters in these economies is increasing in (cid:133)rms(cid:146)s productivity, this additional export-driven labor demand in the N1-economy results in a higher relative demand for more skilled workers, which is inconsistent with labor market clearing. Accordingly, the wages of these workers must be higher in the N1-economy.38 The extensive-margin channel captures the impact on relative wages of allowing the fraction of exporters to adjust, i.e. the e⁄ects on wages of replacing (cid:11)1((cid:30)) with [1+F rd;l((cid:30))(cid:28)1 (cid:27)=(cid:27)fx n(cid:28)1 (cid:27)] in l(cid:0) l(cid:0) the BVP associated with N1. Little can be said about these e⁄ects at this level of generality. In (cid:133)gure (cid:0) (cid:1) 2, which illustrates only one of the many possibilities, the impact of this channel is given by the di⁄erence between N1 and Nl, the dashed and solid blue lines, respectively. In this example, the weight of some middle-productivity (cid:133)rms among exporters in the post-liberalization economy is larger than in the ancillary N1-economy. Then, the change in the set of exporters drives up the relative demand for some middle-skill workers, pushing up their wages relative to those of workers with lower and higher skill levels. That said, the impact of this channel could take other forms depending on the distribution function of (cid:133)xed export costs, F, including a pervasive rise and a pervasive decline in wage inequality. Moreover, the e⁄ects of this channel can be strong enough to o⁄set the impact of the other two channels in some parts of the wage distribution, as shown by the crossing of Nh and Nl in (cid:133)gure 2. Proposition 3.iii presents a set of su¢ cient conditions on the distribution of (cid:133)xed exports costs, F, that guarantee that a trade liberalization always leads to a pervasive rise in wage inequality. When the condition on the function (cid:17)F is satis(cid:133)ed, reducing variable trade costs while keeping the activity cuto⁄ 1 unchanged in the BVP of the open economy (that allows the set of exporters to change) always leads to pervasively higher wage inequality. In addition, when the condition on (cid:17)F is satis(cid:133)ed, increasing the 0 activitycuto⁄whilekeepingvariabletradecostsconstantinsaidBVPalsoleadstoapervasiveriseinwage dispersion. Accordingly, when both conditions are met, wage inequality increases pervasively following a liberalization, as the e⁄ect on relative wages of changes in the set of exporters (extensive-margin channel) never o⁄sets the combined impact of the selection-into-activity and intensive-margin channels. Although these restrictions on F may appear very restrictive to some readers, one should bear in mind that they are su¢ cient conditions under all paramerizations of the model.39 Regardingtheimpactofatradeliberalizationonthelevel of wages, theanalysisandconclusionsofthe previous section also apply to this case. A liberalization increases real income and average real wages, but the least productive workers in the economy could see their real wage decline in some parameterizations of the model. 38Formally, the result follows from a direct application of lemma 4.i in the appendix, with (cid:11)1 taking the role of (cid:11)a in the lemma. 39ForaParetodistribution,theconditionon(cid:17)F isalwayssatis(cid:133)ed,whilethaton(cid:17)F issatis(cid:133)edwhentheshapeparameter 0 1 is small enough. Moroever, a su¢ ciently small shape parameter typically precludes the crossing of the matching function even when the condition on (cid:17)F is not satis(cid:133)ed. 1 23

5.3 Trade Openness and Wage Dispersion in Alternative Frameworks The three-channel decomposition of the e⁄ects of higher trade openness on wage inequality described above can be a useful tool to analyze di⁄erences in the implications of alternative frameworks in the literature. For illustration purposes, I compare the e⁄ects of trade on wage inequality in the no-freeentry model in this paper with those in Helpman, Itskhoki, and Redding (2010), henceforth HIR. In the HIR model, (cid:133)rms screen workers to improve the composition of their labor forces as worker ability is not directly observable. As larger (cid:133)rms have higher returns from screening, they do so more intensively and have workforces of higher average ability than smaller (cid:133)rms. This mechanism generates a wage-size premium, implying that both productivity and exporting positively a⁄ect the average wages paid by a (cid:133)rm. In this setting, HIR show that wage inequality increases after an economy opens to trade only when there is selection into exporting (only some (cid:133)rms export), but is unchanged when all (cid:133)rms become exporters. Through the lenses of the decomposition analysis above, the intensive-margin channel of trade does not a⁄ect wage dispersion in the HIR model, as changes in the activity cuto⁄ do not modify the relative size of (cid:133)rms. In addition, trade a⁄ects wage inequality through the extensive-margin channel only when it changes the relative size of (cid:133)rms in the economy, i.e. only when some but not all (cid:133)rms export. In contrast, trade always leads to higher wage inequality in the no-free-entry model of this section. Although trade does not a⁄ect wage inequality through the extensive-margin channel when all (cid:133)rms export (as in HIR), it always drives up wage dispersion through the intensive-margin channel. 6 The Free-Entry Model In the model outlined above, the mass of (cid:133)rms in the industry is (cid:133)xed at an exogenous level. Although this assumption may be a good approximation to the (cid:133)rm-entry dynamics in the short-run, it does not capture the change in the number of (cid:133)rms through endogenous entry and exit over time. In this section, I relax this assumption by allowing (cid:133)rms to enter the industry for a cost, making the mass of (cid:133)rms in the industry, M, an additional endogenous variable. Speci(cid:133)cally, I assume that there is an unbounded pool of prospective (cid:133)rms that can enter the industry by incurring a (cid:133)xed entry cost of feV(s) units of each skill s S: Accordingly, the aggregate expenditure on entry costs is Mfe when a mass M of (cid:133)rms enters 2 the industry.40 Upon entry, (cid:133)rms obtain their productivity as independent draws from the distribution G, as explained in section 2.2. All the other primitives of the model remain unchanged. Below, I brie(cid:135)y describetheopeneconomyequilibriuminthefree-entrymodel, relegatingtotheappendixamoredetailed exposition. The new assumptions above do not a⁄ect the basic structure of the model described in section 2, so equations (1)-(7) continue to hold. Conditional on the mass of (cid:133)rms, M, the equilibrium analysis in section 4 applies almost unchanged to the free-entry model, with the caveat that equilibrium conditions now re(cid:135)ect the labor demand derived from the presence of (cid:133)xed entry costs, i.e. L must be replaced with L feM throughouttheanalysis. Thenewfree-entryassumptionimpliesthat,inequilibrium,prospective (cid:0) 40The numeraire assumption, w=1, yileds M sfew(s)V (s)ds=Mfe. s R 24

entrantsmustbeindi⁄erentbetweenenteringandnotenteringtheindustry. Accordingly, expectedpro(cid:133)ts from entering the industry must equal the cost of entry, [1 G((cid:30) )] (cid:25)d+(cid:25)x = fe, where (cid:25)d and (cid:25)x are, (cid:3) (cid:0) respectively, the average domestic and export pro(cid:133)ts of active (cid:133)rms.41 Per the optimal pricing rule, this (cid:2) (cid:3) free-entry condition can be written as follows, rd((cid:30))(cid:28)1 (cid:27) (cid:30) rd((cid:30)) f g((cid:30))d(cid:30)+ (cid:30) (cid:27)fx (cid:0) n rd((cid:30))(cid:28)1 (cid:0) (cid:27) fxy dF (y)g((cid:30))d(cid:30) = fe. (24) (cid:27) (cid:0) (cid:27) (cid:0) Z (cid:30)(cid:3) (cid:20) (cid:21) Z (cid:30)(cid:3) Z0 (cid:20) (cid:21) The last equation completes the description of the open-economy equilibrium in the free-entry model, prompting the formal de(cid:133)nition in appendix A.5.2. Lemma6oftheappendixprovidesacharacterizationofthefree-entryequilibriumoftheopeneconomy that is analogous to the one given in section 4.1 for the no-free-entry model. In particular, given the activity cuto⁄, (cid:30) , the price, domestic-revenue and inverse-matching functions, p;rd;H , solve a BVP (cid:3) that di⁄ers from that of the no-free-entry model in lemma 3.iii. only in that L is replaced by L feM (cid:8) (cid:9) (cid:0) in the equation de(cid:133)ning the slope of the inverse-matching function. Moreover, the discussion in section 4.2 implies that conditional on (cid:30) , the BVPs of the no-free-entry and free-entry models have the same (cid:3) parametrization in terms of the general BVP (22), so they share the same solution functions rd and H. The equilibrium value for (cid:30) in free-entry model is pinned down by the free entry condition (24).42 (cid:3) The observations above have important implications. First, all the conclusions reached in section 4.2 about the dependence of rd;H on the activity cuto⁄(cid:30) continue to hold in the free-entry model. (cid:3) Accordingly, many results, such as the existence and uniqueness of the equilibrium in the free-entry (cid:8) (cid:9) model, can be derived in a similar way.43 Second, the only relevant di⁄erence between the no-free-entry and free-entry models regarding the determination of the equilibrium matching function is given by the equation that pins down the activity cuto⁄in these models, equations (21) and (24), respectively. In the remainder of this section I explore how this di⁄erence a⁄ects the impact of increased trade openness on wage inequality. 6.1 Autarky vs. Trade in the Free-entry Model In contrast to the case of the no-free entry model, trade may lead to a rise or a fall in the activity cuto⁄ in the free-entry model, with ambiguous e⁄ects on wage inequality through the selection-into-activity channel. Despite this ambiguity at this level of generality, the overall e⁄ects of trade on wage dispersion can lead to only two situations, a pervasive increase in wage inequality or wage polarization, as formally stated in proposition 4. 41Note that (cid:25)x is not the average export pro(cid:133)ts among exporters, but among all active (cid:133)rms. 42This is the case because (cid:30)(cid:3) and rd are the only endogenous variables appearing in equation (24). Note that using the analog of equation (21) for the free-entry model to determine the activity cuto⁄(cid:30)(cid:3) would only give us (cid:30)(cid:3) as a function of the endogenous mass of (cid:133)rms M. 43Asrd((cid:30))dependsnegativelyontheactivitycuto⁄,theleft-handsideofequation(24)isstrictlydecreasingin(cid:30)(cid:3),implying that there is unique free-entry equilibrium if entry costs are not too high. 25

Proposition 4 Let (cid:30) ;Na and (cid:30) ;N(cid:28) be the activity cuto⁄s and matching functions corresponding (cid:3)a (cid:3)(cid:28) f g f g to the free-entry equilibrium of the closed and open economies, respectively. Then (cid:30) could be lower or (cid:3)(cid:28) higher than (cid:30) depending on the model(cid:146)s parameters. (cid:3)a (i) If (cid:30) (cid:30) ; then N(cid:28) (s) > Na(s) on s (s;s), so opening to trade leads to pervasively higher wage (cid:3)(cid:28) (cid:3)a (cid:21) 2 inequality. The selection-into-activity channel leads to a pervasive rise (no change) in wage inequality if (cid:30) > (=)(cid:30) . The extensive-margin channel always leads to a pervasive rise in wage inequality. (cid:3)(cid:28) (cid:3)a (ii) If (cid:30) < (cid:30) ; then N(cid:28) (s) and Na(s) intersect exactly once on (s;s), so opening to trade leads to wage (cid:3)(cid:28) (cid:3)a polarization. The selection-into-activity and extensive-margin channels lead, respectively, to pervasively lower and pervasively higher wage inequality. Atrade-induceddeclineintheactivitycuto⁄isatheoreticalpossibilitythathasimportantimplications for the e⁄ect of trade on wage inequality. As this possibility is not present in the no-free-entry model in this paper or even in standard free-entry models with homogeneous workers, such as Melitz (2003), I start the analysis by discussing the elements of the free-entry model above that allow for such an occurrence. The di⁄erent equilibrium conditions that determine the activity cuto⁄in the free-entry and no-freeentry models in this paper imply that trade can lead to a decline in said cuto⁄ in the former but not in the latter. These di⁄erences are better understood by comparing the impact that trade has on these equilibrium conditions when the set of active (cid:133)rms and the revenue of the least productive ones are assumed to remain unchanged, rd((cid:30) ) = (cid:27)f: As discussed in section 5, in this scenario, trade leads to a (cid:3)a rise in the implied total wages paid to production and non-production workers, as total (cid:133)rms(cid:146)revenue and (cid:133)xed export costs increase. Accordingly, equation (21) implies that a higher activity cuto⁄is required in the open economy of the no-free-entry model. In contrast, in the free-entry model, total (cid:133)rms(cid:146)revenue and (cid:133)xed export costs enter with opposite signs on the left-hand side of the free-entry condition (24), with an ambiguous net e⁄ect, so a lower activity cuto⁄may be required in the open economy. Relative to standard free-entry models with homogeneous workers, a trade-induced decline in the activity cuto⁄is possible in the free-entry model because of the endogenous changes in the matching of heterogeneous workers to (cid:133)rms.44 As before, it is instructive to compare the impact that trade has on the free-entry condition in these models under the same assumptions described in the previous paragraph. In such a scenario, trade increases export pro(cid:133)ts from zero (in autarky) to some strictly positive number in both models. With domestic pro(cid:133)ts remaining unchanged in the homogeneous-workers model (before adjusting the activity cuto⁄), average/expected pro(cid:133)ts necessarily increase, so the free-entry condition requires a higher activity cuto⁄in the open economy. In contrast, in the free-entry model of this paper, trade may lead to a decline in aggregate pro(cid:133)ts due to changes in the matching function. Speci(cid:133)cally, as the matching function N shifts up (H shifts down) in the scenario considered, domestic revenues and pro(cid:133)tsdeclinefor(cid:133)rmswithproductivityabove(cid:30) .45 Forsomeparametervalues, thedeclineinaggregate (cid:3)a domestic pro(cid:133)ts more than o⁄sets the rise in export pro(cid:133)ts, so the free-entry condition (24) requires a lower activity cuto⁄in the open economy. 44The stochastic modeling of (cid:133)xed costs is another di⁄erence between the free-entry model in this paper and standard Melitz-type models. However, said di⁄erence alone cannot produce a trade-induced declined in the activity cuto⁄. 45See discussion associated to equation (14). 26

Figure 3: The E⁄ects of Trade on the Matching Function in Free-entry Model Note: The solid red and blue lines represent, respectively, the matching functions of the closed (Na) and open (N(cid:28)) economies. The dashed red line depicts the matching function of an ancillary autarkic economy (N0)whichisobtainedbysolvingtheBVPinlemma1.iiwith(cid:30)(cid:3)=(cid:30)(cid:3)(cid:28) . Thedi⁄erencebetweenNa andN0 capturestheimpactoftheselection-into-activitychannel,whilethedi⁄erencebetweenN0 andN(cid:28) captures the impact of the extensive-margin channel. The (cid:133)gure depicts the case in which trade induces a decline in the activity cuto⁄. Per proposition 4, conditional on its impact on the activity cuto⁄, trade has a unique qualitative e⁄ect on the dispersion of wages, with an unambiguous e⁄ect through the selection-into-activity and extensivemargin channels. The case in proposition 4.i, (cid:30) (cid:30) , is essentially the same situation considered in (cid:3)(cid:28) (cid:3)a (cid:21) section 5.1 for the no-free-entry model. If (cid:30) > (cid:30) , then the situation is identical to that depicted in (cid:3)(cid:28) (cid:3)a (cid:133)gure 1, so the corresponding analysis applies here as well. When (cid:30) = (cid:30) , the only di⁄erence is that the (cid:3)(cid:28) (cid:3)a selection-into-activity channel has no e⁄ect on wage dispersion. The case in proposition 4.ii, (cid:30) < (cid:30) , requires some additional explanation. As I discuss in the (cid:3)(cid:28) (cid:3)a appendix, the matching function of the open economy, N(cid:28), cannot remain completely below that of the closed economy, Na, on [s;s). Otherwise, per lemma 2.ii, expected domestic pro(cid:133)ts in the open economy would be strictly higher than in autarky, implying a violation of the free-entry condition (24). Then, N(cid:28) and Na must intersect at least once on (s;s). Moreover, adapting the analysis of the extensive-margin channel in section 5.1 to assess the relative position of Na and N(cid:28) to the right of the (cid:133)rst intersection, it can be shown that Na must remain below N(cid:28) there, so the matching functions must intersect exactly once on (s;s).46 The situation is depicted in (cid:133)gure 3, where the solid red and blue lines represent Na and N(cid:28), respectively. Asbefore, thedashedredlineisthematchingfunctionofanancillaryautarkiceconomy, N0, that is obtained by changing the activity cuto⁄in the BVP corresponding to Na from (cid:30) to (cid:30) . As (cid:3)a (cid:3)(cid:28) 46Formally, to the right of the (cid:133)rst intersection point, the matching functions of the closed and open economies can be conceivedassolutionstoparticularparameterizationsofthegeneralBVP(22)withK =0thatdi⁄eronlyintheparameter 1 function(cid:11)((cid:30)),whichisconstantintheformerandincreasinginthelatter. Theresultthenfollowsfrom adirectapplication of lemma 4.i in the appendix. 27

discussed in section 5.1, the e⁄ects of trade on wage inequality through the selection-into-activity and extensive-margin channels are captured, respectively, by the di⁄erence between the pairs Na;N0 and N0;N(cid:28) . While the selection-into-activity channel pervasively reduces wage inequality, the extensive- (cid:8) (cid:9) margin channel pervasively increases it, with the former channel dominating to the left of the interior (cid:8) (cid:9) intersection point of Na and N(cid:28); and the latter dominating to the right. As a result, workers with skill level corresponding to this (interior) intersection point see their wages decline relative to those of all other workers, i.e. trade leads to wage polarization. Turning to the e⁄ects of trade on the level of real wages, the results obtained for the no-free-entry model generally go through in the free-entry model. First, the average real wage is always higher in the open economy. As before, the result follows from the (constrained) e¢ ciency of the equilibrium. Second, trade may induce a decline in the real wage of the least skilled workers in the economy, although in the free-entry model this possibility is fully determined by the impact of trade on the activity cuto⁄. As the free-entry condition implies that the economy(cid:146)s total income and expenditure is given by total labor income, E = wL, rearranging equation (23) yields wi(s)=Pi = ((cid:27) (cid:0) (cid:27) 1) A(s;(cid:30) (cid:3)i )[L=(cid:27)f](cid:27) (cid:0) 1 1 for i = a;(cid:28), i.e. trade rises the real wage of even the least skilled workers in the economy if and only if it rises the activity cuto⁄. Note that this observation, together with proposition 4, implies that no worker loses from trade only if wage inequality increases pervasively. 6.2 Trade Liberalization in the Free-entry Model The e⁄ects of a trade liberalization on the wage distribution in the free-entry-model can derived by resorting to the results in propositions 2 to 4, as they largely cover the range of possible outcomes in this case. For the same reasons behind the corresponding result in proposition 4, a trade liberalization could lead to a rise or a fall in the activity cuto⁄. If the activity cuto⁄ increases, then the situation is identical to that considered in proposition 3 so all the results go through. If the activity cuto⁄declines, then the pre- and post-liberalization matching functions must intersect at least once on (s;s) to avoid a violation of the free entry condition as discussed in the case of proposition 4.ii. However, in the case of a trade liberalization, more than one crossing on (s;s) cannot be ruled out even when the conditions on the functions (cid:17)F (t;(cid:21)) and (cid:17)F (t;(cid:21)) in proposition 3are satis(cid:133)ed. 0 1 7 Conclusion In this paper, I develop a general equilibrium trade model with a large number of skill-groups that emphasizes the within-industry reallocation of workers across heterogenous (cid:133)rms as the mechanism through which international trade a⁄ects the wage distribution. Strong complementarities in production between worker skill and (cid:133)rm productivity lead to positive assortative matching in equilibrium, while heterogeneous (cid:133)xed export costs imply that the productivity distributions of exporters and non-exporters overlap. As a result, the cross-sectional structure of model captures several features of the data identi(cid:133)ed by the trade and labor literatures. More productive (cid:133)rms tend to be larger, have workforces of higher average ability and pay higher average wages, and there is an imperfect correlation between (cid:133)rm size, wages and 28

export status. I consider two versions of the model corresponding to two alternative assumptions about (cid:133)rm entry, no free entry a-lÆ Chaney (2008) and free entry a-lÆ Melitz (2003). I use the model to study the theoretical e⁄ects of higher trade openness on the wage distribution. In the no-free-entry model, opening to trade always leads to pervasively higher wage inequality. By contrast, a trade liberalization necessarily increases inequality at the lower end of the wage distribution, but may reduce it elsewhere. In the free-entry model, opening to trade leads to pervasively higher inequality (wage polarization) if low-productivity (cid:133)rms exit (enter) the market. In the case of a trade liberalization, all the previous possibilities could arise without additional restrictions on primitives. In all cases, higher trade openness never leads to a pervasive decline in wage inequality. In addition, to gain more insight into the elements driving of these results, I decompose the overall impact of trade on the wage distribution into those associated with the selection-into-activity, the intensive-margin and extensive-margin channels of trade. The analysis highlights the importance of new exporters (extensive margin) in shaping the aggregate relative demand for skills and relative wages, a channel that is controlled by the distribution of (cid:133)xed export costs in the model. Finally, I also contribute methodologically to the analysis of assignment problems. In addition to presenting existence and uniqueness results for a general BVP that encompasses those in this paper and others in the literature, I derive general results about the dependence of the solution to this BVP on parameters. These results can be used to analyze comparative statics exercises beyond those considered in this paper. References Acemoglu, D. (2002). Technical Change, Inequality, and the Labor Market. Journal of Economic Literature 40(1), 7(cid:150)72. Agarwal,R.andD.O(cid:146)Regan(2008a).An Introduction to Ordinary Di⁄erential Equations.Universitext. Springer New York. Agarwal, R. and D. O(cid:146)Regan (2008b). Ordinary and Partial Di⁄erential Equations: With Special Functions, Fourier Series, and Boundary Value Problems. Universitext. Springer New York. Amiti, M. and D. R. Davis (2012). Trade, Firms, and Wages: Theory and Evidence. The Review of Economic Studies 79(1), 1(cid:150)36. Antr(cid:224)s, P., L. Garicano, and E. Rossi-Hansberg (2006). O⁄shoring in a Knowledge Economy. The Quarterly Journal of Economics 121(1), 31(cid:150)77. Atkinson, A.B.(1970).OntheMeasurementofInequality.Journal of Economic Theory 2(3), 244(cid:150)263. Attanasio,O.,P.K.Goldberg,andN.Pavcnik(2004).TradeReformsandWageInequalityinColombia. Journal of Development Economics 74(2), 331(cid:150)366. Autor, D. H., L. F. Katz, and M. S. Kearney (2008). Trends in U.S. Wage Inequality: Revising the Revisionists. The Review of Economics and Statistics 90(2), 300(cid:150)323. 29

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A Theoretical Appendix A.1 Section 3 A.1.1 Proof of Lemma 1 ExistenceofamatchingfunctionN. Istartbyde(cid:133)ningsomenotation. LetS((cid:30)) s S : l(s;(cid:30)) > 0 (cid:17) f 2 g andlet(cid:8)(s) = (cid:30) (cid:30) ;(cid:30) : l(s;(cid:30)) > 0 . Toclarifytheexpositionofthispartoftheproof,Iwillproceed (cid:3) 2 in a series of steps. (cid:8) (cid:2) (cid:3) (cid:9) STEP 1: (cid:8)(s) = for all s S and S((cid:30)) = for all (cid:30) (cid:30) ;(cid:30) . (cid:3) 6 ; 2 6 ; 2 The full employment condition (11) and V (s) > 0 directly imply (cid:8)(s) = for all s S. Now (cid:2) (cid:3) 6 ; 2 suppose that we have an equilibrium in which there is (cid:30) (cid:30) ;(cid:30) such that S((cid:30)) = . Then from (3) we (cid:3) 2 ; have q((cid:30)) = 0 and this is incompatible with the demand g (cid:2) iven i (cid:3) n (1), since for any p((cid:30)) R+ we have 2 q((cid:30)) > 0. Then in any equilibrium we must have S((cid:30)) = . 6 ; STEP 2: S(:) and (cid:8)(:) satisfy the following properties: (i) if s S((cid:30)), s S (cid:30) and (cid:30) > (cid:30); then 0 0 0 2 2 s s; and (ii) if (cid:30) (cid:8)(s), (cid:30) (cid:8)(s) and s > s ,then (cid:30) (cid:30). 0 0 0 0 0 (cid:0) (cid:1) (cid:21) 2 2 (cid:21) (i) Suppose that this is not true and so let s < s. Notice that (7) implies that s S((cid:30)) if and only 0 2 if s argmin w(z)=A(z;(cid:30)). Then w(s)=A(s;(cid:30)) w(s)=A(s;(cid:30)). In a similar way, s S (cid:30) z 0 0 0 0 2 (cid:20) 2 implies w(s)=A s;(cid:30) w(s)=A s;(cid:30) . Combining both inequalities we get A s;(cid:30) A(s;(cid:30)) 0 0 0 0 0 0 (cid:0) (cid:1) (cid:20) (cid:20) A(s;(cid:30))A s;(cid:30) , but this contradicts the log-supermodularity of A (remember that (cid:30) > (cid:30) and s > s). 0 0 (cid:0) (cid:1) (cid:0) (cid:1) 0(cid:0) (cid:1) 0 Then we must have s s. (cid:0) (cid:1) 0 (cid:21) (ii) Suppose that this is not true and so let (cid:30) < (cid:30). Then (cid:30) (cid:8)(s) s S((cid:30)) and (cid:30) (cid:8)(s) 0 0 0 2 ) 2 2 ) s S (cid:30) . Then we have (cid:30) < (cid:30); s S((cid:30)), s S (cid:30) and by STEP 2.i this implies s s, which is a 0 0 0 0 0 0 2 2 2 (cid:21) contradiction. Then we must have (cid:30) (cid:30). (cid:0) (cid:1) 0 (cid:0) (cid:1) (cid:21) STEP 3: (i) S((cid:30)) is an interval for all (cid:30) ;(cid:30) and S((cid:30)) S (cid:30) 1 for any two di⁄erent (cid:30);(cid:30) (cid:3) 0 0 j \ j (cid:20) 2 (cid:30) ;(cid:30) ; (ii) (cid:8)(s) is an interval for all s S and (cid:8)(s) (cid:8)(s) 1 for any two di⁄erent s;s S. (cid:3) (cid:2) (cid:3) 0 (cid:0) (cid:1) 0 2 j \ j (cid:20) 2 (i) I will prove the (cid:133)rst part by contradiction. Suppose there is (cid:30) (cid:30) ;(cid:30) such that S((cid:30)) is not an (cid:2) (cid:3) (cid:3) 2 interval. Then there we can (cid:133)nd s;s S((cid:30)), with s < s, and some s (s;s) such that s = S((cid:30)). 0 0 (cid:2)00 (cid:3) 0 00 2 2 2 FromSTEP1weknowthat(cid:8)(s )isnonemptyandsotheremustbea(cid:30) (cid:30) ;(cid:30) suchthats S (cid:30) . 00 00 (cid:3) 00 00 2 2 We have only two possibilities: (cid:30) > (cid:30) and (cid:30) < (cid:30). If (cid:30) > (cid:30), then STEP 2.i implies s s which is a 00 00 00 (cid:2) (cid:3) 00 0 (cid:0) (cid:1) (cid:21) contradiction. If (cid:30) < (cid:30), then STEP 2.i implies s s which is also a contradiction. Then S((cid:30)) is an 00 00 (cid:21) interval for all (cid:30) ;(cid:30) . (cid:3) LetusnowshowthatS((cid:30)) S (cid:30) isatmostasingletonandasbeforeIwillproceedbycontradiction. (cid:2) (cid:3) 0 \ Suppose that the claim is not true. Then there must be (cid:30);(cid:30) (cid:30) ;(cid:30) such that s;s S((cid:30)) S (cid:30) with (cid:0) (cid:1) 0 (cid:3) 0 0 2 2 \ s = s. Without loss of generality assume (cid:30) > (cid:30) and s > s. Then we have (cid:30) > (cid:30); s S((cid:30)); s S (cid:30) 0 0 0 (cid:2) (cid:3) 0 0 (cid:0) (cid:1) 0 6 2 2 and so STEP 2.i implies s s which is a contradiction. This concludes part i. 0 (cid:0) (cid:1) (cid:21) (ii) I prove this by contradiction. Suppose there is s S such that (cid:8)(s) is not an interval. Then 2 there we can (cid:133)nd (cid:30);(cid:30) (cid:8)(s), with (cid:30) < (cid:30), and some (cid:30) (cid:30);(cid:30) such that (cid:30) = (cid:8)(s). From STEP 1 0 0 00 0 00 2 2 2 we know that S (cid:30) is nonempty and so there must be a s S such that (cid:30) (cid:8)(s ). We have only 00 0(cid:0)0 (cid:1) 00 00 2 2 two possibilities: s > s and s < s. If s > s, then STEP 2.ii implies (cid:30) (cid:30) which is a contradiction. (cid:0) 00(cid:1) 00 00 00 0 (cid:21) 32

