# Trade Costs, Generalized Demand and Firm Selection

Badis Tabarki

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CES

Centre d'Économie de la Sorbonne
UMR 8174

Trade Costs, Generalized Demand and Firm Selection

Badis TABARKI

2025.23
# Trade Costs, Generalized Demand, and Firm Selection

Badis Tabarki*

Centre d'Économie de La Sorbonne

(UMR 8174 CNRS-University Paris 1)

October 24, 2025

## Abstract

This paper extends the canonical Melitz-Chaney framework beyond the CES to characterize the selection effect à la Melitz (2003) under generalized demand. It uses a Gorman-Pollak demand system that nests direct and indirect separability and keeps demand curvature unrestricted. The selection effect is always operative and remains the dominant welfare channel in this gravity-consistent environment. The model generalizes the ACR formula (Arkolakis et al., 2012) and reveals that by shaping export participation patterns, demand curvature governs the intensity of the competitive pressure trade exerts on domestic firms forcing them to exit. Super-convex demand yields an upper bound for the probability of exporting and the average export productivity and thus magnifies trade-induced firm exit. Hence, by magnifying the selection effect, it yields an upper bound for the gains from trade. These patterns are reversed under sub-convex demand. Within these bounds, CES demand delivers an intermediate outcome. This paper thus uncovers new implications of demand curvature previously unexplored in the trade literature.

*key words:* International trade; Melitz-Chaney model; Firm heterogeneity; Selection effect; Demand curvature; Generalized separability; Variable markups; Structural gravity.

*JEL Codes:* F12, D11, L11

<footnote>*I am very grateful to Lionel Fontagné, Gianmarco Ottaviano, Mathieu Parenti, Ariell Reshef, Monika Mrazova and Thomas Chaney for insightful suggestions and helpful comments on the previous version of this paper.</footnote>
# 1 Introduction

The canonical Melitz-Chaney framework (Melitz, 2003; Chaney, 2008) is the cornerstone of the modern theory of trade with heterogeneous firms. It is the workhorse of recent theoretical trade models and also widely applied in empirical studies. Yet it is fair to say that the assumption of CES demand in this framework imposes major restrictions. First, despite their heterogeneity in productivity, firms set a common constant markup because they face the same constant price elasticity of demand. Such a rigid pricing behavior runs against mixed empirical evidence. For example, Loecker and Warzynski (2012) find that more productive firms set higher markups, whereas a very recent work by Mertens and Mottironi (2023) observes that they rather set lower markups. A common point of these polar cases is that they both contradict the CES. Second, the selection effect à la Melitz (2003), whereby trade leads to the self-selection of more productive firms into exporting while forcing the least productive ones to exit the market, is the unique welfare channel in general equilibrium.<fnref n="1" /> Third, the CES imposes a complete pass-through, which implies that cross-firm differences in prices, revenues, and profits stem solely from productivity heterogeneity. Such rigidity rules out any influence of consumer preferences on the toughness of firm selection into markets and their partitioning by export status, and thus on the probability of exporting and the average productivity of exporters. As preferences do not affect export participation, they also have no impact on the intensity of trade-induced competition forcing domestic firms to exit. Consequently, the selection effect—and hence the gains from trade— in this canonical model depend only on the domestic trade share and the shape parameter of the Pareto distribution of firm productivity, as illustrated by the ACR formula (Arkolakis et al., 2012).

Thus, I find it both meaningful and important to extend the Melitz-Chaney framework beyond the CES to allow the price elasticity of demand to vary across firms. The key idea of this model is to introduce a flexible demand system that takes the CES as a point of departure while nesting empirically relevant alternatives. The CES is thus a special case in the current setting to assess how new results depart from those obtained in the canonical model. This generalized model is particularly well suited to the current context of retreat from globalization, as it enables the welfare effects of small changes in trade exposure to be studied under realistic demand conditions. By allowing consumer preferences to be more flexible, this model uncovers their role in determining the toughness of firm selection into markets, the degree of firm partitioning by export status, the magnitude of the selection effect and its prominence as unique welfare channel. Developing such a model and revealing new implications of generalized demand for key results in the Melitz-Chaney framework

<footnote n="1">¹In this canonical framework, variety effects à la Krugman (1980) are ambiguous since newly imported varieties displace domestic ones (Feenstra, 2010, 2018). Moreover, the CES imposes constant markups, which are thus unresponsive to the competitive pressure of trade. The selection effect à la Melitz (2003) leads to an increase in the average productivity at the industry level and stands thus as the unique source of welfare gains from trade.</footnote>
is the main objective of this paper. To achieve this goal, this paper considers a generalized Gorman-Pollak demand system (Gorman, 1995; Pollak, 1972) that nests directly and indirectly separable preferences without imposing a restriction on the curvature of demand.

While the type of separability determines whether the argument of the price elasticity of demand is the normalized price in absolute or relative terms, the curvature of demand determines whether the price elasticity of demand increases or decreases with its argument. The homothetic CES demand stands thus as a special case where the price elasticity of demand becomes exogenous and constant. Though still restrictive, I show that separable preferences are rich enough to yield a range of predictions that is much wider than the CES. In particular, this setting enables us to deal with different curvatures of demand through the *superelasticity of demand*, that is, the elasticity of the price elasticity of demand with respect to its argument. Specifically, the sign of this superelasticity determines the curvature of demand, thus allowing for various patterns of price sensitivity that are empirically relevant. When the superelasticity is strictly positive, demand is sub-convex, and more productive firms face less elastic demand and thus set higher markups. By contrast, when the superelasticity of demand is strictly negative, demand is super-convex, and more productive firms face more elastic demand and thus set lower markups. CES demand is a special case which arises when the superelasticity of demand is null, and restricts firms to face the same demand elasticity and thus to set a common constant markup.

Though still restrictive on the supply side, this paper leverages increased generality on the demand side to extend trade theory with heterogeneous firms along two lines. First, this model shows that while markups (if responsive) may decrease or increase with the competitive pressure of trade and variety effects remain ambiguous due to firm exit, the selection effect à la Melitz (2003) is always operative regardless of demand conditions. It also shows that welfare per capita is always the most sensitive to the selection effect. This paper generalizes thus Melitz (2003)'s result by showing that whether demand is CES or not, the selection effect remains the dominant welfare channel in general equilibrium. Second, this paper derives a generalization of the ACR formula which provides a parsimonious measure of the magnitude of the selection effect through the elasticity of the domestic productivity cutoff with respect to the trade exposure index. It replicates the ACR result under the CES while uncovering new welfare predictions once demand curvature departs from CES demand. This new formula reveals that by shaping export participation patterns, demand curvature governs the intensity of the competitive pressure trade exerts on domestic firms forcing the least productive among them to exit the market. By jointly delivering an upper bound for the toughness of firm selection into markets and a lower bound for the degree of firm partitioning by export status, super-convex demand yields an upper bound for the probability of exporting and the average export productivity. This, in turn, intensifies trade-induced firm exit and thus magnifies the selec-
tion effect. Since this latter is the dominant welfare channel in general equilibrium, super-convex demand yields thus an upper bound for the welfare gains from higher exposure to trade. These patterns are reversed under sub-convex demand. Within these bounds, CES demand delivers an intermediate outcome.

The generalized Gorman-Pollak demand system used in this paper has many attractive features that makes it suitable for the Melitz-Chaney model. First, Marshallian demand for any variety depends only on a few variables, namely variety's own price, consumer income, and two aggregate demand shifters which are both a function of a single aggregator that firms take as given. It retains then a key practical property of "generalized separability" (Pollak, 1972): the existence of a single aggregator which itself is a function of consumer income and the schedule of prices, and thus captures all cross-price effects. This single aggregator stands as a unique index of the toughness of competition at the industry level, which is very convenient for welfare analysis under monopolistic competition. Second, it imposes no restriction on the type of preferences. Instead, using a simple parametrization of demand shifters, it flexibly nests directly and indirectly separable preferences. Preferences are thus generally non-homothetic, with the CES case as a homothetic exception.<fnref n="2" /> Third, this demand system imposes no restriction on demand curvature and allows demand to be CES, sub-convex, or super-convex.

Given the presence of fixed costs in the current environment, increased generality on the demand side entails tractability challenges. To address this, I develop a new and simple approach—the Exponent Elasticity Method (hereafter EEM)—which delivers explicit solutions for key variables in general equilibrium, including the single price aggregator Λ, the domestic productivity cutoff $\varphi_d^*$, and the export productivity cutoff $\varphi_x^*$. The central idea of the EEM is to locally approximate (around the trade equilibrium) any implicit function that has a finite number of arguments by a unique multiplicative equivalent, whereby the exponent of each argument captures the function's total elasticity with respect to it.<fnref n="3" />

The findings of this paper are related to, and have implications for, a large number of theoretical and empirical papers in the international trade literature. Many recent papers have studied the welfare impact of tariffs; see e.g. Ignatenko et al. (2025), Baqaee and Malmberg (2025), Feenstra and Hong (2024), Caliendo and Feenstra (2024), Demidova et al. (2024), and Cuñat and Zymek (2023). Theoretical modeling, empirical methodologies, data sources, and conclusions vary, but a common feature of most of the aforementioned papers is their exclusive focus on the small open economy (SOE). To avoid the limit case of an infinitesimally small open economy having zero domestic trade share and experiencing infinite gains from trade, the authors allow trade costs to go to infinity.

<footnote n="2">²As explained in the next section, the parametrization needed to nest directly and indirectly separable preferences rules out the case of homothetic preferences with single aggregator (HSA) introduced by Matsuyama and Ushchev (2017). Thus, the CES stands as the only case where directly and indirectly separable preferences are homothetic.</footnote>

<footnote n="3">³A detailed exposition of the EEM method is provided in section A.2 of the online appendix.</footnote>
By contrast, the current paper considers the case of a finite number of large economies which are symmetric and separated by symmetric and finite trade costs. Other less recent theoretical papers by (Zhelobodko et al., 2012; Bertoletti and Epifani, 2014; Mrázová and Neary, 2017; Bertoletti et al., 2018) are also closely related to this paper. To ensure tractability despite added flexibility on the demand side, a common concession made by these authors is that they abstract from trade costs either fully or partially by considering only variable trade costs.<fnref n="4" /> This constrains trade openness to be modeled as a market size expansion, which implies that the selection effect á la Melitz (2003) strongly hinges on demand conditions. A common feature in Zhelobodko et al. (2012) and Mrázová and Neary (2017) is that they use directly separable preferences and assume frictionless trade. They both model trade as an increase in the market size of the global economy and find that the selection effect is operative only when demand is sub-convex.

Bertoletti et al. (2018) also model trade openness as an increase in population size while abstracting from fixed trade costs and assuming indirectly separable preferences. A peculiar property of this class of preferences is that population size has no impact on the intensity of competition in general equilibrium. The selection effect is then absent in their model and trade yields only variety gains á la Krugman (1980). By modeling trade liberalization as a reduction in trade costs (variable or fixed), Bertoletti and Epifani (2014) show that while the selection effect is always operative under the CES, it becomes more fragile beyond the CES. The modeling approach proposed in the current paper differs from the aforementioned papers in three important ways. First, it considers a highly flexible demand system while incorporating both variable and fixed trade costs. Second, it models marginal changes in exposure to trade as a small variation in trade costs or in the number of trading countries. Third, it shows that the selection effect á la Melitz (2003) is always operative regardless of demand conditions, and further examines the role that demand curvature plays in determining its magnitude.

The rest of this article is organized as follows. Section 2 describes the theoretical framework. Section 3 characterizes the trade equilibrium. Section 4 studies the role that demand curvature plays in shaping export participation patterns. Section 5 examines the welfare impact of small changes in trade exposure and derives a generalization of the ACR formula. Section 6 concludes. A detailed exposition of the EEM Method and all proofs are provided in the online appendix.

<footnote n="4">⁴With Bertoletti and Epifani (2014) as an exception in this regard since they incorporate both variable and fixed trade costs.</footnote>
# 2 Theoretical Framework

Consider a world economy comprising $N$ symmetric countries, one factor of production, labor, and one sector supplying a continuum of product varieties, where each variety accounts for a measure zero of aggregate expenditures. Each country is populated by $L$ identical agents (both workers and consumers), each one with labor endowment $\ell$ which is inelastically supplied at a competitive wage $w$. All workers are perfectly mobile across the production of different varieties and immobile across countries. Throughout the model, labor is chosen as numéraire.

## 2.1 Consumers

All consumers have the same preferences and the same income, $y$, which derives from their wages and the net profits of firms in their country (if any). Each consumer entirely spends her income on a compact set of potential varieties that is included in $[0, \bar{M}]$. Without loss of generality, it is sufficient to assume that $\bar{M}$ is large enough and that producing a new variety involves a fixed cost for $\bar{M}$ not to be binding at equilibrium.

### 2.1.1 Generalized Gorman-Pollak Demand System

If the representative consumer with income $y$ faces a schedule of prices $\boldsymbol{p} \equiv \{p_i\}, i \in [0, \bar{M}]$, her Marshallian demand for any variety $i$ takes a Gorman-Pollak functional form and is given by:

$$q_i(p_i, \boldsymbol{p}, y) = \frac{D_i(F(\Lambda(\boldsymbol{p}, y))^{\frac{p_i}{y}})}{H(\Lambda(\boldsymbol{p}, y))} \quad (1)$$

where $D_i$, $F$ and $H$ are mappings from $\mathbb{R}_+$ to $\mathbb{R}_+$, and where $\Lambda(\boldsymbol{p}, y)$ is implicitly determined by the budget constraint:

$$\int_{0}^{\bar{M}} p_i \frac{D_i(F(\Lambda(\boldsymbol{p}, y))^{\frac{p_i}{y}})}{H(\Lambda(\boldsymbol{p}, y))} di = y \quad (2)$$

As stressed by Fally (2022), an attractive feature of this demand system is that it retains a key practical property of "generalized additive separability" (Pollak, 1972): the existence of a single aggregator $\Lambda(\boldsymbol{p}, y)$ which captures all cross-price effects. $\Lambda(\boldsymbol{p}, y)$ stands thus as a unique index of the toughness of competition at the industry level, which is very convenient for welfare analysis under monopolistic competition. *Integrability.*– the demand system introduced above is integrable if there exists a differentiable utility function $U(x)$ such that marginal utility $\frac{\partial U}{\partial \tilde{q}_i}$ evaluated at $\tilde{q}_i$ (for an income $y$ and a schedule of prices $\boldsymbol{p}$) is proportional to prices $p_i$ across varieties $i$. This is equivalent to restricting the Slutsky substitution matrix to be symmetric. This can be guaranteed by the functional form of our demand (Gorman, 1995) upon imposing regularity conditions on the elasticities of functions $H$, $F$ and $D_i$, as in Fally (2022). Such regularity restrictions ensure that demand for any variety is strictly decreasing in the single price aggregator, and that this latter
always adjusts in order to satisfy the budget constraint.<fnref n="5" /> **Rationalization.**– while integrability requires the existence of a differentiable utility function $U(x)$ which yields a well-defined marginal utility, rationalization further requires $U(x)$ to be quasi-concave. Put differently, while integrability restricts the Slutsky substitution matrix to be symmetric, rationalization further restricts the Slutsky matrix to be semi-definite negative. As shown in section A.1 of the online appendix, in order to prove that our demand system can be rationalized, I extend the approach of Matsuyama and Ushchev (2017) to the non-homothetic case. This consists in invoking regularity conditions to check that the Slutsky substitution matrix is symmetric and semi-definite negative, and then applying the Hurwicz and Uzawa (1971) theorem. In doing so, I show that our demand system can be derived from the maximization of the following well-behaved utility function:

**Proposition 1.** If $H$, $F$ and $D_i$ satisfy the regularity conditions, the demand system described in equations (1) and (2) can be derived from a rational maximization program of a continuous quasi-concave utility:

$$U(\boldsymbol{x}) = \int_{0}^{\bar{M}} u_i(\boldsymbol{x}) \, d\boldsymbol{i} ; \quad u_i(\boldsymbol{x}) = \int_{x_{0i}}^{x=H(\Lambda(\boldsymbol{q}))} D_i^{-1}(\boldsymbol{x}) \, d\boldsymbol{x}$$

where $\Lambda(\boldsymbol{q})$ is now defined as an implicit function of the schedule of quantities $\boldsymbol{q} = \{q_i\}, i \in [0, \bar{M}]$ such as it satisfies the equivalent of the budget constraint (equation 2) that is written using the inverse demand, and $x_{0i} \ge 0$ is a constant. The corresponding indirect utility $V(\boldsymbol{p}, \boldsymbol{y})$ is then given by:

$$V(\boldsymbol{p}, \boldsymbol{y}) = \int_{0}^{\bar{M}} v_i(z) \, d\boldsymbol{i} + F(\Lambda(\boldsymbol{p}, \boldsymbol{y}))H(\Lambda(\boldsymbol{p}, \boldsymbol{y})) - k_0 ; \quad v_i(z) = - \int_{z_{0i}}^{z=F(\Lambda(\boldsymbol{p}, \boldsymbol{y}))} \frac{p_i}{y} D_i(z) \, dz$$

where $\Lambda(\boldsymbol{p}, \boldsymbol{y})$ is an implicit function of income $y$ and the schedule of prices $\boldsymbol{p} = \{p_i\}, i \in [0, \bar{M}]$ such as it satisfies the budget constraint in equation (2) and $k_0 \ge 0$ and $z_{0i} \ge 0$ are constant terms.

**Well-defined marginal utility.**– Whether expressed as an implicit function of income $y$ and the schedule of prices $\boldsymbol{p}$, or as an implicit function of the schedule of quantities $\boldsymbol{q}$, the fact that the aggregator $\Lambda(\cdot)$ satisfies the budget constraint can be thought of as a first-order condition ensuring that consumption profiles are at their optimal level. This, in turn, implies that both direct and indirect utility functions $U(x)$ and $V(p, y)$ have zero derivatives in $\Lambda(\cdot)$. As such, marginal utility takes a simple form:

$$\left\{ \begin{array}{l} \frac{\partial U}{\partial q_i} = \frac{\partial u_i(x)}{\partial q_i} = H(\Lambda(\boldsymbol{q})) D_i^{-1}(x) \\ (-\frac{\partial V}{\partial p_i}) = -\frac{\partial v_i(z)}{\partial p_i} = \frac{F(\Lambda(\boldsymbol{p}, \boldsymbol{y}))}{y} D_i(z) \end{array} \right.$$

**Nesting directly- and indirectly-separable preferences.**– An attractive feature of our Generalized Gorman-Pollak demand system is that demand for any variety $i$ depends only on a few variables,

<footnote n="5">⁵As this section builds heavily on Fally (2022) and also for expositional simplicity, regularity conditions are relegated to section A.1 of the online appendix.</footnote>
namely a variety's own price $p_i$, consumer income $y$, and two aggregate demand shifters $F(\Lambda(\mathbf{p}, y))$ and $H(\Lambda(\mathbf{p}, y))$ which are both a function of a single price aggregator $\Lambda(\mathbf{p}, y)$ that firms will take as given in subsequent sections. Inspection of equation (1) reveals that whereas $H(\Lambda(\mathbf{p}, y))$ only affects the level of demand, $F(\Lambda(\mathbf{p}, y))$ affects both the level and the elasticity of demand, which will have implications for firm-level markups. Our demand system corresponds then to directly-additive utility when $H(\Lambda(\mathbf{p}, y))$ is constant, and to indirectly-additive utility when $F(\Lambda(\mathbf{p}, y))$ is constant. This nesting can be shown in a simple way using the following parameterization of functions H and F, as in Fally (2019):

**A2. [Parametric restrictions]** on functions H and F:

$$i) H(\Lambda(\mathbf{p}, y)) = [\Lambda(\mathbf{p}, y)]^{\varepsilon_H}, \text{ with } \varepsilon_H \in \{0, 1\}$$

$$ii) F(\Lambda(\mathbf{p}, y)) = [\Lambda(\mathbf{p}, y)]^{\varepsilon_F}, \text{ with } \varepsilon_F \in \{0, 1\}$$

As such, the case ($\varepsilon_H=0$ and $\varepsilon_F=1$) corresponds to directly-separable preferences, while indirectly-separable preferences correspond to ($\varepsilon_H=1$ and $\varepsilon_F=0$). **Homotheticity vs Non-Homotheticity.**- another important feature of the Generalized Gorman-Pollak form is that it implies a tight linkage between price and income effects, such as an affine relationship exists between price and income elasticities of demand across varieties $i$, and for any income level $y$. For instance, using our three elasticities $\varepsilon_H$, $\varepsilon_F$, and $\varepsilon_{D_i}$, the income elasticity can be written as:

$$\frac{\partial \log q_i}{\partial \log y} = 1 + \frac{(\varepsilon_H + \varepsilon_F)(\varepsilon_{D_i} - \bar{\varepsilon}_D)}{(\varepsilon_H + \varepsilon_F \bar{\varepsilon}_D)}$$

where $\bar{\varepsilon}_D = \int_0^{\bar{M}} \chi_i \varepsilon_{D_i} di$, is the weighted average price elasticity of demand at the industry level, with $\chi_i = \frac{p_i q_i}{y}$ is the expenditure share of variety $i$. Inspection of the above equation reveals that there are two theoretically possible cases of homotheticity in our environment. The first restricts the price elasticity of demand to be identical across varieties: $\forall i, \varepsilon_{D_i} = \bar{\varepsilon}_D = \varepsilon_D$, which corresponds to the case of preferences with constant elasticity of substitution (CES). The second is a special case which requires $H(\Lambda(\mathbf{p}, y)) = \frac{1}{F(\Lambda(\mathbf{p}, y))}$, so as $\varepsilon_H = (-\varepsilon_F)$. This latter corresponds to homothetic preferences with single aggregator (HSA) introduced by Matsuyama and Ushchev (2017). Since our parametric restrictions A2 on functions H and F imply that $\varepsilon_H \neq (-\varepsilon_F)$, the case of homothetic single aggregator demand (HSA) is ruled out. Hence, CES preferences stand as the only directly separable and indirectly separable preferences that are homothetic. Another key property of our demand system is that the own-price elasticity of demand $\varepsilon_{D_i} = - \frac{\partial \log D_i(F(\Lambda(\mathbf{p}, y)) \frac{p_i}{y})}{\partial \log p_i}$ is allowed to vary with the variety's normalized price either in absolute terms ($\frac{p_i}{y}$) when preferences are indirectly separable, or in relative terms ($F(\Lambda(\mathbf{p}, y)) \frac{p_i}{y}$) when preferences are directly separable, except under the homothetic CES case which imposes $\varepsilon_D$ to be exogenous and identical across varieties. This, in turn, will generate variable markups under monopolistic competition, with CES utility as
a unique exception that imposes constant markups. For future derivations, it is convenient to write the demand function from equation (1) in a way that makes explicit the symmetry across varieties as well as the way in which the aggregate demand shifters, $H(\Lambda(p,y))$ and $F(\Lambda(p,y))$, affect the demand for all varieties. Thus, $q_i(p_i, p, y) = q(\frac{p_i}{y}, F(\Lambda(p, y)), H(\Lambda(p, y)))$ can be rewritten as:

$$q(\frac{p_i}{y}, \Lambda) = \frac{D(F(\Lambda) \frac{p_i}{y})}{H(\Lambda)}$$

This implies that the variability of markups across monopolistically competitive firms is a direct consequence of their heterogeneity in productivity levels. As before, the homothetic CES stands as a special case where firms face the same demand elasticity $\varepsilon_D$ despite their heterogeneity in productivity and thus set the same constant markup $(\frac{\varepsilon_D}{\varepsilon_D-1}) > 1$, as in Melitz (2003).

### 2.1.2 Elasticity, Convexity, and Curvature of Demand

At this stage of the paper, the flexibility of our demand system is reflected in how departing from homothetic CES preferences allows the price elasticity of demand to vary with normalized prices either in absolute terms under indirect separability, or in relative terms under direct separability. Such flexibility raises now the following question: under which conditions the price elasticity of demand increases, decreases, or ceases to vary with normalized prices? As mentioned before, the key feature of our demand system is that it allows for flexible price effects while retaining the practical property that all cross-price effects go through a single price aggregator $\Lambda$. This is very convenient under monopolistic competition as this single industry-wide price indicator uniquely determines the locus of the demand curve for a variety with respect to its own price. Accordingly, our starting point is the fact that a monopolistically competitive firm sets its profit-maximizing price, $p$, by taking the direct demand it perceives as given.<fnref n="6" /> I can now address the above question in two steps. First, I measure the elasticity, the convexity, and the curvature of our direct demand function in a unit-free way. Second, I graphically localize demand curvature in the space of elasticity and convexity, as in Mrázová and Neary (2017). Let us start by recalling that the price elasticity of demand is given by:

$$\varepsilon_D(z) = -\frac{z D'(z)}{D(z)} = -\frac{z v''(z)}{v'(z)} > 1$$

where $z = F(\Lambda) \frac{p}{y}$ is the argument of direct demand and the indirect sub-utility function, which is proportional to a firm's own price $p$. The convexity of demand can be also measured in a simple way using the elasticity of the slope of direct demand:

$$\zeta(z) = -\frac{z D''(z)}{D'(z)} = -\frac{z v'''(z)}{v''(z)} > 0$$

<footnote n="6">⁶This fits within Mrázová and Neary (2017)'s approach that they call "a firm's eye view of demand".</footnote>
Notice that the above measures of demand elasticity $\varepsilon_D(z)$ and convexity $\zeta(z)$ suggest that the direct demand function $D(z)$ is proportional to the absolute value of the indirect marginal utility ($-v'(z)$). This clear relationship follows by inspection of the expressions of marginal utility and invoking the assumption that demand is symmetric across varieties. Now in order to measure demand curvature in a unit-free way, I work with the superelasticity of demand, $S$, which I define as the elasticity of the price elasticity of demand with respect to its argument:

$$S = \frac{z \varepsilon'_{D}(z)}{\varepsilon_{D}(z)} = [(1 + \varepsilon_{D}(z)) - \zeta(z)] \le 0$$

Clearly, the sign of the superelasticity of demand is pinned down by the relationship between the elasticity $\varepsilon_D(z)$ and the convexity $\zeta(z)$ of the direct demand function, the so-called "demand manifold" by Mrázová and Neary (2017). This superelasticity of demand is reminiscent of Kimball (1995)'s superelasticity and acts as a sufficient statistic for demand curvature. Specifically, its sign determines whether demand is CES, sub-convex, or super-convex:<fnref n="7" />

$$\text{demand is} \quad \begin{cases} \text{sub-convex} & \text{if } S > 0 \\ \text{CES} & \text{if } S = 0 \\ \text{super-convex} & \text{if } S < 0 \end{cases}$$

A peculiar property of CES demand is that it exhibits an exogenous demand elasticity, which is reflected by zero superelasticity of demand. It is then convenient to take the homothetic CES as a benchmark to characterize both alternative curvatures of demand in a simple way. Following Mrázová and Neary (2017), a demand function is locally sub-convex if for the same level of demand elasticity $\varepsilon_D(z)$, it exhibits a lower degree of convexity $\zeta(z)$ as compared to the CES case. Similarly, a demand function is super-convex if it is more convex than the CES at a given level of demand elasticity. It is now possible to graphically illustrate these three possible cases of demand curvature in the $(\zeta(z), \varepsilon_D(z))$ space. To do this in a precise way, I need to identify an admissible region in the space of elasticity and convexity. This requires imposing restrictions on the values of $\varepsilon_D(z)$ and $\zeta(z)$, which can be done using the first- and second-order conditions of profit maximization as follows. Let us consider the optimization problem of a firm which hires $f > 0$ units of labor to start producing its own variety at a marginal cost $c$ and selling it in any country with market size, $Y = yL$, and competition index $\Lambda$. As this firm is monopolistically competitive, it takes the price aggregator $\Lambda$ as given and chooses its profit-maximizing price $p$:

$$\pi(c, y, L, \Lambda, w, f) = \max_{p}\{(p - c)q(\frac{p}{y}, \Lambda)L - wf\}$$

<footnote n="7">⁷The superelasticity of Kimball (1995) corresponds to the elasticity of the price elasticity of demand with respect to price.</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1: Localising Demand Curvature in the space of Elasticity and Convexity.</figcaption></figure>

The associated first-order condition is:

$$ (p-c)q' + q = 0 \Rightarrow \frac{(p-c)}{p} = \frac{1}{\varepsilon_D(z)} \Rightarrow \varepsilon_D(z) > 1 $$

which states that the markup rate is inversely related to the price elasticity of demand. This latter must be greater than one for the price-cost margin to be positive. From the second-order condition, decreasing marginal revenues requires that the degree of convexity of demand must be smaller than twice the demand elasticity:

$$ 2q' + (p-c)q'' < 0 \Rightarrow \zeta(z) < 2\varepsilon_D(z) $$

Clearly, these restrictions on the values of the elasticity and the convexity of demand imply an admissible region in the $(\zeta(z), \varepsilon_D(z))$ space, as shown by the shaded region in Figure 1, panel A.<fnref n="8" /> From panel B of Figure 1, we can see that the CES line (whose equation is given by $\zeta = 1 + \varepsilon_D$) divides the admissible region in two. While points located above the CES line correspond to the case of super-convex demand, points below the CES line correspond to the case of sub-convex demand. Within the sub-convex region, the superelasticity of demand is strictly positive ($S > 0$), which implies that the price elasticity of demand increases with the normalized price when demand is sun-convex.<fnref n="9" /> By contrast, the super-convex region is characterized with a strictly negative

<footnote n="8">⁸Ideally, one would borrow an upper bound estimate of the demand elasticity from the empirical literature, and thus concentrates on a more realistic part of the admissible region.</footnote>

<footnote n="9">⁹As previously mentioned, whether the normalized price is expressed in absolute or relative terms depends on the type of separability. When preferences are directly separable, the argument of the price elasticity of demand is the relative normalized price, given by $z = F(\Lambda)^{\frac{1}{\varepsilon_D}}$. However, when preferences are indirectly separable, $F(\Lambda)$ is a constant fixed to unity ($\varepsilon_F = 0 \Rightarrow F(\Lambda) = 1$), and thus the argument of the price elasticity of demand boils down to the absolute normalized price: $z = \frac{1}{\varepsilon_D}$.</footnote>
superelasticity of demand ($S < 0$). This implies that the price elasticity of demand decreases with the normalized price when demand is super-convex. CES demand stands thus as a boundary case under which the price elasticity of demand ceases to vary with the normalized price, as reflected by the zero superelasticity of demand ($S = 0$). Hence, moving along the CES line only changes the values of the price elasticity of demand while preserving its exogenous nature.

## 2.2 Firms

Firms compete under monopolistic competition. Entry may be restricted or free. Under restricted entry, each country has an exogenous mass of firms, $\tilde{M}_e$, with the right to enter the industry. Under free entry, there is an endogenous mass of ex ante identical firms that hire $f_e > 0$ units of labor to enter the industry. Firms then endogenously enter up to the point at which aggregate profits net of the sunk fixed entry costs, $wf_e M_e$, are zero. Hereafter, the mass of entrants in any country is denoted by $M_e$. As in Melitz (2003), upon entry, firms draw their initial productivity level $\varphi$ from a common distribution $g(\varphi)$. This latter has positive support over $[1, +\infty]$ and has a continuous cumulative distribution $G(\varphi)$. As in Chaney (2008), I assume that $G(\varphi)$ is Pareto with the same shape parameter $\theta > 0$ around the world:

**A3. [Unbounded Pareto]** $\forall \varphi \in [1, +\infty], G(\varphi) = 1 - \varphi^{-\theta}$, with $\theta > 0$.

Following Melitz (2003), I assume that each firm must incur an overhead production cost, $wf$, to start producing for the domestic market. Production of any variety is subject then to increasing returns to scale. Serving foreign markets is only possible via exporting, and is more costly than operating on the domestic market. For instance, exporting involves two types of costs: a fixed cost of accessing foreign markets, $wf_x$, and a variable trade cost, $\tau$, modeled in the standard iceberg formulation whereby $\tau > 1$ units of a variety must be shipped for 1 unit to arrive at destination. Accordingly, for a firm with productivity $\varphi$, the constant marginal costs of delivering its own variety to domestic and foreign consumers are respectively given by $c_d = \frac{w}{\varphi}$, and $c_x = \tau \frac{w}{\varphi}$. As mentioned earlier, our demand system does not exhibit a choke price. Thus, the selection of firms into markets is mainly driven by the presence of fixed costs, as in Melitz (2003).

## 3 Trade Equilibrium

In this section, I characterize the trade equilibrium for arbitrary values of trade costs. I proceed in three steps. I first show how the flexible demand system introduced in the previous section shapes firm-level variables. I then describe how firm-level decisions aggregate up to determine bilateral trade flows and the equilibrium mass of entrants. Finally, I provide explicit solutions for the price aggregator and the domestic and export productivity cutoffs in general equilibrium.
# 3.1 Firm-level Variables

## 3.1.1 Firm-level markups

To start with, let us work with $m = \frac{p}{c}$ as a measure of firm-level markups. Using the first-order condition of profit maximization, we can express $m$ as the implicit solution of

$$m = \frac{\varepsilon_D(m/e)}{\varepsilon_D(m/e) - 1} \qquad (3)$$

where, as before, $\varepsilon_D(z) = - \frac{\partial \log D(z)}{\partial \log z}$ measures the price elasticity of demand, and $e = \frac{y}{cF(\Lambda)}$ can be thought of as a measure of firm-level efficiency, either in absolute terms when preferences are indi- rectly separable ($\varepsilon_F = 0 \Rightarrow F(\Lambda) = 1$) or in relative terms when preferences are directly separable ($\varepsilon_F = 1 \Rightarrow F(\Lambda) = \Lambda$). As mentioned in section 2.1, the price aggregator $\Lambda(p, y)$ is an index of the tough- ness of competition at the industry level and is then increasing in the average productivity of active firms in a given market. Thus, under direct-separability, $e = \frac{y}{cF(\Lambda)}$ measures the efficiency of a firm relative to other firms serving the same market. From equation (3), we can see that the aggregate demand shifter $F(\Lambda)$ captures all indirect effects that may lead a firm to change its markup in a given market. Now in order to show that this equation has a unique explicit solution $m \equiv \mu(e)$, I proceed in three steps. First, I rewrite firm-level markup, $m$, as follows:

$$m \equiv [\varepsilon_D(m/e)]^{-\delta} \qquad (4)$$

where $\delta = - \frac{\partial \log m}{\partial \log \varepsilon_D(m/e)} = \frac{1}{\varepsilon_D(m/e) - 1} \in ]0, 1[$ is the absolute value of the elasticity of firm-level markup with respect to the price elasticity of demand. Second, using the definition of the superelasticity of demand, $S = \frac{\partial \log \varepsilon_D(z)}{\partial \log z}$, the price elasticity of demand $\varepsilon_D(m/e)$ can be rewritten as:

$$\varepsilon_D(m/e) \equiv (m/e)^S \qquad (5)$$

Finally, by plugging the above expression in equation (4) and rearranging, I obtain the following explicit solution for firm-level markups:

$$m \equiv \mu(e) \equiv (e)^{\rho(S)} \qquad (6)$$

where $\rho(S) = \frac{\delta S}{1+\delta S} \in ]0, 1[$ measures the elasticity of markups with respect to firm-level efficiency. Now with the aid of the second-order condition of profit maximization, I can impose the following restriction on the value of the superelasticity of demand, defined as $S = [(1 + \varepsilon_D(z)) - \zeta(z)] \le 0$,

$$\zeta(z) < 2\varepsilon_D(z) \Rightarrow S > 1 - \varepsilon_D(z)$$
This ensures that $(1 + \delta S)$ is always strictly positive, with the appealing consequence that the sign of the markup elasticity $\rho(S)$ hinges on the sign of the superelasticity $S$:

$$ \rho(S) = \frac{\partial \log \mu(e)}{\partial \log e} = \frac{\delta S}{1 + \delta S} \begin{cases} > 0 & \text{if } S > 0 \\ = 0 & \text{if } S = 0 \\ < 0 & \text{if } S < 0 \end{cases} \qquad (7) $$

Therefore, whether markups are constant, or monotonically increasing or decreasing in firm-level efficiency crucially depends on the curvature of demand. When demand is sub-convex ($S > 0$), markups are increasing in firm-level efficiency, and so more efficient firms charge higher markups. By contrast, when demand is super-convex, markups are decreasing in firm-level efficiency, and so more efficient firms charge lower markups. CES demand stands again as a special case where markups cease to vary with firm-level efficiency ($\rho(S) = 0$) and are exogenously fixed to $m = \frac{\epsilon_D}{\epsilon_D - 1}$ for all active firms.

### 3.1.2 Firm-level sales revenues and profits

In any given market, the price charged by a firm with marginal cost $c$ and efficiency $e$ is given by $p(c, e) = c\mu(e)$. Given this pricing rule, the revenues earned by this firm from domestic sales and export sales are respectively given by:

$$ \begin{cases} r_d(c_d, e_d, H(\Lambda), L) = c_d \mu(e_d) \frac{D(\mu(e_d)/e_d)}{H(\Lambda)} L \\ r_x(c_x, e_x, H(\Lambda), L) = c_x \mu(e_x) \frac{D(\mu(e_x)/e_x)}{H(\Lambda)} L \end{cases} \qquad (8) $$

where $c_d = \frac{w}{\phi}$, $c_x = \tau \frac{w}{\phi}$, $e_d = \frac{y}{c_d F(\Lambda)}$, and $e_x = \frac{y}{c_x F(\Lambda)}$. At this stage, the symmetry assumption at the country level has a key implication that is worth emphasizing. For instance, by ensuring that all markets are identical both in terms of size, $Y = yL$, and intensity of competition, which is captured by the price aggregator $\Lambda$, it implies that the difference between domestic and export revenues is solely driven by the presence of the variable trade cost $\tau$: $c_x/c_d = \tau$, and $e_x/e_d = \tau^{-1}$. In turn, the operating profits that this firm reaps from domestic and export sales are respectively given by:

$$ \begin{cases} \pi_d^o(c_d, e_d, H(\Lambda), L) = \dot{\mu}(e_d) r_d(c_d, e_d, H(\Lambda), L) \\ \pi_x^o(c_x, e_x, H(\Lambda), L) = \dot{\mu}(e_x) r_x(c_x, e_x, H(\Lambda), L) \end{cases} \qquad (9) $$

where $\dot{\mu}(e_d) = \frac{\mu(e_d) - 1}{\mu(e_d)} \in ]0, 1[$, and $\dot{\mu}(e_x) = \frac{\mu(e_x) - 1}{\mu(e_x)} \in ]0, 1[$ correspond to firm-level markup rates respectively on the domestic and the export market. Using the Lerner index, the operating profit of a firm serving a given market boils down to its sales revenues multiplied by its markup rate.
The added flexibility here is that markup rates are allowed to vary across firms, except under the homothetic CES case where they are restricted to be identical across firms and exogenously given by $\dot{\mu} = \frac{m-1}{m} = \frac{1}{\varepsilon_D} < 1$. Finally, taking into account the existence of fixed costs in the current setting, net profits of a firm from serving the domestic and the export market are respectively given by:

$$ egin{cases} 
\pi_d(c_d, e_d, H(\Lambda), L, w, f) = \pi_d^o(c_d, e_d, H(\Lambda), L) - w f \\ 
\pi_x(c_x, e_x, H(\Lambda), L, w, f_x) = \pi_x^o(c_x, e_x, H(\Lambda), L) - w f_x 
\end{cases} 
\quad (10) $$

## 3.2 Selection and Partitioning of firms by export status

### 3.2.1 Firm Selection into markets

As in Melitz (2003), since operating profits are monotonically increasing in firm productivity $\varphi$, the presence of fixed costs of accessing domestic and foreign markets, $wf$ and $wf_x$, implies the existence of two productivity cutoffs. The first is the domestic productivity cutoff, denoted by $\varphi_d^*$, and defined as the minimum productivity level required to make non-negative net domestic profits and successfully enter the domestic market. The second is the export productivity cutoff, denoted by $\varphi_x^*$, and defined as the minimum productivity level required to make at least zero net export profits and start serving foreign markets. By definition, the domestic cutoff must then satisfy the zero profit condition on the domestic market $(ZPC)_d: \pi_d(\varphi_d^*) = 0$. Similarly, the export cutoff must satisfy the zero profit condition on the export market $(ZPC)_x: \pi_x(\varphi_x^*) = 0$. Using these two equilibrium conditions, these productivity cutoffs can be identified implicitly by:

$$ egin{cases} 
(ZPC)_d \Rightarrow \varphi_d^* : \pi_d^o(c_d, e_d, H(\Lambda), L) = w f \\ 
(ZPC)_x \Rightarrow \varphi_x^* : \pi_x^o(c_x, e_x, H(\Lambda), L) = w f_x 
\end{cases} 
\quad (11) $$

### 3.2.2 Partitioning of firms by export status

For an arbitrary value of the iceberg trade cost $\tau \ge 1$, it is sufficient to assume that $f_x > f$ to ensure that firms are partitioned by export status, as in Melitz (2003). That is, among successful entrants in the domestic market (firms with productivity $\varphi \ge \varphi_d^*$), only more productive firms (a fringe of firms with productivity $\varphi \ge \varphi_x^* > \varphi_d^*$) find it profitable to export.

## 3.3 Aggregate Variables

### 3.3.1 Aggregate sales revenues, profits, and income

The value of domestic trade in any country, $X_d$, and the value of bilateral trade flows for any country
pair, $X_x$, are respectively given by:

$$ \begin{cases} X_d = M_e [1 - G(\varphi_d^*)] \int_{\varphi_d^*}^{+\infty} r_d(\varphi, w, y, L, F(\Lambda), H(\Lambda)) \gamma_d(\varphi) d\varphi \\ X_x = M_e [1 - G(\varphi_x^*)] \int_{\varphi_x^*}^{+\infty} r_x(\varphi, \tau, w, y, L, F(\Lambda), H(\Lambda)) \gamma_x(\varphi) d\varphi \end{cases} \quad (12) $$

where $\gamma_d(\varphi)$ and $\gamma_x(\varphi)$ correspond to the productivity distribution conditionally on successful entry, respectively, in the domestic and the export market:

$$ \begin{cases} \gamma_d(\varphi) = \frac{g(\varphi)}{[1-G(\varphi_d^*)]} \quad \forall \varphi \ge \varphi_d^* \\ \gamma_x(\varphi) = \frac{g(\varphi)}{[1-G(\varphi_x^*)]} \quad \forall \varphi \ge \varphi_x^* \end{cases} $$

Given the presence of fixed costs in the current environment, increased generality on the demand side entails tractability challenges. To address this, I develop a new and simple approach—the Exponent Elasticity Method (hereafter EEM)—which delivers explicit solutions for key variables in general equilibrium, including the single price aggregator $\Lambda$, the domestic productivity cutoff $\varphi_d^*$, and the export productivity cutoff $\varphi_x^*$. The central idea of the EEM is to locally approximate (around the trade equilibrium) any implicit function that has a finite number of arguments by a unique multiplicative equivalent, whereby the exponent of each argument captures the function's total elasticity with respect to it.<fnref n="10" /> Now by applying the EEM, I first obtain an explicit solution for domestic trade $X_d$, and bilateral trade flows $X_x$ in partial equilibrium:

$$ \begin{cases} X_d \equiv M_e(w) \varepsilon_w^{X_d} (f) \varepsilon_f^{X_d} (y) \varepsilon_y^{X_d} (L) \varepsilon_L^{X_d} (\Lambda) \varepsilon_\Lambda^{X_d} \\ X_x \equiv M_e(w) \varepsilon_w^{X_x} (\tau) \varepsilon_\tau^{X_x} (f_x) \varepsilon_{f_x}^{X_x} (y) \varepsilon_y^{X_x} (L) \varepsilon_L^{X_x} (\Lambda) \varepsilon_\Lambda^{X_x} \end{cases} \quad (13) $$

where $\varepsilon_w^{X_d}, \varepsilon_f^{X_d}, \varepsilon_L^{X_d}, \varepsilon_y^{X_d}$, and $\varepsilon_\Lambda^{X_d}$ correspond to the respective total elasticities of domestic trade $X_d$ with respect to the cost of labor $w$, the domestic fixed cost $f$, population size $L$, individual income $y$, and the price aggregator $\Lambda$, and are given by:

$$ \left\{ \begin{aligned} \varepsilon_w^{X_d} &= \frac{d \log X_d}{d \log w} = \varepsilon_w^r (\bar{e}_d) - [\theta - \varepsilon_\varphi^r (\bar{e}_d)] \frac{[1 - \varepsilon_w^{\pi^0} (e_d^*)]}{\varepsilon_\varphi^{\pi^0} (e_d^*)} < 0 \\ \varepsilon_f^{X_d} &= \frac{d \log X_d}{d \log f} = -\frac{[\theta - \varepsilon_\varphi^r (\bar{e}_d)]}{\varepsilon_\varphi^{\pi^0} (e_d^*)} < 0; \quad \varepsilon_L^{X_d} = \frac{d \log X_d}{d \log L} = 1 + \frac{[\theta - \varepsilon_\varphi^r (\bar{e}_d)]}{\varepsilon_\varphi^{\pi^0} (e_d^*)} > 0 \\ \varepsilon_y^{X_d} &= \frac{d \log X_d}{d \log y} = \varepsilon_y^r (\bar{e}_d) + [\theta - \varepsilon_\varphi^r (\bar{e}_d)] \frac{\varepsilon_y^{\pi^0} (e_d^*)}{\varepsilon_\varphi^{\pi^0} (e_d^*)} > 0 \\ \varepsilon_\Lambda^{X_d} &= \frac{d \log X_d}{d \log \Lambda} = [\varepsilon_F \varepsilon_{F(\Lambda)}^r (\bar{e}_d) - \varepsilon_H] + [\theta - \varepsilon_\varphi^r (\bar{e}_d)] \frac{[\varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^0} (e_d^*) - \varepsilon_H]}{\varepsilon_\varphi^{\pi^0} (e_d^*)} < 0 \end{aligned} \right. \quad (14) $$

Inspection of the above expressions clearly shows that they depend on three exogenous parameters, $(\theta, \varepsilon_F, \varepsilon_H)$, and two sets of firm-specific elasticities. The first set is: (i) $\{\varepsilon_\varphi^r (\bar{e}_d), \varepsilon_w^r (\bar{e}_d), \varepsilon_y^r (\bar{e}_d), \varepsilon_{F(\Lambda)}^r (\bar{e}_d)\}$,

<footnote n="10">¹⁰This requires a four-step procedure that I expose in detail in section A.2 of the online appendix.</footnote>
and the second set is given by: (ii) {$\epsilon_{\phi}^{\pi^0}(e_d^*), \epsilon_{w}^{\pi^0}(e_d^*), \epsilon_{y}^{\pi^0}(e_d^*), \epsilon_{F(\Lambda)}^{\pi^0}(e_d^*)$}. The exogenous parameters correspond to the shape parameter of the Pareto $\theta > 0$, and the elasticities of the aggregate demand shifters $F(\Lambda)$ and $H(\Lambda)$ with respect to the price aggregator $\Lambda$, $\epsilon_F = \frac{\partial \log F}{\partial \log \Lambda} \in \{0,1\}$ and $\epsilon_H = \frac{\partial \log H}{\partial \log \Lambda} \in \{0,1\}$, which are restricted to be dummies by assumption A2. The first set of firm-specific elasticities, (i), corresponds to the partial elasticities of firm-level domestic revenues, respectively, with respect to firm productivity $\phi$, the cost of labor $w$, individual income $y$, and the aggregate demand shifter $F(\Lambda)$, which are faced by the domestic firm with an average level of efficiency ($\bar{e}_d$). The second set of firm-specific elasticities, (ii), corresponds to the partial elasticities of firm-level domestic operating profits, respectively, with respect to firm productivity $\phi$, the cost of labor $w$, individual income $y$, and the aggregate demand shifter $F(\Lambda)$, which are faced by the cutoff domestic producer ($e_d^*$). Their explicit expressions are respectively given by:

(i) Firm-specific elasticities faced by the domestic firm with average efficiency $\bar{e}_d$:

$$ 
\left\{
\begin{aligned}
\epsilon_{\phi}^{r}(\bar{e}_{d}) &= \frac{\partial \log r(\bar{e}_{d})}{\partial \log \phi} = (1 - \rho(S))(\epsilon_{D}(\bar{e}_{d}) - 1) > 0; \quad \epsilon_{w}^{r}(\bar{e}_{d}) = \frac{\partial \log r(\bar{e}_{d})}{\partial \log w} = (1 - \rho(S))(1 - \epsilon_{D}(\bar{e}_{d})) < 0 \\
\epsilon_{y}^{r}(\bar{e}_{d}) &= \frac{\partial \log r(\bar{e}_{d})}{\partial \log y} = (1 - \rho(S))\epsilon_{D}(\bar{e}_{d}) + \rho(S) > 0 \\
\epsilon_{F(\Lambda)}^{r}(\bar{e}_{d}) &= \frac{\partial \log r(\bar{e}_{d})}{\partial \log F(\Lambda)} = -[(1 - \rho(S))\epsilon_{D}(\bar{e}_{d}) + \rho(S)] < 0
\end{aligned}
\right.
\qquad (15) $$