If s < s, then STEP 2.ii implies (cid:30) (cid:30) which is also a contradiction. Then (cid:8)(s) is an interval for all 00 00 (cid:21) s S. 2 Let us now show that (cid:8)(s) (cid:8)(s) is at most a singleton and as before I will proceed by contradiction. 0 \ Suppose that the claim is not true. Then there must be s;s S such that (cid:30);(cid:30) (cid:8)(s) (cid:8)(s) with 0 0 0 2 2 \ (cid:30) = (cid:30). Without loss of generality assume (cid:30) > (cid:30) and s > s. Then we have s > s; (cid:30) (cid:8)(s); (cid:30) (cid:8)(s) 0 0 0 0 0 0 6 2 2 and so STEP 2.ii implies (cid:30) (cid:30) which is a contradiction. This concludes part ii. 0 (cid:21) STEP 4: S((cid:30)) is a singleton for all but a countable subset of (cid:30) ;(cid:30) . (cid:3) I show this by contradiction. Let (cid:8) = (cid:30) (cid:30) ;(cid:30) : S((cid:30)) > 1 and suppose (cid:8) is uncountable. 0 (cid:3) (cid:2) (cid:3) 0 2 j j Notice that STEP 3.i implies that S((cid:30)) is a nondegenerate interval for all (cid:30) (cid:8) . Then for each (cid:30) (cid:8) (cid:8) (cid:2) (cid:3) (cid:9) 0 0 2 2 we can pick a rational skill r((cid:30)) intS((cid:30)) and given that S((cid:30)) S (cid:30) 1 for any two di⁄erent (cid:30);(cid:30) 0 0 2 j \ j (cid:20) we must have r((cid:30)) = r (cid:30) 0 when (cid:30) = (cid:30) 0 . Then the function r : (cid:8) 0 (cid:0) (cid:1)Q S de(cid:133)ned before is injective 6 6 ! \ and so it is a contradiction since (cid:8) is uncountable. (cid:0) (cid:1) 0 STEP 5: (cid:8)(s) is a singleton for all but a countable subset of S. This follows from the same arguments as in STEP 4. STEP 6: S((cid:30)) is a singleton for all (cid:30) (cid:30) ;(cid:30) . (cid:3) 2 I proceed by contradiction. Suppose there is (cid:30) (cid:30) ;(cid:30) such that S((cid:30)) is not a singleton. Then (cid:2) (cid:3) (cid:3) 2 STEP 3.i implies that S((cid:30)) is an interval. By STEP 5 (cid:8)(s) = (cid:30) for all but a countable subset of S((cid:30)). (cid:2) (cid:3) f g Then l(s;(cid:30)) = V (s)(cid:14) 1 I for almost all s S((cid:30)) S((cid:30)) (cid:0) 2 (cid:2) (cid:3) where (cid:14) is the Dirac delta function. But then q((cid:30)) = A(s;(cid:30))l(s;(cid:30))ds = , and this is incoms S((cid:30)) 1 2 patible with an equilibrium (as de(cid:133)ned above). In other words, if S((cid:30)) is not a singleton, then we would R have a positive mass of workers producing in a single type of productivity (cid:133)rms which are of mass zero, and this cannot happen in equilibrium. STEP 7: (cid:8)(s) is a singleton for all s S: 2 I proceed by contradiction. Suppose there is an s S such that (cid:8)(s) is not a singleton. Then STEP 2 3.ii implies that (cid:8)(s) is an interval. By STEP 6 S((cid:30)) = s for all (cid:30) (cid:8)(s). Now let (cid:8) (cid:8)(s) be the 0 f g 2 (cid:18) set of productivity levels that are assign a strictly positive conditional47 mass of s-skill workers. I will showthat(cid:8) isatmostcountable. Thetotalconditionalmassofs-skillworkersallocatedtoproductivities 0 in (cid:8) can be expressed as 0 (cid:30) l(s;(cid:30))d(cid:30) = k((cid:30))(cid:14)[1 I ]d(cid:30) (cid:0) (cid:8)0 Z(cid:8)0 Z (cid:30)(cid:3) where (cid:14) is the Dirac delta function and k((cid:30)) is the conditional mass of worker at productivity (cid:30) 2 (cid:8) . Notice that (cid:8) = (cid:30) (cid:8) : k((cid:30)) 1=n and because of the full employment condition each 0 0 [ 1n=1f 2 0 (cid:21) g 47Remeberthatthemassofworkersofaparticularskillsiszero. However,conditionalontheskill,wecanthinkofl(s;(cid:30)) as the density that represents the distribution of workers with skill s among the (cid:133)rms indexed by the productivity level. Then conditional on skill s, all s-skill workers have a total mass V (s)>0. Then I say that a set A (cid:30)(cid:3);(cid:30) has possitive (cid:18) conditional mass if (cid:2) (cid:3) l(s;(cid:30))d(cid:30)>0: Z(cid:30) 2 A 33

(cid:30) (cid:8) : k((cid:30)) 1=n must be (cid:133)nite. Then (cid:8) is at most countable. This means a zero conditional mass 0 0 f 2 (cid:21) g of s-skill workers are allocated to almost all (cid:30) (cid:8)(s), which in turn means that q((cid:30)) = 0 for almost all 2 (cid:30) (cid:8)(s). However this is incompatible with equilibrium since for any p((cid:30)) R+ , the demand of variety 2 2 (cid:30) (according to (1)) is strictly positive. Steps1,6,7implythatthereisabijectionN : S (cid:30) ;(cid:30) suchthatl(s;(cid:30)) > 0ifandonlyif(cid:30) = N (s) (cid:3) ! and by STEP 2 it must be strictly increasing. (cid:2) (cid:3) Conditions i-iii. Consider a no-free-entry equilibrium of the closed economy with activity cuto⁄ (cid:30) , wage schedule w(s), price function p((cid:30)), domestic revenue function rd((cid:30)) and matching function (cid:3) N (s). The cost minimization condition (5) and the existence of the matching function N imply that w(s) w(s+ds) w(s+ds) w(s) s = argmin w(z)=A(z;N (s)), so and . Combining z A(s;N(s)) (cid:20) A(s+ds;N(s)) A(s+ds;N(s+ds)) (cid:20) A(s;N(s+ds)) these inequalities yields A(s+ds;N (s)) w(s+ds) A(s+ds;N (s+ds)) ; A(s;N (s)) (cid:20) w(s) (cid:20) A(s;N (s+ds)) from which we can obtain the di⁄erentiability of w(s) and equation (12), after taking logs, dividing by ds and taking limits as ds 0.48 This proves condition i. ! The pricing rule (7) and the existence of H imply (cid:30) = argmax p((cid:13))A(H((cid:30));(cid:13)), so (cid:13) p((cid:30))A(H((cid:30));(cid:30)) p((cid:30)+d(cid:30))A(H((cid:30));(cid:30)+d(cid:30)); (cid:21) p((cid:30)+d(cid:30))A(H((cid:30)+d(cid:30));(cid:30)+d(cid:30)) p((cid:30))A(H((cid:30)+d(cid:30));(cid:30)); (cid:21) Combining both inequalities yields A(H((cid:30));(cid:30)+d(cid:30)) p((cid:30)) A(H((cid:30)+d(cid:30));(cid:30)+d(cid:30)) : A(H((cid:30));(cid:30)) (cid:20) p((cid:30)+d(cid:30)) (cid:20) A(H((cid:30)+d(cid:30));(cid:30)) The di⁄erentiability of p((cid:30)) and condition (13) are obtained taking logs, dividing by ds and taking limits as ds 0 in the last expression. Having established the di⁄erentiability of p((cid:30)), the di⁄erentiability of ! rd((cid:30)) and condition (14) follow from the de(cid:133)nition of rd((cid:30)) in (8). The pricing rule (7) implies that the variable production cost of a (cid:133)rm equals a fraction ((cid:27) 1)=(cid:27) (cid:0) of its revenue. Then, the total wages paid to production workers employed at (cid:133)rms with productivity weakly lower than (cid:30) must be equal to a fraction ((cid:27) 1)=(cid:27) of the total revenue generated by those (cid:133)rms, (cid:0) H((cid:30)) (cid:30) w(s)V (s)[L (cid:0) fM]ds = ((cid:27) (cid:0) (cid:27) 1) rd (cid:30) 0 g (cid:30) 0 d(cid:30) 0 M for all (cid:30) 2 (cid:30) (cid:3) ;(cid:30) : (25) s (cid:30) Z Z (cid:0) (cid:1) (cid:0) (cid:1) (cid:2) (cid:3) Due to the continuity of the revenue function rd((cid:30)), the right hand side of (25) is a di⁄erentiable function of the limit of integration (cid:30). Then, the left hand side must also be a di⁄erentiable function of (cid:30), which together with the continuity of V (s) and w(s), implies that H((cid:30)) is di⁄erentiable. Di⁄erentiating (25) 48The limits are well de(cid:133)ned since all the functions involved are continuous. 34

with respect to (cid:30) and using the pricing rule (7) to substitute for the wage w(s) yields condition (15). Concluding the proof of condition ii, the boundary conditions on H follow from the de(cid:133)nition of the matching function, while the initial condition on rd((cid:30)) is just the the zero-pro(cid:133)t condition for (cid:133)rms (cid:133)rms with productivity (cid:30) . Finally, condition iii follows from equation (25), evaluated at (cid:30) = (cid:30), and the (cid:3) s numeraire assumption, w(s)V (s)ds = 1. s Let us turn to the su¢ cient conditions for an equilibrium stated in the last part of the lemma. R Suppose that (cid:30) ;p;rd;H satisfy conditions (ii)-(iii) and de(cid:133)ne N H 1, M [1 G((cid:30) )]M, w(s) (cid:3) (cid:0) (cid:3) (cid:17) (cid:17) (cid:0) (cid:17) (cid:27) 1A(s;N (s))p(N (s)), q((cid:30)) r((cid:30)) , and l(s;(cid:30)) V (s)[L fM](cid:14)((cid:30) N (s)); where (cid:14)(x) is the Dirac- (cid:0)(cid:27) (cid:8) (cid:9) (cid:17) p((cid:30)) (cid:17) (cid:0) (cid:0) delta function. In what follows I show that M;(cid:30) ;w;p;q;l is a no-free-entry equilibrium of the closed (cid:3) f g economy. The de(cid:133)nitions of w(s), M, and l(s;(cid:30)) above immediately imply that the pricing rule (7), condition (10) and the labor market clearing condition (11) are satis(cid:133)ed. The de(cid:133)nition of q((cid:30)) and equation (15) yield an expression for q((cid:30)) in terms of H and primitives of the model. The same expression is obtained computing the right hand sideof (3) using the labor allocationl(s;(cid:30))constructed here, socondition (3) is satis(cid:133)ed. The initial condition on the function rd((cid:30)) in point ii of the lemma implies that the zero-pro(cid:133)t condition (9) holds. Using the de(cid:133)nition of w above to substitute for p in equation (15), we arrive at (25) after rearranging and integrating on both sides. Evaluating (25) at (cid:30) = (cid:30) and using condition iii of the s lemma yields w(s)V (s)ds = 1, so the numeraire condition holds. Finally, the construction of q((cid:30)) s implies that the consumer(cid:146)s budget constraint is satis(cid:133)ed and, together with conditions (13) and (14), R (cid:27) implies q (cid:30) 0 =q((cid:30)) = p (cid:30) 0 =p((cid:30)) (cid:0) , so conditions (1) and (2) hold. This concludes the proof of the lemma. (cid:0) (cid:1) (cid:2) (cid:0) (cid:1) (cid:3) A.1.2 Matching function and Lorenz dominance Consider two economies A and B with matching functions NA;NB and suppose that NB(s) > NA(s) for all s [s ;s ] [s;s]. As discussed in the main text, the strict log-supermodularity implies 0 1 2 (cid:18) wA(s)=wA(s) < wB(s)=wB(s), for all s > s in [s ;s ]. 0 0 0 0 1 In this context, the poorest (cid:26) fraction of workers in the interval [s ;s ] is associated with a skill s((cid:26)) 0 1 given by s((cid:26)) s1 (cid:26) = V (s)ds V (s)ds: Z s0 , Z s0 The Lorenz Curve is then s((cid:26)) w(s) L((cid:26)) s((cid:26)) w(s)V (s)ds s1 w(s)V (s)ds = s0 w(s((cid:26))) V (s)ds (cid:17) Z s0 , Z s0 s s 0 ((cid:26)) w w (s ( ( s (cid:26) ) )) RV (s)ds+ s s ( 1 (cid:26)) w w (s ( ( s (cid:26) ) )) V (s)ds R R It is readily seen that this implies that LA((cid:26)) > LB((cid:26)) for all (cid:26) (0;1): Finally, from Atkinson (1970) 2 we know that Lorenz dominance is equivalent to Normalized Second-Order Stochastic Dominance. 35

A.2 Section 4 A.2.1 De(cid:133)nition of Equilibrium De(cid:133)nition 2 A no-free-entry equilibrium of the open economy is an activity cuto⁄ (cid:30) , a mass of active (cid:3) (cid:133)rms M > 0; a mass of exporters Mx((cid:30)) > 0 for each productivity level (cid:30) (cid:30) , output functions (cid:3) (cid:21) qd;qx : [(cid:30) (cid:3) ;(cid:30)] R+ , labor allocations functions ld;lx : S [(cid:30) (cid:3) ;(cid:30)] R+ , a price function p : [(cid:30) (cid:3) ;(cid:30)] R+ ! (cid:2) ! ! and a wage schedule w : S R+ such that the following conditions hold, ! (i) consumers behave optimally, equations (1) and (2); (ii) (cid:133)rms behave optimally given their technology, equations (3), (7), (9), (10) and (18); (iii) goods and labor markets clear, equations (8), (17) and (19); (iv) the numeraire assumption holds, w = 1. A.2.2 Characterization of Equilibrium Lemma 3 Inano-free-entryequilibriumoftheopeneconomywithactivitycuto⁄ (cid:30) ((cid:30);(cid:30))thefollowing (cid:3) 2 conditions hold. (i) There exists a continuous and strictly increasing matching function N : S [(cid:30) ;(cid:30)], (with inverse (cid:3) ! function H) such that (i) ld(s;(cid:30))+lx(s;(cid:30)) > 0 if and only if N (s) = (cid:30), (ii) N (s) = (cid:30) , and N (s) = (cid:30). (cid:3) (ii) The wage schedule w is continuously di⁄erentiable and satis(cid:133)es (12) (iii) The price, domestic revenue and matching functions, p;rd;N ; are continuously di⁄erentiable. Given (cid:30) , the triplet p;rd;H solves the BVP comprising the system of di⁄erential equations {(13), (cid:3) (cid:8) (cid:9) (14), (20)} and the boundary conditions rd((cid:30) ) = (cid:27)f, H((cid:30) ) = s, H (cid:30) = s. (cid:8) (cid:9) (cid:3) (cid:3) (iv) The activity cuto⁄ (cid:30) and the revenue function rd satisfy (21). (cid:3) (cid:0) (cid:1) Moreover, if a number (cid:30) (cid:3) ((cid:30);(cid:30)), and functions p;rd : [(cid:30) (cid:3) ;(cid:30)] R+ and H : [(cid:30) (cid:3) ;(cid:30)] S satisfy the 2 ! ! conditions(iii)-(iv), thentheyare, respectively, theactivitycuto⁄, thepricefunction, thedomesticrevenue function, and the inverse of the matching function of a no-free-entry equilibrium of the open economy. Proof. Adapt arguments in the proof of lemma 1. A.2.3 Proof of Lemma 2 Existence. My approach to prove the existence of a solution to the BVP (22) relies on (cid:133)xed-point methods. The (cid:133)rst step in such an approach is to recast the BVP under consideration as a (cid:133)xed point problem of some functional operator. To that end, I de(cid:133)ne the functional (cid:9), mapping the space of continuous functions into itself, as follows (cid:30) (cid:30) h(t;y(t))e (cid:27) (cid:30) t 0 @lnA @ (y (cid:30) (u);u)du 1+F K 0 e ((cid:27) (cid:0) 1) (cid:30) t 0 @lnA @ (y (cid:30) (u);u) dt K 1 dt 0 R " R ! # (cid:9)(y)((cid:30)) s +[s s ] ; 0 1 0 R (cid:17) (cid:0) (cid:30) (cid:30) 1h(t;y(t))e (cid:27) (cid:30) t 0 @lnA @ (y (cid:30) (u);u)du 1+F K 0 e ((cid:27) (cid:0) 1) (cid:30) t 0 @lnA @ (y (cid:30) (u);u) dt K 1 dt 0 R " R ! # R (26) 36

where A(s ;(cid:30) ) V (s ) g(t) (cid:11)(t) h(t;y(t)) 0 0 0 : (27) (cid:17) A(y(t);t)V (y(t))g((cid:30) )(cid:11)((cid:30) ) 0 0 The following lemma states that the problem of (cid:133)nding a solution to the BVP (22) is equivalent to the problem of (cid:133)nding a (cid:133)xed point of the functional (cid:9). Claim 1 A function (cid:0) belongs to a triplet z;x;(cid:0) solving BVP (22) if and only if it is a (cid:133)xed point of f g the functional (cid:9) : C[(cid:30) ;(cid:30) ] C[(cid:30) ;(cid:30) ] de(cid:133)ned in (26)-(27). 0 1 0 1 ! Proof. Let us start with the "only if" part of the lemma. Let z;x;(cid:0) be a solution to the BVP (22). f g It can be shown that each of the functions in the solution triplet must be strictly positive, that x and (cid:0) must be strictly increasing, and that z must be strictly decreasing. Then, equation (22c) implies that for any t ((cid:30) ;(cid:30) ] we can write 0 1 2 x(t)z((cid:30) )[1+F (K x(t))K ] (cid:0) (t) = (cid:0) ((cid:30) )h(t;(cid:0)(t)) 0 0 1 (cid:30) (cid:30) 0 x((cid:30) )z(t)[1+F (K x((cid:30) ))K ] 0 0 0 1 h(t;(cid:0)(t)) (cid:27) t @lnA((cid:0)(u);u)du = (cid:0) (cid:30) ((cid:30) 0 ) [1+F (K )K ] e (cid:30)0 @(cid:30) [1+F (K 0 x(t))K 1 ]; 0 1 R where the second line is obtained using equations (22a)-(22b) and x((cid:30) ) = 1. Integrating (cid:0) (t) between 0 (cid:30) (cid:30) and (cid:30) yields 0 (cid:30) h(t;(cid:0)(t)) (cid:27) t @lnA((cid:0)(u);u)du (cid:0)((cid:30)) = (cid:0)((cid:30) 0 )+(cid:0) (cid:30) ((cid:30) 0 ) [1+F (K )K ] e (cid:30)0 @(cid:30) [1+F (K 0 x(t))K 1 ]dt: (cid:30) 0 1 R Z 0 Evaluating the last expression at (cid:30) = (cid:30) , using the boundary conditions on (cid:0) and solving for (cid:0) ((cid:30) ) we 1 (cid:30) 0 get [s s ] 1 0 (cid:0) ((cid:30) ) = (cid:0) : (cid:30) 0 (cid:30) h(t;(cid:0)(t)) (cid:27) t @lnA((cid:0)(u);u)du (cid:30) 0 1 [1+F(K0)K1] e R (cid:30)0 @(cid:30) [1+F (K 0 x(t))K 1 ]dt R ((cid:27) 1) t @lnA(H(t);t) dt The last two expressions, x(t) = e (cid:0) (cid:30)0 @(cid:30) , and the de(cid:133)nition of (cid:9) in (26) yield (cid:0) = (cid:9)((cid:0)), R i.e. (cid:0) is a (cid:133)xed point of (cid:9). Let us turn to the "if" part of the lemma. Let (cid:0) be a (cid:133)xed point of (cid:9). If we de(cid:133)ne x((cid:30)) = e ((cid:27) (cid:0) 1) (cid:30) t 0 @lnA @ ((cid:0) (cid:30) (u);u)du and z((cid:30)) = [1+F(K0)K1](cid:11)((cid:30) 0 )g((cid:30) 0 )M e(cid:0) (cid:30) t 0 @lnA @ ((cid:0) (cid:30) (u);u)du , then it is easy to verify that R A(s0;(cid:30) 0 )V(s0)(cid:0)(cid:30)((cid:30) 0 ) R z;x;(cid:0) is a solution to BVP (22). f g Having recasted the BVP (22) as the problem of (cid:133)nding a (cid:133)xed point of the functional (cid:9) de(cid:133)ned in (26)-(27), the next step is to establish certain properties of this functional that permit the application of some (cid:133)xed point theorem in the literature. I do so in the next lemma, in which I state that (cid:9) is a compact self-map on some closed and convex subset of Banach space. Claim 2 Let K be the convex and closed subset of C[(cid:30) ;(cid:30) ] given by 0 1 K y C[(cid:30) ;(cid:30) ] : s y((cid:30)) s for all (cid:30) [(cid:30) ;(cid:30) ] ; (28) 0 1 0 1 0 1 (cid:17) f 2 (cid:20) (cid:20) 2 g 37

and let (cid:9) be the functional de(cid:133)ned in (26)-(27). If V;g;(cid:11) are continuous and A is continuously di⁄erf g entiable, then (cid:9) is a compact self-map on K. Proof. By de(cid:133)nition of (cid:9), (cid:9)(y)((cid:30)) is a strictly increasing function with (cid:9)(y)((cid:30) ) = s and (cid:9)(y)((cid:30) ) = 0 0 1 s , so (cid:9)(y) K, i.e. (cid:9) is a self-map on K. To show that (cid:9) is compact we have to show that (cid:9)(K) 1 2 is relatively compact. Per the Arzela-Ascoli theorem, it enough to show that (cid:9)(K) is bounded and equicontinuous. Let us start by showing that (cid:9)(K) is bounded. To simplify notation, let(cid:146)s de(cid:133)ne the following constants: h max h((cid:30);y); h min h((cid:30);y); (cid:17) (cid:30);y 2 [(cid:30) 0 ;(cid:30) 1 ] (cid:2) [s0;s1] (cid:17) (cid:30);y 2 [(cid:30) 0 ;(cid:30) 1 ] (cid:2) [s0;s1] @lnA(y;(cid:30)) @lnA(y;(cid:30)) r max ; r min (cid:17) (cid:30);y 2 [(cid:30) 0 ;(cid:30) 1 ] (cid:2) [s0;s1] @(cid:30) (cid:17) (cid:30);y 2 [(cid:30) 0 ;(cid:30) 1 ] (cid:2) [s0;s1] @(cid:30) Since A;V;g;(cid:11) are continuous and strictly positive on (cid:8) S [(cid:30) ;(cid:30) ] [s ;s ], then the constants 0 1 0 1 f g (cid:2) (cid:19) (cid:2) h and h are well-de(cid:133)ned and are bounded away from zero. Similarly, the assumptions on A implies that @lnA(y;(cid:30)) is strictly positive and continuous on (cid:8) S, so r and r are also well-de(cid:133)ned and bounded away @(cid:30) (cid:2) from zero. Then for any y K we have 2 [s s ] h h j (cid:9)(y)((cid:30)) j (cid:20) s 0 + ((cid:30) 1 (cid:0) (cid:30) 0 )h e(cid:27)r((cid:30) 1(cid:0) (cid:30) 0 )(1+K 1 )((cid:30) (cid:0) (cid:30) 0 ) (cid:20) s 0 +[s 1 (cid:0) s 0 ] h e(cid:27)r((cid:30) 1(cid:0) (cid:30) 0 )(1+K 1 ): 1 0 (cid:0) The last result implies k (cid:9)(y) k (cid:20) s 0 + [s 1 (cid:0) s 0 ] h h e(cid:27)r((cid:30) 1(cid:0) (cid:30) 0 )(1+K 1 ); and given that the selection of 1 y K was arbitrary, we conclude that (cid:9)(K) is bounded. 2 Let us now show that (cid:9)(K) is equicontinuous. For any y K and (cid:30) > (cid:30) we have 0 2 (cid:30) (cid:30)0h(t;(cid:0)(t))e (cid:27) (cid:30) t 0 @lnA @ ((cid:0) (cid:30) (u);u)du 1+F K 0 e ((cid:27) (cid:0) 1) (cid:30) t 0 @lnA @ ((cid:0) (cid:30) (u);u) dt K 1 dt R " R ! # (cid:9)(y) (cid:30) (cid:9)(y)((cid:30)) [s s ] 0 1 0 R (cid:12) (cid:0) (cid:1) (cid:0) (cid:12) (cid:20) (cid:0) (cid:30) (cid:30) 1h(t;H(t))e (cid:27) (cid:30) t 0 @lnA @ ((cid:0) (cid:30) (u);u)du 1+F K 0 e ((cid:27) (cid:0) 1) (cid:30) t 0 @lnA @ ((cid:0) (cid:30) (u);u) dt K 1 dt (cid:12) (cid:12) 0 R " R ! # R [s s ] h 1 (cid:0) 0 e(cid:27)r((cid:30) 1(cid:0) (cid:30) 0 )(1+K 1 ) (cid:30) 0 (cid:30) : (cid:20) ((cid:30) (cid:30) )h (cid:0) 1 0 (cid:0) (cid:12) (cid:12) (cid:12) (cid:12) Given that the selection of y K was arbitrary, the last inequality implies that (cid:9)(K) is equicontinuous 2 on [(cid:30) ;(cid:30) ]. 0 1 Per the last two claims, the existence of a solution to the BVP (22) can be obtained as a direct application of the Schauder (cid:133)xed point theorem (SFPT).49 A function (cid:0) belongs to a triplet z;x;(cid:0) f g solving BVP (22) if and only if (cid:0) is a (cid:133)xed point of the functional (cid:9) de(cid:133)ned in (26)-(27). In addition, this functional is a compact self-map on the closed and convex set K de(cid:133)ned in (28), so the SFPT implies that (cid:9) has a (cid:133)xed point on K. Then, this (cid:133)xed point is part of a solution to the BVP (22). Finally, the 49For a statement of the SFPT see O(cid:146)Reagan (1997), OK (20xx), Granas, Gunther and Lee (1985). 38

continuity of A;V;g;(cid:11);F and (22c) implies that (cid:0) is continuously di⁄erentiable. f g Uniqueness. I start by proving an intermediate result that is used later. The continuous di⁄erentiability of V;g;(cid:11);F and the twice continuous di⁄erentiability of A imply that the right-hand side of f g equations (22a)-(22c) are locally Lipschitz continuous with respect to z;x;(cid:0) , as the relevant partial f g derivatives are bounded on bounded sets. Then, the initial value problem (IVP) given by the di⁄erential equations (22a)-(22c) and the initial conditions x((cid:30) ) = 1, (cid:0)((cid:30) ) = s , z((cid:30) ) = z ; has at most one 0 0 0 0 0 solution. LetusturntotheuniquenessofthesolutiontotheBVP(22). Iproceedbycontradiction. Supposethat there are two di⁄erent solutions z ;x;(cid:0) and z;x;(cid:0) to the BVP (22). Then, the uniqueness result 0 0 0 f g f g in the previous paragraph implies that z ((cid:30) ) = z((cid:30) ), which, together with equation (22c), implies 0 0 0 6 (cid:0) ((cid:30) ) = (cid:0) ((cid:30) ). Without loss of generality suppose (cid:0) ((cid:30) ) < (cid:0) ((cid:30) ), i.e. (cid:0)((cid:30)) > (cid:0) ((cid:30)) in some 0(cid:30) 0 6 (cid:30) 0 0(cid:30) 0 (cid:30) 0 0 neighborhood ((cid:30) ;c); with c > (cid:30) . By assumption, we know that the functions (cid:0) and (cid:0) must intersect 0 0 0 at least once again on ((cid:30) ;(cid:30) ], since (cid:0)((cid:30) ) = (cid:0) ((cid:30) ). Let (cid:30)+ be the (cid:133)rst value to the right of (cid:30) at 0 1 1 0 1 0 which the functions (cid:0) and (cid:0) intersect, i.e. (cid:30)+ inf (cid:30) ((cid:30) ;(cid:30) ] : (cid:0) ((cid:30)) = (cid:0)((cid:30)) , and notice that (cid:30)+ is 0 0 1 0 (cid:17) f 2 g well-de(cid:133)ned since (cid:0) and (cid:0) are continuous. Given our assumptions, (cid:0)((cid:30)) > (cid:0) ((cid:30)) for (cid:30) (cid:30) ;(cid:30)+ , which 0 0 0 2 together with (cid:0) (cid:30)+ = (cid:0) (cid:30)+ , implies that (cid:0) (cid:30)+ (cid:0) (cid:30)+ . This and (cid:0) ((cid:30) ) < (cid:0) ((cid:30) ) imply 0 0(cid:30) (cid:21) (cid:30) 0(cid:30) 0 (cid:30) (cid:0) 0 (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:30)+ =(cid:0) ((cid:30) ) 0(cid:30) 0(cid:30) 0 > 1. (29) (cid:0) (cid:30)+ =(cid:0) ((cid:30) ) (cid:30)(cid:0) (cid:1) (cid:30) 0 (cid:0) (cid:1) As discussed above, (cid:0) and (cid:0) are (cid:133)xed points of the functional (cid:9) de(cid:133)ned in (26), Z((cid:30)) = (cid:9)(Z)((cid:30)), 0 for Z = (cid:0);(cid:0), so Z ((cid:30)) can be obtained di⁄erentiating the right-hand side of (26). Doing so yields, 0 (cid:30) ((cid:27) 1) (cid:30)+ @lnA(Z(u);u) du 1+F K 0 e (cid:0) (cid:30)0 @(cid:30) K 1 Z (cid:30) (cid:30)+ =Z (cid:30) ((cid:30) 0 ) = h (cid:30)+;Z (cid:30)+ e (cid:27) (cid:30) (cid:30) 0 + @lnA( @ Z (cid:30) (u);u)du" [1+F R (K )K ] ! # ; R 0 1 (cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1) for Z = (cid:0);(cid:0). Combining the last expression for both functions yields 0 ((cid:27) 1) (cid:30)+ @lnA((cid:0) 0 (u);u) du (cid:0) 0(cid:30) (cid:30)+ =(cid:0) 0(cid:30) ((cid:30) 0 ) = e (cid:27) (cid:30) (cid:30) 0 + @lnA( @ (cid:0) (cid:30) 0(u);u) (cid:0) @lnA @ ((cid:0) (cid:30) (u);u) du" 1+F K 0 e (cid:0) R (cid:30)0 @(cid:30) ! K 1 # < 1; (cid:0) (cid:30)(cid:0) (cid:30)+ (cid:1) =(cid:0) (cid:30) ((cid:30) 0 ) R (cid:20) (cid:21) ((cid:27) 1) (cid:30)+ @lnA((cid:0)(u);u) du 1+F K 0 e (cid:0) (cid:30)0 @(cid:30) K 1 (cid:0) (cid:1) " R ! # (30) where in the last expression I used the fact that (cid:0) (cid:30)+ = (cid:0) (cid:30)+ , so h (cid:30)+;(cid:0) (cid:30)+ = h (cid:30)+;(cid:0) (cid:30)+ . 0 0 The log-supermodularity of A, (cid:0)((cid:30)) > (cid:0) ((cid:30)) for (cid:30) (cid:30) ;(cid:30)+ and the fact that F strictly increasing 0 (cid:0) (cid:1) 0 (cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:0) (cid:1)(cid:1) 2 imply that each of the terms on the right-hand side of the last expression is strictly less than 1. However, (cid:0) (cid:1) note that equation (30) contradicts equation (29), so it must be the case that there is only one solution to the BVP (22). 39