(ii) Firm-specific elasticities faced by the cutoff domestic firm $e_d^*$:

$$ 
\left\{
\begin{aligned}
\epsilon_{\phi}^{\pi^0}(e_d^*) &= \frac{\partial \log \pi^0(e_d^*)}{\partial \log \phi} = (1 - \rho(S))(\epsilon_D(e_d^*) - 1) + \eta \rho(S) > 0 \\
\epsilon_{w}^{\pi^0}(e_d^*) &= \frac{\partial \log \pi^0(e_d^*)}{\partial \log w} = (1 - \rho(S))(1 - \epsilon_D(e_d^*)) - \eta \rho(S) < 0 \\
\epsilon_{y}^{\pi^0}(e_d^*) &= \frac{\partial \log \pi^0(e_d^*)}{\partial \log y} = (1 - \rho(S))\epsilon_D(e_d^*) + (1 + \eta)\rho(S) > 0 \\
\epsilon_{F(\Lambda)}^{\pi^0}(e_d^*) &= \frac{\partial \log \pi^0(e_d^*)}{\partial \log F(\Lambda)} = -[(1 - \rho(S))\epsilon_D(e_d^*) + (1 + \eta)\rho(S)] < 0
\end{aligned}
\right.
\qquad (16) $$

where $\eta = \frac{\partial \log \dot{\mu}(e)}{\partial \log \mu(e)} > 0$ is the elasticity of the markup rate with respect to the markup. Similarly, the elasticities of bilateral trade $X_x$ with respect to its arguments are given by:

$$ 
\left\{
\begin{aligned}
\epsilon_w^{X_x} &= \frac{d \log X_x}{d \log w} = \epsilon_w^r(\bar{e}_x) - [\theta - \epsilon_\phi^r(\bar{e}_x)] \frac{[1 - \epsilon_\eta^\pi^0(e_x^*)]}{\epsilon_\phi^\pi^0(e_x^*)} < 0 \\
\epsilon_\tau^{X_x} &= \frac{d \log X_x}{d \log \tau} = -\theta < 0; \quad \epsilon_{f_x}^{X_x} = \frac{d \log X_x}{d \log f_x} = -\frac{[\theta - \epsilon_\phi^r(\bar{e}_x)]}{\epsilon_\phi^\pi^0(e_x^*)} < 0 \\
\epsilon_y^{X_x} &= \frac{d \log X_x}{d \log y} = \epsilon_y^r(\bar{e}_x) + [\theta - \epsilon_\phi^r(\bar{e}_x)] \frac{\epsilon_y^\pi^0(e_x^*)}{\epsilon_\phi^\pi^0(e_x^*)} > 0 \\
\epsilon_L^{X_x} &= \frac{d \log X_x}{d \log L} = 1 + \frac{[\theta - \epsilon_\phi^r(\bar{e}_x)]}{\epsilon_\phi^\pi^0(e_x^*)} > 0 \\
\epsilon_\Lambda^{X_x} &= \frac{d \log X_x}{d \log \Lambda} = [\epsilon_F \epsilon_{F(\Lambda)}^r(\bar{e}_x) - \epsilon_H] + [\theta - \epsilon_\phi^r(\bar{e}_x)] \frac{[\epsilon_F \epsilon_{F(\Lambda)}^\pi^0(e_x^*) - \epsilon_H]}{\epsilon_\phi^\pi^0(e_x^*)} < 0
\end{aligned}
\right.
\qquad (17) $$

As before, the above total elasticities of bilateral trade flows $X_x$ depend on the same exogenous
parameters $(\theta, \varepsilon_F, \varepsilon_H)$. The only difference is that they now depend on firm-level elasticities faced by the average productivity exporter ($\bar{e}_x$), and by the cutoff exporter ($e_x^*$):

(i) Firm-specific elasticities faced by the average productivity exporter, with efficiency measure $\bar{e}_x$:

$$ 
\left\{ 
\begin{aligned}
\varepsilon_{\varphi}^{r}(\bar{e}_{x}) &= \frac{\partial \log r(\bar{e}_{x})}{\partial \log \varphi} = (1 - \rho(S)) (\varepsilon_{D}(\bar{e}_{x}) - 1) > 0 \\
\varepsilon_{w}^{r}(\bar{e}_{x}) &= \frac{\partial \log r(\bar{e}_{x})}{\partial \log w} = (1 - \rho(S)) (1 - \varepsilon_{D}(\bar{e}_{x})) < 0 \\
\varepsilon_{y}^{r}(\bar{e}_{x}) &= \frac{\partial \log r(\bar{e}_{x})}{\partial \log y} = (1 - \rho(S)) \varepsilon_{D}(\bar{e}_{x}) + \rho(S) > 0 \\
\varepsilon_{F(\Lambda)}^{r}(\bar{e}_{x}) &= \frac{\partial \log r(\bar{e}_{x})}{\partial \log F(\Lambda)} = -[(1 - \rho(S)) \varepsilon_{D}(\bar{e}_{x}) + \rho(S)] < 0
\end{aligned}
\right. 
\qquad (18) 
$$

(ii) Firm-specific elasticities faced by the firm at the export cutoff, with efficiency measure $e_x^*$:

$$ 
\left\{ 
\begin{aligned}
\varepsilon_{\varphi}^{\pi^0}(e_x^*) &= \frac{\partial \log \pi^0(e_x^*)}{\partial \log \varphi} = (1 - \rho(S)) (\varepsilon_D(e_x^*) - 1) + \eta \rho(S) > 0 \\
\varepsilon_{w}^{\pi^0}(e_x^*) &= \frac{\partial \log \pi^0(e_x^*)}{\partial \log w} = (1 - \rho(S)) (1 - \varepsilon_D(e_x^*)) - \eta \rho(S) < 0 \\
\varepsilon_{y}^{\pi^0}(e_x^*) &= \frac{\partial \log \pi^0(e_x^*)}{\partial \log y} = (1 - \rho(S)) \varepsilon_D(e_x^*) + (1 + \eta) \rho(S) > 0 \\
\varepsilon_{F(\Lambda)}^{\pi^0}(e_x^*) &= \frac{\partial \log \pi^0(e_x^*)}{\partial \log F(\Lambda)} = -[(1 - \rho(S)) \varepsilon_D(e_x^*) + (1 + \eta) \rho(S)] < 0
\end{aligned}
\right. 
\qquad (19) 
$$

Let us note that $\theta$ is assumed to be large enough to exceed the productivity elasticity of firm-level revenues faced by the average productivity firm on both markets, $\theta > \varepsilon_{\varphi}^r(\bar{e}_d)$ and $\theta > \varepsilon_{\varphi}^r(\bar{e}_x)$, for the sign of the aggregate elasticities in equations (14) and (17) to be well-behaved. Under the CES case, this is equivalent to assuming that $\theta > (\varepsilon_D - 1)$, as in Chaney (2008). Now using equations (9) and (10), and invoking the Pareto assumption A3, aggregate net profits on domestic and export markets respectively boil down to:<fnref n="11" />

$$ 
\left\{ 
\begin{aligned}
\Pi_d &= \dot{\mu}(\bar{e}_d) X_d - M_e [1 - G(\varphi_d^*)] w f \\
\Pi_x &= \dot{\mu}(\bar{e}_x) X_x - M_e [1 - G(\varphi_x^*)] w f_x
\end{aligned}
\right. 
\qquad (20) 
$$

where $\dot{\mu}(\bar{e}_d)$ and $\dot{\mu}(\bar{e}_x)$ correspond to average markup rates on the domestic and the export market, and are respectively given by:

$$ 
\left\{ 
\begin{aligned}
\dot{\mu}(\bar{e}_d) &= 1 - \left[ \frac{y}{\bar{e}_d F(\Lambda)} \right]^{-\rho(S)}, & \text{with } \bar{e}_d &= \left( \frac{\theta-1}{\theta} \right) w (\varphi_d^*)^{-1} \\
\dot{\mu}(\bar{e}_x) &= 1 - \left[ \frac{y}{\bar{e}_x F(\Lambda)} \right]^{-\rho(S)}, & \text{with } \bar{e}_x &= \left( \frac{\theta-1}{\theta} \right) w \tau (\varphi_x^*)^{-1}
\end{aligned}
\right. 
\qquad (21) 
$$

Finally, let $Y = yL$ denote aggregate income in any country. It is equal to the sum of labor payments

<footnote n="11">¹¹See section A.2 of the online appendix for details.</footnote>
and net profits, which must add up to the total sales revenues of local firms

$$Y = X_d + (N - 1)X_x \quad (22)$$

### 3.3.2 Measures of entrants, wage, and per-capita income

In any country, the mass of firms that enter the industry can be either exogenously fixed to $\tilde{M}_e$ or endogenously determined depending on the nature of entry:

$$Me = \begin{cases} \tilde{M}_e & \text{if entry is restricted} \\ \frac{[\Pi_d+(N-1)\Pi_x]}{w f_e} & \text{if entry is free} \end{cases}$$

In all countries, the labor market is perfectly competitive. Accordingly, the nominal wage is such that the total cost of labor equates total labor payments:

$$w\ell L = \begin{cases} (X_d - \Pi_d^o) + M_e[1 - G(\varphi_d^*)]wf + (N-1)[(X_x - \Pi_x^o) + M_e[1 - G(\varphi_x^*)]wf_x] & \text{(restricted entry)} \\ (X_d - \Pi_d^o) + M_e[1 - G(\varphi_d^*)]wf + (N-1)[(X_x - \Pi_x^o) + M_e[1 - G(\varphi_x^*)]wf_x] + M_e w f_e & \text{(free entry)} \end{cases}$$

Together with equations (20) and (21), the two previous expressions imply

$$Me = \begin{cases} \tilde{M}_e & \text{if entry is restricted} \\ \frac{\dot{\mu}(\bar{e}) Y}{w [1 - G(\varphi_d^*)] f + (N-1) [1 - G(\varphi_x^*)] f_x + f_e} & \text{if entry is free} \end{cases} \quad (23)$$

$$w\ell L = \begin{cases} [1 - \dot{\mu}(\bar{e})]Y + wM_e [1 - G(\varphi_d^*)]f + (N-1)[1 - G(\varphi_x^*)]f_x & \text{if entry is restricted} \\ Y & \text{if entry is free} \end{cases} \quad (24)$$

where $\dot{\mu}(\bar{e}) = (\frac{Y}{Y_W}) \dot{\mu}(\bar{e}_d) + (\frac{Y_{RoW}}{Y_W}) \dot{\mu}(\bar{e}_x)$ is the weighted average markup rate at the industry level.<fnref n="12" /> Under free entry, since total net profits must equate total sunk entry costs, total sales revenues are exactly equal to the total wage bill, $Y = w\ell L$.<fnref n="13" /> However, under restricted entry, total net profits must be then redistributed. As in Chaney (2008), I assume that each agent owns one share of a national fund. This fund collects total net profits reaped by local firms and equally redistributes them in units of the numéraire (labor) to its shareholders. Accordingly, dividend per capita in any country, evaluates to:

$$\bar{d} = \begin{cases} \frac{[\Pi_d+(N-1)\Pi_x]}{L} & \text{if entry is restricted} \\ 0 & \text{if entry is free} \end{cases}$$

<footnote n="12">¹²Where $\frac{Y}{Y_W} = \frac{1}{N}$ is the GDP share of any country, and $\frac{Y_{RoW}}{Y_W} = \frac{N-1}{N}$ is the GDP share of the rest of the world.</footnote>

<footnote n="13">¹³This implies that per-capita income $y = \frac{Y}{L}$ is equal to the nominal wage $w$ times the individual labor endowment $\ell$: $y = w\ell$.</footnote>
Hence, depending on the nature of entry, per-capita income in any country boils down to

$$y = \begin{cases} w\ell + \bar{d} & \text{if entry is restricted} \\ w\ell & \text{if entry is free} \end{cases} \qquad (25)$$

### 3.3.3 Summary

A trade equilibrium with symmetric countries corresponds to a price schedule $p$, a mass of entrants $M_e$, a nominal wage $w$ and an individual income $y$, such that [1] any firm with productivity $\varphi$ sets a profit-maximizing price in the market it serves:

$$\begin{cases} p_d(\varphi) = \left(\frac{w}{\varphi}\right) \mu\left(\frac{y}{F(\Lambda)}\frac{\varphi}{w}\right), & \text{for all } \varphi \ge \varphi_d^* \\ p_x(\varphi) = \left(\frac{\tau w}{\varphi}\right) \mu\left(\frac{y}{F(\Lambda)}\frac{\varphi}{\tau w}\right), & \text{for all } \varphi \ge \varphi_x^* \end{cases} \qquad (26)$$

where [2] domestic and export productivity cutoffs ($\varphi_d^*, \varphi_x^*$) are implicitly determined by the zero profit conditions on these respective markets in equation (11); [3] the mass of entrants in any country is given by equation (23); [4] the nominal wage is consistent with the labor market clearing condition in equation (24) and is hereafter normalized to unity ($w = 1$) by choice of labor as numéraire; [5] per-capita income is determined by equation (25), with the single price aggregator $\Lambda$ determined by the budget constraint in equation (2), and aggregate revenues from domestic and export sales are given by equation (12). Finally, we can note that country-level symmetry ensures trade balance.

## 3.4 Explicit solutions for ($\Lambda$, $\varphi_d^*, \varphi_x^*$) in general equilibrium

Now by applying the EEM method and normalizing wage to unity ($w = 1$), I obtain a tractable solution for the general equilibrium price aggregator $\Lambda$:<fnref n="14" />

$$\Lambda \equiv (M_e)^{\frac{1}{(-\epsilon_{\Lambda}^X)}} (y)^{\frac{(\epsilon_y^X - 1)}{(-\epsilon_{\Lambda}^X)}} (L)^{\frac{(\epsilon_L^X - 1)}{(-\epsilon_{\Lambda}^X)}} (f)^{\chi_d \frac{\epsilon_d^{\chi_d}}{(-\epsilon_{\Lambda}^X)}} [T(N, \tau, f_x)]^{\frac{(1-\chi_d)}{(-\epsilon_{\Lambda}^X)}} \qquad (27)$$

where $X = X_d + (N - 1)X_x$ stands for aggregate expenditure in any country, and its elasticities with respect to per-capita income $y$, population size $L$, and the price aggregator $\Lambda$ are respectively given by:

$$\begin{cases} \epsilon_y^X = \chi_d \epsilon_y^{X_d} + (1 - \chi_d) \epsilon_y^{X_x} > 0 \\ \epsilon_L^X = \chi_d \epsilon_L^{X_d} + (1 - \chi_d) \epsilon_L^{X_x} > 0 \\ \epsilon_{\Lambda}^X = \chi_d \epsilon_{\Lambda}^{X_d} + (1 - \chi_d) \epsilon_{\Lambda}^{X_x} < 0 \end{cases}$$

<footnote n="14">¹⁴See section A.2 of the online appendix for details.</footnote>
where $\chi_d = \frac{X_d}{X}$ is the domestic trade share in any country, and $(1 - \chi_d) = \frac{(N-1)X_x}{X}$ corresponds to the import penetration rate. Moreover, $T(N, \tau, f_x) = N \tau^{-\theta} (f_x)^{\epsilon_{f_x}^{X_x}}$ can be thought of as an index measuring an economy's degree of exposure to trade. That is, a decrease (increase) in trade costs or an increase (decrease) in the number of trading partners yields a higher (lower) level of $T(N, \tau, f_x)$, which implies that an economy is more (less) exposed to trade. Given the strictly positive sign of $(-\epsilon_{\Lambda}^X) > 0$, inspection of equation (27) clearly shows that the general equilibrium price aggregator is strictly increasing in the trade exposure index. Hence, when an economy is more exposed to trade, competition intensifies on its domestic market. However, when it is less exposed to trade, competition becomes more relaxed on its domestic market. As the EEM method relies on a local approximation around the trade equilibrium, this ensures the uniqueness of the explicit solution it delivers for the price aggregator in general equilibrium. Nevertheless, such a local approximation allows us to study only the impact of small deviations from the trade equilibrium. Accordingly, I only examine marginal changes in an economy's exposure to trade which may occur through small variations in the variable trade cost, the fixed trade cost, or the number of trading partners. Now by rewriting the zero profit conditions on domestic and export markets in equation (11) more explicitly, and using the general equilibrium expression of the price aggregator from equation (27), I first solve for the domestic productivity cutoff $\varphi_d^*$ in general equilibrium:

$$ \varphi_d^* = (f)^{\epsilon_{f_d}^*} (y)^{\epsilon_{y_d}^*} (L)^{\epsilon_{L_d}^*} (M_e)^{\epsilon_{M_e}^*} [T(N, \tau, f_x)]^{\epsilon_{T_d}^*} \quad (28) $$

Studying the impact of changes in the mass of entrants $M_e$ or in market size (occurring through changes in individual income $y$ or population size $L$) is out of the scope of this paper. Instead, the current paper squarely focuses on the general equilibrium effect of small changes in an economy's degree of exposure to trade, which is captured by the trade exposure index $T(N, \tau, f_x)$.<fnref n="15" /> The two strictly positive elasticities below convey two important messages. ¹⁶First, the domestic productivity cutoff $\varphi_d^*$ is strictly increasing in the domestic fixed cost $f$. This shows that firm-level domestic operating profits are monotonically increasing in firm productivity, which in turn ensures the existence of a single productivity cutoff separating firms that exit from those that serve the domestic market.<fnref n="17" /> This confirms thus that the so called "conventional first-order selection effect" by

<footnote n="15">¹⁵For expositional simplicity, the explicit expressions of elasticities of the domestic productivity cutoff with respect to the mass of entrants, individual income, and population size are relegated to the online appendix. They are precisely provided in equation (57) in section A.2</footnote>

<footnote n="16">¹⁶Notice that the explicit expressions of the firm-level elasticities $\{\epsilon_{\varphi}^{\pi^0}(e_d^*), \epsilon_{y}^{\pi^0}(e_d^*), \epsilon_{F(\Lambda)}^{\pi^0}(e_d^*)\}$ present in the below elasticities of the domestic cutoff with respect to the fixed domestic cost, and the trade exposure index are already provided by equation (16).</footnote>

<footnote n="17">¹⁷We can first notice the presence of a negative general equilibrium effect in the second term on the RHS of $\epsilon_{f_d}^{\varphi^*}$. In deed, all components of this term are strictly positive, except the elasticity of domestic trade with respect to the fixed production cost, which is strictly negative: $\epsilon_{f_d}^{X_d} < 0$. Despite the presence of this negative general equilibrium effect, $\epsilon_{f_d}^{\varphi^*}$ is always strictly positive since the domestic trade share is always strictly lower than one ($\chi_d < 1$) at the trade equilibrium.</footnote>
Mrázová and Neary (2019) is robust on the domestic market.

$$ \left\{ \begin{aligned} \varepsilon_{f}^{\varphi_{d}^{*}} &= \frac{d \log \varphi_{d}^{*}}{d \log f} = \frac{1}{\varepsilon_{\varphi}^{\pi^{0}}(e_{d}^{*})} + \frac{[\varepsilon_{H} - \varepsilon_{F} \varepsilon_{F(\Lambda)}^{\pi^{0}}(e_{d}^{*})]}{\varepsilon_{\varphi}^{\pi^{0}}(e_{d}^{*})} \frac{\chi_{d} \varepsilon_{f}^{\chi_{d}}}{(-\varepsilon_{\Lambda}^{\chi_{d}})} &> 0 \\ \varepsilon_{T}^{\varphi_{d}^{*}} &= \frac{d \log \varphi_{d}^{*}}{d \log T(N, \tau, f_{x})} = \frac{[\varepsilon_{H} - \varepsilon_{F} \varepsilon_{F(\Lambda)}^{\pi^{0}}(e_{d}^{*})]}{\varepsilon_{\varphi}^{\pi^{0}}(e_{d}^{*})} \frac{(1 - \chi_{d})}{(-\varepsilon_{\Lambda}^{\chi_{d}})} &> 0 \end{aligned} \right. \qquad (29) $$

Second, the domestic cutoff $\varphi_d^*$ is strictly increasing in the trade exposure index $T(N, \tau, f_x)$. This clearly shows that whether demand is CES or not, the selection effect à la Melitz (2003) is always operative in the current environment. Regardless of the scenario though which it occurs, higher exposure to trade always exerts a competitive pressure on domestic firms. This forces the least productive among them to exit the market and thus leads to an increase in the domestic productivity cutoff. By contrast, when an economy gets less exposed to trade, competition on its domestic market becomes more relaxed. This, in turn, triggers entry of infra-marginal firms and lowers the domestic productivity cutoff. Now using the same procedure, I obtain the following expression of the general equilibrium export productivity cutoff $\varphi_x^*$:

$$ \varphi_x^* \equiv \tau (f_x)^{\varepsilon_{f_x}^{\varphi_x^*}} (y)^{\varepsilon_{y}^{\varphi_x^*}} (L)^{\varepsilon_{L}^{\varphi_x^*}} (M_e)^{\varepsilon_{M_e}^{\varphi_x^*}} [T(N, \tau, f_x)]^{\varepsilon_{T}^{\varphi_x^*}} \qquad (30) $$

where the partial elasticity of the export cutoff with respect to the fixed export cost $\varepsilon_{f_x}^{\varphi_x^*}$, and its elasticity with respect to the trade exposure index $\varepsilon_{T}^{\varphi_x^*}$ are respectively given by:<fnref n="18" />

$$ \left\{ \begin{aligned} \varepsilon_{f_x}^{\varphi_x^*} &= \frac{\partial \log \varphi_x^*}{\partial \log f_x} = \frac{1}{\varepsilon_{\varphi}^{\pi^0}(e_x^*)} &> 0 \\ \varepsilon_{T}^{\varphi_x^*} &= \frac{d \log \varphi_x^*}{d \log T(N, \tau, f_x)} = \frac{[\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^0}(e_x^*)]}{\varepsilon_{\varphi}^{\pi^0}(e_x^*)} \frac{(1 - \chi_d)}{(-\varepsilon_{\Lambda}^{\chi_d})} &> 0 \end{aligned} \right. \qquad (31) $$

As it was the case for the domestic market, the above strictly positive (partial) elasticity of the export productivity cutoff with respect to the fixed export cost $\varepsilon_{f_x}^{\varphi_x^*}$ ensures that the "conventional first-order selection effect" à la Mrázová and Neary (2019) is also robust on the export market.<fnref n="19" /> It is then sufficient to assume that the fixed export cost is larger than the fixed domestic cost, $f_x > f$, to ensure that firms are partitioned by export status as in Melitz (2003): while more productive firms select into exporting, less productive ones serve the domestic market only.<fnref n="20" /> Moreover, the strictly positive elasticity of the export cutoff with respect to the trade exposure index $\varepsilon_{T}^{\varphi_x^*}$ captures the competitive effect that higher exposure to trade exerts on active exporters in general equilibrium.

<footnote n="18">¹⁸As mentioned before, examining the impact of changes in the mass of entrants $M_e$ or in market size (occurring through changes in individual income $y$ or population size $L$) is out of the scope of this paper. Hence, for expositional simplicity, the explicit expressions of the elasticities of the export productivity cutoff with respect to these three variables are relegated to the online appendix. They are precisely provided in equation (59) in section A.2.</footnote>

<footnote n="19">¹⁹As before, the explicit expressions of the firm-level elasticities $\{\varepsilon_{\varphi}^{\pi^0}(e_x^*), \varepsilon_{y}^{\pi^0}(e_x^*), \varepsilon_{F(\Lambda)}^{\pi^0}(e_x^*)\}$ present in the above elasticities are provided by equation (19).</footnote>

<footnote n="20">²⁰Far from being restrictive, this assumption simply ensures that the partitioning of firms by export status always holds even in the limit case where the iceberg trade cost reaches unity, $\tau = 1$.</footnote>
Hence, upon a small reduction in trade costs (variable or fixed), the initial decrease in the export productivity cutoff is partially offset by this general equilibrium effect. While the current environment ensures the robustness of the first-order selection effect on both markets as in Mrázová and Neary (2019) and the partitioning of firms by export status as in Melitz (2003), the key novelty lies in the new research question it examines. As I will show in the next section, the current paper leverages increased generality on the demand side to reveal new implications of demand curvature for the toughness of firm selection into markets and for the degree of firm partitioning by export status.

## 3.5 Structural Gravity

Now by plugging the general equilibrium price aggregator $\Lambda$ from equation (27) in the partial equilibrium expression of bilateral trade flows in equation (13), I obtain the following structural gravity equation:<fnref n="21" />

$$X_x = \underbrace{M_e}_{(1)} \underbrace{\tau^{-\theta} (f_x)^{\varepsilon_{fx}^{X_x}}}_{(2)} \underbrace{(y)^{(1-\varepsilon_y^X + \varepsilon_y^{X_x})} (L)^{(1-\varepsilon_L^X + \varepsilon_L^{X_x})}}_{(3)} \Psi \quad (32)$$

where $\Psi = [M_e [T(N, \tau, f_x)]^{(1-\chi_d)} [(f)^{\varepsilon_f^{X_d}}]^{\chi_d}]^{-1}$ is reminiscent of the inward multilateral resistance term is Anderson and Van Wincoop (2003), and can also be seen as the symmetric version of the remoteness index in Chaney (2008) augmented by a domestic component which is strictly increasing in the fixed cost for serving the domestic market $f$. That is, a higher fixed domestic cost implies a higher domestic productivity cutoff (ceteris paribus). This, in turn, lowers the probability of successful entry for domestic producers, which induces more relaxed competition on the domestic market and makes any trading economy easier to penetrate for foreign exporters. Hence, by determining domestic trade, higher fixed domestic costs foster bilateral trade. As previously mentioned, $\varepsilon_{fx}^{X_x} = \frac{d\log X_x}{d\log f_x} < 0$ is the elasticity of partial equilibrium bilateral trade flows with respect to the fixed export cost. Its expression initially provided by equation (17) can be rewritten more explicitly using firm-level elasticities from equations (18) and (19) as follows:

$$\varepsilon_{fx}^{X_x} = - \frac{[\theta - \varepsilon_{\phi}^{r}(\bar{e}_x)]}{\varepsilon_{\phi}^{\pi^0}(e_x^*)} = - \frac{[\theta - (1 - \rho(S))(\varepsilon_D(\bar{e}_x) - 1)]}{[(1 - \rho(S))(\varepsilon_D(e_x^*) - 1) + \eta \rho(S)]} < 0$$

The terms (1), (2) and (3) on the RHS of equation (32) respectively correspond to the origin-specific, the pair-specific, and the destination-specific components of the gravity equation. Inspection of this equation clearly shows that regardless of the curvature of demand and of the type of separability, unbounded Pareto distribution of firm productivity always gives rise to a constant trade elasticity: $\varepsilon_{\tau}^{X_x} = \frac{d\log X_x}{d\log \tau} = -\theta < 0$, as in (Chaney, 2008; Melitz and Ottaviano, 2008; Arkolakis et al., 2019).<fnref n="22" />

<footnote n="21">²¹Further details are provided in section A.3 of the online appendix.</footnote>

<footnote n="22">²²Notice that this result is obtained upon ignoring the general equilibrium effect (operating through the inward mul-</footnote>
Therefore, two of the three macro-level restrictions imposed in Arkolakis et al. (2012) still hold in the current theoretical environment: trade balance and constant trade elasticity.

# 4 Demand Curvature, Firm Selection, and the Partitioning of Firms by Export Status

As mentioned in the previous section, the current environment ensures the robustness of the first-order selection effect à la Mrázová and Neary (2019) on both markets (domestic and export). By simply assuming that trade involves variable costs and relatively higher fixed costs ($f_x > f$), it ensures that the partitioning of firms by export status as in Melitz (2003) also holds. The key novelty here is that I examine the role of demand curvature in determining the toughness of firm selection into markets and the degree of firm partitioning by export status. This, in turn, has crucial implications for the probability of exporting and the average export productivity.

## 4.1 Demand Curvature and Firm Selection into markets

I proceed here in three steps. First, I show how demand curvature determines the nature of the elasticity of a firm's operating profits with respect to its productivity level: whether it is constant, increasing or decreasing in firm productivity. Then, I graphically illustrate the result in a compact way and infer a novel implication of demand curvature for firm selection. Finally, I show that this result holds in general equilibrium and I explain the economic mechanism driving this result.

### 4.1.1 Constant vs Variable productivity elasticity of operating profits

Let us consider a firm which, upon entering the industry, draws a productivity level $\varphi$. Then, it starts producing its own variety at a marginal cost $c$ and engages in monopolistic competition. Using equation (9), the operating profit of this firm can be written as:

$$ \pi^0(c, e, H(\Lambda), L) = \dot{\mu}(e) r(c, e, H(\Lambda), L) $$

where, as before, $e$ is a measure of firm-level efficiency given by $e = \frac{y}{cF(\Lambda)}$. Given that a firm's marginal cost is inversely related to its productivity level, the elasticity of operating profits with respect to firm productivity boils down to:

$$ \varepsilon_{\varphi}^{\pi^0}(e) = \frac{\partial \log \pi^0(e)}{\partial \log \varphi} = -\frac{\partial \log \pi^0(e)}{\partial \log c} = (1 - \rho(S))(\varepsilon_D(e) - 1) + \eta \rho(S) > 0 \quad (33) $$

<footnote>tilateral resistance term $\Psi$). It is then equivalent to the elasticity of partial equilibrium bilateral trade flows with respect to the variable trade cost $\tau$, which is provided by equation (17).</footnote>
This strictly positive elasticity shows that regardless of demand curvature, firm-level operating profits are monotonically increasing in firm productivity. As previously mentioned, this is a sufficient condition ensuring that the first-order selection effect à la Mrázová and Neary (2019) is robust on both markets (domestic and export). Yet, the key novelty here is that demand curvature plays a critical role in determining whether the pace at which the logarithm of operating profits increases with the logarithm of firm productivity is constant or variable. Clearly, the homothetic CES case is very special: since the price elasticity of demand is exogenous ($\forall e, \epsilon_D(e) = \epsilon_D$), the superelasticity of demand $S$ collapses to zero ($S = 0$), and the markup is constant ($\rho(S) = 0$). This yields a constant elasticity of (the logarithm of) operating profits with respect to (the logarithm of) firm productivity: $\log \pi^0(\varphi)$ always increases with $\log \varphi$ at a constant pace regardless of the firm's productivity level ($\forall e, \epsilon_\varphi^{\pi^0}(e) = \epsilon_D - 1$). Therefore, departing from CES demand allows for a variable elasticity of operating profits: $\epsilon_\varphi^{\pi^0}(e)$ may then increase or decrease in firm productivity $\varphi$. Hence, depending on whether demand is super-convex or sub-convex, the pace at which $\log \pi^0(\varphi)$ increases with $\log \varphi$ may be faster or slower as compared with the CES case. To show this in a simple way, let us define $\xi(S)$ as the elasticity of the productivity elasticity of operating profits with respect to firm productivity:

$$ \xi(S) = \frac{\partial \log \epsilon_{\varphi}^{\pi^0}(e)}{\partial \log \varphi} = \frac{\partial \log \epsilon_{\varphi}^{\pi^0}(e)}{\partial \log \epsilon_D(e)} \frac{\partial \log \epsilon_D(e)}{\partial \log e} \frac{\partial \log e}{\partial \log \varphi} \quad (34) $$

After simplification, the final expression of $\xi(S)$ boils down to:

$$ \xi(S) = (-S) a(e, S) \quad (35) $$

where $a(e, S)$ is a strictly positive term, the sign of $\xi(S)$ thus hinges on the sign of the superelasticity of demand $S$.<fnref n="23" /> Therefore, whether the elasticity of firm-level operating profits with respect to firm productivity is increasing, decreasing, or invariant to firm productivity crucially depends on demand curvature:

$$ \xi(S) = \begin{cases} < 0 & \text{if } S > 0 \text{ (sub-convex demand)} \\ = 0 & \text{if } S = 0 \text{ (CES demand)} \\ > 0 & \text{if } S < 0 \text{ (super-convex demand)} \end{cases} \quad (36) $$

### 4.1.2 A novel implication of demand curvature for firm selection: A graphical illustration

Inspection of equation (35) shows that $\log \pi^0(\varphi)$ is convex (concave) in $\log \varphi$ when demand is super-convex (sub-convex). Here, CES demand arises again as a special case which imposes a linear relationship between $\log \pi^0(\varphi)$ and $\log \varphi$, as illustrated by the upward sloping "CES" line in

<footnote n="23">²³For expositional clarity, the explicit expression of $a(e, S)$ is relegated to section A.4 of the online appendix.</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 2: Demand Curvature and Toughness of Firm Selection into markets.</figcaption></figure>

Figure 2, panel A. The underlying economics are simple: when demand is super-convex, more productive firms face more elastic demand and set lower markups ($\rho(S) < 0$). Higher productivity induces a more than proportional reduction in price, $(1 - \rho(S)) > 1$, thus magnifying a firm's initial level of price competitiveness (given by its productivity level) by lower markups.<fnref n="24" /> This, in turn, magnifies the responsiveness of firm-level operating profits to changes in firm productivity. As illustrated in Figure 2, panel A, when demand is super-convex, the responsiveness of operating profits to increases in firm productivity is relatively faster as compared with CES demand.<fnref n="25" />

These patterns are reversed when demand is sub-convex: now more productive firms face less elastic demand and set higher markups ($\rho(S) > 0$). Higher productivity induces a less than proportional reduction in price, $(1 - \rho(S)) < 1$, thus dampening a firm's initial level of price competitiveness (given by its productivity level) by higher markups. This, in turn, dampens the responsiveness of firm-level operating profits to changes in firm productivity. As illustrated in Figure 2, panel A, when demand is sub-convex, the responsiveness of operating profits to increases in firm productivity is relatively slower as compared with CES demand. As illustrated in Figure 2, panel A, the profile of operating profits across firms is linear under CES demand. This clearly shows that the CES delivers an intermediate outcome in the current environment. By determining the nature of the elasticity of operating profits with respect to firm productivity, demand curvature has an immediate implication for the toughness of firm selection. As illustrated in Figure 2, panel B,

<footnote n="24">²⁴Let us recall that for any firm with marginal cost $c$ and efficiency measure $e$, the optimal pricing rule is given by $p(c, e) = c \mu(e)$. The relative cost-price pass-through ($\frac{d\log p(c,e)}{d\log c}$) evaluates then to $(1 - \rho(S)) \le 1$. Hence, the relative pass-through may be equal to 100 percent under CES demand ($\rho(S) = 0$), smaller than 100 percent under sub-convex demand ($\rho(S) > 0$), or larger than 100 percent under super-convex demand ($\rho(S) < 0$).</footnote>

<footnote n="25">²⁵Under super-convex demand, more productive firms set lower markups and this has two opposite effects on their operating profits. The first is positive: by magnifying their price competitiveness, this pricing behavior strongly increases their revenues. The second is negative: lower markups imply lower markup rates. The fact that operating profits are monotonically increasing in firm productivity ensures that the positive effect on revenues dominates the negative effect on markup rates.</footnote>
for any level of fixed cost of accessing a given market, $wf$, the productivity cutoff reaches its lowest level when demand is sub-convex and its highest level when demand is super-convex. Hence, while super-convex demand yields an upper bound for the toughness of firm selection, sub-convex demand yields a lower bound. Within these bounds, the CES delivers an intermediate outcome.

### 4.1.3 General equilibrium behavior and Economic intuition

Now I apply the above result on firm selection for the case of two markets: domestic and export. Access to these markets involves fixed costs which are respectively given by $wf$ and $wf_x$.<fnref n="26" /> Using equations (28) and (30), I prove that this novel result holds for both markets in general equilibrium, as clearly stated in the following theorem:<fnref n="27" />

**Theorem 1.** Under Generalized Gorman-Pollak demand, unbounded Pareto distribution of firm productivity, country-level symmetry and the presence of fixed costs of accessing markets, super-convex demand provides an upper bound for the degree of toughness of firm selection in any market in general equilibrium, while sub-convex demand provides a lower bound. Within these bounds, CES demand delivers an intermediate outcome.

$$ \forall f > 0, \varphi_d^*[sub - convex] < \varphi_d^*[CES] < \varphi_d^*[super - convex] $$

$$ \forall f_x > f, \varphi_x^*[sub - convex] < \varphi_x^*[CES] < \varphi_x^*[super - convex] $$

The economic force behind this result can be explained as follows. When demand is super-convex, the initial level of price competitiveness of more productive firms (given by their initially high productivity levels) is magnified by lower markups. In addition to that, consumers are more sensitive to price variations of varieties supplied by this category of firms. Hence, as compared with the CES benchmark, the super-convexity of demand reinforces the allocation of larger market shares to more productive firms. This induces then a relatively tougher competitive environment for low productivity firms. As compared to the CES case, setting higher markups makes these less productive firms even less price competitive. This implies then additional difficulty in capturing enough market shares to successfully enter the market, which is reflected by higher domestic and export productivity cutoffs under super-convex demand. All these patterns are reversed when demand is sub-convex.

## 4.2 Demand Curvature and Partitioning of Firms by Export status

In order to examine the role that demand curvature plays in determining the degree of partitioning of firms by export status, I proceed in three steps. First, I show how demand curvature determines

<footnote n="26">²⁶Access to the export market involves both variable and fixed trade costs. For expositional simplicity, I only focus on fixed costs and assume that exporting is costlier than serving the domestic market ($f_x > f$)</footnote>

<footnote n="27">²⁷See section A.6 of the online appendix for a detailed proof of theorem 1.</footnote>
whether the negative impact of the variable trade cost $\tau$ on export operating profits is identical or heterogeneous across firms. Then, I graphically illustrate the result in a compact way and uncover a new implication of demand curvature for the partitioning of firms by export status. Finally, I show that this novel result holds in general equilibrium and I explain the underlying economic intuition.

### 4.2.1 Homogeneous vs Heterogeneous effect of the variable trade cost on firm-level export operating profits

As before, our starting point is a monopolistically competitive firm which, upon entering the industry, draws a productivity level $\phi$. To start producing and selling its own variety in the domestic market, this firm must incur a fixed cost $wf$. However, serving foreign markets is more costly and involves not only, a higher fixed cost $w f_x > wf$, but also a variable trade cost $\tau \ge 1$. From equation (9), firm-level operating profits in the domestic and export markets are respectively given by:

$$ 
\left\{ 
\begin{aligned}
\pi_d^o(c_d, e_d, H(\Lambda), L) &= \dot{\mu}(e_d) r_d(c_d, e_d, H(\Lambda), L) \\
\pi_x^o(c_x, e_x, H(\Lambda), L) &= \dot{\mu}(e_x) r_x(c_x, e_x, H(\Lambda), L)
\end{aligned}
\right. 
\qquad (37) 
$$

where $c_d = \frac{w}{\phi}$, $c_x = \tau \frac{w}{\phi}$, $e_d = \frac{y}{c_d F(\Lambda)}$, and $e_x = \frac{y}{c_x F(\Lambda)}$. Since the marginal cost of delivering one variety to the export market is proportional to the iceberg trade cost $\tau \ge 1$, the elasticity of export operating profits with respect to the variable trade cost evaluates to:

$$ \varepsilon_{\tau}^{\pi^{o}}(e_{x}) = \frac{\partial \log \pi^{o}(e_{x})}{\partial \log \tau} = \frac{\partial \log \pi^{o}(e_{x})}{\partial \log c_{x}} = (1 - \rho(S))(1 - \varepsilon_{D}(e_{x})) - \eta \rho(S) < 0 \quad (38) $$

Now to see whether this negative effect of the variable trade cost on firm-level export operating profits is identical or heterogeneous across firms, we can work the elasticity of the absolute value of $\varepsilon_{\tau}^{\pi^{o}}(e_x)$ with respect to firm productivity: $\frac{\partial \log(-\varepsilon_{\tau}^{\pi^{o}}(e_x))}{\partial \log \phi}$. Inspection of equations (33) and (38) clearly shows that $(-\varepsilon_{\tau}^{\pi^{o}}(e_x)) = \varepsilon_{\phi}^{\pi^{o}}(e)$. Using equations (34) and (35), $\frac{\partial \log(-\varepsilon_{\tau}^{\pi^{o}}(e_x))}{\partial \log \phi}$ boils down to:

$$ \frac{\partial \log(-\varepsilon_{\tau}^{\pi^{o}}(e_x))}{\partial \log \phi} = \zeta(S) = (-S) a(e, S) \qquad (39) $$

where $a(e, S)$ is a strictly positive term.<fnref n="28" /> The above expression of $\zeta(S)$ reveals that the sign of the elasticity of $(-\varepsilon_{\tau}^{\pi^{o}}(e_x))$ with respect to firm productivity hinges on the sign of the superelasticity of demand $S$. Thus, whether the negative impact of the variable trade cost is identical or heterogeneous

<footnote n="28">²⁸As previously mentioned, the explicit expression of $a(e, S)$ is relegated to section A.4 of the online appendix for expositional clarity.</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 3: Demand Curvature and Partitioning of Firms by Export status.</figcaption></figure>

neous across firms crucially depends on whether demand is CES or not:

$$ \frac{\partial \log(-\varepsilon \pi_{\tau}^{0}(e_{x}))}{\partial \log \varphi} = \zeta(S) \begin{cases} < 0 & \text{if } S > 0 \text{ (sub-convex demand)} \\ = 0 & \text{if } S = 0 \text{ (CES demand)} \\ > 0 & \text{if } S < 0 \text{ (super-convex demand)} \end{cases} \qquad (40) $$

### 4.2.2 A graphical illustration of the role of demand curvature:

Country-level symmetry implies that the difference between domestic and export operating profits is solely driven by the existence of the variable trade cost $\tau \ge 1$: $\forall \varphi, \frac{c_x}{c_d} = \tau$, and $\frac{e_x}{e_d} = \tau^{-1}$.29 The profile of export operating profits $\pi_x^0(\varphi)$ in Figure 3 can be thus interpreted as response to a negative trade shock ($\tau = 1 \rightarrow \tau' > 1$). Whether this response is firm-specific or not crucially depends on demand curvature, as illustrated in Figure 3. Inspection of equation (39) reveals that the absolute value of the elasticity of firm-level export operating profits with respect to the variable trade cost is increasing in firm productivity when demand is super-convex, whereas it is decreasing in firm productivity when demand is sub-convex. This implies that under super-convex demand, the negative impact of the variable trade cost on export operating profits is magnified for more productive firms and dampened for less productive ones. By contrast, when demand is sub-convex, this negative effect is rather magnified for less productive firms and dampened for more productive ones. Here, CES demand stands again as a special case which restricts the negative effect of the variable trade cost on firm-level export operating profits to be identical across firms ($S = 0$). By determining whether this effect is identical or heterogeneous across firms, demand curvature plays a crucial role in determining the degree of firm partitioning by export status. As illustrated in Panel A of Figure 3, when demand is super-convex, the distance between the export and the

<footnote>29By symmetry, all countries have an identical market size, $Y = yL$, and display the same intensity of competition, which is measured by the price aggregator $\Lambda$.</footnote>
domestic productivity cutoff [$\varphi_x^* - \varphi_d^*$] is the smallest, thus yielding a lower bound for the degree of firm partitioning by export status. These patterns are reversed under sub-convex demand which yields an upper bound for firm partitioning (Panel C). Within these bounds, CES demand delivers an intermediate outcome (Panel B).

### 4.2.3 General equilibrium behavior and Economic intuition

The concept of firm partitioning by export status has been initially introduced by Melitz (2003) showing how more productive firms self-select into exporting while less productive ones serve the domestic market only. The key novelty here is that I examine the role that demand curvature plays in determining the degree of this partitioning. Now using equations (28) and (30), I prove that this novel result holds in general equilibrium, as clearly stated in the following theorem:<fnref n="30" />

**Theorem 2.** Under Generalized Gorman-Pollak demand, unbounded Pareto distribution of firm productivity, country-level symmetry and the presence of fixed costs of accessing markets, super-convex demand provides a lower bound for the degree of partitioning of firms by export status in general equilibrium, while sub-convex demand provides an upper bound. Within these bounds, CES demand delivers an intermediate outcome.

$$ \forall f_x > f, \frac{\varphi_x^*}{\varphi_d^*} [\text{super-convex}] < \frac{\varphi_x^*}{\varphi_d^*} [\text{CES}] < \frac{\varphi_x^*}{\varphi_d^*} [\text{sub-convex}] $$

The underlying economics are simple and can be explained as follows. Departing from the homothetic CES benchmark, when demand is super-convex, selection is relatively tougher and only very productive firms successfully enter the domestic market. Thus, a relatively large subset of these very productive firms can export. Put differently, these firms are enough productive to successfully enter the domestic market despite tougher competitive conditions implied by the super-convexity of demand. A large fringe of these firms is thus enough price competitive to penetrate the export market. Therefore, when demand is super-convex, the partitioning of firms by export status is less pronounced as compared with the homothetic CES case. These patterns are reversed when demand is sub-convex.

# 5 Analysis of the Welfare Impact of Trade Exposure

## 5.1 Measuring the Welfare Impact of Trade Exposure

### 5.1.1 Three key variables for welfare analysis

In section 3, I have provided explicit solutions for the main equilibrium variables, including the mass of entrants $M_e$, the price aggregator $\Lambda$, the domestic productivity cutoff $\varphi_d^*$, and the export

<footnote n="30">³⁰See section A.7 of the online appendix for a detailed proof.</footnote>
productivity cutoff $\varphi_x^*$. Now I use their general equilibrium expressions to solve for three key variables for welfare analysis: (i) the total mass of firms competing in a single country $M$; (ii) the weighted average marginal cost of these firms $\bar{c}$; and (iii) the weighted average markup they set $\mu(\bar{e})$. I proceed as follows. Let $M_d$ first denote the equilibrium mass of domestic firms. Using equations (23) and (28), $M_d$ can be written as: $M_d = M_e [1 - G(\varphi_d^*)]$. Similarly, the equilibrium mass of exporting firms in any country is given by $M_x = M_e [1 - G(\varphi_x^*)]$. The total mass of firms competing in any country (or, alternatively the total mass of available varieties in any country) is then given by:

$$M = M_d + M_m \quad (41)$$

where $M_m = (N - 1)M_x$ is the total mass of imported varieties in any country. Now let $\bar{c}$ denote the weighted average marginal cost of all firms (both domestic and foreign exporters) competing in a single country. Upon normalizing wage to unity, this is given by:

$$\bar{c} = \left(\frac{Y}{Y_W}\right) (\bar{\varphi}_d)^{-1} + \left(\frac{Y_{RoW}}{Y_W}\right) \tau (\bar{\varphi}_x)^{-1} \quad (42)$$

where, as before, $\frac{Y}{Y_W} = \frac{1}{N}$ is the GDP share of any country, and $\frac{Y_{RoW}}{Y_W} = \frac{N-1}{N}$ is the GDP share of the rest of the world. Finally, let $\mu(\bar{e})$ denote the weighted average markup set by all firms participating in a single market. Using equation (6), this boils down to:

$$\mu(\bar{e}) = \left[ \frac{y}{F(\Lambda)} \right]^{\rho(S)} \left[ \left( \frac{Y}{Y_W} \right) (\bar{\varphi}_d)^{\rho(S)} + \left( \frac{Y_{RoW}}{Y_W} \right) (\tau)^{-\rho(S)} (\bar{\varphi}_x)^{\rho(S)} \right] \quad (43)$$

where $\bar{\varphi}_d$ is the average productivity of domestic firms and $\bar{\varphi}_x$ is the average productivity of foreign exporters. These are respectively given by:

$$\left\{ \begin{array}{l} \bar{\varphi}_d = \int_{\varphi_d^*}^{+\infty} \varphi \, \gamma_d(\varphi) \, d\varphi \\ \bar{\varphi}_x = \int_{\varphi_x^*}^{+\infty} \varphi \, \gamma_x(\varphi) \, d\varphi \end{array} \right. \quad (44)$$

where $\gamma_d(\varphi)$ and $\gamma_x(\varphi)$ correspond to the productivity distribution conditionally on successful entry, respectively, on the domestic and the export market:

$$\left\{ \begin{array}{l} \gamma_d(\varphi) = \frac{g(\varphi)}{[1-G(\varphi_d^*)]} \quad \forall \varphi \ge \varphi_d^* \\ \gamma_x(\varphi) = \frac{g(\varphi)}{[1-G(\varphi_x^*)]} \quad \forall \varphi \ge \varphi_x^* \end{array} \right. $$

### 5.1.2 A sufficient statistic for welfare per worker

As previously mentioned in Section 2, an attractive feature of our generalized Gorman-Pollak demand system is that it allows for flexible price effects while retaining the key property that the
level of intensity of competition in any country is captured by a single industry-wide indicator $\Lambda$. Accordingly, I work with the price aggregator $\Lambda$ as a sufficient statistic for welfare per worker $W$. Using the budget constraint from equation (2) and equations (42, 43, 44) and invoking the unbounded Pareto assumption A3, I can write welfare per worker $W$ as a function of solely the total number of available varieties $M$, and the domestic productivity cutoff $\varphi_d^{*31}$

$$W = \Lambda = \Omega(.) \underbrace{(M \frac{1}{\psi(\varepsilon_F, \varepsilon_H, \varepsilon_D(\bar{e}), \rho(S))})}_{(1)Variety \ effect} \underbrace{(\varphi_d^* \frac{(\varepsilon_D(\bar{e})-1)}{\psi(\varepsilon_F, \varepsilon_H, \varepsilon_D(\bar{e}), \rho(S))})}_{(2)Selection \ effect} \underbrace{(\varphi_d^*)^{-\rho(S)} \frac{(\varepsilon_D(\bar{e})-1)}{\psi(\varepsilon_F, \varepsilon_H, \varepsilon_D(\bar{e}), \rho(S))}}_{(3)Markup \ effect} \quad (45)$$

where $\Omega(.)$ is a composite function and $\psi(\varepsilon_F, \varepsilon_H, \varepsilon_D(\bar{e}), \rho(S))$ is a strictly positive composite elasticity given by:<fnref n="32" />

$$\psi(\varepsilon_F, \varepsilon_H, \varepsilon_D(\bar{e}), \rho(S)) = \varepsilon_F [(1 - \rho(S)) \varepsilon_D(\bar{e}) + \rho(S)] + \varepsilon_H \geq 1$$

where $\varepsilon_D(\bar{e}) = (\frac{\gamma}{\gamma_W}) \varepsilon_D(\bar{e}_d) + (\frac{\gamma_{RoW}}{\gamma_W}) \varepsilon_D(\bar{e}_x)$ is the weighted average price elasticity of demand at the industry level, $\rho(S)$ is the markup elasticity, and $\varepsilon_F$ and $\varepsilon_H$ are both dummies capturing the type of separability as previously described in section 2.