Condition i. Let zi;xi;(cid:0)i be the unique solution to BVP (22) with K = 0 and s = si, for 1 0 0 i = a;b and sa > sb. To prove the result we show that if (cid:0)a and (cid:0)b intersect at some point (cid:30)+ ((cid:30) ;(cid:30) ), 0 0 (cid:8) (cid:9) 2 0 1 then there are functions yi and wi for i = a;b, such that wa;ya;(cid:0)a and wb;yb;(cid:0)b solve the same IVP f g on [(cid:30) ;(cid:30) ] given by the system (22a)-(22c) and the same initial value at any (cid:30) (cid:30) ;(cid:30) . Then, the 0 1 (cid:8) (cid:9) + 1 2 uniqueness result proved at the beginning of the previous section implies that ya;wa;(cid:0)a = yb;wb;(cid:0)b (cid:0) (cid:1) f g on [(cid:30) ;(cid:30) ], contradicting the initial initial assumption sa > sb. 0 1 0 0 (cid:8) (cid:9) Suppose that there is a (cid:30)+ ((cid:30) ;(cid:30) ) and (cid:0)a (cid:30)+ = (cid:0)b (cid:30)+ s+. If we de(cid:133)ne the functions 0 1 2 (cid:17) yi;wi : [(cid:30) 0 ;(cid:30) 1 ] R+ as yi((cid:30)) = zi((cid:30))=xi (cid:30)+ , wi (cid:0) = x (cid:1) i((cid:30))=xi (cid:0) (cid:30)+ (cid:1) , it is readily seen that on (cid:30)+;(cid:30) 1 ! and for i = a;b; yi;wi;(cid:0)i is a solution to the BVP given by the system of di⁄erential equations (22a)- (cid:0) (cid:1) (cid:0) (cid:1) (cid:2) (cid:3) (22c) and boundary conditions w((cid:30)+) = 1; (cid:0)((cid:30)+) = s+, (cid:0)((cid:30) ) = s . As this BVP is just a particular case (cid:8) (cid:9) 1 1 of BVP (22), it has a unique solution, implying that ya;wa;(cid:0)a = yb;wb;(cid:0)b on (cid:30)+;(cid:30) . Moreover, 1 f g this result implies that wa;ya;(cid:0)a and wb;yb;(cid:0)b solve the same IVP on [(cid:30) ;(cid:30) ] given by the system (cid:8) 0(cid:9) 1 (cid:2) (cid:3) f g (22a)-(22c) and the same initial conditions at any (cid:30) (cid:30) ;(cid:30) , which is the desired result. The no- (cid:8) (cid:9) + 1 2 crossing result related to the inverses of (cid:0)i can be establish in a similar way. (cid:0) (cid:1) Condition ii. Let (cid:30)a > (cid:30)b and suppose that xa((cid:30)) x((cid:30);(cid:30)a) x (cid:30);(cid:30)b xb((cid:30)) for some (cid:30) on 0 0 0 0 (cid:17) (cid:21) (cid:17) [(cid:30)a;(cid:30) ]. From their de(cid:133)nitions, it is clear that xa((cid:30)a) < xb((cid:30)a), so let (cid:30) be the (cid:133)rst productivity level 0 1 0 0 0(cid:0) (cid:1) such that xa((cid:30)) = xb((cid:30)). Notice that (cid:30) is well de(cid:133)ned due to the continuity of the functions involved 0 and due to our initial assumption. By de(cid:133)nition of (cid:30), we have xa (cid:30) = xb (cid:30) and xa((cid:30)) < xb((cid:30)) for 0 0 0 (cid:30) < (cid:30). This means that xa((cid:30)) is catching up to xb((cid:30)), so this and the log-supermodularity of A imply 0 (cid:0) (cid:1) (cid:0) (cid:1) that there is a (cid:30) < (cid:30), such that (cid:0)a((cid:30)) > (cid:0)b((cid:30)) on (cid:30) ;(cid:30) , i.e. (cid:0)a and (cid:0)b must intersect at least once 00 0 00 0 strictly to left of (cid:30). Let (cid:30) denote the productivity level corresponding to the (cid:133)rst intersection of (cid:0)a and 0 (cid:0) (cid:1) (cid:0) (cid:0)b that is strictly to the left of (cid:30). Notice that (cid:30) is well de(cid:133)ned (cid:151)due to the continuity of the functions 0 (cid:0) (cid:0)i and the fact that (cid:0)a((cid:30)a) < (cid:0)b((cid:30)a)(cid:151) and that (cid:30) < (cid:30). Similarly, let (cid:30) denote the productivity level 0 0 0 + (cid:0) corresponding to the (cid:133)rst intersection of (cid:0)a and (cid:0)b that is weakly to the right of (cid:30). Notice that (cid:30) is 0 + also well de(cid:133)ned and that (cid:30) (cid:30). + 0 (cid:21) From the de(cid:133)nitions above we have (cid:0)a (cid:30) = (cid:0)b (cid:30) , (cid:0)a((cid:30)) > (cid:0)b((cid:30)) on (cid:30) ;(cid:30) and (cid:0)a (cid:30) = + + (cid:0) (cid:0) (cid:0) (cid:0)b (cid:30) . Then (cid:0)a (cid:30) (cid:0)b (cid:30) and (cid:0)a (cid:30) (cid:0)b (cid:30) , so + (cid:30) (cid:0) (cid:21) (cid:30) (cid:0) (cid:30) (cid:0)+ (cid:1) (cid:20) (cid:30) (cid:0)+ (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0)a((cid:30) )=(cid:0)a((cid:30) ) (cid:30) + (cid:30) (cid:0) 1. (31) (cid:0)b((cid:30) )=(cid:0)b((cid:30) ) (cid:20) (cid:30) + (cid:30) (cid:0) As discussed above, we can di⁄erentiate the right-hand side of (26) to get (cid:0)i (cid:30) (cid:30) + = hi (cid:30) ;(cid:30) e (cid:27) (cid:30) (cid:30)+ @lnA( @ (cid:0) (cid:30) i(t);t) dt 1+F K 0 xi (cid:30) + K 1 for i = a;b; (cid:0)i (cid:30)(cid:0) (cid:30) (cid:1) (cid:0) + R (cid:0) (cid:2) 1+F (cid:0) K 0 xi (cid:0) (cid:30) (cid:1)(cid:1) K 1(cid:3) (cid:0) (cid:0) (cid:1) (cid:0) (cid:0) (cid:1) (cid:2) (cid:0) (cid:0) (cid:1)(cid:1) (cid:3) 40

where h is de(cid:133)ned in (27). This implies (cid:0)a((cid:30) )=(cid:0)a((cid:30) ) (cid:27) (cid:30)+ @lnA((cid:0)a(t);t) @lnA((cid:0)b(t);t) dt 1+F K xa (cid:30) K = 1+F K xa (cid:30) K (cid:30) + (cid:30) (cid:0) = e (cid:30) (cid:0)" @(cid:30) (cid:0) @(cid:30) # 0 + 1 0 (cid:0) 1 (cid:0)b((cid:30) )=(cid:0)b((cid:30) ) R 1+F K xb (cid:30) K = 1+F K xb (cid:30) K (cid:30) + (cid:30) (cid:2) (cid:0) 0 (cid:0) +(cid:1)(cid:1) 1(cid:3) (cid:2) (cid:0) 0 (cid:0) (cid:1)(cid:1) 1(cid:3) (cid:0) (cid:0) xa (cid:30) + =xa (cid:30) 1+F K 0 xa (cid:30)(cid:2)+ K 1(cid:0) = 1+(cid:0) F(cid:1)(cid:1)K 0 x(cid:3) a (cid:30)(cid:2) K(cid:0)1 (cid:0) (cid:1)(cid:1) (cid:3) > (cid:0) (cid:0) ; (32) xb (cid:30) =xb (cid:30) 1+F K xb (cid:30) K = 1+F K xb (cid:30) K (cid:0) +(cid:1) (cid:0) (cid:1)(cid:2) (cid:0) 0 (cid:0) +(cid:1)(cid:1) 1(cid:3) (cid:2) (cid:0) 0 (cid:0) (cid:1)(cid:1) 1(cid:3) (cid:0) (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) (cid:2) (cid:0) (cid:0) (cid:1)(cid:1) (cid:3) (cid:2) (cid:0) (cid:0) (cid:1)(cid:1) (cid:3) wherethesecondlineisobtainedmultiplyingtheright-handsidebyexp (cid:30) + @lnA((cid:0)b(t);t) @lnA((cid:0)a(t);t) < (cid:30) @(cid:30) (cid:0) @(cid:30) (cid:0) (cid:20) (cid:21) 1. Per our de(cid:133)nitions we have xa (cid:30) = xb((cid:30)), xa (cid:30) xb((cid:30) ) (R(cid:0)a((cid:30)) (cid:0)b((cid:30)) on (cid:30);(cid:30) ), and 0 0 + + 0 + (cid:21) (cid:21) xa (cid:30) < xb((cid:30) ), which together with (32), imply (cid:0) (cid:1) (cid:0) (cid:1) (cid:2) (cid:3) (cid:0) (cid:0) (cid:0) (cid:1) (cid:0)a((cid:30) )=(cid:0)a((cid:30) ) (cid:30) + (cid:30) (cid:0) > 1, (cid:0)b((cid:30) )=(cid:0)b((cid:30) ) (cid:30) + (cid:30) (cid:0) contradicting (31). Then, it must be the case that xa((cid:30)) < xb((cid:30)) for all (cid:30) [(cid:30)a;(cid:30) ], which is the desired 0 1 2 result. A.2.4 Proof of Proposition 1 The proof of the existence and uniqueness of the equilibrium in the closed and open economies was laid out in the text. Here, I prove the (constrained) e¢ ciency of the equilibrium, starting with the simpler closed-economy case. E¢ ciency of the Equilibrium of the Closed Economy Below I show that an allocation is an equilibrium of the closed economy if and only if it is a solution to the planner(cid:146)s problem (cid:30) (cid:27) 1 max q((cid:30)) (cid:0)(cid:27) g((cid:30))Md(cid:30) (cid:30)(cid:3);q((cid:30));H((cid:30)) (cid:30)(cid:3) R e e subjeect to (33) (cid:30) q((cid:30)0) g (cid:30) d(cid:30)M = H((cid:30)) V(s)ds L f[1 G((cid:30) )]M for all (cid:30) (cid:30) ;(cid:30) ; (cid:30)(cid:3) A(H((cid:30)0);(cid:30)0) 0 0 s (cid:0) (cid:0) (cid:3) 2 (cid:3) R e (cid:0) (cid:1) R eH((cid:30) (cid:3) ) = s;(cid:2)H((cid:30)) = s; (cid:3) (cid:2) (cid:3) e where the left- and right-hand sides of the integral equation represent, respectively, the total mass of e e workers required to produce q (cid:30) units of each variety with productivity below (cid:30), and the total mass of 0 workers employed in the production of said varieties. Di⁄erentiating both sides of the integral equation (cid:0) (cid:1) above with respect to (cid:30) yieldes the the following ordinary di⁄erential equation (ODE) for all (cid:30) (cid:30) ;(cid:30) , (cid:3) 2 (cid:2) (cid:3) q((cid:30))g((cid:30))M H ((cid:30)) = hH((cid:30) ;q((cid:30));H((cid:30));(cid:30)): (34) (cid:30) (cid:3) A(H((cid:30));(cid:30))V(H((cid:30))) L f[1 G((cid:30) )]M (cid:17) (cid:3) (cid:0) (cid:0) e e (cid:2) (cid:3) e e e e 41

Moreover, if (34) is satis(cid:133)ed for all (cid:30) (cid:30) ;(cid:30) , then we can recover the integral equation above by moving (cid:3) 2 V(H((cid:30)))[L f[1 G((cid:30) )]] to the left-hand side before integrating both sides of the resulting expression (cid:3) (cid:2) (cid:3) (cid:0) (cid:0) between (cid:30) ;(cid:30) for each (cid:30). That is, the integral equation is equivalent to the ODE in (34), with the (cid:3) 0 0 e latter being the version of the constraint I consider below. (cid:2) (cid:3) Following chapter 9 of Luenberger (1969), if (cid:30) ;q;H solves problem (33), then there is a function (cid:3) f g of bounded variation, (cid:21)H, and a real number; (cid:22)H, such that the Lagrangian, e e L((cid:30) (cid:3) ;q;H) = (cid:30) (cid:30) (cid:3) q((cid:30)) (cid:27) (cid:0)(cid:27) 1 g((cid:30))d(cid:30)M+ (cid:30) (cid:30) (cid:3) H((cid:30)) (cid:0) s (cid:0) (cid:30) (cid:30) (cid:3) hH((cid:30) (cid:3) ;q (cid:30) 0 ;H (cid:30) 0 ;(cid:30) 0 )d(cid:30) 0 d(cid:21)H((cid:30))+(cid:22)H H (cid:30) (cid:0) s h i h i R R R (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) is stat e ioneary at f (cid:30) e(cid:3) ;q;H g . Integrating by eparts the term involvin e g a doueble integral and using the faect that (cid:21)H is di⁄erentiable, the Lagrangian can be expressed as50 e e L((cid:30) (cid:3) ;q;H) = (cid:30) (cid:30) (cid:3) q((cid:30)) (cid:27) (cid:0)(cid:27) 1 g((cid:30))d(cid:30)M + (cid:30) (cid:30) (cid:3) H((cid:30))(cid:21)H (cid:30) ((cid:30))d(cid:30)+(cid:21)H((cid:30) (cid:3) )s (cid:0) H (cid:30) (cid:21)H((cid:30))+ (cid:1)(cid:1)(cid:1) 8 < R e (cid:1)(cid:1)(cid:1) (cid:30) (cid:30) (cid:3) hH((cid:30) (cid:3) ;qR((cid:30)); e H((cid:30));(cid:30))(cid:21)H((cid:30))d(cid:30)+(cid:22)H H e (cid:30)(cid:0) (cid:0) (cid:1)s e e R (cid:2) (cid:0) (cid:1) (cid:3) The stationarity co:ndition, together with the econstraeints of the problem, yields the following (cid:133)rst order necessary conditions for an optimum H ((cid:30)) = hH((cid:30) ;q((cid:30));H((cid:30));(cid:30)) (cid:30) (cid:3) hH((cid:30) ;q((cid:30));H((cid:30));(cid:30))(cid:21)H((cid:30))+(cid:21)H((cid:30)) = 0 H (cid:3)e e e (cid:30) (cid:27) (cid:0)(cid:27) 1q((cid:30)) (cid:0)(cid:27) 1 g( e (cid:30))M + e hH q ((cid:30) (cid:3) ;q((cid:30));H((cid:30));(cid:30))(cid:21)H((cid:30)) = 0 (35) (cid:22)H (cid:21)H (cid:30) = 0 e (cid:0) e e H(cid:2)((cid:30) ) = s; (cid:0)H(cid:1)((cid:3)(cid:30)) = s (cid:3) (cid:30) (cid:30) (cid:3) hH (cid:30)(cid:3) ((cid:30) (cid:3) ;q((cid:30));H((cid:30));(cid:30))(cid:21)H((cid:30))d(cid:30) e = q((cid:30) (cid:3) ) (cid:27) (cid:0)(cid:27) 1 e g((cid:30) (cid:3) )M +hH((cid:30) (cid:3) ;q((cid:30));H((cid:30));(cid:30))(cid:21)H((cid:30) (cid:3) ): R The (cid:133)rst (cid:133)ve linees in (e35) are the standardenecessary conditions of opetimal econtrol theory and re(cid:135)ect the constraints of the problem and the implications of stationarity of the Lagrangian with respect to H;q . The last line in (35) follows from the stationarity with respect to (cid:30) . Below I show that if (cid:3) f g (cid:30) ;q;H;(cid:21)H satis(cid:133)es (35), then we can de(cid:133)ne functions p((cid:30));r((cid:30)) such that (cid:30) ;p((cid:30));r((cid:30));H (cid:3) (cid:3) f sa e tisefy the co g nditions of lemma 1, proving that a solution t f o the planne g r(cid:146)s problem f is an equilibrium o g f the celos e ed economy. e e e e e Let (cid:30) ;q;H;(cid:21)H satisfy the conditions in (35). For some (still unde(cid:133)ned) positive constant p , de(cid:133)ne (cid:3) 0 f g e e p((cid:30)) p (cid:27) (cid:0) (cid:21)H((cid:30)) ; (36) (cid:17) 0(cid:27) (cid:0) 1 A(H((cid:30));(cid:30))V(H((cid:30))) 50See section 9.5 of Luenberger (1969)efor a derivation of the di⁄erentiability of (cid:21)H. e e 42

which, together with (35), implies @lnA(H((cid:30));(cid:30)) p ((cid:30)) = p((cid:30)) (37) (cid:30) (cid:0) @(cid:30) e Using (36) in the third line of (35) yeields e q((cid:30)) = p(cid:27) L f[1 G((cid:30) )]M (cid:27) p((cid:30)) (cid:27); 0 (cid:0) (cid:0) (cid:3) (cid:0) so de(cid:133)ning(cid:2) (cid:3) e e r((cid:30)) q((cid:30))p((cid:30)) (38) (cid:17) implies e e e r((cid:30)) = p(cid:27) L f[1 G((cid:30) )]M (cid:27) p((cid:30))1 (cid:27); 0 (cid:0) (cid:0) (cid:3) (cid:0) (39) (cid:27) 1 r((cid:30)) = p 0(cid:2) L f[1 G((cid:30) (cid:3) )]M (cid:3) q((cid:30)) (cid:0)(cid:27) (cid:0) (cid:0) e e and (cid:2) (cid:3) e @lnA(H(e(cid:30));(cid:30)) r ((cid:30)) = ((cid:27) 1)r ((cid:30)) (40) (cid:30) (cid:30) (cid:0) @(cid:30) e With these de(cid:133)nitions, the the (cid:133)rst line of (35) can be expressed as e e r((cid:30))g((cid:30))M H ((cid:30)) = : (41) (cid:30) A(H((cid:30));(cid:30))V(H((cid:30)))p((cid:30)) L fM [1 G((cid:30) )] (cid:3) (cid:0) (cid:0) e e (cid:2) (cid:3) Finally, noting that the third line in e (35) implie e s (cid:27) (cid:0)(cid:27) 1eq((cid:30)) (cid:27) (cid:0)(cid:27) 1 g((cid:30))M = (cid:0) H((cid:30))(cid:21)H((cid:30)), the last line in (35) can be expressed as (cid:30) (cid:30) (cid:3)(cid:0) H((cid:30))(cid:21)H((cid:30))d(cid:30) [L f f [1 Mg G ((cid:30) ((cid:30) (cid:3) (cid:3) ) )]M] = q((cid:30) (cid:3) ) (cid:27) (cid:0)(cid:27) 1 g((cid:30) (cid:3) )M e +H((cid:30) (cid:3) )(cid:21)H((cid:30) (cid:3) ); e (cid:0) (cid:0) (cid:27) 1 R (cid:30) (cid:30) (cid:3) q q ( e ( (cid:30) (cid:30) (cid:3) ) ) (cid:0)(cid:27) g((cid:30))d(cid:30)(cid:27)fM = (cid:27) (cid:0) (cid:27) 1 L (cid:0) e f[1 (cid:0) G((cid:30) (cid:3) )]M ; e R h e i (cid:2) (cid:3) e (cid:30) r((cid:30)) (cid:27) (cid:27)f g((cid:30))d(cid:30)M = L f[1 G((cid:30) )]M ; (42) (cid:3) r((cid:30) ) (cid:27) 1 (cid:0) (cid:0) Z (cid:30)(cid:3) (cid:3) (cid:0) e (cid:2) (cid:3) where the derivation uses (39). If we choose the constant p in (36) such that r((cid:30) ) = (cid:27)f, then the last e 0 (cid:3) equation can be expressed as e (cid:30) (cid:27) r((cid:30))g((cid:30))d(cid:30)M = [L f[1 G((cid:30) )]] (43) (cid:3) (cid:27) 1 (cid:0) (cid:0) Z (cid:30)(cid:3) (cid:0) e Note that conditions {(37),(40),(41),(43)}, H((cid:30) ) = s;H((cid:30)) = s , and r((cid:30) ) = (cid:27)f are identical to those (cid:3) (cid:3) f g in lemma 1, so p;r;H are the price, revenue and inverse matching functions corresponding to the closed economy equilib f rium. g e e e On the otheredeire e ction, let (cid:30) ;p;rd;H be the activity cuto⁄, price, revenue and inverse matching (cid:3)a functionsoftheclosedeconomyequilibrium,withoutputfunctionqd((cid:30)) = rd((cid:30))=p((cid:30)). As p;rd;H sat- (cid:8) (cid:9) isfytheODE(15),then qd;H satisfythe(cid:133)rstconditionin(35). De(cid:133)ne(cid:21) (cid:21) (cid:27) 1p((cid:30))A(H((cid:30));(cid:30))V(H((cid:30))) (cid:17) (cid:0) 0 (cid:0)(cid:27) (cid:8) (cid:9) (cid:8) (cid:9) 43

for some positive constant (cid:21) . Log-di⁄erentiating (cid:21), together with equilibrium condition (13), yields 0 (cid:0) the second line in (35). Using these de(cid:133)nitions in the third condition of (35) yields (cid:27) 1 qd((cid:30) (cid:3)a ) (cid:27) (cid:0)(cid:27) 1 (cid:20) q q d d ( ( (cid:30) (cid:30) (cid:3)a ) ) (cid:21) (cid:0)(cid:27) (cid:0) L (cid:0) f[1 r (cid:0) d( G (cid:30)) ((cid:30) (cid:3)a )]M (cid:21) 0 = 0: (cid:2) (cid:3) Recalling that the CES demand system implies rd((cid:30)) = Bqd((cid:30)) (cid:27) (cid:0)(cid:27) 1 for some constant B, the last expression holds for L f[1 G((cid:30) (cid:3)a )]M qd((cid:30) (cid:3) ) (cid:27) (cid:0)(cid:27) 1 (cid:21) = (cid:0) (cid:0) : 0 rd((cid:30) ) (cid:2) (cid:3)a (cid:3) Finally, the derivations above imply that qd;H;(cid:21);(cid:30) satisfy the last line in (35) if and only if they (cid:3)a satisfy equation (42), a fact that follows from the zero pro(cid:133)t condition rd((cid:30) ) = (cid:27)f and the numeraire (cid:8) (cid:9) (cid:3)a condition (16). Accordingly, qd;H;(cid:21);(cid:30) solves the planner(cid:146)s problem. (cid:3)a (cid:8) (cid:9) E¢ ciency of the Equilibrium of the Open Economy In this section, I show that an allocation is an equilibrium of the open economy if and only if it is a solution to the planner(cid:146)s problem max (cid:30) qd((cid:30)) (cid:27) (cid:0)(cid:27) 1 g((cid:30))Md(cid:30)+n (cid:30) qx((cid:30)) (cid:27) (cid:0)(cid:27) 1 F (y((cid:30)))g((cid:30))Md(cid:30) (cid:30)(cid:3);qd;qx;H;y (cid:30)(cid:3) (cid:30)(cid:3) R R e e e e e subject to e e (44) (cid:30) qd((cid:30)0) g (cid:30) d(cid:30)M +nf (cid:30) qx((cid:30)0)(cid:28) F y (cid:30) g (cid:30) Md(cid:30) = (cid:30)(cid:3) A(H((cid:30)0);(cid:30)0) 0 0 x (cid:30)(cid:3) A(H((cid:30)0);(cid:30)0) 0 0 0 (cid:1)(cid:1)(cid:1) R e (cid:0) (cid:1) H((cid:30)) V(s)dsLpRw((cid:30) ;ey) for all (cid:30)(cid:0) (cid:0) (cid:30)(cid:1)(cid:1);(cid:30)(cid:0); (cid:1) e (cid:1)(cid:1)(cid:1) s (cid:3) e e2 (cid:3) e R H((cid:30) ) = s; H((cid:30)) = s: (cid:2) (cid:3) (cid:3) e where Lpw((cid:30) ;y) represents the mass of peroduction woerkers, (cid:3) e (cid:30) y((cid:30)0) Lpw((cid:30) ;y) = L f[1 G((cid:30) )]M nf ydF(y)g (cid:30) Md(cid:30) (cid:3) (cid:3) x 0 0 (cid:0) (cid:0) (cid:0) " Z (cid:30)(cid:3) Z ye # (cid:0) (cid:1) e As explained in the case of the planner(cid:146)s problem for the closed economy, the integral constraint is equivalent to the following ODE, H ((cid:30)) = hH((cid:30) ;qd((cid:30));qx((cid:30));H((cid:30));y((cid:30));(cid:30)); (cid:30) (cid:3) qd((cid:30))+qx((cid:30))F (y((cid:30)))(cid:28)n g((cid:30))M (45) hHe( ;(cid:30)) e e e e : (cid:1)(cid:1)(cid:1) (cid:17) A(H((cid:30));(cid:30))V(H((cid:30)))Lpw((cid:30) ;y) (cid:2) (cid:3) (cid:3) e e e Following chapter 9 of Luenberger (1969), if (cid:30)e (cid:3) ;qd;qx;H;ye solves probleem (44), then there is a function f g of bounded variation, (cid:21)H, and a real number; (cid:22)H, such that the Lagrangian, e e e e 44