### 5.1.3 A sufficient statistic for the magnitude of the welfare impact of trade exposure

Inspection of the above expression of welfare per capita $W$ reveals that the selection effect is the dominant welfare channel in general equilibrium. In fact, welfare per capita $W$ is most sensitive to the selection effect (2), relatively less sensitive to the markup effect (3) (if any: $\rho(S) \neq 0$), and least sensitive to the variety effect (1):

$$\underbrace{\frac{(\varepsilon_D(\bar{e}) - 1)}{\psi(\varepsilon_F, \varepsilon_H, \varepsilon_D(\bar{e}), \rho(S))}}_{(2)Selection} > \underbrace{| - \rho(S) \frac{(\varepsilon_D(\bar{e}) - 1)}{\psi(\varepsilon_F, \varepsilon_H, \varepsilon_D(\bar{e}), \rho(S))}|}_{(3)Markup (\forall \rho(S) \neq 0)} > \underbrace{\frac{1}{\psi(\varepsilon_F, \varepsilon_H, \varepsilon_D(\bar{e}), \rho(S))}}_{(1)Variety}$$

Moreover, the general equilibrium expression of the domestic cutoff $\varphi_d^*$ in equation (28) clearly shows that $\varphi_d^*$ is strictly increasing in the trade exposure index $T(N, \tau, f_x)$. That is, when an economy becomes more exposed to trade, this always leads to an increase in the domestic productivity cutoff $\varphi_d^*$, which in turn leads to an increase in the weighted average productivity at the industry level. Therefore, by intensifying competition on the domestic market and forcing the least productive domestic firms to exit, higher exposure to trade always generates a welfare gain. Conversely, protectionist measures lowering an economy's exposure to trade always induce a decrease in the domestic productivity cutoff $\varphi_d^*$, which leads to a decrease in the weighted average productivity

<footnote n="31">³¹See section A.5 of the online appendix for details.</footnote>

<footnote n="32">³²For expositional clarity, the explicit expression of $\Omega(.)$ is relegated to section A.5 of the online appendix.</footnote>
at the industry level. Hence, by relaxing competition on the domestic market and triggering entry of the least productive domestic firms, lower exposure to trade always yields a welfare loss. Since the selection effect is the dominant welfare channel, we can conclude that higher (lower) exposure to trade is always welfare improving (deteriorating) regardless of its impact on the total mass of available varieties and on the weighted average markup at the industry level.<fnref n="33" />

Since the selection effect à la Melitz (2003) is always operative and is the dominant welfare channel in the current environment, I restrict my focus on it and I measure its magnitude with the elasticity of the domestic cutoff $\varphi_d^*$ with respect to the trade exposure index $T(N, \tau, f_x)$. This simple measure captures the magnitude of the competitive pressure that higher exposure to trade exerts on domestic firms forcing the least productive among them to to exit the market (hereafter CPD). Given the dominance of the selection effect in general equilibrium, the CPD can be taken as a proxy for the magnitude of the welfare impact of small changes in trade exposure.<fnref n="34" />

$$CPD = \frac{\partial \log \varphi_d^*}{\partial \log T(N, \tau, f_x)} > 0 \qquad (46)$$

For expositional simplicity, I consider hereafter the case of a small increase in an economy's exposure to trade. The implications of the opposite case are inferred symmetrically by reversing the corresponding mechanisms and outcomes.<fnref n="35" /> As previously mentioned in section 3, an attractive feature of this multi-country environment incorporating both variable and fixed trade costs is that it gives rise to the trade exposure index $T(N, \tau, f_x)$ which is a summary statistic capturing trade-induced competition in general equilibrium. This model is thus well suited for two types of welfare analysis. The first is *generic*, whereby higher exposure to trade is modeled as increase in $T(N, \tau, f_x)$ regardless of the scenario through which it occurs. The second is *granular*, whereby the three different scenarios that lead to higher exposure to trade are examined separately. That is, (i) a small reduction in the variable trade cost ($-\tau$); (ii) a small reduction in the fixed export cost ($-f_x$); and (iii) a small increase in the number of trading countries ($+N$). Accordingly, when the welfare analysis is generic, the magnitude of the gains from higher exposure to trade is given by equation (46). When the welfare analysis gets more granular, the magnitude of the gains from higher exposure to

<footnote n="33">³³Since welfare per capita is most sensitive to the selection effect, this paper squarely focuses on it. Examining the net variety effect is then out of the scope of this paper, and the analysis of the response of firm-level markups to small changes in trade exposure is relegated to section A.10 of the online appendix.</footnote>

<footnote n="34">³⁴CPD: Competitive Pressure on Domestic firms.</footnote>

<footnote n="35">³⁵Under the opposite scenario, this strictly positive elasticity in equation (46) equally captures the magnitude of the anti-competitive effect of lower exposure to trade on domestic firms, which triggers entry of least productive infra-marginal firms.</footnote>
trade occurring through each specific scenario is then given by:

$$ \left\{ \begin{aligned} CPD(-\tau) &= \frac{\partial \log \varphi_d^*}{\partial \log T(N, \tau, f_x)} \left( -\frac{\partial \log T(N, \tau, f_x)}{\partial \log \tau} \right) > 0 \\ CPD(-f_x) &= \frac{\partial \log \varphi_d^*}{\partial \log T(N, \tau, f_x)} \left( -\frac{\partial \log T(N, \tau, f_x)}{\partial \log f_x} \right) > 0 \\ CPD(+N) &= \frac{\partial \log \varphi_d^*}{\partial \log T(N, \tau, f_x)} \left( \frac{\partial \log T(N, \tau, f_x)}{\partial \log N} \right) > 0 \end{aligned} \right. \quad (47) $$

By going beyond the CES, the key idea I explore here is how alternative demand conditions affect the magnitude of the selection effect and thus of the welfare gains from higher exposure to trade. Specifically, I disentangle the role of demand curvature from that of the type of separability in determining the magnitude of the competitive pressure that higher exposure to trade exerts on domestic firms forcing the least productive among them to exit (CPD). In doing so, I assess how new results depart from those obtained under the CES in the canonical model of Melitz-Chaney.

## 5.2 Magnitude of the CPD under Generalized Demand: A Generic Analysis

In this subsection, I show that while demand curvature plays a crucial role in determining the magnitude of the competitive pressure that higher exposure to trade exerts on domestic firms forcing the least productive among them to exit the market (CPD), the type of separability plays a secondary role. To do so, I proceed in two steps. I first derive a general formula for the welfare impact of small changes in exposure to trade. This simple formula corresponds to the explicit expression of the CPD measure in equation (46) and is presented in Theorem 3. Then, I connect Theorem 3 to Theorems 1 and 2 and I provide an industry-level explanation for this novel result. Specifically, I show that by governing the toughness of firm selection and the degree of their partitioning by export status, demand curvature plays a crucial role in determining the magnitude of the CPD.

### 5.2.1 A simple welfare formula

**Theorem 3.** Under Generalized Gorman-Pollak demand, unbounded Pareto distribution of firm productivity, country-level symmetry and the presence of fixed costs of accessing markets, the magnitude of the welfare gains (losses) from higher (lower) exposure to trade is given by

$$ CPD = \frac{(1 - \chi_d)}{\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))} > 0 $$

where $\chi_d = \frac{\chi_d}{X}$ is the domestic trade share in any country, and $\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))$ is a composite function given by:

$$ \Gamma(.) = \varepsilon_H[\chi_d C_d^H + (1 - \chi_d) C_x^H \Phi^H(R_{xd}^*)] + \varepsilon_F[\chi_d C_d^F + (1 - \chi_d) C_x^F \Phi^F(R_{xd}^*)] \quad (48) $$
where $\epsilon_H$ and $\epsilon_F$ are both dummies capturing the type of separability. As previously described in section 2, preferences are directly separable when $\epsilon_H = 0$ and $\epsilon_F = 1$, whereas they are indirectly separable when $\epsilon_H = 1$ and $\epsilon_F = 0$ instead. Accordingly, for each type of separability, the above composite function $\Gamma(.)$ has three components: (i) a domestic component, denoted by $C_d$, (ii) an export component, denoted by $C_x$, and (iii) a strictly decreasing function $\Phi(R_{xd}^*)$. These are respectively given by:<fnref n="36" />

* Type 1: Direct separability ($\epsilon_H = 0$; $\epsilon_F = 1$):

$$ 
\left\{ 
\begin{aligned}
C_d^F &= \left[ \theta + \left( 1 - \frac{1}{\left[ (1-\rho(S))\epsilon_D(e_d^*) + (1+\eta)\rho(S) \right]} \right) - \frac{(1-\rho(S))(\epsilon_D(\bar{e}_d) - 1)}{\left[ (1-\rho(S))\epsilon_D(e_d^*) + (1+\eta)\rho(S) \right]} \right] \\
C_x^F &= \left[ \theta + \left( 1 - \frac{1}{\left[ (1-\rho(S))\epsilon_D(e_x^*) + (1+\eta)\rho(S) \right]} \right) - \frac{(1-\rho(S))(\epsilon_D(\bar{e}_x) - 1)}{\left[ (1-\rho(S))\epsilon_D(e_x^*) + (1+\eta)\rho(S) \right]} \right] \\
\Phi^F(R_{xd}^*) &= \frac{\left[ \frac{1}{\epsilon_{\phi}^{\pi^0}(e_d^*)} + R_{xd}^* \right]}{\left[ \frac{1}{\epsilon_{\phi}^{\pi^0}(e_d^*)} + 1 \right]} (R_{xd}^*)^{-1}, \quad \text{with} \quad R_{xd}^* = \frac{\epsilon_{\phi}^{\pi^0}(e_x^*)}{\epsilon_{\phi}^{\pi^0}(e_d^*)} = \frac{(1-\rho(S))(\epsilon_D(e_x^*) - 1) + \eta \rho(S)}{(1-\rho(S))(\epsilon_D(e_d^*) - 1) + \eta \rho(S)}
\end{aligned}
\right. 
\quad (49) 
$$

* Type 2: Indirect separability ($\epsilon_H = 1$; $\epsilon_F = 0$):

$$ 
\left\{ 
\begin{aligned}
C_d^H &= \left[ \theta + (1 - \rho(S))(\epsilon_D(e_d^*) - \epsilon_D(\bar{e}_d)) + \eta \rho(S) \right] \\
C_x^H &= \left[ \theta + (1 - \rho(S))(\epsilon_D(e_x^*) - \epsilon_D(\bar{e}_x)) + \eta \rho(S) \right] \\
\Phi^H(R_{xd}^*) &= (R_{xd}^*)^{-1}, \quad \text{with} \quad R_{xd}^* = \frac{\epsilon_{\phi}^{\pi^0}(e_x^*)}{\epsilon_{\phi}^{\pi^0}(e_d^*)} = \frac{(1-\rho(S))(\epsilon_D(e_x^*) - 1) + \eta \rho(S)}{(1-\rho(S))(\epsilon_D(e_d^*) - 1) + \eta \rho(S)}
\end{aligned}
\right. 
\quad (50) 
$$

where $\theta$ is the shape parameter of the Pareto, $\epsilon_D(e_d^*)$ is the price elasticity of demand faced by the cutoff domestic firm, and $\epsilon_D(\bar{e}_d)$ is the price elasticity of demand faced by the average productivity domestic firm. Similarly, $\epsilon_D(e_x^*)$ is the price elasticity of demand faced by the cutoff exporter, and $\epsilon_D(\bar{e}_x)$ is the price elasticity of demand faced by the average productivity exporter. $\eta$ is a strictly positive parameter which corresponds the elasticity of the markup rate with respect to the markup. The last parameter is the elasticity of markups with respect to firm efficiency $\rho(S) = \frac{\delta S}{1+\delta S} \le 0$, which sign is pinned down by the sign of the superelasticity of demand $S$. This latter captures the curvature of demand. That is, if $S = 0$, demand is CES; if $S > 0$, demand is sub-convex; and if $S < 0$, demand is super-convex, as previously described in Section 2.

The simple welfare formula described in Theorem 3 offers a parsimonious generalization of previous welfare formulas derived by (Arkolakis et al., 2012, 2019; Fally, 2019) in two respects. First, it encompasses three possible scenarios through which an economy's exposure to trade may change. Second, as compared with Arkolakis et al. (2019); Fally (2019), here the novel aspect of this formula is that it allows, not only the type of separability, but also the curvature of demand to play a role in determining the magnitude of the welfare gains from higher exposure to trade. Whether the

<footnote n="36">³⁶See section A.8 of the online appendix for details.</footnote>
homothetic CES is considered as a special case of direct or indirect separability, it always imposes a null superelasticity of demand ($S = 0$), a constant price elasticity of demand ($\forall e, \varepsilon_D(e) = \varepsilon_D$), and thus a constant markup ($\rho(S) = 0$). By inspection of the welfare formula in Theorem 3, ACR's result (Arkolakis et al., 2012) is replicated under the CES case: $CPD = \frac{(1-\chi_d)}{\theta} > 0$.<fnref n="37" />

When demand is sub-convex ($S > 0$), the price elasticity of demand decreases in firm productivity and thus the markup elasticity is positive ($\rho(S) > 0$). This has two immediate implications. The first is that in any market (domestic or export), the firm at the productivity cutoff faces a higher demand elasticity than the average productivity firm: $\varepsilon_D(e_d^*) > \varepsilon_D(\bar{e}_d)$ and $\varepsilon_D(e_x^*) > \varepsilon_D(\bar{e}_x)$. The second is that the firm at the domestic cutoff faces a higher demand elasticity than the firm at the export cutoff: $\varepsilon_D(e_d^*) > \varepsilon_D(e_x^*)$. Inspection of equations (49) and (50) reveals that under both types of separability (direct and indirect), sub-convex demand yields higher values of the domestic component $C_d$, the export component $C_x$, and the function $\Phi(R_{xd}^*)$, and thus a relatively higher value of the composite function $\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))$ as compared with the case of CES demand:

$$ \Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S)) [\text{sub-convex}] > \Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S)) [\text{CES}] = \Gamma(\theta) = \theta $$

The welfare formula in Theorem 3 indicates that the magnitude of the competitive pressure of higher exposure to trade on domestic firms (CPD) is inversely related to the composite function $\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))$. Hence, regardless of the type of separability, sub-convex demand yields a milder magnitude of the selection effect (CPD) and thus smaller gains from trade as compared with CES demand. This result is reversed when demand is super-convex ($S < 0$): now in any market, the cutoff productivity firm faces a lower price elasticity of demand than the average productivity firm: $\varepsilon_D(e_d^*) < \varepsilon_D(\bar{e}_d)$ and $\varepsilon_D(e_x^*) < \varepsilon_D(\bar{e}_x)$, and the cutoff domestic producer faces a lower demand elasticity than the cutoff exporter: $\varepsilon_D(e_d^*) < \varepsilon_D(e_x^*)$. Inspection of equations (49) and (50) reveals that under both types of separability, super-convex demand implies lower values of the domestic component $C_d$, the export component $C_x$, and the function $\Phi(R_{xd}^*)$, and thus a relatively lower value of the composite function $\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))$ as compared with the case of CES demand:

$$ \Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S)) [\text{super-convex}] < \Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S)) [\text{CES}] = \Gamma(\theta) = \theta $$

Therefore, regardless of the type of separability, super-convex demand yields a stronger magnitude of the selection effect (CPD) and thus larger gains from trade as compared with CES demand.

### 5.2.2 An industry-level explanation

Theorems 1 and 2 reveal that super-convex demand yields an upper bound for the toughness of firm

<footnote n="37">³⁷See section A.8 of the online appendix for details.</footnote>
selection into markets and a lower bound for the degree of firm partitioning by export status. The underlying economics are simple and can be explained as follows. Under super-convex demand, while more productive firms face more elastic demand and set lower markups, less productive ones face less elastic demand and set higher markups. Such a flexible pricing behavior simultaneously makes more productive firms even more price competitive (tanks to lower markups) and less productive ones even less price competitive (due to higher markups). This implies tough competitive conditions which make survival hard for less productive firms and lead to the self-selection of high productivity firms into serving the domestic market. A large subset of these highly productive firms find it thus profitable to export. In any trading country, domestic firms are thus exposed to a fierce foreign competition from a large number of exporters which are on average highly productive. Thus, super-convex demand magnifies the competitive pressure that higher exposure to trade exerts on domestic firms forcing the least productive among them to exit (CPD). Therefore, by magnifying the selection effect à la Melitz (2003), super-convex demand yields an upper bound for the welfare gains from higher exposure to trade (Theorem 3). These patterns are reversed under sub-convex demand. The CES imposes a rigid pricing behavior whereby firms set a common constant markup despite their heterogeneity in productivity, and thus yields an intermediate outcome.

## 5.3 Magnitude of the CPD under Generalized Demand: A Granular Analysis

In this section, the welfare analysis becomes more granular in two respects. First, I separately examine three different scenarios leading to a marginal increase in an economy's trade exposure: (i) a reduction in the variable trade cost ($-\tau$), (ii) a reduction in the fixed export cost ($-f_x$), or (iii) an increase in the number of trading partners ($+N$). Second, under each scenario, I provide a more granular explanation highlighting the key role that demand curvature plays in driving the result.

### 5.3.1 A small reduction in the variable trade cost ($-\tau$)

Here I examine the magnitude of welfare gains from higher exposure to trade occurring through a small reduction in the variable trade cost $\tau$. I proceed in two steps. I first show that while demand curvature plays a crucial role in determining the magnitude of the selection effect (CPD), the type of separability plays only a secondary role. Then, I provide a finer explanation at the firm-level.

**A simple formula.**— Using the simple measures from equation (47), I show that under generalized demand, the magnitude of the competitive pressure on domestic firms driven by a small reduction in the variable trade cost takes a simple form:

$$CPD(-\tau) = \frac{(1 - \chi_d) \theta}{\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))} > 0 \quad (51)$$

where $\theta$ is the shape parameter of the Pareto distribution, $\chi_d$ is the domestic trade share in any
country, and $\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))$ is a composite function defined by equation (48) and the explicit expressions of its components are provided by equations (49) and (50). Interpretation.- As previously shown in subsection 5.2.1, whether it is taken as knife-edge case of direct or indirect separability, the homothetic CES is a special case under which the price elasticity of demand is exogenous and identical across firms ($\forall e, \varepsilon_D(e) = \varepsilon_D$), and so firm-level markups are constant ($\rho(S) = 0$). This immediately implies that $\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))$ solely varies with the shape parameter of the Pareto and evaluates to $\Gamma(\theta) = \theta$. When demand is sub-convex ($S > 0$), more productive firms face less elastic demand and set higher markups ($\rho(S) > 0$). As explained in subsection 5.2.1, under both types of separability, sub-convex demand leads to a positive deviation of the composite function $\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))$ from its CES level:

$$ \Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S)) [\text{sub-convex}] > \Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S)) [\text{CES}] = \Gamma(\theta) = \theta $$

This immediately implies a smaller magnitude of the competitive pressure of higher exposure to trade on domestic firms (CPD) as compared with the case of CES demand. Hence, by dampening the magnitude of the selection effect, sub-convex demand yields a lower bound for the welfare gains from a small reduction in the variable trade cost. This result is reversed when demand is super-convex ($S < 0$): now more productive firms face more elastic demand and set lower markups ($\rho(S) < 0$). As shown in subsection 5.2.1, under both types of separability, super-convex demand leads to a negative deviation of the composite function $\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))$ from its CES level:

$$ \Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S)) [\text{super-convex}] < \Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S)) [\text{CES}] = \Gamma(\theta) = \theta $$

This immediately implies a larger magnitude of the competitive pressure of higher exposure to trade on domestic firms (CPD) as compared with the case of CES demand. Hence, by magnifying the magnitude of the selection effect, super-convex yields an upper bound for the welfare gains from a small reduction in the variable trade cost. A finer explanation at the firm-level.- I proceed in three steps. I first study the effect of a small reduction in the variable trade cost on the profile of operating profits of active exporters. Then, I show how this initial effect is transmitted to purely domestic firms through the import competition channel. Finally, I show how demand curvature plays a crucial role in shaping both effects, thus giving rise to a wide range of new predictions.

i) The impact of trade on active exporters.- The impact of a small reduction in the variable trade cost $\tau$ on operating profits of any active exporter (with productivity $\varphi \ge \varphi_x^*$) can be derived using the absolute value of the total elasticity of firm-level export operating profits in general equilibrium with respect to $\tau$. As shown in section A.9 of the online appendix, this elasticity can be simply
written as:

$$ \left(-\frac{d\log \pi^0(e_x)}{d\log \tau}\right) = \underbrace{\left[(1 - \rho(S))(\varepsilon_D(e_x) - 1) + \eta \rho(S)\right]}_{(I^+)} - \underbrace{\left[\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^0(e_x)\right]}_{(1)} \underbrace{\left[\frac{(1 - \chi_d)\theta}{(-\varepsilon_\Lambda^X)}\right]}_{(2)} \underbrace{\left.\left(\frac{d\log \pi^0(e_x)}{d\log \tau}\right)\right]_{(C^-)}}_{(C^-)} \quad (52) $$

The net outcome depends on a horse race between two opposite effects. The first is positive, given by $(I^+)$, and can be called *the initial positive effect*. That is, a reduction in the variable trade cost makes any exporter more price competitive, and thus increases its revenues and operating profits at the initial level of competition. The second is negative, given by $(C^-)$, and can be thought of as a *competition effect*. In fact, such a reduction in the variable trade cost makes every active exporter more price competitive and also induces an increase in the number of exporting firms. This, in turn, intensifies competition on the export market. As illustrated in Table 1, the net effect is always positive and heterogeneous across active exporters except under the case of CES demand. This special case imposes an identical net effect for all exporting firms regardless of their productivity level. Beyond the CES, whether this net effect is relatively stronger or milder for the least or the most productive exporters crucially depends on demand curvature.

<table><caption>Table 1: Heterogeneous vs Identical Impact of Trade on Active Exporters</caption><thead><tr><th>Exporters Category</th><th>Measure</th><th>Sub-convex</th><th>CES</th><th>Super-convex</th></tr></thead><tbody><tr><td>Least productive: $\varphi \to \varphi_x^*$</td><td>$-\frac{d \log \pi^0(e_x)}{d \log \tau}$</td><td>&gt;&gt; 0 (&gt; CES)</td><td>$\chi_d (\varepsilon_D - 1) &gt; 0$</td><td>&gt; 0 (&lt; CES)</td></tr><tr><td>Most productive: $\varphi \to +\infty$</td><td>$-\frac{d \log \pi^0(e_x)}{d \log \tau}$</td><td>&gt; 0 (&lt; CES)</td><td>$\chi_d (\varepsilon_D - 1) &gt; 0$</td><td>&gt;&gt; 0 (&gt; CES)</td></tr></tbody></table>

As shown in Table 1, upon a small reduction in the variable trade cost $\tau$, operating profits rise strongly for most productive exporters, whereas they rise mildly for least productive ones when demand is super-convex. This is illustrated in Figure 5, panel A, where the solid locus $\pi_x$ denotes the initial profile of export operating profits across firms, while the dashed locus $\pi'_x$ denotes the post-variable trade cost reduction profile when demand is super-convex. The outcome exhibits a strong *Matthew Effect*: while most productive exporters (which are initially the most profitable) experience a strong increase in their operating profits, least productive ones (which are initially the least profitable) experience a mild increase in their operating profits.<fnref n="38" /> As shown in Panel A of Figure 4, this result is reversed when demand is sub-convex. CES demand yields a net increase in operating profits that is identical for all active exporters and thus delivers an intermediate outcome.

<footnote n="38">³⁸In the words of Mrázová and Neary (2017), the *Matthew Effect* refers to the case where *to those who have more, more shall be given*.</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 5: Impact of Trade on Active Exporters and Domestic Firms under Super-convex Demand.</figcaption></figure>

<figure><img src="image_6.png" /></figure>
This result can be summarized with the following proposition:<fnref n="39" />

**Proposition 2.** Whether preferences are directly or indirectly separable, the impact of a small reduction in the variable trade cost on the profile of operating profits across active exporters crucially depends on demand curvature. When demand is super-convex, profits rise strongly for more productive exporters, whereas those of less productive ones rise mildly. By contrast, when demand is sub-convex, profits rise strongly for less productive exporters, whereas those of more productive ones rise mildly. CES demand implies that profits of all active exporters rise with the same magnitude and thus delivers an intermediate outcome.