L((cid:30) (cid:3) ;qd;qx;H;y) = (cid:30) (cid:30) (cid:3) qd((cid:30)) (cid:27) (cid:0)(cid:27) 1 g((cid:30))Md(cid:30)+n (cid:30) (cid:30) (cid:3) qx((cid:30)) (cid:27) (cid:0)(cid:27) 1 F (y((cid:30)))g((cid:30))Md(cid:30)+ (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:30) (cid:30) (cid:3)e H e ((cid:30)) e(cid:0)e s (cid:0) R (cid:30) (cid:30) (cid:3) h e H( (cid:1)(cid:1)(cid:1) ;(cid:30) 0 )d(cid:30) 0 d(cid:21)H((cid:30))+R(cid:22)H e H (cid:30) (cid:0) s e h i h i is s R tationeary at (cid:30) (cid:3) ; R qd;qx;H((cid:30));y . Integrating by paert (cid:0) s t (cid:1) he term involving a double integral and using f g the fact that (cid:21)H is di⁄erentiable, the Lagrangian can be expressed as e e e e (cid:30) qd((cid:30)) (cid:27) (cid:0)(cid:27) 1 g((cid:30))Md(cid:30)+n (cid:30) qx((cid:30)) (cid:27) (cid:0)(cid:27) 1 F (y((cid:30)))g((cid:30))Md(cid:30)+ (cid:30)(cid:3) (cid:30)(cid:3) (cid:1)(cid:1)(cid:1) L((cid:30) (cid:3) ;qd;qx;H;y) = 8 > > < R e (cid:1)(cid:1)(cid:1) R (cid:30) (cid:30) (cid:3) H( (cid:30) (cid:30)) h (cid:21) H H (cid:30) ( ((cid:30)) ; d (cid:30) (cid:30) ) R (cid:21) + H (cid:21) ( e (cid:30) H ) ( d (cid:30) (cid:30) (cid:3) ) + s (cid:0) (cid:22)H H H e (cid:0) (cid:30) (cid:1)(cid:30) (cid:21)H((cid:30) s )+ (cid:1)(cid:1)(cid:1) e e e e (cid:1)(cid:1)(cid:1)e (cid:30)(cid:3) (cid:1)(cid:1)(cid:1) e (cid:0) > > R (cid:2) (cid:0) (cid:1) (cid:3) The stationarity condition :and the constraints of the problem yield the following (cid:133)rst order necessary conditions for an optimum H ((cid:30)) = hH( ;(cid:30)) (cid:30) (cid:1)(cid:1)(cid:1) hH( ;(cid:30))(cid:21)H((cid:30))+(cid:21)H((cid:30)) = 0 H (cid:1)(cid:1) e (cid:1) (cid:30) (cid:27) (cid:0)(cid:27) 1qd((cid:30)) (cid:0)(cid:27) 1 g((cid:30))M +hH qd ( (cid:1)(cid:1)(cid:1) ;(cid:30))(cid:21)H((cid:30)) = 0 (cid:27) (cid:0)(cid:27) 1qx((cid:30)) (cid:0)e (cid:27) 1 (cid:28)nF (y((cid:30)))g((cid:30))M +hH qx ( (cid:1)(cid:1)(cid:1) ;(cid:30))(cid:21)H((cid:30)) = 0 nqx((cid:30)) (cid:27) (cid:0)(cid:27) 1 F y (ey((cid:30)))g((cid:30))Mde(cid:30)+ [1 F + y F (y ( ( y (cid:30) ( ) (cid:30) ) ) (cid:28) ) 1 (cid:28) (cid:0) 1 (cid:27)n (cid:27)n] hH( (cid:1)(cid:1)(cid:1) ;(cid:30))(cid:21)H((cid:30))+ (cid:1)(cid:1)(cid:1) (46) (cid:0) (cid:30) hH( ;(cid:30))(cid:21)H((cid:30))d(cid:30) e e (cid:1)(cid:1) e (cid:1) R (cid:30)(cid:3) Lp (cid:1) w (cid:1)(cid:1) ((cid:30)(cid:3);y) nf x y((cid:30)) e F y (y((cid:30)))g((cid:30))M = 0 (cid:22)H (cid:21)H (cid:30) = 0 e (cid:0) e e H(cid:2)((cid:30) ) = s; (cid:0)H(cid:1)((cid:3)(cid:30)) = s (cid:3) (cid:30) (cid:30) (cid:3) hH (cid:30)(cid:3) ( (cid:1)(cid:1)(cid:1) ;(cid:30))(cid:21)H((cid:30))d(cid:30) = qd((cid:30) (cid:3) ) (cid:27) (cid:0)(cid:27) 1 e +nqx((cid:30) (cid:3) ) (cid:27) e (cid:0)(cid:27) 1 F (y((cid:30) (cid:3) )) g((cid:30) (cid:3) )M +hH( (cid:1)(cid:1)(cid:1) ;(cid:30) (cid:3) )(cid:21)H((cid:30) (cid:3) ): h i R The (cid:133)rst seven lines in (46e) are the staendard necessaery conditions of optimal control theory and re(cid:135)ect the constraints of the problem and the implications of stationarity of the Lagrangian with respect to H;qd;qx;y . The last line in (46) follows from the stationarity with respect to (cid:30) . Below, (cid:3) f g I show that if (cid:30) ;H;qd;qx;y;(cid:21)H satis(cid:133)es (46), then we can de(cid:133)ne functions p((cid:30));r((cid:30)) such that (cid:3) (cid:30) ;p((cid:30));r e ((cid:30)e) f ;He esatisfy the con g ditions of lemma 3 in the appendix, proving f that a solu g tion to the (cid:3) f planner(cid:146)s problem g is e aneeqeuiliebrium of the open economy. e e Leet (cid:30)e;H;qd e ;qx;y;(cid:21)H satisfy the conditions in (46). The third and fourth lines in (46) yield (cid:3) f g qx((cid:30)) = qd((cid:30))(cid:28) (cid:27): For some (still unde(cid:133)ned) positive constant p , de(cid:133)ne (cid:0) 0 e e e e (cid:21)H((cid:30)) e e p((cid:30)) p (cid:27) (cid:0) ; (47) (cid:17) 0(cid:27) (cid:0) 1 A(H((cid:30));(cid:30))V(H((cid:30))) e e e 45

which, together with the second line in (46), implies @lnA(H((cid:30));(cid:30)) p ((cid:30)) = p((cid:30)) (48) (cid:30) (cid:0) @(cid:30) e Using (47) in the third condition in (e46) yieldse q((cid:30)) = p(cid:27)Lpw((cid:30) ;y)(cid:27)p((cid:30)) (cid:27): 0 (cid:3) (cid:0) Accordingly, de(cid:133)ning e e e rd((cid:30)) qd((cid:30))p((cid:30)) (49) (cid:17) we get e e e rd((cid:30)) = p(cid:27)(Lpw((cid:30) ;y))(cid:27)p((cid:30))1 (cid:27); 0 (cid:3) (cid:0) (50) rd((cid:30)) = p 0 Lpw((cid:30) (cid:3) ;y)qd((cid:30)) (cid:27) (cid:0)(cid:27) 1 e e e and e @elneA(H((cid:30));(cid:30)) r ((cid:30)) = ((cid:27) 1)r ((cid:30)) (51) (cid:30) (cid:30) (cid:0) @(cid:30) e Noting that the third condition in (46) yields e e (cid:27) (cid:0)(cid:27) 1qd((cid:30)) (cid:27) (cid:0)(cid:27) 1 1+F (y((cid:30)))(cid:28)1 (cid:0) (cid:27)n g((cid:30))M = (cid:0) hH( (cid:1)(cid:1)(cid:1) ;(cid:30))(cid:21)H((cid:30)); the (cid:133)fth line i(cid:2)n (46) implies (cid:3) e e rd((cid:30))(cid:28)1 (cid:27) (cid:0) y((cid:30)) = ; (cid:27)f C x 0 (52) C 0 (cid:17) (cid:27) (cid:0)(cid:27) 1 R e(cid:30) (cid:30) (cid:3) rd((cid:30))[1 e +F L ( p y w ( ( (cid:30) (cid:30) ) (cid:3) )(cid:28) ;y 1 ) (cid:0) (cid:27)n]g((cid:30))Md(cid:30) : e e With the derivations above in mind, the (cid:133)rst condition in (46) becomes e rd((cid:30)) 1+F rd((cid:30))(cid:28)1 (cid:0) (cid:27) (cid:28)1 (cid:27)n g((cid:30))M H ((cid:30)) = (cid:27)fxC0 (cid:0) (53) (cid:30) A(H((cid:30)) h ;(cid:30))V(H (cid:16)e ((cid:30)))p((cid:30)) (cid:17) Lpw((cid:30) i ; rd((cid:30))(cid:28)1 (cid:0) (cid:27) ) e (cid:3) (cid:27)fxC0 e e Finally, using the previous observateions and e e (cid:30) hH( ;(cid:30))(cid:21)H((cid:30))d(cid:30) = (cid:30) hH( ;(cid:30))(cid:21)H((cid:30)) fg((cid:30)(cid:3))+nfx y y((cid:30)(cid:3)) ydF(y)g((cid:30)(cid:3))M d(cid:30) (cid:30)(cid:3) (cid:30)(cid:3) (cid:1)(cid:1)(cid:1) (cid:0) (cid:30)(cid:3) (cid:1)(cid:1)(cid:1) Lpw(R(cid:30)(cid:3) e;rd (cid:27) ((cid:30) f ) x (cid:28) C 1 0 (cid:0) (cid:27) ) R R e in the last condition of (46) yields rd((cid:30)(cid:3))(cid:28)1 (cid:0) (cid:27) 2 (cid:27)f+n(cid:27)fx y (cid:27)fxC0 ydF(y) 3 e 6 4 [1+F R (y((cid:30)(cid:3)))(cid:28)1 (cid:0) (cid:27)n] 7 5 (cid:30) (cid:30) (cid:3) r r d d ( ( (cid:30) (cid:30) (cid:3) ) ) 1+F rd (cid:27) ((cid:30) fx )(cid:28) C 1 0 (cid:0) (cid:27) (cid:28)1 (cid:0) (cid:27)n g((cid:30))Md(cid:30) = (cid:27) (cid:0) (cid:27) 1 Lpw((cid:30) (cid:3) ; rd (cid:27) ((cid:30) fx )(cid:28) C 1 0 (cid:0) (cid:27) ): If we choose the constant p 0R in (e50) shuch tha(cid:16)te (cid:17) i e e e 46

rd((cid:30)(cid:3))(cid:28)1 (cid:0) (cid:27) (cid:27)f +n(cid:27)f (cid:27)fxC0 ydF(y) x ye " # rd((cid:30) ) = ; (54) (cid:3) R 1+F rd((cid:30)(cid:3))(cid:28)1 (cid:0) (cid:27) (cid:28)1 (cid:27)n (cid:27)fxC0 (cid:0) e h (cid:16)e (cid:17) i then the previous condition becomes (cid:30) (cid:27) rd((cid:30)) 1+F rd((cid:30))(cid:28)1 (cid:0) (cid:27) (cid:28)1 (cid:27)n g((cid:30))Md(cid:30) = Lpw((cid:30) ; rd((cid:30))(cid:28)1 (cid:0) (cid:27) ) (55) (cid:27)fx (cid:0) (cid:27) 1 (cid:3) (cid:27)fx Z (cid:30)(cid:3) h (cid:16)e (cid:17) i (cid:0) e e as (52) yields C = 1. Note that conditions (48), (51), (53),(55), C = 1 and H((cid:30) ) = s;H((cid:30)) = 0 0 (cid:3) f s , imply that (cid:30) ;p;rd;H satisfy all the conditions in lemma 3 with the exception of rd((cid:30) ) = (cid:27)f. (cid:3) (cid:3) A g ccordingly, per f condition ( g 54), p;rd;H is an equilibrium of the open economy only e if F rd((cid:30)(cid:3))(cid:28)e1 (cid:0) (cid:27) = rd((cid:30)(cid:3))(cid:28)1 (cid:0) (cid:27) e e e f g (cid:16) e e (cid:27)fx (cid:17) ye (cid:27)fxC0 ydF(y) = 0, a conditioenethaet is satis(cid:133)ed when the restriction on parameters assumed in the paper holds, f(cid:28)1 (cid:27) f . As in the case of the closed economy, we can walk back on this derivations R (cid:0) x (cid:20) to show that given a triplet p;rd;H corresponding to an equilibrium of the open economy, then qd f g f (cid:17) rd ;H is a solution to the planner(cid:146)s problem above when f(cid:28)1 (cid:27) f . p g (cid:0) (cid:20) x When the restriction on parameters f(cid:28)1 (cid:27) f is not satis(cid:133)ed, the equivalence between equilibria (cid:0) x (cid:20) of the open economy and solutions to problem (44) no longer holds. Intuitively, if f(cid:28)1 (cid:27) > f , then (cid:0) x the planner is willing to accept some "negative domestic pro(cid:133)ts", rd((cid:30) ) < (cid:27)f, because they are more (cid:3) rd((cid:30)(cid:3))(cid:28)1 (cid:0) (cid:27) than o⁄set by positive export pro(cid:133)ts, rd((cid:30) )F rd((cid:30)(cid:3))(cid:28)1 (cid:0) (cid:27) n(cid:28)1 (cid:27) > (cid:27)f (cid:27)fx ydF(y). However, (cid:3) (cid:27)fx (cid:0) e x y e by changing slightly the arguments above, it can(cid:16)ebe shown(cid:17)that when f(cid:28)1 R(cid:0) (cid:27) > f x , the equilibria of the open economy are equivalent to solutioens to constrained planner(cid:146)s problems that feature the following additional constraint (cid:27) 1 (cid:30) qd((cid:30)) (cid:0)(cid:27) (cid:27) (cid:27)f 1+F (y((cid:30)))(cid:28)1 (cid:27)n g((cid:30))Md(cid:30) = Lpw((cid:30) ;y((cid:30))): qd((cid:30) ) (cid:0) (cid:27) 1 (cid:3) Z (cid:30)(cid:3) (cid:20) (cid:3) (cid:21) (cid:0) e (cid:2) (cid:3) e e Accordingly, (cid:25)the equilibrium is constrained e¢ cient in this case. e A.3 Additional Results related to BVP (22) In this section, I present some results related to BVP (22) that are used in the text and in the proof of other results. Lemma 4 Fori = a;b, let zi;xi;(cid:0)i betheuniquesolutiontotheBVP(22)withparameters (cid:11)i((cid:30));Ki;Ki 0 1 and boundary conditions xi((cid:30) ) = 1, (cid:0)i((cid:30) ) = s and (cid:0)i((cid:30) ) = s . (cid:8) 0 (cid:9) 0 0 1 1 (cid:8) (cid:9) (i) Suppose that Ki = 0, (cid:11)a((cid:30)0) (cid:11)b((cid:30)0) for all (cid:30) > (cid:30) [(cid:30) ;(cid:30) ], and (cid:11)a((cid:30)0) (cid:11)b((cid:30)0) for all (cid:30) > (cid:30) on 1 (cid:11)a((cid:30)) (cid:21) (cid:11)b((cid:30)) 0 2 0 1 (cid:11)a((cid:30)) (cid:21) (cid:11)b((cid:30)) 0 some subinterval [(cid:30) ;(cid:30) ] [(cid:30) ;(cid:30) ]. Then (cid:0)a((cid:30)) < (cid:0)b((cid:30)) for all (cid:30) ((cid:30) ;(cid:30) ) and (cid:0)a((cid:30) ) < (cid:0)b ((cid:30) ) and l h (cid:18) 0 1 2 0 1 (cid:30) 0 (cid:30) 0 47

(cid:0)a((cid:30) ) > (cid:0)b ((cid:30) ). (cid:30) 1 (cid:30) 1 (ii) Suppose that Ki = K ; (cid:11)i((cid:30)) = (cid:11)((cid:30)) and Kb < Ka. Then (cid:0)a((cid:30) ) < (cid:0)b ((cid:30) ), so there is a 0 0 1 1 (cid:30) 0 (cid:30) 0 (cid:30)+ ((cid:30) ;(cid:30) ] such that (cid:0)a (cid:30)+ = (cid:0)b (cid:30)+ and (cid:0)a((cid:30)) < (cid:0)b((cid:30)) for all (cid:30) ((cid:30) ;(cid:30)+). 0 1 0 2 2 (iii) Let (cid:8)i (cid:30) 1xi((cid:30)) [1+(cid:0)F(K(cid:1) 0 ixi((cid:30)))K(cid:0) 1 i] (cid:11)(cid:1)i((cid:30)) g((cid:30))d(cid:30): If (cid:0)a((cid:30)) < (cid:0)b((cid:30)) for (cid:30) ((cid:30) ;(cid:30) ); then (cid:8)a > (cid:8)b. (cid:17) (cid:30) 0 [1+F(K 0 i)K 1 i] (cid:11)i((cid:30) 0 ) 2 0 1 (iv) If (cid:11)i((cid:30)) R = (cid:11)((cid:30)); Kb = (cid:21)Ka and Kb = (cid:21)Ka for (cid:21) > 1, then xb((cid:30))(cid:21) > xa((cid:30)) for all for all 0 0 1 1 (cid:30) [(cid:30) ;(cid:30) ]: 0 1 2 (v) Let (cid:14)i((cid:30)) 1+F Kixi((cid:30)) Ki (cid:11)i((cid:30)). If (cid:0)a = (cid:0)b and, (cid:14)a((cid:30)) < (cid:14)b((cid:30)) for all (cid:30) [(cid:30) ;(cid:30) ], then (cid:17) 0 1 6 2 0 1 (cid:2) (cid:0) (cid:1) (cid:3) (cid:30) (cid:30) 1 1 xa((cid:30))(cid:14)a((cid:30))g((cid:30))d(cid:30) < xb((cid:30))(cid:14)b((cid:30))g((cid:30))d(cid:30): (56) (cid:30) (cid:30) Z 0 Z 0 (vi) Suppose that (cid:11)i((cid:30));K 1 i = f (cid:11)((cid:30));K 1 g , K 0 i;K 1 2 R++ and K 0 a > K 0 b. If the function (cid:17) 0 (t;(cid:21)) (cid:17) Fy(t(cid:21))(cid:21)K1 is strictly decreasing (increasing) in (cid:21) for all t [Kb;Kbxb((cid:30) )], then (cid:0)a((cid:30)) > (<)(cid:0)b((cid:30)) on [1+F(t(cid:21))K1] (cid:8) (cid:9) 2 0 0 1 ((cid:30) ;(cid:30) ), with (cid:0)a((cid:30) ) > (<)(cid:0)b ((cid:30)). 0 1 (cid:30) 0 (cid:30) (vii) Suppose that (cid:11)i((cid:30)) = (cid:11)((cid:30)); K 0 i;K 1 i 2 R++ and K i a = (cid:21)K i b for (cid:21) > 1. If the function (cid:17) 1 (t;(cid:21)) (cid:17) Fy(t(cid:21))(cid:21)2K 1 b is strictly increasing (decreasing) in (cid:21) for all t [Kb;Kbxb((cid:30) )], then (cid:0)a((cid:30)) < (>)(cid:0)b((cid:30)) on [1+F(t(cid:21))(cid:21)Kb] 2 0 0 1 1 ((cid:30) ;(cid:30) ) with (cid:0)a((cid:30) ) < (>)(cid:0)b ((cid:30) ). 0 1 (cid:30) 0 (cid:30) 0 Proof. Lemma 4.i. I proceed in steps. STEP 1: Under the assumptions of the lemma, (cid:0)a((cid:30)) (cid:0)b((cid:30)) for all (cid:30) ((cid:30) ;(cid:30) ). 0 1 (cid:20) 2 Suppose to the contrary that there is a (cid:30) ((cid:30) ;(cid:30) ) such that (cid:0)a (cid:30) > (cid:0)b (cid:30) . Let (cid:30) be the (cid:133)rst 0 0 1 0 0 2 (cid:0) time the functions (cid:0)a and (cid:0)b intersect to the left of (cid:30) and let (cid:30) be the (cid:133)rst time they intersect to the 0 + (cid:0) (cid:1) (cid:0) (cid:1) right of (cid:30), i.e. (cid:30) max (cid:30) (cid:30) : (cid:0)a((cid:30)) = (cid:0)b((cid:30)) and (cid:30) = inf (cid:30) (cid:30) : (cid:0)a((cid:30)) = (cid:0)b((cid:30)) . Note that 0 0 + 0 (cid:0) (cid:17) (cid:20) (cid:21) (cid:30) and(cid:30) arewellde(cid:133)nedduetothecontinuityofthefunctions(cid:0)a and(cid:0)b andthefactthatthefunctions + (cid:8) (cid:9) (cid:8) (cid:9) (cid:0) intersect at least once to the left and to the right of (cid:30) (at (cid:30) and at (cid:30) ). Also note that (cid:0)a((cid:30)) > (cid:0)b((cid:30)) 0 0 1 for (cid:30) (cid:30) ;(cid:30) . The continuity of (cid:0)a and (cid:0)b, implies (cid:0)a((cid:30) ) (cid:0)b((cid:30) ) and (cid:0)a((cid:30) ) (cid:0)b((cid:30) ), i.e. 2 (cid:0) + (cid:30) (cid:30) (cid:30) (cid:0) (cid:21) (cid:30) (cid:0) (cid:30) + (cid:20) (cid:30) + (cid:0) (cid:1) (cid:0)a((cid:30) )=(cid:0)a((cid:30) ) (cid:30) + (cid:30) (cid:0) 1. (57) (cid:0)b((cid:30) )=(cid:0)b((cid:30) ) (cid:20) (cid:30) + (cid:30) (cid:0) Di⁄erentiating the right-hand side of (26) yields (cid:0)i (cid:30) (cid:30) + = hi (cid:30) ;(cid:30) e (cid:27) (cid:30) (cid:30)+ @lnA( @ (cid:0) (cid:30) i(u);u) du ; (58) (cid:0)i (cid:30)(cid:0) (cid:30) (cid:1) (cid:0) + R (cid:0) (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) where hi (cid:30) ;(cid:30) is given by (27) with (cid:11) = (cid:11)i. By assumption, we have (cid:0)a (cid:30) = (cid:0)b (cid:30) and + (cid:0)a (cid:30) + =(cid:0)(cid:0) (cid:0) b (cid:30) +(cid:1), which together with the de(cid:133)nition of hi, imply h h a b( ( (cid:30) (cid:30) (cid:0) ; ; (cid:30) (cid:30) + + ) ) = (cid:11) (cid:11) a b( ( (cid:30) (cid:30) (cid:0) + + ) ) = = (cid:0) (cid:11) (cid:11) (cid:1) b a ( ( (cid:30) (cid:30) (cid:0)) ) . C(cid:0)om (cid:0) b(cid:1)ining (cid:0) (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) 48

this result and (58) yields (cid:0)a((cid:30) )=(cid:0)a((cid:30) ) (cid:27) (cid:30)+ @lnA((cid:0)a(u);u) @lnA((cid:0)b(u);u) du(cid:11)a((cid:30) )=(cid:11)a((cid:30) ) (cid:30) + (cid:30) (cid:0) = e (cid:30) (cid:0)" @(cid:30) (cid:0) @(cid:30) # + (cid:0) : (59) (cid:0)b((cid:30) )=(cid:0)b((cid:30) ) R (cid:11)b((cid:30) )=(cid:11)b((cid:30) ) (cid:30) + (cid:30) + (cid:0) (cid:0) Thestrictlog-supermodularityofAandthefactthat(cid:0)a((cid:30)) > (cid:0)b((cid:30))for(cid:30) (cid:30) ;(cid:30) implythatthe(cid:133)rst + 2 (cid:0) term of the last expression is strictly greater than 1. In addition, the assumption about relative values (cid:0) (cid:1) (cid:0)a((cid:30) )=(cid:0)a((cid:30) ) of (cid:11)a and (cid:11)b on [(cid:30) ;(cid:30) ] implies that the second term is weakly greater than one, i.e. (cid:30) + (cid:30) > 1. 0 1 (cid:0)b((cid:30) )=(cid:0)b((cid:30) (cid:0) ) (cid:30) + (cid:30) This result contradicts (57), so it must be that (cid:0)a((cid:30)) (cid:0)b((cid:30)) for (cid:30) [(cid:30) ;(cid:30) ]. (cid:0) 0 1 (cid:20) 2 STEP 2: Under the assumptions in the lemma, (cid:0)a((cid:30)) and (cid:0)b((cid:30)) cannot satisfy (cid:0)a((cid:30)) = (cid:0)b((cid:30)) on any non-degenerate interval I [(cid:30) ;(cid:30) ]. l h (cid:18) Suppose to the contrary that (cid:0)a((cid:30)) = (cid:0)b((cid:30)) for some non-degenerate interval I [(cid:30) ;(cid:30) ] and let l h (cid:18) (cid:30) < (cid:30) be two interior points of I. Notice that (cid:0)a((cid:30)) = (cid:0)b((cid:30)) on I implies that (cid:0)a((cid:30)) = (cid:0)b ((cid:30)) on the + (cid:30) (cid:30) (cid:0) interior of I, so (cid:0)a((cid:30) )=(cid:0)a((cid:30) ) (cid:30) + (cid:30) (cid:0) = 1. (60) (cid:0)b((cid:30) )=(cid:0)b((cid:30) ) (cid:30) + (cid:30) (cid:0) In addition, equation (59) must also hold in this case, which under the current assumptions yields (cid:0)a((cid:30) )=(cid:0)a((cid:30) ) (cid:11)a((cid:30) )=(cid:11)a((cid:30) ) (cid:30) + (cid:30) (cid:0) = + (cid:0) > 1; (cid:0)b((cid:30) )=(cid:0)b((cid:30) ) (cid:11)b((cid:30) )=(cid:11)b((cid:30) ) (cid:30) + (cid:30) + (cid:0) (cid:0) where the strict inequality follows from (cid:30) ;(cid:30) [(cid:30) ;(cid:30) ] and the assumption about relative values of (cid:11)a + l h (cid:0) 2 and (cid:11)b on this interval. The last expression contradicts (60). Then it must be the case that (cid:0)a and (cid:0)b cannot be equal on any non-degenerate interval I [(cid:30) ;(cid:30) ]. l h (cid:18) Figure 4: Solutions to the General BVP ( 22), (cid:0) Note: The (cid:133)gure depicts solutions to alternative parametrizations of the general BVP (22). The BVPs corresponding to (cid:0)a and (cid:0)b di⁄eronly in the parameterfunction (cid:11)((cid:30))asindicated in lemma 4.iRestricted to [(cid:30)0;(cid:30) 1 ],the BVPs corresponding to (cid:0)b and (cid:0) b di⁄eronly in theirinitialconditions. 49

STEP 3: Under the assumptions in the lemma, (cid:0)a((cid:30)) < (cid:0)b((cid:30)) for all (cid:30) ((cid:30) ;(cid:30) ) 0 1 2 Steps 1 and 2 imply that there is a (cid:30) ((cid:30) ;(cid:30) ) [(cid:30) ;(cid:30) ] such that (cid:0)a (cid:30) < (cid:0)b (cid:30) . The situation 0 l h 0 1 0 0 2 (cid:18) is depicted in (cid:133)gure 4. Now, I prove that (cid:0)a((cid:30)) < (cid:0)b((cid:30)) on [(cid:30);(cid:30) ). To establish this result I show 0 1 (cid:0) (cid:1) (cid:0) (cid:1) that there exists a function (cid:0) b : (cid:30);(cid:30) S (dashed blue line), such that (cid:0)a((cid:30)) (cid:0) b ((cid:30)) < (cid:0)b((cid:30)) 0 1 ! (cid:20) for all (cid:30) [(cid:30);(cid:30) ). Letting s (cid:0)i (cid:30) for i = a;b, if we de(cid:133)ne on (cid:30);(cid:30) , wb((cid:30)) xb((cid:30))=xb (cid:30) 2 0 1 0i (cid:17) (cid:2) 0(cid:3) 0 1 (cid:17) 0 and yb((cid:30)) zb((cid:30))=xb (cid:30) , then yb;wb;(cid:0)b is the unique solution to the BVP (22) on (cid:30);(cid:30) with 0 (cid:0) (cid:1) (cid:2) (cid:3) 0 1 (cid:0) (cid:1) (cid:17) f g parameters (cid:11)b((cid:30));Kb;Kb and boundary conditions w (cid:30) = 1, (cid:0)b (cid:30) = s and (cid:0)b((cid:30) ) = s .51 0(cid:0) (cid:1)1 0 0 0b (cid:2) 1 (cid:3) 1 Now, let zb;xb;(cid:0) b be the unique solution to the BVP (22) on (cid:30);(cid:30) with the same parameters and (cid:8) (cid:9) (cid:0) (cid:1) 0 1 (cid:0) (cid:1) f g boundary conditions xb (cid:30) = 1, (cid:0) b (cid:30) = s < s and (cid:0) b ((cid:30) ) = s . It is readily seen that zb;xb;(cid:0) b 0 0 0a 0b 1 (cid:2) 1 (cid:3) f g and yb;wb;(cid:0)b satisfy the conditions of the no-crossing result in lemma 2.ii with (cid:0) b ((cid:30)) < (cid:0)b((cid:30)), so (cid:0) (cid:1) (cid:0) (cid:1) 0 0 (cid:0) b ((cid:30)) < (cid:0)b((cid:30)) on [(cid:30);(cid:30) ). De(cid:133)ning wa and ya on (cid:30);(cid:30) from xa and za as I did above implies that (cid:8) (cid:9) 0 1 0 1 ya;wa;(cid:0)a is the unique solution to the BVP (22) on (cid:30);(cid:30) with parameters (cid:11)a((cid:30));Ka;Ka and f g (cid:2) (cid:3)0 1 f 0 1g boundary conditions wa (cid:30) = 1, (cid:0)a (cid:30) = s and (cid:0)a((cid:30) ) = s . Then, wa;ya;(cid:0)a and zb;xb;(cid:0) b 0 0 0a (cid:2) 1 (cid:3) 1 f g f g satisfy the conditions of step 1 above, so (cid:0)a((cid:30)) (cid:0) b ((cid:30)) on (cid:30);(cid:30) as depicted in the (cid:133)gure. (cid:0) (cid:1) (cid:0) (cid:1) 0 1 (cid:20) The argument in the last paragraph can be easily adapted to show that there is a function (cid:0)b : (cid:2) (cid:3) (cid:30) ;(cid:30) S, such that (cid:0)a((cid:30)) (cid:0)b((cid:30)) < (cid:0)b((cid:30)) for all (cid:30) ((cid:30) ;(cid:30)], completing the proof of step 3. 0 0 0 0 ! (cid:20) 2 Of note, this part of the argument requires slightly di⁄erent version of the no-crossing in theorem 2.ii. (cid:2) (cid:3) Speci(cid:133)cally, in the notation of theorem 2, it can be shown that if we consider the solution to BVP (22) as a function of (s ;s ), then (cid:0)((cid:30);s ;sa) < (cid:0) (cid:30);s ;sb on ((cid:30) ;(cid:30) ] if sa < sb. 0 1 0 1 0 1 0 1 1 1 STEP 4: Under the same assumptions made in step 3, (cid:0)a((cid:30) ) < (cid:0)b ((cid:30) ) and (cid:0)a((cid:30) ) > (cid:0)b ((cid:30) ). (cid:0) (cid:1) (cid:30) 0 (cid:30) 0 (cid:30) 1 (cid:30) 1 Let (cid:30) ((cid:30) ;(cid:30) ) and the triplets of functions ya;wa;(cid:0)a , zb;xb;(cid:0) b and yb;wb;(cid:0)b on (cid:30);(cid:30) be 0 0 1 0 1 2 f g f g f g de(cid:133)nedasinstep3. Giventhat(cid:0)a((cid:30)) (cid:0) b ((cid:30))on (cid:30);(cid:30) , thenitmustbethecasethat(cid:0)a((cid:30) ) (cid:0) b ((cid:30) ); (cid:20) 0 1 (cid:30) 1 (cid:2) (cid:21) (cid:30) (cid:3)1 otherwise (cid:0)a((cid:30)) > (cid:0) b ((cid:30)) on some neighborhood of (cid:30) . In a similar way, (cid:0) b ((cid:30)) < (cid:0)b((cid:30)) on on [(cid:30);(cid:30) ) (cid:2) 1 (cid:3) 0 1 implies (cid:0) b ((cid:30)) (cid:0)b ((cid:30)). Moreover, if (cid:0) b ((cid:30)) = (cid:0)b ((cid:30)), then yb;wb;(cid:0)b (cid:151)with yb((cid:30)) = yb((cid:30)) xb((cid:30) ) and (cid:30) (cid:21) (cid:30) (cid:30) (cid:30) wb((cid:30) 1 ) 1 wb((cid:30)) = wb((cid:30)) xb((cid:30) )(cid:151)and zb;xb;(cid:0) b satisfy the same I(cid:8)VP with i(cid:9)nitial condition at (cid:30) , so (cid:0) b = (cid:0)b wb((cid:30) 1 ) 1 f g 1 on [(cid:30);(cid:30) ], contradicting our earlier results. Then it must be the case that (cid:0) b ((cid:30) ) > (cid:0)b ((cid:30) ). Putting 0 1 (cid:30) 1 (cid:30) 1 together these results we get (cid:0)a((cid:30) ) (cid:0) b ((cid:30) ) > (cid:0)b ((cid:30) ). The other part of the claim can be proved (cid:30) 1 (cid:21) (cid:30) 1 (cid:30) 1 making only minor adjustments to this argument. Lemma 4.ii. I proceed in steps. STEP 1: Under the assumptions of the lemma, there is no (cid:30) ((cid:30) ;(cid:30) ] such that (cid:0)a((cid:30)) (cid:0)b((cid:30)) for 0 0 1 2 (cid:21) all (cid:30) ((cid:30) ;(cid:30)]. 0 0 2 Suppose to the contrary that there is such a value (cid:30) ((cid:30) ;(cid:30) ]. Let (cid:30) be the (cid:133)rst time the functions 0 0 1 + 2 (cid:0)a and (cid:0)b intersect to the right of (cid:30), i.e. (cid:30) = inf (cid:30) (cid:30) : (cid:0)a((cid:30)) = (cid:0)b((cid:30)) . Note (cid:30) is well de(cid:133)ned 0 + 0 + (cid:21) due to the continuity of the functions (cid:0)a and (cid:0)b and the fact that the functions intersect at least once (cid:8) (cid:9) to the right of (cid:30) (at (cid:30) ). Also note that (cid:0)a((cid:30)) (cid:0)b((cid:30)) for (cid:30) (cid:30) ;(cid:30) . The continuity of (cid:0)a and (cid:0)b, 0 1 (cid:21) 2 0 + (cid:30) (cid:30) 51Note that we are using the same notation to denote the restriction of a fu(cid:0)nction t(cid:1)o a subset of its domain. 50

implies (cid:0)a((cid:30) ) (cid:0)b((cid:30) ) and (cid:0)a((cid:30) ) (cid:0)b((cid:30) ), i.e. (cid:30) 0 (cid:21) (cid:30) 0 (cid:30) + (cid:20) (cid:30) + (cid:0)a((cid:30) )=(cid:0)a((cid:30) ) (cid:30) + (cid:30) 0 1. (61) (cid:0)b((cid:30) )=(cid:0)b((cid:30) ) (cid:20) (cid:30) + (cid:30) 0 Di⁄erentiating the right-hand side of (26) yields (cid:0)i (cid:30) (cid:30) + = hi (cid:30) ;(cid:30) e (cid:27) (cid:30) (cid:30)+ @lnA( @ (cid:0) (cid:30) i(u);u) du 1+F K 0 xi((cid:30)) K 1 i ; (62) (cid:0)i ((cid:30) ) 0 + (cid:0) 1+F (K )Ki (cid:30)(cid:0) 0(cid:1) R (cid:2) (cid:0) 0 (cid:1)1 (cid:3) (cid:0) (cid:1) (cid:2) (cid:3) where hi (cid:30) ;(cid:30) is given by (27). By assumption, we have (cid:0)a((cid:30) ) = (cid:0)b((cid:30) ) and (cid:0)a (cid:30) = (cid:0)b (cid:30) , 0 + 0 0 + + which together with the de(cid:133)nition of hi, imply ha (cid:30) ;(cid:30) = hb (cid:30) ;(cid:30) . Combining this result with (62) (cid:0) (cid:1) 0 + 0 + (cid:0) (cid:1) (cid:0) (cid:1) for i = a;b yields (cid:0) (cid:1) (cid:0) (cid:1) (cid:0)a((cid:30) )=(cid:0)a((cid:30) ) (cid:27) (cid:30)+ @lnA((cid:0)a(u);u) @lnA((cid:0)b(u);u) du 1+F K xa (cid:30) Ka =[1+F (K )Ka] (cid:30) + (cid:30) 0 = e (cid:30) (cid:0)" @(cid:30) (cid:0) @(cid:30) # 0 + 1 0 1 : (63) (cid:0)b((cid:30) )=(cid:0)b((cid:30) ) R 1+F K xb (cid:30) Kb = 1+F (K )Kb (cid:30) + (cid:30) 0 (cid:2) (cid:0) 0 (cid:0) +(cid:1)(cid:1) 1(cid:3) 0 1 (cid:2) (cid:0) (cid:0) (cid:1)(cid:1) (cid:3) (cid:2) (cid:3) The strict log-supermodularity of A and the fact that (cid:0)a((cid:30)) (cid:0)b((cid:30)) for (cid:30) (cid:30) ;(cid:30) imply that the 0 + (cid:21) 2 (cid:133)rst term of the right-hand side of the last expression is weakly greater than 1. In addition, note that we (cid:0) (cid:1) can write [1+F(K0xi((cid:30)))K 1 i] = 1 + F(K0)K 1 i F(K0xi((cid:30))) ; [1+F(K0)K 1 i] [1+F(K0)K 1 i] [1+F(K0)K 1 i] F(K0) so xa((cid:30)) xb((cid:30)) for (cid:30) (cid:30) ;(cid:30) ((cid:0)a((cid:30)) (cid:0)b((cid:30))) and Ka > Kb imply that the second term of the (cid:21) 2 0 + (cid:21) 1 1 (cid:0)a((cid:30) )=(cid:0)a((cid:30) ) right-hand side of (63) is strictly higher than one. Accordingly, (cid:30) + (cid:30) 0 > 1; contradicting (61). (cid:0) (cid:1) (cid:0)b((cid:30) )=(cid:0)b((cid:30) ) (cid:30) + (cid:30) 0 STEP 2: Under the assumptions of the lemma, (cid:0)a((cid:30) ) < (cid:0)b ((cid:30) ), immediately proving the lemma. (cid:30) 0 (cid:30) 0 The result of step 1 immediately yields that (cid:0)a((cid:30) ) (cid:0)b ((cid:30) ). Otherwise, (cid:0)a((cid:30) ) > (cid:0)b ((cid:30) ) implies (cid:30) 0 (cid:20) (cid:30) 0 (cid:30) 0 (cid:30) 0 that there is a (cid:30) ((cid:30) ;(cid:30) ] such that (cid:0)a((cid:30)) > (cid:0)b((cid:30)) on ((cid:30) ;(cid:30)], contradicting the result in step 1. 0 0 1 0 0 2 Suppose then that (cid:0)a((cid:30) ) = (cid:0)b ((cid:30) ) = (cid:13) : Note that the (same) boundary conditions of the BVPs under (cid:30) 0 (cid:30) 0 0 consideration imply (cid:0)i((cid:30) ) = s , xi((cid:30) ) = 1: In turn, these observations and equations (22a)-(22b) imply 0 0 0 xi (cid:30) ((cid:30)) = ((cid:27) (cid:0) 1)@l @ n (cid:30) A(s0;(cid:30) 0 ) and z z (cid:30) i i( ( (cid:30) (cid:30) 0 0 ) ) = (cid:0) @lnA @ ( (cid:30) s0;(cid:30) 0 ) . Log-di⁄erentiating both sides of equation (22c) and evaluating at (cid:30) yields 0 (cid:0)i (cid:30)(cid:30) ((cid:30) 0 ) = xi (cid:30) ((cid:30) 0 ) + Fy (K0xi((cid:30) 0 ))K 1 iK0xi (cid:30) ((cid:30) 0 ) + (cid:11)(cid:30)((cid:30) 0 ) + g(cid:30)((cid:30) 0 ) [ @lnA((cid:0)i((cid:30) 0 );(cid:30) 0 ) (cid:0)i ((cid:30) ) + @lnA((cid:0)i((cid:30) 0 );(cid:30) 0 ) + (cid:0)i (cid:30) ((cid:30) 0 ) xi((cid:30) 0 ) [1+F(K0xi((cid:30) 0 ))K 1 i] (cid:11)((cid:30) 0 ) g((cid:30) 0 ) (cid:0) @s (cid:30) 0 @(cid:30) Vs ((cid:0)i((cid:30) 0 )) (cid:0)i ((cid:30) )+ z (cid:30) i((cid:30) 0 ) ]; V((cid:0)i((cid:30) )) (cid:30) 0 zi((cid:30) ) 0 0 (cid:0)i (cid:30) (cid:13) (cid:30) 0 ((cid:30)) = ((cid:27) (cid:0) 1)@l @ n (cid:30) A(s0;(cid:30) 0 ) + Fy(K0)K [1 1 i + K F 0 ( ( (cid:27) K (cid:0) 0 1 ) ) K @ 1 l i @ n ] (cid:30) A(s0;(cid:30)0) + (cid:11) (cid:11) (cid:30) ( ( (cid:30) (cid:30) 0 0 ) ) + g g (cid:30) ( ( (cid:30) (cid:30) 0 0 ) ) (cid:0) [ @lnA @ ( s s0;(cid:30) 0 ) (cid:13) 0 + @lnA @ ( (cid:30) s0;(cid:30) 0 ) + Vs(s0) (cid:13) @lnA(s0;(cid:30) 0 ) ]; V(s0) 0 (cid:0) @(cid:30) i.e., (cid:0)a (cid:30)(cid:30) ((cid:30) 0 ) (cid:0) (cid:0)b (cid:30)(cid:30) ((cid:30) 0 ) = Fy F (K (K 0 0 )K ) 0((cid:27) (cid:0) 1)@l @ n (cid:30) A(s0;(cid:30) 0 ) (cid:13) 0 (cid:26) [1+ F F (K (K 0) 0 K )K 1 a 1 a] (cid:0) [1+ F F (K (K 0) 0 K )K 1 b 1 b] (cid:27) > 0, where the inequality follows from Ka > Kb. The last expression implies that there is some (cid:30) ((cid:30) ;(cid:30) ] 1 1 0 2 0 1 such that (cid:0)a((cid:30)) > (cid:0)b ((cid:30)) on ((cid:30) ;(cid:30)], which yields a contradiction of step 1. Accordingly, we must have (cid:30) (cid:30) 0 0 51

(cid:0)a((cid:30) ) < (cid:0)b ((cid:30) ). (cid:30) 0 (cid:30) 0 Finally, (cid:0)a((cid:30) ) < (cid:0)b ((cid:30) ) implies (cid:0)a((cid:30)) < (cid:0)b((cid:30)) on some (small enough) interval (cid:30) ;(cid:30) , so (cid:30)+ (cid:30) 0 (cid:30) 0 0 00 described in the lemma is the (cid:133)rst time (cid:0)a and (cid:0)b intersect to the right of (cid:30) . 00 (cid:0) (cid:1) Lemma 4.iii. The idea of the proof is to show that (cid:0)a and (cid:0)b can be thought of as the inverse of the matching functions of two arti(cid:133)cial economies, and then use this additional information to prove the result. Let zi;xi;(cid:0)i be the solution to the BVP in the statement of the lemma and consider the following arti(cid:133)cial economy. In this economy there are no (cid:133)xed costs of production and no (cid:133)xed costs to export but the (cid:8) (cid:9) set of active (cid:133)rms and the set of exporters are (cid:133)xed. In particular, the set of active (cid:133)rms are those with productivityintherange[(cid:30) ;(cid:30) ], whilethefractionof(cid:133)rmsthatexportateachproductivitylevelisgiven 0 1 by F Kixi((cid:30)) . The set of available workers are those with skills in the range [s ;s ]. The distribution 0 0 1 of skills is given by the restriction of V to [s ;s ] and the mass of workers is s1V (s)dsL. The total (cid:0) (cid:1) 0 1 s0 mass of (cid:133)rms with productivity (cid:30) is given by g((cid:30))(cid:11)i((cid:30))M, so the total mass of (cid:133)rms is (cid:30) 1g((cid:30))(cid:11)i((cid:30))M: R (cid:30) 0 Finally, (cid:28) is set to satisfy Ki (cid:28)1 (cid:27). i 1 (cid:17) i(cid:0) R Now I show that if pi, rd;i and Hi denote the price, domestic revenue and inverse-matching functions of the economy described above, then Hi = (cid:0)i. An argument similar to the one in section 4 implies that pi;rd;i;Hi satisfy the di⁄erential equations (13), (14) and (cid:8) (cid:9) rd;i((cid:30)) 1+F Kixi((cid:30)) Ki g((cid:30))(cid:11)i((cid:30))M Hi ((cid:30)) = 0 1 ; (64) (cid:30) A(Hi((cid:30));(cid:30))V (Hi((cid:30)))pi((cid:30))L (cid:2) (cid:0) (cid:1) (cid:3) with boundary conditions Hi((cid:30) ) = s and Hi((cid:30) ) = s . Note that we don(cid:146)t have a boundary condition 0 0 1 1 onthedomesticrevenuefunctionrd;i,asthezero-pro(cid:133)tconditionfor(cid:133)rmswithproductivity(cid:30) isnolonger 0 an equilibrium condition (no (cid:133)xed costs of production). As a result, the levels of the functions rd;i and pi cannot be determined without an additional condition (provided below). However, these conditions are enough to pin down Hi. To see this, let pi;rd;i;Hi be any triplet of functions satisfying the equilibrium conditionsdescribedabove, andde(cid:133)ne(cid:14)i((cid:30)) 1+F Kixi((cid:30)) Ki (cid:11)i((cid:30)), vi((cid:30)) rd;i((cid:30))=rd;i((cid:30) )and (cid:8) (cid:17) (cid:9) 0 1 (cid:17) 0 yi((cid:30)) pi((cid:30))L=rd;i((cid:30) )M. Then, it is readily seen that yi;vi;Hi is the unique solution to the BVP 0 (cid:2) (cid:0) (cid:1) (cid:3) (cid:17) (22) with parameter K = 0 and (cid:11) = (cid:14)i.52 However, note that, by construction, zi;xi;(cid:0)i is also a 1 (cid:8) (cid:9) solution to this parametrization of the BVP (22), so it must be the case that Hi = (cid:0)i. (cid:8) (cid:9) Let us now derive an additional condition to pin down the revenue function of this arti(cid:133)cial economy. In equilibrium, the total revenue of (cid:133)rms with productivity less or equal than (cid:30) must equal a constant 0 fraction of the total wages paid to workers employed at those (cid:133)rms, (cid:30)0 1+F Kixi((cid:30)) Ki (cid:11)i((cid:30)) rd;i((cid:30) )(cid:11)i((cid:30) ) 1+F Ki Ki xi((cid:30)) 0 1 g((cid:30))Md(cid:30) = (65) 0 0 0 1 1+F Ki Ki (cid:11)i((cid:30) ) Z (cid:30) 0 (cid:2) (cid:0) 0 (cid:1)1 (cid:3) 0 (cid:2) (cid:27) (cid:0) L (cid:1) Hi((cid:30) (cid:3) 0) wi Hi((cid:30)) V (cid:2) Hi((cid:30) (cid:0) ) d (cid:1) s, fo (cid:3) r i = a;b: (cid:1)(cid:1)(cid:1) (cid:27) 1 (cid:0) Z s0 (cid:0) (cid:1) (cid:0) (cid:1) 52With K =0, the value of K is irrelevant. 1 0 52

Di⁄erentiating the left- and right hand sides of the last expression with respect to (cid:30); and evaluating the 0 resulting expressions at (cid:30) = (cid:30) yields 0 0 (cid:27) rd;i((cid:30) )(cid:11)i((cid:30) ) 1+F Ki Ki g((cid:30) )M = Lwi(s )V (s )Hi ((cid:30) ) for i = a;b. (66) 0 0 0 1 0 (cid:27) 1 0 0 (cid:30) 0 (cid:0) (cid:2) (cid:0) (cid:1) (cid:3) The last expression, together with the numeraire assumption, s1wi(s)V (s)ds = 1; and the inverse s0 matching function Hi, can be used to pin down the value of ri ((cid:30) ). To see this, note that Hi determines d R0 the growth rate of wages along the skill dimension (condition 12), while the numeraire assumption pins down their levels, i.e. the wage schedule is fully determined. Then, equation (66) can be used to pin down ri ((cid:30) ); the only remaining endogenous variable. d 0 With previous results we are ready to prove the lemma. As Ha((cid:30)) < Hb((cid:30)) for (cid:30) [(cid:30) ;(cid:30) ] by 0 1 2 assumption, wages grow faster along the skill dimension in economy a than in economy b, so the numeraire assumption implies wa(s ) < wb(s ). In addition, Ha((cid:30)) < Hb((cid:30)) for (cid:30) [(cid:30) ;(cid:30) ] also im- 0 0 0 1 2 plies that Ha((cid:30) ) Hb((cid:30) ). These observations and (66) imply rd;a((cid:30) )(cid:11)a((cid:30) )[1+F (Ka)Ka] < (cid:30) 0 (cid:20) (cid:30) 0 0 0 0 1 rd;b((cid:30) )(cid:11)b((cid:30) ) 1+F Kb Kb . Finally, the last inequality, expression (65) evaluated at (cid:30) = (cid:30) for 0 0 0 1 0 1 i = a;b, and the numeraire assumption yield the desired result. (cid:2) (cid:0) (cid:1) (cid:3) Lemma 4.iv. As (cid:0)i is a (cid:133)xed point of the functional (cid:9)i de(cid:133)ned in (26) with parameters (cid:11)i((cid:30));Ki;Ki , (cid:0)i((cid:30)) = 0 1 (cid:9)i (cid:0)i ((cid:30)), (cid:0)i ((cid:30)) can be obtained di⁄erentiating the right-hand side of (26). Doing so yields, (cid:30) (cid:8) (cid:9) (cid:0) (cid:1) hi (cid:30);(cid:0)i((cid:30)) xi((cid:30))(cid:27) (cid:27) 1 1+F Kixi((cid:30)) Ki (cid:0)i ((cid:30)) = [s s ] (cid:0) 0 1 ; (67) (cid:30) 1 (cid:0) 0 (cid:30) (cid:30) 1h (cid:0) i(t;(cid:0)i(t) (cid:1) )xi(t)(cid:27) (cid:0) (cid:27) 1 (cid:2) 1+F (cid:0) K 0 ixi(t) (cid:1) K 1 i (cid:3) dt 0 R (cid:2) (cid:0) (cid:1) (cid:3) @lnA((cid:0)i(u);u) where in the last expression I used the fact that xi((cid:30)) = e ((cid:27) (cid:0) 1) (cid:30) (cid:30) 0 @(cid:30) du . The last expression R plays a central role in the proof. Speci(cid:133)cally, I show that if the claim of the lemma is not satis(cid:133)ed, then it ispossibletoderivecontradictingimplicationsregardingthevaluesofthedenominatorsontheright-hand side of (67), Demi for i = a;b: Throughout the proof, I denote the numerator on the right hand side of (67) by Numi((cid:30)). Suppose the claim of the lemma is not true and xb((cid:30))(cid:21) xa((cid:30)) for some (cid:30) [(cid:30) ;(cid:30) ]. Noting that 0 1 (cid:20) 2 xa((cid:30) ) < (cid:21)xb((cid:30) ), let (cid:30)~ > (cid:30) be the lowest productivity value at which xb((cid:30))(cid:21) = xa((cid:30)).53 Clearly, 0 0 0 xa((cid:30)) must be catching up to xb((cid:30))(cid:21) to the left of (cid:30)~, so equation (22b) implies that (cid:0)b((cid:30)) < (cid:0)a((cid:30)) on some interval ((cid:30);(cid:30) ); with (cid:30) < (cid:30)~ (cid:30) and (cid:0)b (cid:30) = (cid:0)a (cid:30) . This situation is depicted in (cid:133)gure 5. 0 00 0 00 00 00 (cid:20) 53Note that (cid:30)~ is well de(cid:133)ned due to the continuity of th(cid:0)e fun(cid:1)ctions x(cid:0)a an(cid:1)d xb. 53

Figure 5: Hypotetical Solutions to the General BVP ( 22), (cid:0) Note: The (cid:133)gure depicts hypothetical solutions to the general BVP (22) with the features implied by the assumption xb((cid:30))(cid:21) xa((cid:30)) given the conditions in lemma 4.iv. as described in the proof. Of note, said (cid:20) assumption implies (cid:30)~ ((cid:30)0;(cid:30)00],with the (cid:133)gure showing one ofmany possibilities. 2 By construction, (cid:0)b (cid:30) (cid:0)a (cid:30) and xb (cid:30) (cid:21) xa (cid:30) , where the latter inequality implies (cid:30) 00 (cid:21) (cid:30) 00 00 (cid:20) 00 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:27) 1 xb (cid:30) 00 (cid:27) (cid:0) 1 1+F K 0 bxb (cid:30) 00 K 1 b = xb (cid:30) 00 (cid:27) (cid:0) 1 xb (cid:30) 00 +F K 0 a(cid:21)xb (cid:30) 00 xb (cid:30) 00 (cid:21)K 1 a (cid:0) (cid:1) h (cid:16) (cid:0) (cid:1) (cid:17) i < xa(cid:0) (cid:30) 00 (cid:1) (cid:27) (cid:0) 1 1 h xa(cid:0) (cid:30) 00 (cid:1) +F (cid:16) K 0 axa (cid:30) (cid:0) 00 (cid:1) (cid:17) xa (cid:30) (cid:0) 00 (cid:1) K 1 a i (cid:27) = xa (cid:0)(cid:30) 00(cid:1)(cid:27) (cid:0) 1 (cid:2)1+(cid:0)F (cid:1)K 0 axa (cid:0) (cid:30) 00 K(cid:0) 1 a (cid:1):(cid:1) (cid:0) (cid:1) (cid:3) (68) (cid:0) (cid:1) (cid:2) (cid:0) (cid:0) (cid:1)(cid:1) (cid:3) In addition, by de(cid:133)nition of hi we have ha (cid:30) ;(cid:0)a (cid:30) = hb (cid:30) ;(cid:0)b (cid:30) , which, together with the last 00 00 00 00 expression, implies that Numb (cid:30) < Numa((cid:30) ). This last result, (cid:0)b (cid:30) (cid:0)a (cid:30) and expression 00 (cid:0) 00(cid:0) (cid:1)(cid:1) (cid:0) (cid:0) (cid:30)(cid:1)(cid:1) 00 (cid:21) (cid:30) 00 (67) yield Demb < Dema: (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Expression (67) implies (cid:0)b (cid:30) (cid:30) 00 =(cid:0)b (cid:30) ((cid:30) 0 ) = xb (cid:30) 00 (cid:27) (cid:0) (cid:27) 1 1+F K 0 bxb (cid:30) 00 K 1 b [1+F (K 0 a)K 1 a] < 1; (cid:0)a (cid:30)(cid:0) (cid:30) 00(cid:1) =(cid:0)a (cid:30) ((cid:30) 0 ) xa (cid:0)(cid:30) 00(cid:1)(cid:27) (cid:0) (cid:27) 1 (cid:2)1+F (cid:0)K 0 axa (cid:0)(cid:30) 00(cid:1)(cid:1)K 1 a (cid:3) [1+F ((cid:21)K 0 a)(cid:21)K 1 a] (cid:0) (cid:1) (cid:0) (cid:1) (cid:2) (cid:0) (cid:0) (cid:1)(cid:1) (cid:3) where the inequality follows from (68) and (cid:21) > 1. The last result and (cid:0)b (cid:30) (cid:0)a (cid:30) imply (cid:0)b ((cid:30) ) > (cid:30) 00 (cid:21) (cid:30) 00 (cid:30) 0 (cid:0)a((cid:30) ),i.e. (cid:0)b((cid:30)) > (cid:0)a((cid:30))onsomeneighborhoodof(cid:30) (excluding(cid:30) ). Let(cid:30) bethelowestproductivity (cid:30) 0 0 0 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) value to the right of (cid:30) such that (cid:0)b (cid:30) = (cid:0)a (cid:30) . As (cid:0)b((cid:30)) > (cid:0)a((cid:30)) on (cid:30) ;(cid:30) , we have (cid:0)b (cid:30) 0 (cid:0) (cid:0) 0 (cid:0) (cid:30) (cid:0) (cid:20) (cid:0)a (cid:30) and xb((cid:30) ) > xa((cid:30) ). Using these results and (67) yields (cid:30) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:0) (cid:0) (cid:0) (cid:1) D D e e m m a b = (cid:0) (cid:0) a (cid:30) b (cid:30)(cid:0) (cid:30) (cid:30) (cid:0) (cid:0)(cid:1) x x b a (cid:0) (cid:30) (cid:30) (cid:0) (cid:0)(cid:1) (cid:27) (cid:0) (cid:27) (cid:27) 1 (cid:0) (cid:27) 1(cid:2) 1 1 + + F F(cid:0) (cid:21) K K 0 0 a a x x b a (cid:0) (cid:30) (cid:30) (cid:0) (cid:1)(cid:1) (cid:21) K K 1 a 1 a (cid:3) > 1, (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) (cid:2) (cid:0) (cid:0) (cid:1)(cid:1) (cid:3) 54