The economic intuition behind this result can be explained as follows. When demand is super-convex, while more productive exporters face more elastic demand, less productive ones face less elastic demand. This means that consumers are most sensitive to price variations of varieties sold by most productive exporters, whereas they are least sensitive to price variations of those supplied by least productive ones. This implies that upon a small reduction in the variable trade cost, the initial positive effect ($I^+$) is so magnified for most productive exporters that it strongly dominates the competition effect ($C^-$). This, in turn, leads to a strong increase in their operating profits. By contrast, the initial positive effect ($I^+$) is so mild for least productive exporters that it only weakly dominates the competition effect ($C^-$). This, in turn, leads to a mild increase in their operating profits. These patterns are reversed under sub-convex demand. While demand curvature plays a crucial role in shaping these patterns, the type of separability solely determines two characteristics of the competition effect ($C^-$). The first consists in its structure: whether a more intense level of competition leads only to a decrease in the level of demand (quantity effect), or also to a reduction in firm-level markups (negative price effect). The second consists in its nature: whether the competition effect is identical across exporters or firm-specific.<fnref n="40" />

ii) *The impact of trade on purely domestic firms.*– As illustrated in panel B of figures 4 and 5, a small reduction in the variable trade cost always leads to an increase in the domestic productivity cutoff. This captures the fact that higher exposure to trade always exerts a competitive pressure on domestic firms which forces the least productive among them to exit the market (CPD). While this selection effect is due to Melitz (2003), the key novelty here is that demand curvature governs its magnitude. As highlighted in Proposition 2, upon a reduction in the variable trade cost, additional market shares are mainly reaped by the most productive exporters under super-convex demand. Since market shares of this category of exporters are initially large, their increase strongly intensifies competition for purely domestic firms. This, in turn, forces a large fringe of least productive domestic firms to exit the market, thus yielding an upper bound for the selection effect. These pat-

<footnote n="39">³⁹See section A.9 of the online appendix for more details.</footnote>

<footnote n="40">⁴⁰As shown in section A.9 of the online appendix, regardless of the nature and the structure of the competition effect ($C^-$), the firm-specific aspect of the initial positive effect ($I^+$) always dominates this of the competition effect. Therefore, while demand curvature plays a crucial role in driving this result, the type of separability plays only a secondary role.</footnote>
terns are reversed when demand is sub-convex. CES demand is a special case where the increase in export market shares is identical across active exporters and thus yields an intermediate magnitude of the selection effect.

### 5.3.2 A small reduction in the fixed trade cost ($-f_x$)

Now I examine the magnitude of welfare gains from higher exposure to trade occurring through a small reduction in the fixed export cost $f_x$. Using the simple measures from equation (47), I show that under generalized demand, the magnitude of the competitive pressure on domestic firms driven by a small reduction in the fixed trade cost takes a simple form:

$$CPD(-f_x) = \frac{(1 - \chi_d)}{\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))} \frac{[\theta - (1 - \rho(S))(\varepsilon_D(\bar{e}_x) - 1)]}{[(1 - \rho(S))(\varepsilon_D(e_x^*) - 1) + \eta \rho(S)]} > 0 \quad (53)$$

As it was the case for the previous scenario, the above formula clearly shows that the magnitude of the selection effect under this scenario, $CPD(-f_x)$, also crucially depends on demand curvature. A small reduction in the trade fixed cost triggers entry of new exporters with productivity $\varphi \in ]\varphi_d^*, \varphi_x^*[$. Whether these new entrants into the export market capture small or large market shares crucially depends on demand curvature. Under super-convex demand, the toughness of firm selection into markets reaches its upper bound. Productivity levels of these new exporters are thus relatively higher as compared with the CES case, which ensures that they capture relatively larger market shares. This, in turn, implies that the magnitude of the competitive pressure on domestic firms forcing them to exit is stronger under super-convex demand than under CES demand. Hence, by magnifying the selection effect, super-convex demand yields an upper bound for the welfare gains from a small reduction in the fixed trade cost. These patterns are reversed under sub-convex demand. Within these bounds, CES demand delivers an intermediate outcome.

### 5.3.3 A small increase in the number of trading countries ($+N$)

Now I examine the magnitude of welfare gains from higher exposure to trade occurring through a small increase in the number of trading countries $N$. Using the simple measures from equation (47), I show that under generalized demand, the magnitude of the competitive pressure on domestic firms driven by a small increase in the number of trading partners takes a simple form:

$$CPD(+N) = \frac{(1 - \chi_d)}{\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))} > 0 \quad (54)$$

As it was the case for the two previous scenarios, the above formula clearly shows that the magnitude of the selection effect under this scenario, $CPD(+N)$, also crucially depends on demand curvature. A small increase in the number of trading partners implies that domestic firms in any
economy face an additional competition from exporters in these newly trading countries. Whether the magnitude of this foreign competition effect is strong or mild crucially depends on demand curvature. Super-convex demand yields an upper bound for the toughness of firm selection into markets and a lower bound for the degree of firm partitioning by export status. Domestic firms in any economy will thus face a fierce competition from a relatively larger mass of foreign exporters which are relatively more productive as compared with the CES case. This, in turn, implies that the magnitude of the competitive pressure on domestic firms forcing them to exit is stronger under super-convex demand than under CES demand. Hence, by magnifying the selection effect, super-convex demand yields an upper bound for the welfare gains from a small increase in the number of trading partners. These patterns are reversed under sub-convex demand. Within these bounds, CES demand delivers an intermediate outcome.

## 6 Conclusion

This paper has extended the canonical Melitz-Chaney framework beyond the CES to investigate how generalized demand shapes the selection effect and welfare gains from trade. By adopting a Gorman-Pollak demand system, the model introduces a flexible structure that accommodates both direct and indirect separability while keeping demand curvature unrestricted. This generalization maintains analytical tractability through the Exponent Elasticity Method (EEM), which provides explicit solutions for key variables in general equilibrium and enables a closed-form characterization of welfare effects. Three central insights emerge. First, the selection effect à la Melitz (2003) is always operative: whether demand is CES or not, trade exposure always reallocates market shares toward more productive exporters, thus forcing least productive domestic firms to exit the market. Notably, despite the presence of variable markups and variety effects in the current environment, the selection effect remains the dominant welfare channel in general equilibrium.

Second, demand curvature governs the strength of this mechanism. Super-convex demand amplifies the toughness of firm selection and magnifies welfare gains by intensifying the competitive pressure that leads to firm exit, whereas sub-convex demand dampens these effects. Within these bounds, the CES delivers an intermediate outcome. Third, the generalized ACR formula derived here provides a parsimonious measure of the magnitude of the selection effect through the elasticity of the domestic productivity cutoff with respect to the trade exposure index. It replicates the ACR result under the CES while uncovering new welfare implications once demand curvature departs from the CES case. Beyond its theoretical contribution, this framework clarifies the fundamental role of preferences in shaping export participation and aggregate welfare. It shows that the welfare consequences of globalization—or its retreat—depend critically on how consumers adjust their sensitivity to price differences across varieties. More broadly, by restoring the influence of demand
curvature in a setting with firm heterogeneity and trade costs, this paper reconnects international trade theory with realistic patterns of price sensitivity. Future research could apply this generalized framework to empirical exercises linking observed demand curvature to measurable welfare outcomes.

## References

Anderson, J. E. and Van Wincoop, E. (2003). Gravity with gravitas: a solution to the border puzzle. *American economic review*, 93(1):170–192.

Arkolakis, C., Costinot, A., Donaldson, D., and Rodríguez-Clare, A. (2019). The elusive pro-competitive effects of trade. *The Review of Economic Studies*, 86(1):46–80.

Arkolakis, C., Costinot, A., and Rodriguez-Clare, A. (2012). New trade models, same old gains? *American Economic Review*, 102(1):94–130.

Baqaee, D. and Malmberg, H. (2025). Long-run effects of trade wars. Technical report, National Bureau of Economic Research.

Bertoletti, P. and Epifani, P. (2014). Monopolistic competition: Ces redux? *Journal of International Economics*, 93(2):227–238.

Bertoletti, P., Etro, F., and Simonovska, I. (2018). International trade with indirect additivity. *American Economic Journal: Microeconomics*, 10(2):1–57.

Caliendo, L. and Feenstra, R. C. (2024). Foundation of the small open economy model with product differentiation. *Journal of International Economics*, 150:103820.

Chaney, T. (2008). Distorted gravity: the intensive and extensive margins of international trade. *American Economic Review*, 98(4):1707–21.

Cuñat, A. and Zymek, R. (2023). Trade barriers and imbalances in small open economies.

Demidova, S., Kucheryavyy, K., Naito, T., and Rodríguez-Clare, A. (2024). The small open economy in a generalized gravity model. *Journal of International Economics*, 152:103997.

Fally, T. (2019). Generalized separability and the gains from trade. *Economics Letters*, 178:8–12.

Fally, T. (2022). Generalized separability and integrability: Consumer demand with a price aggregator. *Journal of Economic Theory*, 203:105471.

Feenstra, R. C. (2010). Measuring the gains from trade under monopolistic competition. *Canadian Journal of Economics/Revue canadienne d'économique*, 43(1):1–28.
Feenstra, R. C. (2018). Restoring the product variety and pro-competitive gains from trade with heterogeneous firms and bounded productivity. *Journal of International Economics*, 110:16–27.

Feenstra, R. C. and Hong, C. (2024). Estimating the regional welfare impact of tariff changes: Application to the united states. Technical report, National Bureau of Economic Research.

Gorman, W. M. (1995). *Collected Works of WM Gorman: Separability and Aggregation*, volume 1. Oxford University Press.

Hurwicz, L. and Uzawa, H. (1971). On the problem of integrability of demand functions. in. chipman, js, et al. (eds.), preferences, utility and demand.

Ignatenko, A., Lashkaripour, A., Macedoni, L., and Simonovska, I. (2025). Making america great again? the economic impacts of liberation day tariffs. Technical report, National Bureau of Economic Research.

Kimball, M. (1995). The quantitative analytics of the basic neomonetarist model. *Journal of Money, Credit and Banking*, 27(4):1241–77.

Krugman, P. (1980). Scale economies, product differentiation, and the pattern of trade. *The American Economic Review*, 70(5):950–959.

Loecker, J. D. and Warzynski, F. (2012). Markups and firm-level export status. *American economic review*, 102(6):2437–2471.

Matsuyama, K. and Ushchev, P. (2017). Beyond ces: Three alternative cases of flexible homothetic demand systems. *Global Poverty Research Lab Working Paper*, (17-109).

Melitz, M. J. (2003). The impact of trade on intra-industry reallocations and aggregate industry productivity. *econometrica*, 71(6):1695–1725.

Melitz, M. J. and Ottaviano, G. I. (2008). Market size, trade, and productivity. *The review of economic studies*, 75(1):295–316.

Mertens, M. and Mottironi, B. (2023). Do larger firms exert more market power? markups and markdowns along the size distribution. Technical report, IWH-CompNet Discussion Papers.

Mrázová, M. and Neary, J. P. (2017). Not so demanding: Demand structure and firm behavior. *American Economic Review*, 107(12):3835–74.

Mrázová, M. and Neary, J. P. (2019). Selection effects with heterogeneous firms. *Journal of the European Economic Association*, 17(4):1294–1334.
Pollak, R. A. (1972). *Generalized separability*. *Econometrica: Journal of the Econometric Society*, pages 431–453.

Zhelobodko, E., Kokovin, S., Parenti, M., and Thisse, J.-F. (2012). *Monopolistic competition: Beyond the constant elasticity of substitution*. *Econometrica*, 80(6):2765–2784.
# Online Appendix for "Trade Costs, Generalized Demand, and Firm Selection"

Badis Tabarki*

Centre d'Économie de La Sorbonne

(UMR 8174 CNRS-University Paris 1)

October 24, 2025

## A.1 Integrability and Rationalization

### A.1.1 Regularity restrictions

Regularity restrictions on functions $D_i$, F and H are given by:

i) $D_i$ is continuously differentiable and sufficiently elastic, $\epsilon_{D_i} > 1$.

ii) H and F are continuously differentiable and $[\epsilon_F(-\epsilon_{D_i}) - \epsilon_H] < 0$ for all $i$, $\Lambda(p, y)$ and $p_i$.

iii) For any variety $i$ and $y>0$, there exists $\Lambda(p, y) \in \mathbb{R}_+$ such that $\int_0^{\bar{M}} \frac{p_i}{y} \frac{D_i(F(\Lambda(p, y)) \frac{p_i}{y})}{H(\Lambda(p, y))} di = 1$.

where $\epsilon_{D_i} = - \frac{\partial \log D_i}{\partial \log z_i}$ is the absolute value of the elasticity of $D_i$ in its argument $z_i = F(\Lambda(p, y)) \frac{p_i}{y}$, and $\epsilon_F = \frac{\partial \log F}{\partial \log \Lambda(p, y)}$ and $\epsilon_H = \frac{\partial \log H}{\partial \log \Lambda(p, y)}$ the elasticity of F and H in $\Lambda(p, y)$. Assumptions i) and ii) ensure that demand for any variety $q_i(p_i, p, y)$ is strictly decreasing in the price aggregator $\Lambda(p, y)$. As for assumption iii), it simply guarantees that the price aggregator $\Lambda(p, y)$ can always adjust in order to satisfy the budget constraint.

### A.1.2 Proof of Proposition 1

To start with, let us recall that, as established in Proposition 1, the single aggregator $\Lambda(.)$ is referred to as an implicit function of the schedule of quantities $q = \{q_i\}$, $i \in [0, \bar{M}]$, to define the direct utility function:

$$U(x) = \int_{0}^{\bar{M}} u_{i}(x) di; u_{i}(x) = \int_{x_{0i}}^{x=H(\Lambda(q))} D_{i}^{-1}(x) dx$$

<footnote>*I am very grateful to Lionel Fontagné, Gianmarco Ottaviano, Mathieu Parenti, Ariell Reshef, Monika Mrazova and Thomas Chaney for insightful suggestions and helpful comments on the previous version of this paper.</footnote>
Then, upon moving from the direct utility function to its corresponding indirect utility function, $\Lambda(.)$ is defined as an implicit function of income $y$ and the schedule of prices $\mathbf{p} \equiv \{p_i\}, i \in [0, \bar{M}]$:

$$U(\mathbf{x}) \equiv V(\mathbf{p}, y) = \int_{0}^{\bar{M}} v_i(z) di + F(\Lambda(\mathbf{p}, y))H(\Lambda(\mathbf{p}, y)) - k_0; v_i(z) = - \int_{z_{0i}}^{z=F(\Lambda(\mathbf{p}, y))} \frac{p_i}{y} D_i(z) dz$$

Now I proceed in four steps. First, I show that the Slutsky substitution matrix is symmetric and negative semi-definite. Second, I apply the Hurwicz and Uzawa (1971) theorem and obtain a well-behaved direct utility function $U(\mathbf{x})$. Third, I show that maximizing $U(\mathbf{x})$ leads to the demand system described in equations (1) and (2) in the main text. Finally, using a simple geometric equality, I obtain the corresponding indirect utility function $V(\mathbf{p}, y)$ and I show that it leads to the demand system described in the text by applying the Roy identity.

## Step 1. Symmetry and Negative Semi-definiteness of the Slutsky Substitution Matrix

Let us consider our generalized Gorman-Pollak demand system:

$$q_i(p_i, \mathbf{p}, y) = \frac{D_i(F(\Lambda(\mathbf{p}, y)) \frac{p_i}{y})}{H(\Lambda(\mathbf{p}, y))} \quad (1)$$

where $\Lambda(\mathbf{p}, y)$ is implicitly defined as a unique solution to the budget constraint which can be rewritten as:

$$\int_{0}^{\bar{M}} \frac{p_i}{y} \frac{D_i(F(\Lambda(\mathbf{p}, y)) \frac{p_i}{y})}{H(\Lambda(\mathbf{p}, y))} di = 1 \quad (2)$$

For any variety pair $(i, j)$, $i \neq j$, the Slutsky substitution coefficient is given by:

$$\Delta_{ij} = \frac{\partial q_i}{\partial p_j} + q_j \frac{\partial q_i}{\partial y}$$

By recalling that $\varepsilon_{D_i} = -\frac{\partial \log D_i}{\partial \log z} > 1$ is the absolute value of the elasticity of the demand function $D_i(z)$ in its argument $z = F(\Lambda(\mathbf{p}, y)) \frac{p_i}{y}$, and rearranging, the Slutsky substitution term boils down to:

$$\Delta_{ij} = \underbrace{\frac{\partial q_i}{\partial \Lambda}}_{(1)} [\underbrace{\frac{\partial \Lambda}{\partial p_j}}_{(2)} + \underbrace{\frac{\partial \Lambda}{\partial y}}_{(3)}] + (\underbrace{\frac{q_i q_j}{y}}_{(3)}) \varepsilon_{D_i} \quad (3)$$

where the above three terms (1), (2), and (3) are respectively given by:

$$\left\{
\begin{array}{l}
\displaystyle \frac{\partial q_i}{\partial \Lambda} = -(\varepsilon_F \varepsilon_{D_i} + \varepsilon_H) \left(\frac{q_i}{\Lambda}\right) \\[2ex]
\displaystyle \frac{\partial \Lambda}{\partial p_j} = [-\chi_j \frac{(\varepsilon_{D_j}-1)}{(\varepsilon_F \varepsilon_{D} + \varepsilon_H)}] \left(\frac{\Lambda}{p_j}\right) \\[2ex]
\displaystyle \frac{\partial \Lambda}{\partial y} = \frac{(\varepsilon_{D}-1)}{(\varepsilon_F \varepsilon_{D} + \varepsilon_H)} \left(\frac{\Lambda}{y}\right)
\end{array}
\right.
\quad (4)$$
where the final expressions of the second and the third term are obtained using the fact that equation (2) has zero derivatives, respectively, in $p_j$ and $y$. $\chi_j = \frac{p_j q_j}{y}$ corresponds to the expenditure share of variety $j$, and $\bar{\epsilon}_D = \int_0^{\bar{M}} \chi_i \epsilon_{D_i} di$ is the weighted average price elasticity of demand at the industry level. Now by plugging the final expressions of the three terms from equation (4) in equation (3) and rearranging, the Slutsky substitution coefficient simplifies to:

$$ \Delta_{ij} = \left(\frac{q_i q_j}{y}\right) \frac{\epsilon_F (\epsilon_{D_i} - \bar{\epsilon}_D) (\epsilon_{D_j} - \bar{\epsilon}_D)}{(\epsilon_F \bar{\epsilon}_D + \epsilon_H)} + \left(\frac{q_i q_j}{y}\right) (\epsilon_{D_i} + \epsilon_{D_j} - \bar{\epsilon}_D) \quad (5) $$

The above expression clearly shows that $\Delta_{ij}$ is symmetric in $i$ and $j$. As in Fally (2022), it is more convenient to rewrite the Slutsky substitution term using $\vartheta_i = \epsilon_F \epsilon_{D_i} + \epsilon_H \ge 1$ as follows:

$$ \Delta_{ij} = \left(\frac{q_i q_j}{y}\right) \frac{(\vartheta_i - \bar{\vartheta})(\vartheta_j - \bar{\vartheta})}{(\epsilon_F \bar{\vartheta})} + \left(\frac{q_i q_j}{y}\right) \frac{(\vartheta_i + \vartheta_j - \bar{\vartheta} - \epsilon_H)}{\epsilon_F} $$

After simplification, $\Delta_{ij}$ boils down to:

$$ \Delta_{ij} = \left(\frac{q_i q_j}{y}\right) \frac{(\vartheta_i \vartheta_j)}{(\epsilon_F \bar{\vartheta})} - \left(\frac{q_i q_j}{y}\right) \frac{(\epsilon_H)}{\epsilon_F} \quad (6) $$

where $\bar{\vartheta} = \int_0^{\bar{M}} \chi_i \vartheta_i di$ is the weighted average across varieties. Now for any variety $i$, the diagonal coefficient of the Slutsky matrix $\Delta_{ii}$ is given by:

$$ \Delta_{ii} = \frac{\partial q_i}{\partial p_i} + q_i \frac{\partial q_i}{\partial y} $$

By recalling that $\epsilon_{D_i} = -\frac{\partial \log D_i}{\partial \log p_i} > 1$ and rearranging, the diagonal term boils down to:

$$ \Delta_{ii} = \left(\frac{q_i}{p_i}\right) (-\epsilon_{D_i}) + \left(\frac{q_i^2}{y}\right) \epsilon_{D_i} + \underbrace{\frac{\partial q_i}{\partial \Lambda}}_{(1)} + \underbrace{\left[\frac{\partial \Lambda}{\partial p_i} + q_i \frac{\partial \Lambda}{\partial y}\right]}_{(2)(3)} \quad (7) $$

where the above three terms (1), (2), and (3) are respectively given by:

$$ \left\{ \begin{array}{l} \frac{\partial q_i}{\partial \Lambda} = -(\epsilon_F \epsilon_{D_i} + \epsilon_H) \left(\frac{q_i}{\Lambda}\right) \\ \\ \frac{\partial \Lambda}{\partial p_i} = \left[-\chi_i \frac{(\epsilon_{D_i} - 1)}{(\epsilon_F \bar{\epsilon}_D + \epsilon_H)}\right] \left(\frac{\Lambda}{p_i}\right) \\ \\ \frac{\partial \Lambda}{\partial y} = \frac{(\bar{\epsilon}_D - 1)}{(\epsilon_F \bar{\epsilon}_D + \epsilon_H)} \left(\frac{\Lambda}{y}\right) \end{array} \right. \quad (8) $$

By plugging the final expressions of the three terms from equation (8) in equation (7), and using $\vartheta_i = \epsilon_F \epsilon_{D_i} + \epsilon_H \ge 1$ and rearranging, the diagonal coefficient of the Slutsky substitution matrix boils
down to:

$$ \Delta_{ii} = -\frac{q_i (\theta_i - \varepsilon_H)}{p_i \varepsilon_F} + \left(\frac{q_i^2}{y}\right) \frac{(\theta_i^2)}{(\varepsilon_F \theta)} - \left(\frac{q_i^2}{y}\right) \frac{(\varepsilon_H)}{(\varepsilon_F)} \quad (9) $$

Thus far, I have obtained the following expressions of the substitution coefficient and the diagonal term of the Slutsky matrix:

$$ \left\{ \begin{aligned} \Delta_{ij} &= \left(\frac{q_i q_j}{y}\right) \frac{(\theta_i \theta_j)}{(\varepsilon_F \theta)} - \left(\frac{q_i q_j}{y}\right) \frac{(\varepsilon_H)}{(\varepsilon_F)} \\ \Delta_{ii} &= -\frac{q_i (\theta_i - \varepsilon_H)}{p_i \varepsilon_F} + \left(\frac{q_i^2}{y}\right) \frac{(\theta_i^2)}{(\varepsilon_F \theta)} - \left(\frac{q_i^2}{y}\right) \frac{(\varepsilon_H)}{(\varepsilon_F)} \end{aligned} \right. \quad (10) $$

Now in order to show that the Slutsky matrix is negative semi-definite, we need to show that for any vector $\mathbf{x}$ we have:

$$ \sum_{i,j} \Delta_{ij} \mathbf{x}_i \mathbf{x}_j \le 0 \quad (11) $$

This is equivalent to showing that $\sum_{i,j} \left(\frac{p_i p_j}{y}\right) \Delta_{ij} \tilde{\mathbf{x}}_i \tilde{\mathbf{x}}_j \le 0$, where for any variety $i$, $\tilde{\mathbf{x}}_i$ corresponds to $\mathbf{x}_i$ normalized by $(\frac{p_i}{\sqrt{y}})$: $\tilde{\mathbf{x}}_i = \frac{\mathbf{x}_i}{(p_i/\sqrt{y})}$, and $\tilde{\mathbf{x}}_j = \frac{\mathbf{x}_j}{(p_j/\sqrt{y})}$. Using equation (10), this is then equivalent to showing the following inequality:

$$ -\sum_i \chi_i \left( \frac{\theta_i - \varepsilon_H}{\varepsilon_F} \right) \tilde{\mathbf{x}}_i^2 + \left( \frac{1}{\varepsilon_F} \right) \sum_{i,j} \frac{\chi_i \theta_i \tilde{\mathbf{x}}_i \chi_j \theta_j \tilde{\mathbf{x}}_j}{\sum_i \chi_i \theta_i} - \left( \frac{\varepsilon_H}{\varepsilon_F} \right) \sum_{i,j} \chi_i \tilde{\mathbf{x}}_i \chi_j \tilde{\mathbf{x}}_j \le 0 \quad (12) $$

By successive rearrangement, the above inequality simplifies to:

$$ \underbrace{\left(-\frac{1}{\varepsilon_F}\right) \left[ \frac{\sum_i \chi_i \theta_i \tilde{\mathbf{x}}_i^2}{\sum_i \chi_i \theta_i} - \left(\frac{\sum_i \chi_i \theta_i \tilde{\mathbf{x}}_i}{\sum_i \chi_i \theta_i}\right)^2 \right]}_{(1)} + \underbrace{\left(\frac{\varepsilon_H}{\varepsilon_F \sum_i \chi_i \theta_i}\right) \left[ \sum_i \chi_i \tilde{\mathbf{x}}_i^2 - \left(\sum_i \chi_i \tilde{\mathbf{x}}_i\right)^2 \right]}_{(2)} \le 0 \quad (13) $$

The terms in brackets can be interpreted as variances and are positive. As indicated in section 2 of the main text, our Generalized Gorman-Pollak demand system nests two types of separability:

* Type 1: Direct separability if $\varepsilon_F = 1$ and $\varepsilon_H = 0$. Inspection of equation (13) reveals that under this case, the second component (2) of the inequality collapses to zero and the first component (1) is strictly negative. This ensures that the Slutsky substitution matrix is semi-definite negative under this case.