contradicting our previous (cid:133)nding, Demb < Dema. Then it must be the case that xb((cid:30))(cid:21) > xa((cid:30)) for all for all (cid:30) [(cid:30) ;(cid:30) ], which is the desired result. 0 1 2 Lemma 4.v. As in the case of lemma 4.iii, the idea of the proof is to show that (cid:0)a and (cid:0)b can be thought of as the inverse matching functions of two arti(cid:133)cial economies, and then use this additional information to prove the result. Moreover, I de(cid:133)ne these arti(cid:133)cial economies here in the same way I did in proof of lemma 4.iii. Let zi;xi;(cid:0)i be the solution to the BVP in the statement of the lemma and consider the following arti(cid:133)cial economy. In this economy there are no (cid:133)xed costs of production and no (cid:133)xed costs (cid:8) (cid:9) to export but the set of active (cid:133)rms and the set of exporters are (cid:133)xed. In particular, the set of active (cid:133)rms are those with productivity in the range [(cid:30) ;(cid:30) ], while the fraction of (cid:133)rms that export of each 0 1 productivity level is given by F Kixi((cid:30)) . The set of available workers are those with skills in the range 0 [s ;s ]. The distribution of skills is given by the restriction of V to [s ;s ] and the mass of workers is 0 1 (cid:0) (cid:1) 0 1 s1V (s)dsL. The total mass of (cid:133)rms with productivity (cid:30) is given by g((cid:30))(cid:11)i((cid:30))M, so the total mass of s0 (cid:133)rms is (cid:30) 1g((cid:30))(cid:11)i((cid:30))M: Finally, I set (cid:28) such that Ki (cid:28)1 (cid:27). R (cid:30) 0 i 1 (cid:17) i(cid:0) The same argument used in the proof of lemma 4.iii implies that if pi; rd;i and Hi are the price, R domestic revenue and inverse-matching functions of the economy described above, then Hi = (cid:0)i. In addition, equation (65) also holds in this economy, which can be di⁄erentiated with respect to the limit of integration to get (cid:27) rd;i((cid:30))(cid:14)i((cid:30))g((cid:30))M = Lwi Hi((cid:30)) V Hi((cid:30)) Hi ((cid:30)) for i = a;b, (69) (cid:27) 1 (cid:30) (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) where (cid:14)i((cid:30)) was de(cid:133)ned in the statement of the lemma. As discussed in the proof of lemma 4.iii, the last expression and the numeraire assumption, s1wi(s)V (s)ds = 1; can be used to pin down the level of the s0 domestic revenue function rd;i. For this reason, the last expression is central in the proof of this lemma, R as the main result is an immediate implication of the values recovered for rd;i((cid:30) ) and equation (65). 0 STEP 1: Let (cid:8) be the set of productivity levels given by (cid:3) (cid:8) = (cid:30) [(cid:30) ;(cid:30) ] : Hb((cid:30)) = Ha((cid:30));Hb((cid:30)) Ha((cid:30)) ; (cid:3) 0 1 (cid:30) (cid:30) 2 (cid:20) n o and let S denote the set of corresponding skill levels, S s [s ;s ] : s = Hi((cid:30)) for some (cid:30) (cid:8) . (cid:3) (cid:3) 0 1 (cid:3) (cid:17) 2 2 Then, wb(s) < wa(s) for some s S . (cid:3) (cid:8) (cid:9) 2 Supposethatthisisnotthecaseandwb(s) wa(s)foralls S andletNi bethematchingfunction (cid:3) (cid:21) 2 of the arti(cid:133)cial economy described above, i.e. Ni is the inverse function of Hi. For any s [s ;s ] S , 0 1 (cid:3) 2 n there are three possibilities, (i) Na(s) = Nb(s), (ii) Na(s) < Nb(s), and (iii) Na(s) > Nb(s). I show that wb(s) > wa(s) in all cases. Let us start with case (i). As s = S , then Hb((cid:30)) > Ha((cid:30)) for (cid:30) = Ni(s), implying Hb (cid:30) < Ha (cid:30) 2 (cid:3) (cid:30) (cid:30) 0 0 on some neighborhood to the left of (cid:30). Let (cid:30) be the (cid:133)rst time Ha and Hb intersect to the left of (cid:30), and (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) let s Hi (cid:30) . By construction, we have (cid:30) (cid:8) (s S ) and Hb((cid:30)) < Ha (cid:30) for all (cid:30) (cid:30) ;(cid:30) (cid:0) (cid:0) (cid:0) (cid:3) (cid:0) (cid:3) 0 0 0 (cid:0) (cid:17) 2 2 2 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) 55

(Nb(s) > Na(s) for all s (s ;s)), so 0 0 0 (cid:0) 2 wb(s) = wb s (cid:0) e s s (cid:0) @lnA( @ t; s Nb(t)) dt > wa s (cid:0) e s s 0 @lnA( @ t; s Na(t)) dt = wa(s); (70) R R (cid:0) (cid:1) (cid:0) (cid:1) where the last inequality in a consequence of the log-supermodularity of A and wb(s ) wa(s ). (cid:0) (cid:0) (cid:21) Turning to case (ii), let s and s+ be the (cid:133)rst time Na and Nb intersect to the left and right of s (cid:0) respectively. These skill levels are well de(cid:133)ned due to the continuity of the functions involved and the fact that Na and Nb intersect at least once to the left and right of s (at s and s ). Letting (cid:30)k Ni sk 0 1 (cid:17) for k = ;+, by construction we have Nb(s) > Na(s) for all s (s ;s+); so Nb(s ) Na(s ) 0 0 0 (cid:0) s (cid:0) s (cid:0) (cid:0)(cid:1) (cid:0) 2 (cid:21) (Hb (cid:30) Ha (cid:30) ), i.e. s S . Then inequality (70) also holds in this case. (cid:30) (cid:0) (cid:20) (cid:30) (cid:0) (cid:0) 2 (cid:3) Let us now turn to case (iii). Let s and s+ be the (cid:133)rst time Na and Nb intersect to the left and (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) right of s respectively. As before, these skill levels are well de(cid:133)ned. Letting (cid:30)k Ni sk for k = ;+, by (cid:17) (cid:0) construction we have Nb(s) < Na(s) for all s (s ;s+); so Nb(s+) Na(s+) (Hb (cid:30)+ Ha (cid:30)+ ), 0 0 0 2 (cid:0) s (cid:21) s (cid:0)(cid:30) (cid:1) (cid:20) (cid:30) i.e. s+ S . This and the log supermodularity of A imply (cid:3) (cid:0) (cid:1) (cid:0) (cid:1) 2 wb(s+) = e s s+ @lnA( @ t; s Nb(t)) dt < e s s+ @lnA( @ t; s Na(t)) dt = wa(s+) . wb(s) wa(s) R R Perourinitialassumptionands+ S wehavewb(s+) wa(s+),whichtogetherwiththelastexpression, (cid:3) 2 (cid:21) yields wb(s) > wa(s). Given that the selection of s [s ;s ] S was arbitrary, we conclude that wb(s) > wa(s) for all 0 1 (cid:3) 2 n s [s ;s ] S . However, notice that wb(s) wa(s) on [s ;s ] and wb(s) > wa(s) on [s ;s ] S imply 0 1 (cid:3) 0 1 0 1 (cid:3) 2 n (cid:21) n wb > wa, which contradicts our numeraire selection. Then it must be the case that wb(s) < wa(s) for some s S . (cid:3) 2 STEP 2: Let S be de(cid:133)ned as before, let s+ S such that wb(s+) < wa(s+) and let (cid:30)+ = Ni(s+). (cid:3) (cid:3) 2 If xb (cid:30)+ xa (cid:30)+ , then rd;b((cid:30) ) < rd;a((cid:30) ). 0 0 (cid:21) By assumption we have wb(s+) < wa(s+), Ha (cid:30)+ = Hb (cid:30)+ and Hb (cid:30)+ Hb (cid:30)+ , which, (cid:0) (cid:1) (cid:0) (cid:1) (cid:30) (cid:20) (cid:30) together with equation (69) evaluated at (cid:30)+, imply (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) rd;b((cid:30) )xb (cid:30)+ (cid:14)b (cid:30)+ < rd;a((cid:30) )xa (cid:30)+ (cid:14)a (cid:30)+ : 0 0 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) The last expression, xb (cid:30)+ xa (cid:30)+ , and the assumption in the of the lemma ((cid:14)b((cid:30)) > (cid:14)a((cid:30))) imply (cid:21) rd;b((cid:30) ) < rd;a((cid:30) ). 0 0 (cid:0) (cid:1) (cid:0) (cid:1) STEP 3: Let S , s+ and (cid:30)+ be de(cid:133)ned as in step 2. If xb (cid:30)+ < xa (cid:30)+ , then rd;b((cid:30) ) < rd;a((cid:30) ). (cid:3) 0 0 (cid:0) (cid:1) (cid:0) (cid:1) 56

The continuity of Hi and of Hi imply that (cid:8) and S are closed sets, so s infS S . As Ha (cid:30) (cid:3) (cid:3) (cid:0) (cid:17) (cid:3) 2 (cid:3) and Hb intersect at (cid:30) and at (cid:30)+, the following equality holds for i = a;b, 0 ln A s+;(cid:30)+ = (cid:30)+ @lnA(Hi(t);t) Hi(t)dt+ (cid:30)+ @lnA(Hi(t);t) dt A(s ;(cid:30) ) @s (cid:30) @(cid:30) (cid:0) 0 0(cid:1) Z (cid:30) 0 Z (cid:30) 0 s+ (cid:30)+ @lnA(u;Ni(u)) @lnA(Hi(t);t) = du+ dt: @s @(cid:30) Z s0 Z (cid:30) 0 Couplingthelastexpressionwithassumptionxb (cid:30)+ < xa (cid:30)+ yields s+ @lnA(u;Nb(u)) du > s+ @lnA(u;Na(u)) du, s0 @s s0 @s which, together with condition ( 12), implies (cid:0) (cid:1) (cid:0) (cid:1) R R wb(s (cid:0) )wb(s+) s+ @lnA(u;Nb(u)) s+ @lnA(u;Na(u)) wa(s (cid:0) )wa(s+) = du > du = : (71) wb(s ) wb(s ) @s @s wa(s ) wa(s ) 0 (cid:0) Z s0 Z s0 0 (cid:0) Now I show that wb(s )=wb(s ) wa(s )=wa(s ). If s = s there is nothing to prove, so let(cid:146)s (cid:0) 0 (cid:0) 0 (cid:0) 0 (cid:20) assume that s > s . First, notice that Nb(s) Na(s) for s [s ;s ]. To see this, suppose to the (cid:0) 0 0 (cid:0) (cid:20) 2 contrary that Nb(s) > Na(s) for some s (s ;s ), and let s be the (cid:133)rst time Nb and Na intersect to 0 (cid:0) 0 2 the left of s. Then we have s < s , Nb(s) = Na(s) and Nb(s) Na(s), i.e. s s with s < s . 0 (cid:0) 0 0 s 0 s 0 0 (cid:3) 0 (cid:0) (cid:21) 2 However, this contradicts the de(cid:133)nition of s , so it must be the case that Nb(s) Na(s) for s [s ;s ]. (cid:0) 0 (cid:0) (cid:20) 2 This result and the log supermodularity of A implies wb(s (cid:0) ) s (cid:0) @lnA(u;Nb(u)) s (cid:0) @lnA(u;Na(u)) wa(s (cid:0) ) = du du = . (72) wb(s ) @s (cid:20) @s wa(s ) 0 Z s0 Z s0 0 The inequalities (71)-(72) and our assumption wb(s+) < wa(s+) imply wb(s ) < wa(s ). Using this (cid:0) (cid:0) result, Hb((cid:30) ) Ha((cid:30) ) and the assumption in the lemma about (cid:14)i((cid:30)) in expression (69) (evaluated at (cid:30) (cid:0) (cid:20) (cid:30) (cid:0) (cid:30) ) yields rd;b (cid:30) < rd;a (cid:30) . If (cid:30) = (cid:30) , we are done, so let us assume (cid:30) > (cid:30) . As discussed above, (cid:0) (cid:0) (cid:0) (cid:0) 0 (cid:0) 0 Nb(s) Na(s) for s [s ;s ] (Hb((cid:30)) Ha((cid:30)) for (cid:30) (cid:30) ;(cid:30) ), implying (cid:0) (cid:1) 0(cid:0) (cid:0)(cid:1) 0 (cid:0) (cid:20) 2 (cid:21) 2 (cid:2) (cid:3) rd;b (cid:30) (cid:0) ((cid:27) 1) (cid:30)(cid:0) @lnA(Hb(t);t) dt ((cid:27) 1) (cid:30)(cid:0) @lnA(Ha(t);t) dt rd;a (cid:30) (cid:0) = e (cid:0) (cid:30)0 @(cid:30) e (cid:0) (cid:30)0 @(cid:30) = . rd;b((cid:30) ) (cid:21) rd;a((cid:30) ) (cid:0) 0(cid:1) R R (cid:0) 0(cid:1) The last expression and rd;b (cid:30) < rd;a (cid:30) imply rd;b((cid:30) ) < rd;a((cid:30) ), which is the desired result. (cid:0) (cid:0) 0 0 STEP 4: Under the assum(cid:0) pti(cid:1)ons of t(cid:0)he L(cid:1)emma, inequality (56) holds. Steps 2 and 3 together imply that rd;b((cid:30) ) < rd;a((cid:30) ), holds for these two arti(cid:133)cial economies. This 0 0 result, the numeraire assumption for these economies and equation (65) evaluated at (cid:30) = (cid:30) imply that 0 1 inequality (56) holds. Lemma 4.vi. I prove the statement for the case in which (cid:17) (t;(cid:21)) is strictly decreasing in (cid:21). 0 STEP 1: Under the assumptions of the lemma, there is no (cid:30) ((cid:30) ;(cid:30) ] such that (cid:0)a((cid:30)) (cid:0)b((cid:30)) for 0 0 1 2 (cid:20) all (cid:30) ((cid:30) ;(cid:30)]. 0 0 2 57

Suppose to the contrary that there is such a value (cid:30) ((cid:30) ;(cid:30) ]. Let (cid:30) be the (cid:133)rst time the functions 0 0 1 + 2 (cid:0)a and (cid:0)b intersect to the right of (cid:30), i.e. (cid:30) = inf (cid:30) (cid:30) : (cid:0)a((cid:30)) = (cid:0)b((cid:30)) . Note (cid:30) is well de(cid:133)ned 0 + 0 + (cid:21) due to the continuity of the functions (cid:0)a and (cid:0)b and the fact that the functions intersect at least once (cid:8) (cid:9) to the right of (cid:30) (at (cid:30) ). Also note that (cid:0)a((cid:30)) (cid:0)b((cid:30)) for (cid:30) (cid:30) ;(cid:30) . The continuity of (cid:0)a and (cid:0)b, 0 1 (cid:20) 2 0 + (cid:30) (cid:30) implies (cid:0)a((cid:30) ) (cid:0)b((cid:30) ) and (cid:0)a((cid:30) ) (cid:0)b((cid:30) ), i.e. (cid:30) 0 (cid:20) (cid:30) 0 (cid:30) + (cid:21) (cid:30) + (cid:0) (cid:1) (cid:0)a((cid:30) )=(cid:0)a((cid:30) ) (cid:30) + (cid:30) 0 1. (73) (cid:0)b((cid:30) )=(cid:0)b((cid:30) ) (cid:21) (cid:30) + (cid:30) 0 Di⁄erentiating the right-hand side of (26) yields (cid:0)i (cid:30) (cid:30) + = hi (cid:30) ;(cid:30) e (cid:27) (cid:30) (cid:30)+ @lnA( @ (cid:0) (cid:30) i(u);u) du 1+F K 0 ixi((cid:30)) K 1 ; (74) (cid:0)i ((cid:30) ) 0 + (cid:0) 1+F Ki K (cid:30)(cid:0) 0(cid:1) R (cid:2) (cid:0) 0 (cid:1)1 (cid:3) (cid:0) (cid:1) (cid:2) (cid:0) (cid:1) (cid:3) where hi (cid:30) ;(cid:30) is given by (27). By assumption, we have (cid:0)a((cid:30) ) = (cid:0)b((cid:30) ) and (cid:0)a (cid:30) = (cid:0)b (cid:30) , 0 + 0 0 + + which together with the de(cid:133)nition of hi, imply ha (cid:30) ;(cid:30) = hb (cid:30) ;(cid:30) . Combining this result with (74) (cid:0) (cid:1) 0 + 0 + (cid:0) (cid:1) (cid:0) (cid:1) for i = a;b yields (cid:0) (cid:1) (cid:0) (cid:1) (cid:0)a (cid:30) ((cid:30) + )=(cid:0)a (cid:30) ((cid:30) 0 ) = e (cid:27) (cid:30) (cid:30) (cid:0) + " @lnA( @ (cid:0) (cid:30) a(u);u) (cid:0) @lnA( @ (cid:0) (cid:30) b(u);u) # du[1+F(K 0 axa((cid:30) + ))K1 ]=[1+F(K 0 a)K1 ] ; (cid:0)b (cid:30) ((cid:30) + )=(cid:0)b (cid:30) ((cid:30) 0 ) R [1+F(K 0 bxb((cid:30) + ))K1 ]=[1+F(K 0 b)K1 ] (75) (cid:0)a (cid:30) ((cid:30) + )=(cid:0)a (cid:30) ((cid:30) 0 ) e (cid:27) (cid:30) (cid:30) (cid:0) + " @lnA( @ (cid:0) (cid:30) a(u);u) (cid:0) @lnA( @ (cid:0) (cid:30) b(u);u) # du[1+F(K 0 b(cid:21)xb((cid:30) + ))K1 ]=[1+F(K 0 b(cid:21))K1 ] ; (cid:0)b (cid:30) ((cid:30) + )=(cid:0)b (cid:30) ((cid:30) 0 ) (cid:20) R [1+F(K 0 bxb((cid:30) + ))K1 ]=[1+F(K 0 b)K1 ] where the second line uses (cid:21) Ka=Kb > 1 and xb (cid:30) xa (cid:30) , with the latter being a consequence of (cid:17) 0 0 + (cid:21) + thestrictlog-supermodularityofAandthefactthat(cid:0)a((cid:30)) (cid:0)b((cid:30))for(cid:30) (cid:30) ;(cid:30) . Anotherimplication (cid:0) (cid:1) (cid:0) (cid:1) 0 + (cid:20) 2 of this the last two observation is that the (cid:133)rst term of the right-hand side of the last expression is weakly (cid:0) (cid:1) lower than 1. Focusing on the second term, note that [1+F(K 0 b(cid:21)xb((cid:30) + ))K1 ] = exp (cid:30) + Fy (K 0 b(cid:21)xb((cid:30)))K 0 bK1(cid:21)xb (cid:30) ((cid:30)) d(cid:30) = exp (cid:30) +(cid:17)0 Kbxb((cid:30));(cid:21) Kbxb ((cid:30))d(cid:30) [1+F(K 0 b(cid:21))K1 ] f (cid:30) 0 [1+F(K 0 b(cid:21)xb((cid:30)))K1 ] g f (cid:30) 0 0 0 (cid:30) g [1+F(K 0 bxb((cid:30) + ))K1 ] = expR (cid:30) + Fy (K 0 bxb((cid:30)))K1K 0 bxb (cid:30) ((cid:30)) d(cid:30) = exp R(cid:30) +(cid:17)0 K(cid:0) bxb((cid:30));1 K(cid:1) bxb ((cid:30))d(cid:30) [1+F(K 0 b)K1 ] f (cid:30) 0 [1+F(K 0 bxb((cid:30)))K1 ] g f (cid:30) 0 0 0 (cid:30) g (76) R R (cid:0) (cid:1) As(cid:17)0 isstrictlydecreasingin(cid:21), thesecondlinein(76)isstrictlygreaterthanthe(cid:133)rst, sothesecondterm on the right-hand side of the second line of (75) is strictly lower than 1, contradicting (73). Accordingly, the statement in step 1 must be true. STEP 2: Under the assumptions of the lemma, (cid:0)a((cid:30) ) > (cid:0)b ((cid:30) ), so there is a (cid:30) ((cid:30) ;(cid:30) ) such (cid:30) 0 (cid:30) 0 + 2 0 1 that (cid:0)a (cid:30) = (cid:0)b (cid:30) and (cid:0)a((cid:30)) > (cid:0)b((cid:30)) on ((cid:30) ;(cid:30) ). + + 0 + The result of step 1 immediately yields that (cid:0)a((cid:30) ) (cid:0)b ((cid:30) ). Otherwise, (cid:0)a((cid:30) ) < (cid:0)b ((cid:30) ) implies (cid:0) (cid:1) (cid:0) (cid:1) (cid:30) 0 (cid:21) (cid:30) 0 (cid:30) 0 (cid:30) 0 that there is a (cid:30) ((cid:30) ;(cid:30) ] such that (cid:0)a((cid:30)) < (cid:0)b((cid:30)) on ((cid:30) ;(cid:30)], contradicting the result in step 1. 0 0 1 0 0 2 Suppose then that (cid:0)a((cid:30) ) = (cid:0)b ((cid:30) ) = (cid:13) : Note that the (same) boundary conditions of the BVPs under (cid:30) 0 (cid:30) 0 0 consideration imply (cid:0)i((cid:30) ) = s , xi((cid:30) ) = 1: In turn, these observations and equations (22a)-(22b) imply 0 0 0 58

xi (cid:30) ((cid:30)) = ((cid:27) (cid:0) 1)@l @ n (cid:30) A(s0;(cid:30) 0 ) and z z (cid:30) i i( ( (cid:30) (cid:30) 0 0 ) ) = (cid:0) @lnA @ ( (cid:30) s0;(cid:30) 0 ) . Log-di⁄erentiating both sides of equation (22c) and evaluating at (cid:30) yields 0 (cid:0)i (cid:30)(cid:30) ((cid:30) 0 ) = xi (cid:30) ((cid:30) 0 ) + Fy (K 0 ixi((cid:30) 0 ))K1K 0 ixi (cid:30) ((cid:30) 0 ) + (cid:11)(cid:30)((cid:30) 0 ) + g(cid:30)((cid:30) 0 ) [ @lnA((cid:0)i((cid:30) 0 );(cid:30) 0 ) (cid:0)i ((cid:30) ) + @lnA((cid:0)i((cid:30) 0 );(cid:30) 0 ) + (cid:0)i (cid:30) ((cid:30) 0 ) xi((cid:30) 0 ) [1+F(K 0 ixi((cid:30) 0 ))K1 ] (cid:11)((cid:30) 0 ) g((cid:30) 0 ) (cid:0) @s (cid:30) 0 @(cid:30) Vs ((cid:0)i((cid:30) 0 )) (cid:0)i ((cid:30) )+ z (cid:30) i((cid:30) 0 ) ]; V((cid:0)i((cid:30) )) (cid:30) 0 zi((cid:30) ) 0 0 (cid:0)i (cid:30) (cid:13) (cid:30) 0 ((cid:30)) = ((cid:27) (cid:0) 1)@l @ n (cid:30) A(s0;(cid:30) 0 ) + Fy (K 0 i)K [1 1 + K F 0 i ( ( K (cid:27) (cid:0) 0 i 1 ) ) K @ 1 l @ n ] (cid:30) A(s0;(cid:30)0) + (cid:11) (cid:11) (cid:30) ( ( (cid:30) (cid:30) 0 0 ) ) + g g (cid:30) ( ( (cid:30) (cid:30) 0 0 ) ) (cid:0) [ @lnA @ ( s s0;(cid:30) 0 ) (cid:13) 0 + @lnA @ ( (cid:30) s0;(cid:30) 0 ) + Vs(s0) (cid:13) @lnA(s0;(cid:30) 0 ) ]; V(s0) 0 (cid:0) @(cid:30) i.e., (cid:0)a (cid:30)(cid:30) ((cid:30) 0 ) (cid:0) (cid:0)b (cid:30)(cid:30) ((cid:30) 0 ) = K 0 b((cid:27) (cid:0) 1)@l @ n (cid:30) A(s0;(cid:30) 0 ) (cid:13) 0 (cid:26) [1 F + y F (K (K 0 b(cid:21) 0 b ) (cid:21) (cid:21) ) K K 1 1 ] (cid:0) [1 F + y F ( ( K K 0 b 0 b ) ) K K 1 1 ] (cid:27) < 0, where the inequality follows from (cid:17)0 Kb;(cid:21) < (cid:17)0 Kb;1 . The last expression implies that there is some 0 0 (cid:30) ((cid:30) ;(cid:30) ] such that (cid:0)a((cid:30)) < (cid:0)b ((cid:30)) on ((cid:30) ;(cid:30)], which yields a contradiction of step 1. Accordingly, we 0 2 0 1 (cid:30) (cid:30) (cid:0) (cid:1)0 0 (cid:0) (cid:1) must have (cid:0)a((cid:30) ) > (cid:0)b ((cid:30) ). (cid:30) 0 (cid:30) 0 Finally, (cid:0)a((cid:30) ) > (cid:0)b ((cid:30) ) implies (cid:0)a((cid:30)) > (cid:0)b((cid:30)) on some (small enough) interval (cid:30) ;(cid:30) , so (cid:30)+ (cid:30) 0 (cid:30) 0 0 00 described in the statement of the step the (cid:133)rst time (cid:0)a and (cid:0)b intersect to the right of (cid:30) . 0(cid:0)0 (cid:1) STEP 3: Under the assumptions of the lemma, (cid:0)a((cid:30)) > (cid:0)b((cid:30)) on ((cid:30) ;(cid:30) ). 0 1 Ishowthat(cid:30) = (cid:30) ;where(cid:30) wasde(cid:133)nedinstep2. Supposeforamomentthat(cid:30) < (cid:30) . Ifwede(cid:133)ne + 0 + + 0 on[(cid:30) ;(cid:30) ];wi((cid:30)) xi((cid:30))=xi (cid:30) andyi((cid:30)) = zi((cid:30))=xi (cid:30) ,thenitisreadilyseenthat yi;wi((cid:30));(cid:0)i + 0 + + (cid:17) solve BVP (22) in said interval, with (cid:11)i((cid:30));Ki = (cid:11)((cid:30));K and parameter K i = Kixi (cid:30) . Per (cid:0) (cid:1) 1 f (cid:0) (cid:1) 1 g 0 (cid:8)0 + (cid:9) step 2 we have xa (cid:30) > xb (cid:30) , so K a > K b . Then, the BVPs associated to yi;wi((cid:30));(cid:0)i satisfy + + (cid:8) 0 0 (cid:9) (cid:0) (cid:1) the conditions of lemma 4.vi, so step 2 implies (cid:0)a((cid:30) ) > (cid:0)b ((cid:30) ). However, (cid:0)a((cid:30)) > (cid:0)b((cid:30)) on (cid:30) ;(cid:30) (cid:0) (cid:1) (cid:0) (cid:1) (cid:30) 0 (cid:30) 0 (cid:8) (cid:9) 0 + implies (cid:0)a (cid:30) (cid:0)b (cid:30) , so assuming (cid:30) < (cid:30) yields a contradiction. (cid:30) + (cid:20) (cid:30) + + 0 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Lemma 4.vii. I prove the statement for the case in which (cid:17) (t;(cid:21)) is strictly increasing in (cid:21). 1 STEP 1: Under the assumptions of the lemma, there is no (cid:30) ((cid:30) ;(cid:30) ] such that (cid:0)a((cid:30)) (cid:0)b((cid:30)) for 0 0 1 2 (cid:21) all (cid:30) ((cid:30) ;(cid:30)]. 0 0 2 Suppose to the contrary that there is such a value (cid:30) ((cid:30) ;(cid:30) ]. Let (cid:30) be the (cid:133)rst time the functions 0 0 1 + 2 (cid:0)a and (cid:0)b intersect to the right of (cid:30), i.e. (cid:30) = inf (cid:30) (cid:30) : (cid:0)a((cid:30)) = (cid:0)b((cid:30)) . Note (cid:30) is well de(cid:133)ned 0 + 0 + (cid:21) due to the continuity of the functions (cid:0)a and (cid:0)b and the fact that the functions intersect at least once (cid:8) (cid:9) to the right of (cid:30) (at (cid:30) ). Also note that (cid:0)a((cid:30)) (cid:0)b((cid:30)) for (cid:30) (cid:30) ;(cid:30) . The continuity of (cid:0)a and (cid:0)b, 0 1 (cid:21) 2 0 + (cid:30) (cid:30) implies (cid:0)a((cid:30) ) (cid:0)b((cid:30) ) and (cid:0)a((cid:30) ) (cid:0)b((cid:30) ), i.e. (cid:30) 0 (cid:21) (cid:30) 0 (cid:30) + (cid:20) (cid:30) + (cid:0) (cid:1) (cid:0)a((cid:30) )=(cid:0)a((cid:30) ) (cid:30) + (cid:30) 0 1. (77) (cid:0)b((cid:30) )=(cid:0)b((cid:30) ) (cid:20) (cid:30) + (cid:30) 0 Di⁄erentiating the right-hand side of (26) yields (cid:0)i (cid:30) (cid:30) + = hi (cid:30) ;(cid:30) e (cid:27) (cid:30) (cid:30)+ @lnA( @ (cid:0) (cid:30) i(u);u) du 1+F K 0 ixi((cid:30)) K 1 i ; (78) (cid:0)i ((cid:30) ) 0 + (cid:0) 1+F Ki Ki (cid:30)(cid:0) 0(cid:1) R (cid:2) (cid:0) 0 (cid:1)1 (cid:3) (cid:0) (cid:1) (cid:2) (cid:0) (cid:1) (cid:3) 59

where hi (cid:30) ;(cid:30) is given by (27). By assumption, we have (cid:0)a((cid:30) ) = (cid:0)b((cid:30) ) and (cid:0)a (cid:30) = (cid:0)b (cid:30) , 0 + 0 0 + + which together with the de(cid:133)nition of hi, imply ha (cid:30) ;(cid:30) = hb (cid:30) ;(cid:30) . Combining this result with (78) (cid:0) (cid:1) 0 + 0 + (cid:0) (cid:1) (cid:0) (cid:1) for i = a;b yields (cid:0) (cid:1) (cid:0) (cid:1) (cid:0)a (cid:30) ((cid:30) + )=(cid:0)a (cid:30) ((cid:30) 0 ) = e (cid:27) (cid:30) (cid:30) (cid:0) + " @lnA( @ (cid:0) (cid:30) a(u);u) (cid:0) @lnA( @ (cid:0) (cid:30) b(u);u) # du[1+F(K 0 axa((cid:30) + ))K 1 a]=[1+F(K 0 a)K 1 a] ; (cid:0)b((cid:30) )=(cid:0)b((cid:30) ) R [1+F(Kbxb((cid:30) ))Kb]=[1+F(Kb)Kb] (cid:30) + (cid:30) 0 0 + 1 0 1 (79) (cid:0)a (cid:30) ((cid:30) + )=(cid:0)a (cid:30) ((cid:30) 0 ) e (cid:27) (cid:30) (cid:30) (cid:0) + " @lnA( @ (cid:0) (cid:30) a(u);u) (cid:0) @lnA( @ (cid:0) (cid:30) b(u);u) # du[1+F(K 0 b(cid:21)xb((cid:30) + ))K 1 b(cid:21)]=[1+F(K 0 b(cid:21))K 1 b(cid:21)] ; (cid:0)b (cid:30) ((cid:30) + )=(cid:0)b (cid:30) ((cid:30) 0 ) (cid:21) R [1+F(K 0 bxb((cid:30) + ))K 1 b]=[1+F(K 0 b)K 1 b] where the second line uses (cid:21) Ka=Kb > 1 and xa (cid:30) xb (cid:30) , with the latter being a consequence (cid:17) i i + (cid:21) + of the strict log-supermodularity of A and the fact that (cid:0)a((cid:30)) (cid:0)b((cid:30)) for (cid:30) (cid:30) ;(cid:30) . Another (cid:0) (cid:1) (cid:0) (cid:1) 0 + (cid:21) 2 implication of the last two observations is that the (cid:133)rst term of the right-hand side of the last expression (cid:0) (cid:1) is weakly greater than 1. Focusing on the second term, note that [1+F(K 0 b(cid:21)xb((cid:30) + ))K 1 b(cid:21)] = exp (cid:30) + Fy (K 0 b(cid:21)xb((cid:30)))K 0 bK 1 b(cid:21)2xb (cid:30) ((cid:30)) d(cid:30) = exp (cid:30) +(cid:17)1 Kbxb((cid:30));(cid:21) Kbxb ((cid:30))d(cid:30) [1+F(K 0 b(cid:21))K 1 b(cid:21)] f (cid:30) 0 [1+F(K 0 b(cid:21)xb((cid:30)))K 1 b(cid:21)] g f (cid:30) 0 0 0 (cid:30) g [1+F(K 0 bxb((cid:30) + ))K 1 b] = expR (cid:30) + Fy (K 0 bxb((cid:30)))K 0 bK 1 bxb (cid:30) ((cid:30)) d(cid:30) = exp (cid:30)R+(cid:17)1 K(cid:0)bxb((cid:30));1 K(cid:1)bxb ((cid:30))d(cid:30) [1+F(K 0 b)K 1 b] f (cid:30) 0 [1+F(K 0 bxb((cid:30)))K 1 b] g f (cid:30) 0 0 0 (cid:30) g (80) R R (cid:0) (cid:1) As (cid:17)1 is strictly increasing in (cid:21), the second line in (80) is strictly lower than the (cid:133)rst, so the second term on the right-hand side of the second line of (79) is strictly greater than 1, contradicting (77). Accordingly, the statement in step 1 must be true. STEP 2: Under the assumptions of the lemma, (cid:0)a((cid:30) ) < (cid:0)b ((cid:30) ), so there is a (cid:30) ((cid:30) ;(cid:30) ) such (cid:30) 0 (cid:30) 0 + 2 0 1 that (cid:0)a (cid:30) = (cid:0)b (cid:30) and (cid:0)a((cid:30)) < (cid:0)b((cid:30)) on ((cid:30) ;(cid:30) ). + + 0 + The result of step 1 immediately yields that (cid:0)a((cid:30) ) (cid:0)b ((cid:30) ). Otherwise, (cid:0)a((cid:30) ) > (cid:0)b ((cid:30) ) implies (cid:0) (cid:1) (cid:0) (cid:1) (cid:30) 0 (cid:20) (cid:30) 0 (cid:30) 0 (cid:30) 0 that there is a (cid:30) ((cid:30) ;(cid:30) ] such that (cid:0)a((cid:30)) > (cid:0)b((cid:30)) on ((cid:30) ;(cid:30)], contradicting the result in step 1. 0 0 1 0 0 2 Suppose then that (cid:0)a((cid:30) ) = (cid:0)b ((cid:30) ) = (cid:13) : Note that the (same) boundary conditions of the BVPs under (cid:30) 0 (cid:30) 0 0 consideration imply (cid:0)i((cid:30) ) = s , xi((cid:30) ) = 1: In turn, these observations and equations (22a)-(22b) imply 0 0 0 xi (cid:30) ((cid:30)) = ((cid:27) (cid:0) 1)@l @ n (cid:30) A(s0;(cid:30) 0 ) and z z (cid:30) i i( ( (cid:30) (cid:30) 0 0 ) ) = (cid:0) @lnA @ ( (cid:30) s0;(cid:30) 0 ) . Log-di⁄erentiating both sides of equation (22c) and evaluating at (cid:30) yields 0 (cid:0)i (cid:30)(cid:30) ((cid:30) 0 ) = xi (cid:30) ((cid:30) 0 ) + Fy (K 0 ixi((cid:30) 0 ))K 1 iK 0 ixi (cid:30) ((cid:30) 0 ) + (cid:11)(cid:30)((cid:30) 0 ) + g(cid:30)((cid:30) 0 ) [ @lnA((cid:0)i((cid:30) 0 );(cid:30) 0 ) (cid:0)i ((cid:30) ) + @lnA((cid:0)i((cid:30) 0 );(cid:30) 0 ) + (cid:0)i (cid:30) ((cid:30) 0 ) xi((cid:30) 0 ) [1+F(K 0 ixi((cid:30) 0 ))K 1 i] (cid:11)((cid:30) 0 ) g((cid:30) 0 ) (cid:0) @s (cid:30) 0 @(cid:30) Vs ((cid:0)i((cid:30) 0 )) (cid:0)i ((cid:30) )+ z (cid:30) i((cid:30) 0 ) ]; V((cid:0)i((cid:30) )) (cid:30) 0 zi((cid:30) ) 0 0 (cid:0)i (cid:30) (cid:13) (cid:30) 0 ((cid:30)) = ((cid:27) (cid:0) 1)@l @ n (cid:30) A(s0;(cid:30) 0 ) + Fy (K 0 i)K [1 1 i + K F 0 i ( ( K (cid:27) (cid:0) 0 i 1 ) ) K @ 1 l i @ n ] (cid:30) A(s0;(cid:30)0) + (cid:11) (cid:11) (cid:30) ( ( (cid:30) (cid:30) 0 0 ) ) + g g (cid:30) ( ( (cid:30) (cid:30) 0 0 ) ) (cid:0) [ @lnA @ ( s s0;(cid:30) 0 ) (cid:13) 0 + @lnA @ ( (cid:30) s0;(cid:30) 0 ) + Vs(s0) (cid:13) @lnA(s0;(cid:30) 0 ) ]; V(s0) 0 (cid:0) @(cid:30) i.e., (cid:0)a (cid:30)(cid:30) ((cid:30) 0 ) (cid:0) (cid:0)b (cid:30)(cid:30) ((cid:30) 0 ) = K 0 b((cid:27) (cid:0) 1)@l @ n (cid:30) A(s0;(cid:30) 0 ) (cid:13) 0 [1 F + y F (K (K 0 b(cid:21) b ) (cid:21) K )(cid:21) 1 b K (cid:21)2 b] (cid:0) [1 F + y F ( ( K K 0 b b ) ) K K 1 b b] > 0, (cid:26) 0 1 0 1 (cid:27) where the inequality follows from (cid:17)1 Kb;(cid:21) > (cid:17)1 Kb;1 . The last expression implies that there is some 0 0 (cid:30) ((cid:30) ;(cid:30) ] such that (cid:0)a((cid:30)) > (cid:0)b ((cid:30)) on ((cid:30) ;(cid:30)], which yields a contradiction of step 1. Accordingly, we 0 2 0 1 (cid:30) (cid:30) (cid:0) (cid:1)0 0 (cid:0) (cid:1) 60

must have (cid:0)a((cid:30) ) < (cid:0)b ((cid:30) ). (cid:30) 0 (cid:30) 0 Finally, (cid:0)a((cid:30) ) < (cid:0)b ((cid:30) ) implies (cid:0)a((cid:30)) < (cid:0)b((cid:30)) on some (small enough) interval (cid:30) ;(cid:30) , so (cid:30)+ (cid:30) 0 (cid:30) 0 0 00 described in the statement of the step the (cid:133)rst time (cid:0)a and (cid:0)b intersect to the right of (cid:30) . 0(cid:0)0 (cid:1) STEP 3: Under the assumptions of the lemma, (cid:0)a((cid:30)) < (cid:0)b((cid:30)) on ((cid:30) ;(cid:30) ). 0 1 I show that (cid:30) = (cid:30) ; where (cid:30) was de(cid:133)ned in step 2. Suppose for a moment that (cid:30) < (cid:30) . If + 0 + + 0 we de(cid:133)ne on [(cid:30) ;(cid:30) ]; wi((cid:30)) xi((cid:30))=xi (cid:30) and yi((cid:30)) = zi((cid:30))=xi (cid:30) , then it is readily seen that + 1 + + (cid:17) yi;wi((cid:30));(cid:0)i solveBVP(22)insaidinterval, with(cid:11)i((cid:30)) = (cid:11)((cid:30))andparameterK i = Kixi (cid:30) . That (cid:0) (cid:1) (cid:0) (cid:1) 0 0 + (cid:8)is, K a 0 = (cid:21) 1 K b 0(cid:9), where (cid:21) 1 (cid:17) (cid:21) x x b a ( ( (cid:30) (cid:30) + ) ) . As the BVPs associated with (cid:0)i satisfy the conditions of(cid:0)lem(cid:1)ma 4.iv + on [(cid:30) ;(cid:30) ], (cid:21) > 1. In addition, step 2 implies xa (cid:30) < xb (cid:30) , so (cid:21) < (cid:21). 0 + 1 + + 1 The previous discussion implies that the BVPs that yi;wi((cid:30));(cid:0)i for i = a;b solve on (cid:30) ;(cid:30) di⁄er (cid:0) (cid:1) (cid:0) (cid:1) + 1 only in the parameters K i ;Ki ; with K a = (cid:21) K b and Ka = (cid:21)Kb. To understand the implication of f 0 1g 0 1 0 (cid:8) 1 1(cid:9) (cid:2) (cid:3) this di⁄erence, it is convenient to consider a third BVP on (cid:30) ;(cid:30) di⁄ering from the previous two only + 1 in the parameters K c ;Kc , with K c = K a = (cid:21) K b and Kc = (cid:21) Kb. Given these de(cid:133)nitions, note that f 0 1g 0 0 1 0 (cid:2)1 1(cid:3)1 the BVPs associated with yb;wb((cid:30));(cid:0)b and yc;wc((cid:30));(cid:0)c satisfy the conditions in lemma 4.vii, so f g (cid:0)c (cid:30) < (cid:0)b (cid:30) . In addition, the BVPs associated to ya;wa((cid:30));(cid:0)a and yc;wc((cid:30));(cid:0)c satisfy the (cid:30) + (cid:30) + (cid:8) (cid:9) f g f g assumptions of 4.ii with Ka > Kc, so (cid:0)a (cid:30) < (cid:0)c (cid:30) . These inequalities yield (cid:0)a (cid:30) < (cid:0)b (cid:30) . (cid:0) (cid:1) (cid:0) (cid:1) 1 1 (cid:30) + (cid:30) + (cid:30) + (cid:30) + However, step 2 implies that (cid:0)a (cid:30) (cid:0)b (cid:30) , which is a contradiction. Then it must be the case that (cid:30) + (cid:21) (cid:30)(cid:0) +(cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:30) = (cid:30) . + 1 (cid:0) (cid:1) (cid:0) (cid:1) This concludes the proof of lemma 4. A.4 Section 5 A.4.1 Proof of Proposition 2 Let us start with the proof of (cid:30) < (cid:30) . For any (cid:30) [(cid:30);(cid:30)], let p(:;(cid:30) );rd(:;(cid:30) );H(;(cid:30) ) denote the (cid:3)a (cid:3)(cid:28) (cid:3) (cid:3) (cid:3) (cid:3) 2 solution to the BVP of the open economy described in lemma 3.iii, and let p(:;(cid:30) );rd(:;(cid:30) );H(:;(cid:30) ) (cid:8) (cid:3) (cid:3)(cid:9) (cid:3) be the solution to the BVP of the closed economy described in lemma 1.ii where the notation emphasizes (cid:8) (cid:9) the dependence of the solution on the parameter (cid:30) . Note that this notation implies pa;rd;a;Ha = (cid:3) p(:;(cid:30) );rd(:;(cid:30) );H(;(cid:30) ) and p(cid:28);rd;(cid:28);H(cid:28) = p(:;(cid:30) );rd(:;(cid:30) );H(;(cid:30) ) , where the superscripts a (cid:3)a (cid:3)a (cid:3)a (cid:3)(cid:28) (cid:3)(cid:28) (cid:3)(cid:28) (cid:8) (cid:9) and (cid:28) denote, respectively, the variables corresponding the autarky and trade equilibria of the economy (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9) underconsideration. Perthediscussionleadingtoproposition1, theleft-handsideofequation(21), which pins down the activity cuto⁄in the open economy, is strictly decreasing in the value of the parameter (cid:30) . (cid:3) Then, the result is proved if we show that the left-hand side of (21) is strictly greater than the right-hand side at (cid:30) = (cid:30) , i.e. if we show (cid:12) rd(:;(cid:30) );(cid:30) > (cid:12)a rd(:;(cid:30) );(cid:30) = L.54 (cid:3) (cid:3)a (cid:3)a (cid:3)a (cid:3)a (cid:3)a First,Ishowthatlemma4.iimpliesthatwhentheBVPsoftheopenandclosedeconomysharethesame (cid:0) (cid:1) (cid:0) (cid:1) boundaryconditions, thentheinversematchingfunction(matchingfunction)correspondingtotheformer lies completely below (above) that of the latter. In particular, for any (cid:30) [(cid:30);(cid:30)], H((cid:30);(cid:30) ) < H((cid:30);(cid:30) ) (cid:3) (cid:3) (cid:3) 2 for all (cid:30) ((cid:30) ;(cid:30)). De(cid:133)ne x((cid:30);(cid:30) ) rd((cid:30);(cid:30) )=(cid:27)f and z((cid:30);(cid:30) ) p((cid:30);(cid:30)(cid:3)) L f[1 G((cid:30) )]M . Then, 2 (cid:3) (cid:3) (cid:17) (cid:3) (cid:3) (cid:17) (cid:27)f (cid:0) (cid:0) (cid:3) 54(cid:12)a(:;:) and (cid:12)(:;:) are de(cid:133)ned in proposition 1 as the functions of rd and (cid:30)(cid:3) de(cid:133)ned(cid:2)by left-hand side of equa(cid:3)tions (16) and (21), respectively. 61

z(:;(cid:30) );x(:;(cid:30) );H(:;(cid:30) ) is the unique solution to BVP (22) with parameters K = 0, (cid:11)((cid:30);(cid:30) ) = 1 and (cid:3) (cid:3) (cid:3) 1 (cid:3) f g boundaryconditionsx((cid:30) ) = 1,H((cid:30) ) = sandH (cid:30) = s. Similarly,ifwede(cid:133)nex((cid:30);(cid:30) ) rd((cid:30);(cid:30) )=(cid:27)f (cid:3) (cid:3) (cid:3) (cid:3) (cid:17) and z((cid:30);(cid:30) ) p((cid:30)) [L fM (cid:30) f fx((cid:30)0;(cid:30)(cid:3))(cid:28)1 (cid:0) (cid:27)=fxydF (y)g (cid:30) Md(cid:30)], then we can think of the (cid:3) (cid:17) (cid:27)f (cid:0) (cid:0) (cid:30)(cid:3) x 0 (cid:0) (cid:1) 0 0 solution to the open economy BVP z(:;(cid:30) );x(:;(cid:30) );H(:;(cid:30) ) as the unique solution to BVP (22) with R R (cid:3) (cid:3) (cid:3) (cid:0) (cid:1) parameters K = 0, (cid:11)((cid:30);(cid:30) ) = 1+F f(cid:28)1 (cid:0) (cid:27) x((cid:30);(cid:30) ) (cid:28)1 (cid:27) and boundary conditions x((cid:30) ) = 1, 1 (cid:3) (cid:8) fx (cid:3) (cid:0)(cid:9) (cid:3) H((cid:30) ) = s and H (cid:30) = s.55 Givehn these(cid:16)de(cid:133)nitions, it(cid:17)is readiily seen that z(:;(cid:30) );x(:;(cid:30) );H(:;(cid:30) ) (cid:3) (cid:3) (cid:3) (cid:3) f g and z(:;(cid:30) );x(:;(cid:30) );H(:;(cid:30) ) satisfytheconditionsoflemma4.i,with (cid:11);1 takingtherolesof (cid:11)a;(cid:11)b , (cid:3) (cid:3)(cid:0) (cid:1) (cid:3) f g respectively. Then, H((cid:30);(cid:30) ) < H((cid:30);(cid:30) ) for all (cid:30) ((cid:30) ;(cid:30)). (cid:8) (cid:3) (cid:9) (cid:3) (cid:3) (cid:8) (cid:9) 2 I now show (cid:12)(r(:;(cid:30) );(cid:30) ) > (cid:12)a rd(:;(cid:30) );(cid:30) = L. The result in the last paragraph implies that (cid:3)a (cid:3)a (cid:3)a (cid:3)a z(:;(cid:30) );x(:;(cid:30) );H(:;(cid:30) ) and z(:;(cid:30) );x(:;(cid:30) );H(:;(cid:30) ) satisfy the conditions of lemma 4.iii, so (cid:3) (cid:3) (cid:3) (cid:0)(cid:3) (cid:3) (cid:1) (cid:3) f g (cid:30) x((cid:30);(cid:30) ) (cid:11)((cid:30);(cid:30)(cid:3)) g((cid:30))d(cid:30) > (cid:30)(cid:8)x((cid:30);(cid:30) )g((cid:30))d(cid:30). (cid:9) (cid:30)(cid:3) (cid:3) (cid:11)((cid:30)(cid:3);(cid:30)(cid:3)) (cid:30)(cid:3) (cid:3) RAn implication of this result aRnd (cid:11)((cid:30) ;(cid:30) ) 1 is that total wages paid to production workers are higher (cid:3) (cid:3) (cid:21) in the open economy if it shares the activity cuto⁄with the closed economy, (cid:27) 1 (cid:30) rd((cid:30);(cid:30) )(cid:11)((cid:30);(cid:30) )g((cid:30))d(cid:30)M = ((cid:27) 1)f(cid:11)((cid:30) ;(cid:30) ) (cid:30) x((cid:30);(cid:30) ) (cid:11)((cid:30);(cid:30)(cid:3)) g((cid:30))d(cid:30)M > (cid:0)(cid:27) (cid:30)(cid:3) (cid:3) (cid:3) (cid:0) (cid:3) (cid:3) (cid:30)(cid:3) (cid:3) (cid:11)((cid:30)(cid:3);(cid:30)(cid:3)) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) ((cid:27)R (cid:0) 1)f (cid:30) (cid:30) (cid:3) x((cid:30);(cid:30) (cid:3) )g((cid:30))d(cid:30)M = (cid:27) (cid:0)(cid:27) 1 (cid:30) (cid:30) (cid:3) rd((cid:30);(cid:30) (cid:3) )g((cid:30))Rd(cid:30)M In addition,Rper de(cid:133)nition we have, R (cid:12)a rd((cid:30);(cid:30) );(cid:30) = (cid:27) 1 (cid:30) rd((cid:30);(cid:30) )g((cid:30))d(cid:30)M +f[1 G((cid:30) )]M (cid:3) (cid:3) (cid:0)(cid:27) (cid:30)(cid:3) (cid:3) (cid:0) (cid:3) (cid:0) (cid:1) R (cid:27) 1 (cid:30) rd((cid:30);(cid:30) )(cid:11)((cid:30);(cid:30) )g((cid:30))d(cid:30)M + (cid:12) rd((cid:30);(cid:30) (cid:3) );(cid:30) (cid:3) = 8 (cid:0)(cid:27) (cid:30)(cid:3) (cid:3) rd((cid:30);(cid:30)(cid:3) (cid:3) )(cid:28)1 (cid:0) (cid:27) (cid:1)(cid:1)(cid:1) > f[1 G((cid:30) )] R M + (cid:30) f (cid:27)fx ydF (y)g (cid:30) Md(cid:30); (cid:0) (cid:1) < (cid:0) (cid:3) (cid:30)(cid:3) x 0 0 0 > R R (cid:0) (cid:1) : For (cid:30) = (cid:30) , these observations imply (cid:3) (cid:3)a (cid:12) rd(:;(cid:30) );(cid:30) > (cid:12)a rd((cid:30);(cid:30) );(cid:30) = L; (cid:3)a (cid:3)a (cid:3)a (cid:3)a (cid:16) (cid:17) (cid:16) (cid:17) which is the desired result.56 Let us now prove the other results in the proposition, i.e. N(cid:28) (s) > Na(s) for all s [s;s) and 2 proposition 2.ii. Let N (s;(cid:30) ) be the inverse function of H((cid:30);(cid:30) ). Following the discussion above, these (cid:3) (cid:3) results can be easily proved by decomposing the total e⁄ect on the matching function into that of the increase in the exit cuto⁄(intensive-margin channel) and that of having an increasing share of exporters at each productivity level in the open economy (extensive-margin channel). Starting with the former, the no-crossing result in lemma 2.i and (cid:30) < (cid:30) imply Na(s) = N (s;(cid:30) ) < N (s;(cid:30) ) on [s;s).57 Bringing (cid:3)a (cid:3)(cid:28) (cid:3)a (cid:3)(cid:28) 55Note that we are considering z(:;(cid:30)(cid:3));x(:;(cid:30)(cid:3));H(:;(cid:30)(cid:3)) as the solution to a di⁄erent parametrization of the BVP (22) than the one considered in section 4.2. (cid:8) (cid:9) 56In this derivation we used (cid:27)fx((cid:30);(cid:30)(cid:3)a )=rd;a((cid:30)) and the fact that equation (16) holds in autarky. 57As N(:;(cid:30)(cid:3)) solves the BVP of the closed economy with activity cuto⁄(cid:30)(cid:3), note that N(s;(cid:30)(cid:3)(cid:28) ) is the matching function of the ancillary autarkic economy described in the paper. 62

the e⁄ects of exporters into the picture, lemma 4.i implies that H((cid:30);(cid:30) ) > H((cid:30);(cid:30) ) = H(cid:28)((cid:30)) on (cid:30) ;(cid:30) , (cid:3)(cid:28) (cid:3)(cid:28) (cid:3)(cid:28) i.e. N (s;(cid:30) ) < N(s;(cid:30) ) = N(cid:28) (s) on (s;s): Combining these observations yield the desired result. (cid:3)(cid:28) (cid:3)(cid:28) (cid:0) (cid:1) A.4.2 Proof of Proposition 3 Proposition 3.i Letusstartwiththeproofof(cid:30) < (cid:30) . Forany(cid:30) [(cid:30);(cid:30)]andi = l;h,let pi(:;(cid:30) );rd;i(:;(cid:30) );H i (;(cid:30) ) (cid:3)h (cid:3)l (cid:3) (cid:3) (cid:3) (cid:3) 2 denote the solution to the BVP of the open economy described in lemma 3n.iii with variable trade costso (cid:28) and productivity exit cuto⁄(cid:30) (the notation emphasizes the dependence of the solution on (cid:28) and (cid:30) .) i (cid:3) i (cid:3) With this notation we have pi;rd;i;Hi = pi(:;(cid:30) );rd;i(:;(cid:30) );H i (;(cid:30) ) , where pi;rd;i;Hi are the (cid:3)i (cid:3)i (cid:3)i equilibrium price, revenue and inverse-matchning functions of an open econoomy with variable trade costs (cid:8) (cid:9) (cid:8) (cid:9) (cid:28) . Let (cid:12)i rd;(cid:30) be the function de(cid:133)ned by the left-hand side of equation (21) in terms of rd and (cid:30) i (cid:3) (cid:3) when variable trade costs are given by (cid:28) .58 Per the discussion leading to proposition 1, (cid:12)i rd;i(:;(cid:30) );(cid:30) (cid:0) (cid:1) i (cid:3) (cid:3) is strictly decreasing in the value of the parameter (cid:30) . Then, to prove the result it is enough to show (cid:3) (cid:0) (cid:1) (cid:12)l rd;l(:;(cid:30) );(cid:30) > (cid:12)h rd;h(:;(cid:30) );(cid:30) = L. (cid:3)h (cid:3)h (cid:3)h (cid:3)h As a (cid:133)rst step, I show that for any (cid:30) [(cid:30);(cid:30)), rd;l((cid:30);(cid:30) )(cid:28)1 (cid:27) > rd;h((cid:30);(cid:30) )(cid:28)1 (cid:27) for all (cid:30) [(cid:30) ;(cid:30)]. (cid:0) (cid:1) (cid:0) (cid:1) (cid:3) 2 (cid:3) l(cid:0) (cid:3) h(cid:0) 2 (cid:3) Letting xi((cid:30);(cid:30) ) rd;i((cid:30);(cid:30) )=(cid:27)f; (cid:3) (cid:3) (cid:17) zi((cid:30);(cid:30) ) pi((cid:30)) [L fM (cid:30) f fxi((cid:30)0;(cid:30)(cid:3))(cid:28)1 i(cid:0) (cid:27)=fxydF (y)g (cid:30) Md(cid:30)]; (cid:3) (cid:17) (cid:27)f (cid:0) (cid:0) (cid:30)(cid:3) x 0 0 0 then zi(:;(cid:30) );xi(:;(cid:30) );H i (:;(cid:30) ) is the uni R que so R lution to BVP (22) with (cid:0) pa (cid:1) rameters Ki = f (cid:28)1 (cid:27), f (cid:3) (cid:3) (cid:3) g 0 fx i(cid:0) Ki = (cid:28)1 (cid:27), (cid:11)i((cid:30);(cid:30) ) = 1 and boundary conditions xi((cid:30) ) = 1, H i ((cid:30) ) =s and H i (cid:30) = s. Noting that 1 i(cid:0) (cid:3) (cid:3) (cid:3) Kl = (cid:21)Kh and Kl = (cid:21)Kh with (cid:21) = ((cid:28) =(cid:28) )1 (cid:27) > 1, it is readily seen that zi(:;(cid:30) );xi(:;(cid:30) );H i (:;(cid:30) ) 0 0 1 1 l h (cid:0) (cid:3)(cid:0) (cid:1) (cid:3) (cid:3) for i = l;k, satisfy the conditions of lemma 4.iv, so rd;l((cid:30);(cid:30) )(cid:28)1 (cid:27) > rd;h((cid:30)n;(cid:30) )(cid:28)1 (cid:27) for all (cid:30) [(cid:30) ;(cid:30)].o (cid:3) l(cid:0) (cid:3) h(cid:0) 2 (cid:3) Let us now show (cid:12)l rd;l(:;(cid:30) );(cid:30) > (cid:12)h rd;h(:;(cid:30) );(cid:30) = L. To economize on space, I de(cid:133)ne the (cid:3)h (cid:3)h (cid:3)h (cid:3)h following notation (cid:0) (cid:1) (cid:0) (cid:1) (cid:14)i((cid:30)) = 1+F Kixi((cid:30);(cid:30) ) Ki 0 (cid:3) 1 (cid:30) Ri((cid:30) ) (cid:2) rd;i (cid:0) ((cid:30);(cid:30) ) 1+F (cid:1) rd(cid:3);i((cid:30);(cid:30)(cid:3)) (cid:28)1 (cid:27) (cid:28)1 (cid:27) g((cid:30))d(cid:30)M; (cid:3) (cid:17) (cid:3) (cid:27)fx i(cid:0) i(cid:0) Z (cid:30)(cid:3) h (cid:16) (cid:17) i (cid:30) rd;i((cid:30);(cid:30)(cid:3))(cid:28)1 (cid:0) (cid:27) FFd((cid:30) ) f[1 G((cid:30) )]M; and FFx;i((cid:30) ) f (cid:27)fx ydF (y)g (cid:30) Md(cid:30), (cid:3) (cid:3) (cid:3) x 0 0 (cid:17) (cid:0) (cid:17) Z (cid:30)(cid:3) Z0 (cid:0) (cid:1) where Ki;Ki;xi were de(cid:133)ned above. These de(cid:133)nitions and the result in the previous paragraph imply f 0 1 g (cid:14)l((cid:30)) > (cid:14)h((cid:30)) for all (cid:30) (cid:30) ;(cid:30) , i.e zi(:;(cid:30) );xi(:;(cid:30) );H i (:;(cid:30) ) satisfy the conditions of lemma 4.v, (cid:3) (cid:3) (cid:3) (cid:3) 2 f g so Rl((cid:30) ) > Rh((cid:30) ). In addition, the result in the last paragraph also implies FFx;l((cid:30) ) > FFx;h((cid:30) ). (cid:3) (cid:3) (cid:2) (cid:3) (cid:3) (cid:3) 58Note that (cid:12)i(:;:) is just the function (cid:12)(:;:) de(cid:133)ned in proposition 1, where the superscript i in the current notation emphasizes the dependence of this function on (cid:28) . i 63