* Type 2: Indirect separability if $\varepsilon_F = 0$ and $\varepsilon_H = 1$. Under this case, it follows by inspection of equations (4) and (8) that:

$$ \left\{ \begin{aligned} \frac{\partial q_i}{\partial \Lambda} &= -(\frac{q_i}{\Lambda}); \quad \frac{\partial \Lambda}{\partial p_i} = -\chi_i (\varepsilon_{D_i} - 1) (\frac{\Lambda}{p_i}) \\ \frac{\partial \Lambda}{\partial p_j} &= -\chi_j (\varepsilon_{D_j} - 1) (\frac{\Lambda}{p_j}); \quad \frac{\partial \Lambda}{\partial y} = (\varepsilon_D - 1) (\frac{\Lambda}{y}) \end{aligned} \right. \quad (14) $$
Using the above expressions, the Slutsky substitution term in equation (3) boils down to:

$$ \Delta_{ij} = -\left(\frac{q_i q_j}{y}\right) \left[\bar{\varepsilon}_D - (\varepsilon_{D_i} + \varepsilon_{D_j})\right] \quad (15) $$

Similarly, the diagonal term of the Slutsky matrix in equation (7) simplifies to:

$$ \Delta_{ii} = \left(\frac{q_i}{p_i}\right) (-\varepsilon_{D_i}) - \left(\frac{q_i^2}{y}\right) (\bar{\varepsilon}_D - 2\varepsilon_{D_i}) \quad (16) $$

As before, in order to prove that the Slutsky substitution matrix is negative semi-definite, we need:

$$ -\sum_i \chi_i \tilde{\chi}_i^2 \varepsilon_{D_i} - \sum_{i,j} \chi_i \tilde{\chi}_i \chi_j \tilde{\chi}_j [\bar{\varepsilon}_D - (\varepsilon_{D_i} + \varepsilon_{D_j})] \le 0 \quad (17) $$

By successive rearrangement, the above inequality simplifies to:

$$ \frac{2 \left(\sum_i \chi_i \tilde{\chi}_i\right) \left(\sum_i \chi_i \tilde{\chi}_i \varepsilon_{D_i}\right)}{\left(\sum_i \chi_i \varepsilon_{D_i}\right)} \le \frac{\sum_i \chi_i \tilde{\chi}_i^2 \varepsilon_{D_i} + \left(\sum_i \chi_i \tilde{\chi}_i\right)^2 \left(\sum_i \chi_i \varepsilon_{D_i}\right)}{\left(\sum_i \chi_i \varepsilon_{D_i}\right)} \quad (18) $$

With further simplifications, the above inequality boils down to:

$$ \left\{ \begin{array}{l} \sum_i \chi_i \varepsilon_{D_i} [\tilde{\chi}_i^2 + (\sum_i \chi_i \tilde{\chi}_i)^2] \ge 2 (\sum_i \chi_i \tilde{\chi}_i) (\sum_i \chi_i \tilde{\chi}_i \varepsilon_{D_i}) \\ \Rightarrow \sum_i \chi_i \varepsilon_{D_i} [\tilde{\chi}_i - (\sum_i \chi_i \tilde{\chi}_i)]^2 + 2 (\sum_i \chi_i \tilde{\chi}_i) (\sum_i \chi_i \tilde{\chi}_i \varepsilon_{D_i}) \ge 2 (\sum_i \chi_i \tilde{\chi}_i) (\sum_i \chi_i \tilde{\chi}_i \varepsilon_{D_i}) \\ \Rightarrow \sum_i \chi_i \varepsilon_{D_i} [\tilde{\chi}_i - (\sum_i \chi_i \tilde{\chi}_i)]^2 \ge 0 \end{array} \right. $$

Which is satisfied since $\varepsilon_{D_i} > 1$ for all varieties.

## Step 2. A Well-behaved Direct Utility Function

After verifying that the Slutsky substitution matrix is symmetric and negative semi-definite in Step 1, I can now apply the Hurwicz and Uzawa (1971) theorem and show that our generalized Gorman-Pollak demand system described in equations (1) and (2) in the main text can be rationalized and obtained from the following continuous quasi-concave utility function:

$$ U(\boldsymbol{x}) = \int_{0}^{\bar{M}} u_{i}(\boldsymbol{x}) d i ; \quad u_{i}(\boldsymbol{x}) = \int_{x_{0 i}}^{x=H(\Lambda(\boldsymbol{q}))} q_{i} D_{i}^{-1}(\boldsymbol{x}) d x \quad (19) $$

where $\Lambda(\boldsymbol{q})$ is now defined as an implicit function of the schedule of quantities $\boldsymbol{q} \equiv \{q_i\}, i \in [0, \bar{M}]$ such that it satisfies the equivalent of the budget constraint (equation 2) that is written using the inverse demand, and $x_{0i} \ge 0$ is a constant.
Step 3. Marshallian Demand and Price Aggregator

Maximizing U(x) under the budget constraint leads to:

$$
\frac{\partial U}{\partial q_i} = \frac{\partial u_i(x)}{\partial q_i} = H(\Lambda(\boldsymbol{q})) D_i^{-1} (H(\Lambda(\boldsymbol{q})) q_i) = \nu p_i \quad (20)
$$

where $v$ hereafter denotes the Lagrange multiplier associated with the budget constraint. Now
integrating over all varieties gives rise to the following expression of the Lagrange multiplier:

$$
\nu = \frac{H(\Lambda(\boldsymbol{q}))}{y} \int_{0}^{\bar{M}} q_i D_i^{-1}(H(\Lambda(\boldsymbol{q})) q_i) di
$$

The fact that Λ(q̂) must satisfy the budget constraint implies that F(Λ(q̂)) = ∫_o^M qᵢ Dᵢ⁻¹(H(Λ(q̂)) qᵢ) di.
Hence, the Lagrange multiplier ν boils down to:

$$
\nu = \frac{H(\Lambda(\boldsymbol{q})) F(\Lambda(\boldsymbol{q}))}{y} \qquad (21)
$$

Now by plugging the above expression of v in the FOC of Utility maximization in equation (20), I
obtain the inverse demand function:

$$
p_i(q_i, y, \Lambda(\boldsymbol{q})) = \frac{D_i^{-1}(H(\Lambda(\boldsymbol{q}))) q_i}{(F(\Lambda(\boldsymbol{q}))/y)} \quad (22)
$$

This latter immediately gives rise to the demand system described in equations (1) and (2) in the
main text:

$$
p_i(q_i, y, \Lambda(\boldsymbol{q})) = \frac{D_i^{-1}(H(\Lambda(\boldsymbol{q}))) q_i}{(F(\Lambda(\boldsymbol{q}))/y)} \Rightarrow q_i(p_i, \boldsymbol{p}, y) = \frac{D_i(F(\Lambda(\boldsymbol{p}, y)))^{\frac{p_i}{y}}}{H(\Lambda(\boldsymbol{p}, y))}
$$

where Λ(*p*, *y*) is implicitly defined as a unique solution to the budget constraint:

$$
\int_{0}^{\bar{M}} p_i \frac{D_i(F(\Lambda(\boldsymbol{p}, y)))^{\frac{p_i}{y}}}{H(\Lambda(\boldsymbol{p}, y))} di = y
$$

**Step 4. From Direct to Indirect Utility**

Following Fally (2022), to obtain the corresponding indirect utility function in a simple way, we can
use the following geometric equality that applies to any strictly monotonic mapping B:

$$
\int_{x_0}^{x_1} B^{-1}(x) dx + B^{-1}(x_0) x_0 = - \int_{z_0}^{z_1} B(z) dz + B(z_1) z_1
$$
with $x_0 = B(z_0)$ and $x_1 = B(z_1)$. Applying this formula to $B = D_i$, $x_1 = H(\Lambda(\boldsymbol{q})) q_i$ and $z_1 = F(\Lambda(\boldsymbol{p}, y)) \frac{p_i}{y}$, we get:

$$ \int_{x_{0i}}^{H(\Lambda) q_i} D_i^{-1}(x) dx = - \int_{z_{0i}}^{F(\Lambda) \frac{p_i}{y}} D_i(z) dz + D_i(F(\Lambda) \frac{p_i}{y}) (F(\Lambda) \frac{p_i}{y}) - z_{0i} x_{0i} $$

Integrating over all varieties, we obtain:

$$ \int_{0}^{\bar{M}} \left( \int_{x_{0i}}^{H(\Lambda) q_i} D_i^{-1}(x) dx \right) di = - \int_{0}^{\bar{M}} \left( \int_{z_{0i}}^{F(\Lambda) \frac{p_i}{y}} D_i(z) dz \right) di + F(\Lambda) \int_{0}^{\bar{M}} \frac{p_i}{y} D_i(F(\Lambda) \frac{p_i}{y}) di - \int_{0}^{\bar{M}} z_{0i} x_{0i} di $$

Since the budget constraint (equation 2 in the main text) implies that $H(\Lambda) = \int_{0}^{\bar{M}} \frac{p_i}{y} D_i(F(\Lambda) \frac{p_i}{y}) di$, the above equality boils down to:

$$ \int_{0}^{\bar{M}} \left( \int_{x_{0i}}^{H(\Lambda) q_i} D_i^{-1}(x) dx \right) di = - \int_{0}^{\bar{M}} \left( \int_{z_{0i}}^{F(\Lambda) \frac{p_i}{y}} D_i(z) dz \right) di + F(\Lambda) H(\Lambda) - k_0 $$

where $k_0 = \int_{0}^{\bar{M}} z_{0i} x_{0i} di$. Now let $u_i(x) = \int_{x_{0i}}^{x=H(\Lambda(\boldsymbol{q}))} D_i^{-1}(x) dx$, and $v_i(z) = - \int_{z_{0i}}^{z=F(\Lambda(\boldsymbol{p}, y))} \frac{p_i}{y} D_i(z) dz$, respectively denote the direct, and indirect sub-utility of a variety $i$. Using $u_i(x)$ and $v_i(z)$, the above geometric equality further simplifies to:

$$ \int_{0}^{\bar{M}} u_i(x) di = \int_{0}^{\bar{M}} v_i(z) di + F(\Lambda(\boldsymbol{p}, y)) H(\Lambda(\boldsymbol{p}, y)) - k_0 $$

Hence, we obtain the following indirect utility function as a function of normalized prices:

$$ U(\boldsymbol{x}) \equiv V(\boldsymbol{p}, y) = \int_{0}^{\bar{M}} v_i(z) di + F(\Lambda(\boldsymbol{p}, y)) H(\Lambda(\boldsymbol{p}, y)) - k_0 \quad (23) $$

Applying the Roy identity and simplifying, individual demand for a variety $i$ evaluates to:

$$ q_i = \frac{\left( -\frac{\partial V}{\partial p_i} \right)}{\frac{\partial V}{\partial y}} = \frac{D_i(z)}{\int_{0}^{\bar{M}} \frac{p_i}{y} D_i(F(\Lambda(\boldsymbol{p}, y)) \frac{p_i}{y}) di} $$

The fact that $\Lambda(\boldsymbol{p}, y)$ must satisfy the budget constraint implies that $H(\Lambda(\boldsymbol{p}, y)) = \int_{0}^{\bar{M}} \frac{p_i}{y} D_i(F(\Lambda) \frac{p_i}{y}) di$. Hence, we obtain the direct demand function described in the main text:

$$ q_i(p_i, \boldsymbol{p}, y) = \frac{D_i(F(\Lambda(\boldsymbol{p}, y)) \frac{p_i}{y})}{H(\Lambda(\boldsymbol{p}, y))} $$

## A.2 The "Exponent Elasticity Method" (EEM)

As stressed in the main text, given the presence of fixed costs in the current environment, increased generality on the demand side entails tractability challenges. To address this, I develop a new
and simple approach—the Exponent Elasticity Method (hereafter EEM)—which delivers explicit solutions for key variables in general equilibrium, including the single price aggregator Λ, the domestic productivity cutoff φ̂<sub>d</sub>, and the export productivity cutoff φ̂<sub>x</sub>. The central idea of the EEM is to locally approximate (around the trade equilibrium ) any implicit function F (a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>n</sub>) having a finite number of arguments n by a unique multiplicative equivalent, whereby the exponent of each argument captures the function's total elasticity with respect to it:

$$F(a_1, a_2, \dots, a_n) \equiv (a_1)^{\varepsilon_{a_1}^F} (a_2)^{\varepsilon_{a_2}^F} \dots (a_n)^{\varepsilon_{a_n}^F}$$

where $\varepsilon_{a_1}^F = \frac{d \log F(.)}{d \log a_1} = \frac{dF}{da_1} \frac{a_1}{F}$, $\varepsilon_{a_2}^F = \frac{d \log F(.)}{d \log a_2}$, ..., and $\varepsilon_{a_n}^F = \frac{d \log F(.)}{d \log a_n}$ respectively denote the total elasticity of function F(.) with respect to each of its arguments. I proceed in four steps. First, I apply the EEM to firm-level revenues and operating profits and I obtain an explicit solution for the domestic and the export productivity cutoff in partial equilibrium. Second, I solve for aggregate revenues from domestic and export sales in partial equilibrium. Third, I rewrite the budget constraint with an explicit expression of aggregate expenditures and I solve for the price aggregator Λ in general equilibrium. Finally, I obtain an explicit solution for the domestic and the export productivity cutoff in general equilibrium (φ̂<sub>d</sub>, φ̂<sub>x</sub>) by plugging the general equilibrium price aggregator Λ in their respective partial equilibrium expressions.

## Step 1. Explicit Solutions for Productivity Cutoffs (φ̂<sub>d</sub>, φ̂<sub>x</sub>) in Partial Equilibrium

From equations (8) and (9) in the main text, for a firm with marginal cost c and a measure of efficiency $e = \frac{y}{cF(\Lambda)}$, the sales revenues and operating profits it earns from serving a given market can be respectively written as:

$$\begin{cases} r(c, \frac{y}{cF(\Lambda)}, H(\Lambda), L) = c\mu(\frac{y}{cF(\Lambda)}) \frac{D(\mu(\frac{y}{cF(\Lambda)})/\frac{y}{cF(\Lambda)})}{H(\Lambda)} \\ \pi^o(c, \frac{y}{cF(\Lambda)}, H(\Lambda), L) = \dot{\mu}(\frac{y}{cF(\Lambda)}) r(c, \frac{y}{cF(\Lambda)}, H(\Lambda), L) \end{cases}$$

where $\mu(\frac{y}{cF(\Lambda)}) = [\frac{y}{cF(\Lambda)}]^{\rho(S)}$, and $\dot{\mu}(\frac{y}{cF(\Lambda)}) = \frac{\mu(\frac{y}{cF(\Lambda)})^{-1}}{\mu(\frac{y}{cF(\Lambda)})} \in ]0, 1[$ respectively correspond to the markup and the markup rate at the firm level. Since the price aggregator Λ is implicitly defined as a unique solution to the budget constraint, I can apply the EEM method and locally approximate the above firm-level variables with a unique multiplicative equivalent as follows:

* Firm-level sales revenues:

$$r(c, \frac{y}{cF(\Lambda)}, H(\Lambda), L) \equiv (c)^{\varepsilon_c^r} (y)^{\varepsilon_y^r} (F(\Lambda))^{\varepsilon_{F(\Lambda)}^r} (H(\Lambda))^{-1} L \quad (24)$$
where the above exponents are respectively given by:

$$
\left\{
\begin{aligned}
\varepsilon_c^r(e) &= \frac{\partial \log r(.)}{\partial \log c} = (1 - \rho(S))(1 - \varepsilon_D(e)) < 0 \\
\varepsilon_y^r(e) &= \frac{\partial \log r(.)}{\partial \log y} = (1 - \rho(S)) \varepsilon_D(e) + \rho(S) > 0 \\
\varepsilon_{F(\Lambda)}^r(e) &= \frac{\partial \log r(.)}{\partial \log F(\Lambda)} = -[(1 - \rho(S)) \varepsilon_D(e) + \rho(S)] < 0
\end{aligned}
\right.
\qquad (25)
$$

• Firm-level operating profits:

$$
\pi^0(c, \frac{y}{cF(\Lambda)}, H(\Lambda), L) \equiv (c)^{\varepsilon_c^{\pi^0}} (y)^{\varepsilon_y^{\pi^0}} (F(\Lambda))^{\varepsilon_{F(\Lambda)}^{\pi^0}} (H(\Lambda))^{-1} L \quad (26)
$$

where the above exponents are respectively given by:

$$
\left\{
\begin{aligned}
\varepsilon_c^{\pi^0}(e) &= \frac{\partial \log \pi^0(.)}{\partial \log c} = (1 - \rho(S))(1 - \varepsilon_D(e)) - \eta \rho(S) < 0 \\
\varepsilon_y^{\pi^0}(e) &= \frac{\partial \log \pi^0(.)}{\partial \log y} = (1 - \rho(S)) \varepsilon_D(e) + \rho(S) + \eta \rho(S) > 0 \\
\varepsilon_{F(\Lambda)}^{\pi^0}(e) &= \frac{\partial \log \pi^0(.)}{\partial \log F(\Lambda)} = -[(1 - \rho(S)) \varepsilon_D(e) + \rho(S)] - \eta \rho(S) < 0
\end{aligned}
\right.
\qquad (27)
$$

where $\eta = \frac{\partial \log \dot{\mu}(e)}{\partial \log \mu(e)} > 0$ is the elasticity of firm-level markup rates with respect to firm-level markups.
The above elasticities of firm-level variables in equations (25) and (27) are firm-specific beyond the
CES case, as stated in the main text. Now by recalling that for a firm with productivity $\varphi$, the
marginal cost of delivering a unit of its own variety to domestic consumers is given by $c_d = \frac{w}{\varphi}$, its
sales revenues and operating profits from serving the domestic market can be rewritten as:

• Firm-level sales revenues from domestic sales:

$$
r_d(\varphi, w, y, \Lambda, L) \equiv (\varphi)^{\varepsilon_{\varphi}^{r}(e_d)} (w)^{\varepsilon_{w}^{r}(e_d)} (y)^{\varepsilon_{y}^{r}(e_d)} (\Lambda)^{\varepsilon_{\Lambda}^{r}(e_d)} L \quad (28)
$$

where the above exponents are respectively given by:

$$
\left\{
\begin{aligned}
\varepsilon_{\varphi}^{r}(e_d) &= -\frac{\partial \log r_d(.)}{\partial \log c_d} = (1 - \rho(S))(\varepsilon_D(e_d) - 1) > 0; \quad \varepsilon_w^r(e_d) = \frac{\partial \log r_d(.)}{\partial \log c_d} = (1 - \rho(S))(1 - \varepsilon_D(e_d)) < 0 \\
\varepsilon_y^r(e_d) &= \frac{\partial \log r_d(.)}{\partial \log y} = (1 - \rho(S)) \varepsilon_D(e_d) + \rho(S) > 0 \\
\varepsilon_{\Lambda}^{r}(e_d) &= \frac{\partial \log r_d(.)}{\partial \log \Lambda} = \frac{\partial \log r_d(.)}{\partial \log F(\Lambda)} \frac{\partial \log F(\Lambda)}{\partial \log \Lambda} + \frac{\partial \log r_d(.)}{\partial \log H(\Lambda)} \frac{\partial \log H(\Lambda)}{\partial \log \Lambda} = \varepsilon_F \varepsilon_{F(\Lambda)}^r(e_d) - \varepsilon_H < 0
\end{aligned}
\right.
\tag{29}
$$

where $\varepsilon_{F(\Lambda)}^r(e_d) = -[(1 - \rho(S)) \varepsilon_D(e_d) + \rho(S)] < 0$.

• Firm-level operating profits:

$$
\pi_d^0(\varphi, w, y, \Lambda, L) \equiv (\varphi)^{\varepsilon_{\varphi}^{\pi^0}(e_d)} (w)^{\varepsilon_w^{\pi^0}(e_d)} (y)^{\varepsilon_y^{\pi^0}(e_d)} (\Lambda)^{\varepsilon_{\Lambda}^{\pi^0}(e_d)} L \quad (30)
$$
where the above exponents are respectively given by:

$$
\left\{
\begin{aligned}
\varepsilon_{\phi}^{\pi_{d}^{0}}(e_{d}) &= -\frac{\partial \log \pi_{d}^{0}(.)}{\partial \log c_{d}} = (1 - \rho(S))(\varepsilon_{D}(e_{d}) - 1) + \eta \rho(S) > 0 \\
\varepsilon_{w}^{\pi_{d}^{0}}(e_{d}) &= \frac{\partial \log \pi_{d}^{0}(.)}{\partial \log c_{d}} = (1 - \rho(S))(1 - \varepsilon_{D}(e_{d})) - \eta \rho(S) < 0 \\
\varepsilon_{y}^{\pi_{d}^{0}}(e_{d}) &= \frac{\partial \log \pi_{d}^{0}(.)}{\partial \log y} = (1 - \rho(S)) \varepsilon_{D}(e_{d}) + (1 + \eta) \rho(S) > 0 \\
\varepsilon_{\Lambda}^{\pi_{d}^{0}}(e_{d}) &= \frac{\partial \log \pi_{d}^{0}(.)}{\partial \log \Lambda} = \frac{\partial \log \pi_{d}^{0}(.)}{\partial \log F(\Lambda)} \frac{\partial \log F(\Lambda)}{\partial \log \Lambda} + \frac{\partial \log \pi_{d}^{0}(.)}{\partial \log H(\Lambda)} \frac{\partial \log H(\Lambda)}{\partial \log \Lambda} = \varepsilon_{F} \varepsilon_{F(\Lambda)}^{\pi_{d}^{0}}(e_{d}) - \varepsilon_{H} < 0
\end{aligned}
\right.
\quad (31)
$$

where $\varepsilon_{F(\Lambda)}^{\pi_d^0}(e_d) = -[(1 - \rho(S)) \varepsilon_D(e_d) + (1 + \eta) \rho(S)] < 0$. Now using the local approximation of firm-level domestic operating profits in equation (30), the zero profit condition on the domestic market (ZPC)$_d$ from equation (11) in the main text can be rewritten as:

$$
\varphi_d^* : \pi_d^0(\varphi_d^*, w, y, \Lambda, L) = w f \Rightarrow (\varphi_d^*)^{\varepsilon_{\varphi}^{\pi_d^0}(e_d^*)} (w)^{\varepsilon_w^{\pi_d^0}(e_d^*)} (y)^{\varepsilon_y^{\pi_d^0}(e_d^*)} (\Lambda)^{[\varepsilon_F \varepsilon_{F(\Lambda)}^{\pi_d^0}(e_d^*) - \varepsilon_H]} L = w f
$$

By simply rearranging, I obtain thus an explicit solution for the domestic cutoff in partial equilib-
rium:

$$
\varphi_d^* \equiv (f)^{\frac{1}{\varepsilon_{\varphi}^{\pi_d^0}(e_d^*)}} (w)^{\frac{[1-\varepsilon_w^{\pi_d^0}(e_d^*)]}{\varepsilon_{\varphi}^{\pi_d^0}(e_d^*)}} (y)^{-\frac{\varepsilon_y^{\pi_d^0}(e_d^*)}{\varepsilon_{\varphi}^{\pi_d^0}(e_d^*)}} (L)^{-\frac{1}{\varepsilon_{\varphi}^{\pi_d^0}(e_d^*)}} (\Lambda)^{\frac{[\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^{\pi_d^0}(e_d^*)]}{\varepsilon_{\varphi}^{\pi_d^0}(e_d^*)}} \quad (32)
$$