These inequalities and the de(cid:133)nition of (cid:12)i yield (cid:12)l rd;l((cid:30);(cid:30) );(cid:30) = (cid:27) 1Rl((cid:30) )+FFd((cid:30) )+FFx;l((cid:30) ) (cid:3)h (cid:3)h (cid:0)(cid:27) (cid:3)h (cid:3)h (cid:3)h (cid:16) (cid:17) > (cid:27) 1Rh((cid:30) )+FFd((cid:30) )+FFx;h((cid:30) ) (cid:0)(cid:27) (cid:3)h (cid:3)h (cid:3)h = (cid:12)h rd;h((cid:30);(cid:30) );(cid:30) = L: (cid:3)h (cid:3)h (cid:16) (cid:17) As discussed above, this result implies (cid:30) < (cid:30) . (cid:3)h (cid:3)l Finally, the continuity of the matching functions and (cid:30) < (cid:30) imply that there is a skill level s (s;s] (cid:3)h (cid:3)l 0 2 such that Nl(s) > Nh(s) on [s;s), i.e. inequality necessarily increases among the least skilled workers 0 of the economy after a trade liberalization. Proposition 3.ii Here I formally derive the impact on relative wages of the selection-into-activity and the intensivemargin channels discussed in the text. I start by de(cid:133)ning some notation. In the sequel, z((cid:30);(cid:30) ;(cid:11)); (cid:3) f x((cid:30);(cid:30) ;(cid:11)); H((cid:30);(cid:30) ;(cid:11)) denotes the unique solution to BVP (22) with constant K = 0; parameter (cid:3) (cid:3) 1 g function (cid:11), and boundary conditions x((cid:30) ) = 1;H((cid:30) ) = s;H (cid:30) = s , where the notation emphasizes (cid:3) (cid:3) f g the dependence of the solution on (cid:30) ;(cid:11) . In addition, I will use N ((cid:30);(cid:30) ;(cid:11)) to denote the inverse of (cid:3) (cid:0) (cid:1) (cid:3) f g H((cid:30);(cid:30) ;(cid:11)). For i = l;h; let (cid:30) ;pi;rd;i;Hi be the activity cuto⁄, price, domestic revenue and inverse- (cid:3) (cid:3)i matching functions of the two open economies in the statement of the proposition (these economies di⁄er (cid:8) (cid:9) only in the variable trade costs they face, with (cid:28) < (cid:28) ). De(cid:133)ning the parameter functions (cid:11)i((cid:30)) l h 1+F (cid:28)1 i(cid:0) (cid:27) rd;i((cid:30)) (cid:28)1 (cid:27) for i = l;h, we can think of the BVPs associated with each Hi as particula (cid:17) r (cid:27)fx i(cid:0) pharame(cid:16)terizations o(cid:17)f BVPi(22) with K 1 = 0 and (cid:11) = (cid:11)i.59 In the notation de(cid:133)ned here, x (cid:30);(cid:30) ;(cid:11)i = rd;i((cid:30);(cid:30) )=(cid:27)f (cid:3)i (cid:3)i z (cid:0) (cid:30);(cid:30) ;(cid:11)i (cid:1) = pi((cid:30)) [L fM (cid:30) f rd;i((cid:30) (cid:27) ;(cid:30) f (cid:3)i x )(cid:28)1 (cid:0) (cid:27) ydF (y)g (cid:30) Md(cid:30)] (cid:3)i (cid:27)f (cid:0) (cid:0) x 0 0 Z (cid:30)(cid:3) Z0 H (cid:0) (cid:30);(cid:30) ;(cid:11)i(cid:1) = Hi (cid:0) (cid:1) (cid:3)i (cid:0) (cid:1) After these preliminaries we are ready to prove the claim. Let us start with the selection-into-activity channel. As discussed in the text, the matching functions N and Nh in (cid:133)gure 2 di⁄er only in their activity cuto⁄s, i.e. N = N (cid:30);(cid:30) ;(cid:11)h and Nh = 0 0 (cid:3)l N (cid:30);(cid:30) ;(cid:11)h . Accordingly, the no-crossing result in lemma 2.i implies N (s) > Nh(s) on [s;s). Note (cid:3)h 0 (cid:0) (cid:1) that by sharing the same parameter function (cid:11)h, the economies associated with N and Nh have the (cid:0) (cid:1) 0 same fraction of exporters at each productivity (among active (cid:133)rms) and face the same variable costs. Accordingly, their di⁄erence captures the e⁄ects of the selection-into-activity channel on relative wages. Let us now turn to the intensive-margin channel. De(cid:133)ne (cid:11)1((cid:30)) 1+F (cid:28)1 h(cid:0) (cid:27) rd;h((cid:30)) (cid:28)1 (cid:27) , (cid:17) (cid:27)fx l(cid:0) (cid:20) (cid:18) (cid:19) (cid:21) so (cid:11)1((cid:30)) di⁄ers from (cid:11)h((cid:30)) only in the value of the variable trade cost outside the function F. In 59See the proof of proposition 2. 64

addition, note that for any pair (cid:30) ;(cid:30) [(cid:30) ;(cid:30)] such that (cid:30) > (cid:30) and F (cid:28)1 (cid:27)rd;h((cid:30) )=(cid:27)f > 0, we have 00 0 2 (cid:3) 00 0 h(cid:0) 00 x (cid:11)1 (cid:30) =(cid:11)1 (cid:30) > (cid:11)h (cid:30) =(cid:11)h (cid:30) . Finally, as discussed in the text, the matching functions N and N 00 0 00 0 (cid:0) (cid:1) 0 1 in (cid:133)gure 2 di⁄er only in their parameter function (cid:11), i.e. N = N (cid:30);(cid:30) ;(cid:11)h and N = N (cid:30);(cid:30) ;(cid:11)1 . (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) 0 (cid:3)l 1 (cid:3)l Accordingly, the BVPs associated with N and N satisfy the conditions of lemma 4.i, so N (s) > N (s) 0 1 (cid:0) (cid:1) 1 (cid:0) 0 (cid:1) on [s;s). Proposition 3.iii ToprovetheresultitisconvenienttobreakthechangesintheBVPoftheopeneconomyintroducedby the liberalization in two parts, the change associated to the decline in variable trade costs and the change associated to the rise in the activity cuto⁄(allowing the set of exporters to adjust in each case). Starting with the former, let N be the matching function resulting from reducing (cid:28) to (cid:28) in the BVP of the 0 h l open economy before the liberalization, keeping the activity cuto⁄unchanged. If the assumption on (cid:17)F is 1 satis(cid:133)ed, then it is readily seen that F and the open-economy BVPs associated with Nh and N0 satisfy the conditions in lemma 4.vii with Kh = f(cid:28)1 (cid:27)=f ; Kh = n(cid:28)1 (cid:27); K0 = (cid:21)Kh, and (cid:21) = ((cid:28) =(cid:28) )1 (cid:27) > 1. 0 h(cid:0) x 1 h(cid:0) i i l h (cid:0) Accordingly, N0(s) > Nh(s) on (s;s) as shown in (cid:133)gure 6. Proof. Figure 6: Trade Liberalization under Su¢ cient Conditions Note: Thesolidredandbluelinesrepresent,respectively,thepre-(Nh)andpost-liberalization(Nl)matching functions. The dashed black line (N0)representsthe solution to the BVP ofthe open economy with (cid:28) =(cid:28)l and (cid:30)(cid:3) =(cid:30)(cid:3)h . When (cid:17)F 1 satis(cid:133)es the su¢ cient conditions in proposition 3.iii, N0 lies above Nh. When (cid:17)F 0 satis(cid:133)es the su¢ cient conditions in said proposition,Nl lies above N0. Now consider the change in the matching function associated with the rise in the activity cuto⁄, i.e. the di⁄erence between N and Nl in (cid:133)gure 6. Suppose that N and Nl intersect on (s;s) with the (cid:133)rst 0 0 intersection occurring at s, namely N (s) = Nl(s) = (cid:30). If we de(cid:133)ne on (cid:30);(cid:30) , wi((cid:30)) rd;i((cid:30)) 0 0 0 0 0 0 (cid:17) rd;0((cid:30)0) and yi((cid:30)) pi((cid:30)) [L fMi (cid:30) f rd;i( (cid:27) (cid:30) f ) x (cid:28)1 l(cid:0) (cid:27) ydF (y)g((cid:30))Md(cid:30)], then w (cid:2) i;yi; (cid:3) Hi is the unique (cid:17) rd;i((cid:30)0)M (cid:0) (cid:0) (cid:30)(cid:3)i x y solution to BVP (22) with parameteRrs (cid:11)i(R(cid:30)) = 1, Ki = rd;i((cid:30)0)(cid:28)1 l(cid:0) (cid:27) , Ki = n(cid:28)1 (cid:27)(cid:8)and boun(cid:9)dary conditions 0 (cid:27)f 1 l(cid:0) 65

wi (cid:30) = 1, Hi (cid:30) = s and Hi (cid:30) = s. In addition, note that the log-supermodularity of A and 0 0 0 Hl((cid:30)) < H ((cid:30)) on [(cid:30) ;(cid:30)) implies rd;0 (cid:30) > rd;l (cid:30) , so K0 > Kl. Accordingly, if the assumption on (cid:0) (cid:1) 0 (cid:0) (cid:1) (cid:3)l 0 (cid:0) (cid:1) 0 0 0 0 (cid:17)F is satis(cid:133)ed, then its is readily seen that F and the open-economy BVPs associated with Nl and N0 0 (cid:0) (cid:1) (cid:0) (cid:1) satisfytheconditionsoflemma4.vion (cid:30);(cid:30) , soHl (cid:30) < H0 (cid:30) . However, Hl((cid:30)) < H0((cid:30))on[(cid:30) ;(cid:30)) 0 (cid:30) 0 (cid:30) 0 (cid:3)l 0 implies Hl (cid:30) H0 (cid:30) , which is a contradiction. Then it must be the case that Nl and N0 do not (cid:30) 0 (cid:21) (cid:30) 0 (cid:2) (cid:3) (cid:0) (cid:1) (cid:0) (cid:1) intersect on (s;s), so Nl lies strictly above N0 on [s;s) as shown in the picture. (cid:0) (cid:1) (cid:0) (cid:1) Combining the last two results we get Nl(s) > Nh(s) on [s;s), so inequality is pervasively higher after the liberalization. This concludes the proof of the proposition. A.5 Section 6 A.5.1 Free-Entry Equilibrium in the Closed Economy In the free-entry model the mass of (cid:133)rms in the industry, M, is an additional endogenous variable. As described in the main text, there is an unbounded pool of prospective (cid:133)rms that can enter the industry by incurringa(cid:133)xedentry-costoffeV(s)unitsofeachskills S. Uponentry, (cid:133)rmsobtaintheirproductivity 2 as independent draws from the distribution G, as explained in section 2.2. Note that the new free-entry assumption does not a⁄ect the basic structure of the model described in section 2, so equations (1)-(7) continue to hold. The analysis of the closed-economy equilibrium in section 3 is valid for any mass of (cid:133)rms, M, so it applies almost unchanged to the free-entry model once M has been determined. In fact, conditional on M, the analysis needs to be modi(cid:133)ed only to account for the presence of (cid:133)xed entry-costs, i.e. L must be replaced with L feM throughout the analysis. A free-entry condition provides the additional (cid:0) equilibrium condition to pin down the mass of (cid:133)rms. In the free-entry model, the labor market clearing condition is given by (cid:30) g((cid:30)) LV (s) = ld(s;(cid:30)) d(cid:30)M +MfV (s)+Mfe for all s S: (81) [1 G((cid:30) )] 2 Z (cid:30)(cid:3) (cid:0) (cid:3) With unrestricted entry, prospective entrants must be indi⁄erent between entering and not entering the industry, i.e. expected pro(cid:133)ts from entering must equal the cost of entry, [1 G((cid:30) )](cid:25)d = fe, where (cid:25)d (cid:3) (cid:0) is the average domestic pro(cid:133)t of active (cid:133)rms. Per the optimal pricing rule , this free-entry condition can be written as follows, (cid:30) rd((cid:30)) f g((cid:30))d(cid:30) = fe. (82) (cid:27) (cid:0) Z (cid:30)(cid:3) (cid:20) (cid:21) De(cid:133)nition 3 A free-entry equilibrium of the closed economy is a mass of (cid:133)rms M > 0, a mass of active (cid:133)rms M > 0, a productivity activity-cuto⁄ , (cid:30) (cid:3) ((cid:30);(cid:30)), an output function qd : [(cid:30) (cid:3) ;(cid:30)] R+ , a labor 2 ! allocationfunctionld : S [(cid:30) (cid:3) ;(cid:30)] R+ , apricefunctionp : [(cid:30) (cid:3) ;(cid:30)] R+ andawageschedulew : S R+ (cid:2) ! ! ! such that the following conditions hold, (i) consumers behave optimally, equations (1) and (2); 66

(ii) (cid:133)rms behave optimally given their technology, equations (3), (7), (9), and (10); (iii) goods and labor markets clear, equations (8) and (81), respectively; (iv) the numeraire assumption holds, w = 1; (v) the free-entry condition holds, equation (82). Given the equilibrium activity cuto⁄, (cid:30) , the price, domestic-revenue and inverse-matching functions, (cid:3) p;rd;H , solve a BVP that is almost identical to the one de(cid:133)ned in lemma 1.ii for the no-free-entry model. The only di⁄erence lies in the slope of the inverse-matching function, which is now given by (cid:8) (cid:9) rd((cid:30))g((cid:30))M H ((cid:30)) = : (83) (cid:30) A(H((cid:30));(cid:30)) L fM feM V (H((cid:30)))p((cid:30)) (cid:0) (cid:0) (cid:2) (cid:3) The discussion in section 4.2 implies that, for a given activity cuto⁄(cid:30) , the functions rd and H that solve (cid:3) this BVP do not depend on the mass of (cid:133)rms nor the mass of production workers. Noting that equations (15) and (83) may di⁄er only in these parameters, the last observation implies that, for a given (cid:30) , the (cid:3) closed-economy BVPs of the no-free-entry and free-entry models share the same solution functions rd and H.60 As the revenue function rd depends only on (cid:30) , the free-entry condition (82) can be used to determine (cid:3) the equilibrium activity cuto⁄, (cid:30) . Finally, combining the equilibrium relationship L = Mfe +Mf + (cid:3) (cid:27) 1Mrd (the counterpart of condition (16) in the no-free-entry model), and free-entry condition we can (cid:0)(cid:27) express the mass of (cid:133)rms as a function of exogenous variables and the activity cuto⁄(cid:30) , (cid:3) L M = : (84) (cid:27)fe+(cid:27)f[1 G((cid:30) )] (cid:3) (cid:0) I summarize this discussion in the following lemma. Lemma 5 In a free-entry equilibrium of the closed economy with activity cuto⁄ (cid:30) ((cid:30);(cid:30)) the following (cid:3) 2 conditions hold. (i) There exists a continuous and strictly increasing matching function N : S [(cid:30) ;(cid:30)], (with inverse (cid:3) ! function H) such that (i) ld(s;(cid:30)) > 0 if and only if N (s) = (cid:30), (ii) N (s) = (cid:30) , and N (s) = (cid:30). (cid:3) (ii) The wage schedule w is continuously di⁄erentiable and satis(cid:133)es (12). (iii) The price, revenue and matching functions, p;rd;N (H) ; are continuously di⁄erentiable. Given (cid:30) , the triplet p;rd;H solves the BVP comprising the di⁄erential equations {(13), (14), (83)} and the (cid:3) (cid:8) (cid:9) boundary conditions rd((cid:30) ) = (cid:27)f, H((cid:30) ) = s, H (cid:30) = s. (cid:8) (cid:9) (cid:3) (cid:3) (iv) The activity cuto⁄ (cid:30) and the revenue function rd satisfy the free-entry condition (82). (cid:3) (cid:0) (cid:1) (v) The mass of (cid:133)rms in the industry, M, is given by (84). Moreover, if a number (cid:30) (cid:3) ((cid:30);(cid:30)), and functions p;rd : [(cid:30) (cid:3) ;(cid:30)] R+ and H : [(cid:30) (cid:3) ;(cid:30)] S satisfy condi- 2 ! ! tions (ii)-(iv), then they are, respectively, the productivity activity-cuto⁄, the price function, the revenue function, and the inverse of the matching function of a free-entry equilibrium of the closed economy. 60These two BVPs are equivalent to the same parametrization of BVP (22). 67

The discussion preceding proposition 1 implies that rd((cid:30)) decreases with (cid:30) , making the left-hand (cid:3) side of (82) strictly decreasing in (cid:30) . If the (cid:133)xed entry costs are not too high, then there is a unique (cid:3) activity cuto⁄(cid:30) that solves (82). In turn, this result implies that there is a unique free-entry equilibrium (cid:3) of the open economy. A.5.2 Free-Entry Equilibrium in the Open Economy The similarities between the analyses of the closed-economy equilibrium in the no-free-entry and freeentry models extend to the open-economy case. In particular, replacing L with L feM throughout (cid:0) the analysis in section 4 yields the characterization of the open-economy equilibrium in the free-entry model, conditional on the mass of (cid:133)rms M. The free-entry condition provides the additional equilibrium condition to determine M. The labor market clearing condition in the open economy is given by (cid:30) [ld(s;(cid:30))g((cid:30))M +lx(s;(cid:30))Mx((cid:30))]d(cid:30)+ LV (s) = fMV (Rs (cid:30) ) (cid:3) +nfx (cid:28)1 (cid:0) (cid:27) (cid:27) f r x d((cid:30)) ydF (y)Mx((cid:30))V (s)+f (cid:1)(cid:1) e (cid:1) MV (s) for all s 2 S: (85) 0 R As before, unrestricted entry implies that expected pro(cid:133)ts from entering the industry must equal the cost of entry, [1 G((cid:30) )] (cid:25)d+(cid:25)x = fe, where (cid:25)d and (cid:25)x are, respectively, the average domestic and export (cid:3) (cid:0) pro(cid:133)t of active (cid:133)rms.61 Per the optimal pricing rule , this free-entry condition can be written as shown (cid:2) (cid:3) in equation (24) in the main text. De(cid:133)nition 4 A free-entry equilibrium of the open economy is a mass of (cid:133)rms M, an activity cuto⁄ (cid:30) , (cid:3) a mass of active (cid:133)rms M > 0; a mass of exporters Mx((cid:30)) > 0 for each productivity level (cid:30) (cid:30) , output (cid:3) (cid:21) functions qd;qx : [(cid:30) (cid:3) ;(cid:30)] R+ , labor allocations functions ld;lx : S [(cid:30) (cid:3) ;(cid:30)] R+ , a price function ! (cid:2) ! p : [(cid:30) (cid:3) ;(cid:30)] R+ and a wage schedule w : S R+ such that the following conditions hold, ! ! (i) consumers behave optimally, equations (1) and (2); (ii) (cid:133)rms behave optimally given their technology, equations (3), (7), (9), (10) and (18); (iii) goods and labor markets clear, equations (8), (17) and (24); (iv) the numeraire assumption holds, w = 1; (v) the free-entry condition holds, equation (24). Given the equilibrium activity cuto⁄, (cid:30) , the price, domestic-revenue and inverse-matching functions, (cid:3) p;rd;H , solve a BVP that is almost identical to the one de(cid:133)ned in lemma 3.iii for the no-free-entry model. The only di⁄erence lies in the slope of the inverse-matching function, which is now given by (cid:8) (cid:9) rd((cid:30))(cid:28)1 (cid:27) rd((cid:30)) 1+F (cid:0) n(cid:28)1 (cid:27) g((cid:30))M (cid:27)fx (cid:0) H (cid:30) ((cid:30)) = (cid:20) (cid:18) (cid:19) rd( (cid:21) (cid:30)0)(cid:28)1 (cid:0) (cid:27) : (86) A(H((cid:30));(cid:30))V(H((cid:30)))p((cid:30)) 2 L (cid:0) fM (cid:0) feM (cid:0) (cid:30) (cid:30) (cid:3) nfx 0 (cid:27)fx ydF(y)g((cid:30)0)Md(cid:30)03 6 R R 7 4 5 61Note that (cid:25)x is not the average export pro(cid:133)ts among exporters, but among all active (cid:133)rms. 68

Noting that equations (20) and (86) may di⁄er only in the mass of (cid:133)rms or the mass of production workers, the discussion in the preceding section implies that, for a given (cid:30) , the open-economy BVPs of (cid:3) the no-free-entry and free-entry models share the same solution functions rd and H. As before, the free-entry condition (24) can be used to determine the equilibrium activity cuto⁄, (cid:30) . (cid:3) rd((cid:30)0)(cid:28)1 (cid:0) (cid:27) Finally,theequilibriumrelationship,L = Mfe+Mf+ (cid:30) nfx (cid:27)fx ydF (y)g (cid:30) Md(cid:30) +(cid:27) 1Mrd+ (cid:30)(cid:3) 0 0 0 (cid:0)(cid:27) (cid:27) 1Mrx, can be combined with the free-entry condition to express the mass of (cid:133)rms in the industry as a (cid:0)(cid:27) R R (cid:0) (cid:1) function of exogenous parameters, the activity cuto⁄(cid:30) and the revenue function rd, (cid:3) L M = : (87) rd((cid:30)0)(cid:28)1 (cid:0) (cid:27) (cid:27) fe+f[1 G((cid:30) )]+ (cid:30) nfx (cid:27)fx ydF (y)g (cid:30) d(cid:30) 2 (cid:0) (cid:3) (cid:30)(cid:3) 0 0 0 3 R R (cid:0) (cid:1) 4 5 I summarize this discussion in the following lemma. Lemma 6 In a free-entry equilibrium of the open economy with activity cuto⁄ (cid:30) ((cid:30);(cid:30)) the following (cid:3) 2 conditions hold. (i) There exists a continuous and strictly increasing matching function N : S [(cid:30) ;(cid:30)], (with inverse (cid:3) ! function H) such that (i) ld(s;(cid:30))+lx(s;(cid:30)) > 0 if and only if N (s) = (cid:30), (ii) N (s) = (cid:30) , and N (s) = (cid:30). (cid:3) (ii) The wage schedule w is continuously di⁄erentiable and satis(cid:133)es (12) (iii) The price, domestic revenue and matching functions, p;rd;N ; are continuously di⁄erentiable. Given (cid:30) , the triplet p;rd;H solves the BVP comprising the system of di⁄erential equations {(13), (cid:3) (cid:8) (cid:9) (14), (86)} and the boundary conditions rd((cid:30) ) = (cid:27)f, H((cid:30) ) = s, H (cid:30) = s. (cid:8) (cid:9) (cid:3) (cid:3) (iv) The activity cuto⁄ (cid:30) and the revenue function rd satisfy the free-entry condition (24). (cid:3) (cid:0) (cid:1) (v) The mass of (cid:133)rms in the industry, M, is given by (87) Moreover, if a number (cid:30) (cid:3) ((cid:30);(cid:30)), and functions p;rd : [(cid:30) (cid:3) ;(cid:30)] R+ and H : [(cid:30) (cid:3) ;(cid:30)] S satisfy 2 ! ! the conditions (iii)-(iv), then they are, respectively, the activity cuto⁄, the price function, the domestic revenue function, and the inverse-matching function of a free-entry equilibrium of the open economy. A.5.3 Proof of Proposition 4 Inthefree-entry model theactivitycuto⁄mayincreaseordecreasewhentheeconomystartstrading. The reasons behind this ambiguity are discussed in the text. In addition, as stated in the text, proposition 4.i considers essentially the same case as proposition 2, so the arguments in the proof of the latter also applies to the former. Here I focus on Proposition 4.ii. Proposition 4.ii Let (cid:30) < (cid:30) . If N(cid:28) (s) < Na(s) for all s [s;s), then lemma 2.ii implies that rd;(cid:28) ((cid:30)) > rd;a((cid:30)) (cid:3)(cid:28) (cid:3)a 2 for all (cid:30) (cid:30) , so domestic pro(cid:133)ts in the open economy are necessarily higher than in autarky. With (cid:3)a (cid:21) strictly positive export pro(cid:133)ts, this observation implies that total average pro(cid:133)ts must be higher in the 69

open economy, violating the free entry condition (24). Accordingly, N(cid:28) (s) must lie above Na(s) for some values of s, implying that N(cid:28) (s) and Na(s) must intersect at least once on (s;s). Next, I show that N(cid:28) (s) and Na(s) intersect exactly once on (s;s). The argument is more easily stated in terms of the inverse functions H(cid:28) and Ha. Let (cid:30) be the (cid:133)rst time that H(cid:28) and Ha intersect 0 on ((cid:30) ;(cid:30)).62 Note that H(cid:28) and Ha are part of the unique solutions to parameterizations of BVP (22) (cid:3)a that di⁄er only in the parameter function (cid:11)i, with Ki = 0 for i = (cid:28);a; (cid:11)(cid:28) ((cid:30)) = 1+F rd;(cid:28)(cid:30)(cid:28)1 (cid:27) and 1 (cid:27)fx (cid:0) (cid:11)a((cid:30)) = 1.63 Then, an immediate application of lemma 4.i yields H(cid:28)((cid:30)) < Ha((cid:30)) on ((cid:30)(cid:16);(cid:30)), so H(cid:17)(cid:28) and 0 Ha (N(cid:28) (s) and Na(s)) intersect exactly once on ((cid:30) ;(cid:30)) ((s;s)) at (cid:30) (s = Hi((cid:30) )). (cid:3)a 0 0 0 The last result implies that, in the open economy, inequality is lower among workers with skill levels below s , but higher among workers with skill level above s . Put another way, opening to trade leads to 0 0 wage polarization. The e⁄ects of the intensive- and extensive-margin channels can be proved by adapting the arguments in proposition 2.ii. 62(cid:30) is well-de(cid:133)ned due to the continuity of Hi, i=(cid:28);a. 0 63See the proof of theorem X (9 I think) for more details. 70