Given the presence of an iceberg trade cost $\tau > 1$, the marginal cost of delivering a variety produced
by a $\varphi$ productivity firm to foreign consumers evaluates to $c_x = \frac{\tau w}{\varphi}$. Using the same procedure
as before, I start solving for the partial equilibrium export cutoff by rewriting firm-level export
revenues and operating profits as follows:

$$
r_x(\varphi, \tau, w, y, \Lambda, L) \equiv (\varphi)^{\varepsilon_{\varphi}^r(e_x)} (\tau)^{\varepsilon_{\tau}^r(e_x)} (w)^{\varepsilon_w^r(e_x)} (y)^{\varepsilon_y^r(e_x)} (\Lambda)^{\varepsilon_{\Lambda}^r(e_x)} L \quad (33)
$$

where the above exponents are respectively given by:

$$
\left\{
\begin{aligned}
\varepsilon_{\varphi}^{r}(e_x) &= -\frac{\partial \log r_x(.)}{\partial \log c_x} = (1 - \rho(S))(\varepsilon_D(e_x) - 1) > 0; \quad \varepsilon_{\tau}^{r}(e_x) = \frac{\partial \log r_x(.)}{\partial \log c_x} = (1 - \rho(S))(1 - \varepsilon_D(e_x)) < 0 \\
\varepsilon_{w}^{r}(e_x) &= \frac{\partial \log r_x(.)}{\partial \log c_x} = (1 - \rho(S))(1 - \varepsilon_D(e_x)) < 0; \quad \varepsilon_{y}^{r}(e_x) = \frac{\partial \log r_x(.)}{\partial \log y} = (1 - \rho(S)) \varepsilon_D(e_x) + \rho(S) > 0 \\
\varepsilon_{\Lambda}^{r}(e_x) &= \frac{\partial \log r_x(.)}{\partial \log \Lambda} = \frac{\partial \log r_x(.)}{\partial \log F(\Lambda)} \frac{\partial \log F(\Lambda)}{\partial \log \Lambda} + \frac{\partial \log r_x(.)}{\partial \log H(\Lambda)} \frac{\partial \log H(\Lambda)}{\partial \log \Lambda} = \varepsilon_F \varepsilon_{F(\Lambda)}^{r}(e_x) - \varepsilon_H < 0
\end{aligned}
\right.
\quad (34)
$$

where $\varepsilon_{F(\Lambda)}^r(e_x) = -[(1 - \rho(S)) \varepsilon_D(e_x) + \rho(S)] < 0$. Notice that since countries are symmetric, the difference between domestic and export revenues is solely driven by the presence of the variable trade cost $\tau \ge 1$, which inflates a firm's marginal cost $c_x = \tau$ and thus, proportionally deflates its measure of efficiency on the export market: $\frac{e_x}{e_d} = \tau^{-1}$.

$$
\pi_x^0(\varphi, \tau, w, y, \Lambda, L) \equiv (\varphi)^{\varepsilon_{\varphi}^{\pi_d^0}(e_x)} (\tau)^{\varepsilon_{\tau}^{\pi_d^0}(e_x)} (w)^{\varepsilon_w^{\pi_d^0}(e_x)} (y)^{\varepsilon_y^{\pi_d^0}(e_x)} (\Lambda)^{\varepsilon_{\Lambda}^{\pi_d^0}(e_x)} L \quad (35)
$$
where the above exponents are respectively given by:

$$
\left\{
\begin{aligned}
\varepsilon_{\varphi}^{\pi^0}(e_x) &= -\frac{\partial \log \pi_x^0(.)}{\partial \log c_x} = (1-\rho(S))(\varepsilon_D(e_x) - 1) + \eta\rho(S) > 0 \\
\varepsilon_{\tau}^{\pi^0}(e_x) &= \frac{\partial \log \pi_x^0(.)}{\partial \log c_x} = (1-\rho(S))(1-\varepsilon_D(e_x)) - \eta\rho(S) < 0 \\
\varepsilon_{w}^{\pi^0}(e_x) &= \frac{\partial \log \pi_x^0(.)}{\partial \log c_x} = (1-\rho(S))(1-\varepsilon_D(e_x)) - \eta\rho(S) < 0 \\
\varepsilon_{y}^{\pi^0}(e_x) &= \frac{\partial \log \pi_y^0(.)}{\partial \log y} = (1-\rho(S))\varepsilon_D(e_x) + (1+\eta)\rho(S) > 0 \\
\varepsilon_{\Lambda}^{\pi^0}(e_x) &= \frac{\partial \log \pi_x^0(.)}{\partial \log \Lambda} = \frac{\partial \log \pi_x^0(.)}{\partial \log F(\Lambda)} \frac{\partial \log F(\Lambda)}{\partial \log \Lambda} + \frac{\partial \log \pi_y^0(.)}{\partial \log H(\Lambda)} \frac{\partial \log H(\Lambda)}{\partial \log \Lambda} = \varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^0}(e_x) - \varepsilon_H < 0
\end{aligned}
\right.
\qquad (36)
$$

where $\varepsilon_{F(\Lambda)}^{\pi^0}(e_x) = -[(1-\rho(S))\varepsilon_D(e_x) + (1+\eta)\rho(S)] < 0$. As before, using the local approximation of firm-level export operating profits in equation (35), the zero profit condition on the export market $(ZPC)_x$ from equation (11) in the main text can be rewritten as:

$$ \varphi_x^* : \pi_x^0(\varphi_x^*, \tau, w, y, \Lambda, L) = w f_x \Rightarrow (\varphi_x^*)^{\varepsilon_{\varphi}^{\pi^0}(e_x^*)} (\tau)^{\varepsilon_{\tau}^{\pi^0}(e_x^*)} (w)^{\varepsilon_{w}^{\pi^0}(e_x^*)} (y)^{\varepsilon_{y}^{\pi^0}(e_x^*)} (\Lambda)^{[\varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^0}(e_x^*) - \varepsilon_H]} L = w f_x $$

By simply rearranging, I obtain the following explicit solution for the export cutoff in partial equilibrium:

$$ \varphi_x^* \equiv \tau \left( f_x \right)^{\frac{1}{\varepsilon_{\varphi}^{\pi^0}(e_x^*)}} \left( w \right)^{\frac{[1-\varepsilon_{w}^{\pi^0}(e_x^*)]}{\varepsilon_{\varphi}^{\pi^0}(e_x^*)}} \left( y \right)^{-\frac{\varepsilon_{y}^{\pi^0}(e_x^*)}{\varepsilon_{\varphi}^{\pi^0}(e_x^*)}} \left( L \right)^{-\frac{1}{\varepsilon_{\varphi}^{\pi^0}(e_x^*)}} \left( \Lambda \right)^{\frac{[\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^0}(e_x^*)]}{\varepsilon_{\varphi}^{\pi^0}(e_x^*)}} \quad (37) $$

## Step 2. Solving for Aggregate Revenues from Domestic and Export Sales in Partial Equilibrium

* Aggregate revenues from export sales: As in the main text, let $X_x$ denote the value of bilateral trade flows for any country pair. This is given by:

$$ X_x = M_e [1 - G(\varphi_x^*)] \int_{\varphi_x^*}^{+\infty} r_x(\varphi, \tau, w, y, L, \Lambda) \gamma_x(\varphi) d\varphi $$

where $\gamma_x(\varphi)$ corresponds to the productivity distribution conditionally on successful entry in the export market and is given by: $\gamma_x(\varphi) = \frac{g(\varphi)}{[1-G(\varphi_x^*)]} \quad \forall \varphi \ge \varphi_x^*$. Now by applying the EEM to the case of the average productivity exporter $\bar{\varphi}_x$, $X_x$ can be rewritten as:

$$ X_x = M_e [1 - G(\varphi_x^*)] (\bar{\varphi}_x)^{\varepsilon_{\varphi}^r(\bar{\varphi}_x)} (\tau)^{\varepsilon_{\tau}^r(\bar{\varphi}_x)} (w)^{\varepsilon_{w}^r(\bar{\varphi}_x)} (y)^{\varepsilon_{y}^r(\bar{\varphi}_x)} (\Lambda)^{[\varepsilon_F \varepsilon_{F(\Lambda)}^r(\bar{\varphi}_x) - \varepsilon_H]} L $$

Then, by invoking the unbounded Pareto assumption A3, we get:

$$ \left\{
\begin{aligned}
&[1 - G(\varphi_x^*)] = (\varphi_x^*)^{-\theta} \\
&\bar{\varphi}_x = \int_{\varphi_x^*}^{+\infty} \varphi \, \gamma_x(\varphi) \, d\varphi = (\theta - 1) \, \varphi_x^*
\end{aligned}
\right.
\qquad (38) $$
Using equation (38), $X_x$ simplifies to

$$X_x = \tilde{\kappa} M_e (\varphi_x^*)^{[\varepsilon_\varphi^r(\bar{\varepsilon}_x) - \theta]} (\tau)^{\varepsilon_\tau^r(\bar{\varepsilon}_x)} (w)^{\varepsilon_w^r(\bar{\varepsilon}_x)} (y)^{\varepsilon_y^r(\bar{\varepsilon}_x)} (\Lambda)^{[\varepsilon_F \varepsilon_{F(\Lambda)}^r(\bar{\varepsilon}_x) - \varepsilon_H]} L$$

where $\tilde{\kappa} = (\frac{\theta}{\theta-1})^{\varepsilon_\varphi^r(\bar{\varepsilon}_x)}$ is a transformation of the shape parameter of the Pareto which I drop hereafter. Now by plugging the partial equilibrium expression of the export cutoff from equation (37) in the above expression of $X_x$ and rearranging, we obtain:

$$X_x(M_e, w, \tau, f_x, y, L, \Lambda) \equiv M_e(w)^{\varepsilon_w^{X_x}} (\tau)^{\varepsilon_\tau^{X_x}} (f_x)^{\varepsilon_{f_x}^{X_x}} (y)^{\varepsilon_y^{X_x}} (L)^{\varepsilon_L^{X_x}} (\Lambda)^{\varepsilon_\Lambda^{X_x}} \quad (39)$$

where the elasticities of bilateral trade flows $X_x$, respectively, with respect to the cost of labor $w$, the variable trade cost $\tau$, the fixed export cost $f_x$, individual income $y$, population size $L$, and the price aggregator $\Lambda$, can be written as:<fnref n="1" />

$$\left\{
\begin{aligned}
\varepsilon_w^{X_x} &= \frac{d \log X_x}{d \log w} = \underbrace{\varepsilon_w^r(\bar{\varepsilon}_x)}_{\text{intensive}} + \underbrace{[\varepsilon_\varphi^r(\bar{\varepsilon}_x) - \theta] \frac{[1 - \varepsilon_w^{\pi^0}(e_x^*)]}{\varepsilon_\varphi^{\pi^0}(e_x^*)}}_{\text{extensive}} \\
\varepsilon_\tau^{X_x} &= \frac{d \log X_x}{d \log \tau} = \underbrace{\varepsilon_\tau^r(\bar{\varepsilon}_x)}_{\text{intensive}} + \underbrace{[\varepsilon_\varphi^r(\bar{\varepsilon}_x) - \theta]}_{\text{extensive}} = -\theta < 0 \\
\varepsilon_{f_x}^{X_x} &= \frac{d \log X_x}{d \log f_x} = \underbrace{0}_{\text{intensive}} + \underbrace{\frac{[\varepsilon_\varphi^r(\bar{\varepsilon}_x) - \theta]}{\varepsilon_\varphi^{\pi^0}(e_x^*)}}_{\text{extensive}} \\
\varepsilon_y^{X_x} &= \frac{d \log X_x}{d \log y} = \underbrace{\varepsilon_y^r(\bar{\varepsilon}_x)}_{\text{intensive}} + \underbrace{[\varepsilon_\varphi^r(\bar{\varepsilon}_x) - \theta] \left(-\frac{\varepsilon_y^{\pi^0}(e_x^*)}{\varepsilon_\varphi^{\pi^0}(e_x^*)}\right)}_{\text{extensive}} \\
\varepsilon_L^{X_x} &= \frac{d \log X_x}{d \log L} = \underbrace{1}_{\text{intensive}} + \underbrace{[\varepsilon_\varphi^r(\bar{\varepsilon}_x) - \theta] \left(-\frac{1}{\varepsilon_\varphi^{\pi^0}(e_x^*)}\right)}_{\text{extensive}} \\
\varepsilon_\Lambda^{X_x} &= \frac{d \log X_x}{d \log \Lambda} = \underbrace{[\varepsilon_F \varepsilon_{F(\Lambda)}^r(\bar{\varepsilon}_x) - \varepsilon_H]}_{\text{intensive}} + \underbrace{[\varepsilon_\varphi^r(\bar{\varepsilon}_x) - \theta] \frac{[\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^0}(e_x^*)]}{\varepsilon_\varphi^{\pi^0}(e_x^*)}}_{\text{extensive}}
\end{aligned}
\right. \quad (40)$$

Inspection of the above expressions clearly shows that the total elasticity of bilateral trade flows $X_x$ with respect to each of its arguments simply evaluates to the sum of the elasticity of the intensive margin (average sales revenues per exporter) and the elasticity of the extensive margin (the mass of exporters) with respect to the argument at question. Let us take the example of the variable trade cost $\tau$:

$$\varepsilon_{\tau}^{X_x} = \frac{d \log X_x}{d \log \tau} = \underbrace{\frac{\partial \log r_x(\bar{\varepsilon}_x)}{\partial \log \tau}}_{\text{intensive}} + \underbrace{\frac{\partial \log X_x}{\partial \log \varphi_x^*} \frac{\partial \log \varphi_x^*}{\partial \log \tau}}_{\text{extensive}} = \underbrace{\varepsilon_{\tau}^r(\bar{\varepsilon}_x)}_{\text{intensive}} + \underbrace{[\varepsilon_{\varphi}^r(\bar{\varepsilon}_x) - \theta]}_{\text{extensive}} = -\theta < 0$$

<footnote n="1">¹Notice that $\varepsilon_{\tau}^{X_x}$ always boils down to $(-\theta)$ since $\varepsilon_{\tau}^r(\bar{\varepsilon}_x) = -\varepsilon_{\varphi}^r(\bar{\varepsilon}_x)$.</footnote>
We can see that the extensive margin always dominates the intensive margin, as in Chaney (2008). This, in turn, always gives rise to a constant trade elasticity ($-\theta$) regardless of whether demand is rigid (under the CES) or flexible (beyond the CES). Now by assuming that the shape parameter $\theta$ is large enough to exceed the productivity elasticity of firm-level revenues faced by the average productivity firm on both markets, $\theta > \varepsilon_{\phi}^r(\bar{e}_d)$ and $\theta > \varepsilon_{\phi}^r(\bar{e}_x)$, we can rewrite the above elasticities in equation (40) so as their sign gets easily visible:<fnref n="2" />

$$ 
\left\{ 
\begin{aligned}
\varepsilon_{w}^{X_x} &= \frac{d\log X_x}{d\log w} = \varepsilon_{w}^{r}(\bar{e}_x) - [\theta - \varepsilon_{\phi}^{r}(\bar{e}_x)] \frac{[1 - \varepsilon_{w}^{\pi^0}(e_x^*)]}{\varepsilon_{\phi}^{\pi^0}(e_x^*)} < 0 \\
\varepsilon_{\tau}^{X_x} &= \frac{d\log X_x}{d\log \tau} = -\theta < 0 \\
\varepsilon_{f_x}^{X_x} &= \frac{d\log X_x}{d\log f_x} = -\frac{[\theta - \varepsilon_{\phi}^{r}(\bar{e}_x)]}{\varepsilon_{\phi}^{\pi^0}(e_x^*)} < 0 \\
\varepsilon_{y}^{X_x} &= \frac{d\log X_x}{d\log y} = \varepsilon_{y}^{r}(\bar{e}_x) + [\theta - \varepsilon_{\phi}^{r}(\bar{e}_x)] \frac{\varepsilon_{y}^{\pi^0}(e_x^*)}{\varepsilon_{\phi}^{\pi^0}(e_x^*)} > 0 \\
\varepsilon_{L}^{X_x} &= \frac{d\log X_x}{d\log L} = 1 + \frac{[\theta - \varepsilon_{\phi}^{r}(\bar{e}_x)]}{\varepsilon_{\phi}^{\pi^0}(e_x^*)} > 0 \\
\varepsilon_{\Lambda}^{X_x} &= \frac{d\log X_x}{d\log \Lambda} = [\varepsilon_F \varepsilon_{F(\Lambda)}^r(\bar{e}_x) - \varepsilon_H] + [\theta - \varepsilon_{\phi}^{r}(\bar{e}_x)] \frac{[\varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^0}(e_x^*) - \varepsilon_H]}{\varepsilon_{\phi}^{\pi^0}(e_x^*)} < 0
\end{aligned}
\right. 
\quad (41) $$

* Aggregate revenues from domestic sales: using the same procedure, I obtain the following explicit solution for the value of domestic trade in any country, $X_d$:

$$ X_d(M_e, w, f, y, L, \Lambda) \equiv M_e(w)^{\varepsilon_w^{X_d}} (f)^{\varepsilon_f^{X_d}} (y)^{\varepsilon_y^{X_d}} (L)^{\varepsilon_L^{X_d}} (\Lambda)^{\varepsilon_{\Lambda}^{X_d}} \quad (42) $$

where the total elasticities of domestic trade $X_d$, respectively, with respect to the cost of labor $w$, the fixed cost of accessing the domestic market $f$, individual income $y$, population size $L$, and the price aggregator $\Lambda$, and are given by:

$$ 
\left\{ 
\begin{aligned}
\varepsilon_w^{X_d} &= \frac{d\log X_d}{d\log w} = \varepsilon_w^r(\bar{e}_d) - [\theta - \varepsilon_\phi^r(\bar{e}_d)] \frac{[1 - \varepsilon_w^{\pi^0}(e_d^*)]}{\varepsilon_\phi^{\pi^0}(e_d^*)} < 0 \\
\varepsilon_f^{X_d} &= \frac{d\log X_d}{d\log f} = -\frac{[\theta - \varepsilon_\phi^r(\bar{e}_d)]}{\varepsilon_\phi^{\pi^0}(e_d^*)} < 0 \\
\varepsilon_y^{X_d} &= \frac{d\log X_d}{d\log y} = \varepsilon_y^r(\bar{e}_d) + [\theta - \varepsilon_\phi^r(\bar{e}_d)] \frac{\varepsilon_y^{\pi^0}(e_d^*)}{\varepsilon_\phi^{\pi^0}(e_d^*)} > 0 \\
\varepsilon_L^{X_d} &= \frac{d\log X_d}{d\log L} = 1 + \frac{[\theta - \varepsilon_\phi^r(\bar{e}_d)]}{\varepsilon_\phi^{\pi^0}(e_d^*)} > 0 \\
\varepsilon_{\Lambda}^{X_d} &= \frac{d\log X_d}{d\log \Lambda} = [\varepsilon_F \varepsilon_{F(\Lambda)}^r(\bar{e}_d) - \varepsilon_H] + [\theta - \varepsilon_\phi^r(\bar{e}_d)] \frac{[\varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^0}(e_d^*) - \varepsilon_H]}{\varepsilon_\phi^{\pi^0}(e_d^*)} < 0
\end{aligned}
\right. 
\quad (43) $$

Now we can also solve for aggregate net profits on both markets in partial equilibrium as follows.

$$ 
\left\{ 
\begin{aligned}
\Pi_d^o &= M_e [1 - G(\varphi_d^*)] \int_{\varphi_d^*}^{+\infty} \pi_d^o(\varphi, w, y, L, F(\Lambda), H(\Lambda)) \gamma_d(\varphi) d\varphi \\
\Pi_x^o &= M_e [1 - G(\varphi_x^*)] \int_{\varphi_x^*}^{+\infty} \pi_x^o(\varphi, \tau, w, y, L, F(\Lambda), H(\Lambda)) \gamma_x(\varphi) d\varphi
\end{aligned}
\right. 
\quad (44) $$

<footnote n="2">²Under the CES case, this is equivalent to assuming that $\theta > (\varepsilon_D - 1)$, as in Chaney (2008).</footnote>
where $\Pi_d^o$ and $\Pi_x^o$ respectively denote aggregate operating profits on domestic and export markets.
Using equation (9) from the main text and invoking the Pareto assumption A3, we get

$$ \left\{ \begin{array}{l} \Pi_d^o = \dot{\mu}(\bar{e}_d) X_d \\ \\ \Pi_x^o = \dot{\mu}(\bar{e}_x) X_x \end{array} \right. \qquad (45) $$

where $\dot{\mu}(\bar{e}_d)$ and $\dot{\mu}(\bar{e}_x)$ correspond to average markup rates respectively on the domestic and the export market, and are given by:

$$ \left\{ \begin{array}{l} \dot{\mu}(\bar{e}_d) = 1 - \left[ \frac{y}{\bar{e}_d F(\Lambda)} \right]^{-\rho(S)}, \quad \text{with } \bar{e}_d = \left( \frac{\theta-1}{\theta} \right) w (\varphi_d^*)^{-1} \\ \\ \dot{\mu}(\bar{e}_x) = 1 - \left[ \frac{y}{\bar{e}_x F(\Lambda)} \right]^{-\rho(S)}, \quad \text{with } \bar{e}_x = \left( \frac{\theta-1}{\theta} \right) w \tau (\varphi_x^*)^{-1} \end{array} \right. \qquad (46) $$

Let $\Pi_d$ and Let $\Pi_x$ now respectively denote aggregate profits on domestic and export markets:

$$ \left\{ \begin{array}{l} \Pi_d = M_e [1 - G(\varphi_d^*)] \int_{\varphi_d^*}^{+\infty} \pi_d(\varphi, w, y, L, F(\Lambda), H(\Lambda), f) \gamma_d(\varphi) d\varphi \\ \\ \Pi_x = M_e [1 - G(\varphi_x^*)] \int_{\varphi_x^*}^{+\infty} \pi_x(\varphi, \tau, w, y, L, F(\Lambda), H(\Lambda), f_x) \gamma_x(\varphi) d\varphi \end{array} \right. \qquad (47) $$

Using equation (10) from the main text and the above equation (45), we get

$$ \left\{ \begin{array}{l} \Pi_d = \dot{\mu}(\bar{e}_d) X_d - M_e [1 - G(\varphi_d^*)] w f \\ \\ \Pi_x = \dot{\mu}(\bar{e}_x) X_x - M_e [1 - G(\varphi_x^*)] w f_x \end{array} \right. \qquad (48) $$

### Step 3. Solving for the Price Aggregator $\Lambda(p, y)$ in General Equilibrium

To start with, let us recall that the price aggregator $\Lambda(p, y)$ is implicitly defined as a unique solution to budget constraint of the representative consumer (equation (2) in the main text) which can be rewritten as:

$$ M_e [1 - G(\varphi_d^*)] \int_{\varphi_d^*}^{+\infty} p_d(c_d, e_d) q_d \left(\frac{p_d}{y}, \Lambda\right) \gamma_d(\varphi) d\varphi + (N - 1) M_e [1 - G(\varphi_x^*)] \int_{\varphi_x^*}^{+\infty} p_x(c_x, e_x) q_x \left(\frac{p_x}{y}, \Lambda\right) \gamma_x(\varphi) d\varphi = y \qquad (49) $$

where $M_e$ is the mass of entrants in any country, $[1 - G(\varphi_d^*)]$ is the probability of successful entry in the domestic market, $[1 - G(\varphi_x^*)]$ is the probability of successful penetration of the export market. As before, for a firm with productivity $\varphi$, its marginal cost $c$ and measure of efficiency $e$ in each market are respectively given by: $c_d = \frac{w}{\varphi}$, $e_d = \frac{y}{c_d F(\Lambda)}$; and $c_x = \frac{\tau w}{\varphi}$, $e_x = \frac{y}{c_x F(\Lambda)}$. Now by multiplying the LHS and the RHS of the above equation with population size $L$, the budget constraint can be
rewritten as:

$$ \underbrace{M_d \int_{\varphi_d^*}^{+\infty} r_d(\varphi, w, y, L, \Lambda) \gamma_d(\varphi) d\varphi}_{X_d(M_e, w, f, y, L, \Lambda)} + (N-1) \underbrace{M_x \int_{\varphi_x^*}^{+\infty} r_x(\varphi, \tau, w, y, L, \Lambda) \gamma_x(\varphi) d\varphi}_{X_x(M_e, w, \tau, f_x, y, L, \Lambda)} = yL \quad (50) $$

where $M_d = M_e[1 - G(\varphi_d^*)]$ and $M_x = M_e[1 - G(\varphi_x^*)]$ respectively denote the equilibrium mass of domestic, and exporting firms. Notice that the LHS of the above equation (50) corresponds to aggregate expenditure in any country: $X = X_d(w, f, y, L, \Lambda) + (N-1) X_x(w, \tau, f_x, y, L, \Lambda)$. Now using the explicit partial equilibrium expressions of bilateral trade flows $X_x$ and domestic trade $X_d$ from equations (39) and (42), the budget constraint in the above equation can be rewritten as:

$$ \begin{gathered} M_e(w)^{\varepsilon_w^{X_d}} (f)^{\varepsilon_f^{X_d}} (y)^{\varepsilon_y^{X_d}} (L)^{\varepsilon_L^{X_d}} (\Lambda)^{\varepsilon_\Lambda^{X_d}} + (N-1) M_e(w)^{\varepsilon_w^{X_x}} (\tau)^{\varepsilon_\tau^{X_x}} (f_x)^{\varepsilon_{f_x}^{X_x}} (y)^{\varepsilon_y^{X_x}} (L)^{\varepsilon_L^{X_x}} (\Lambda)^{\varepsilon_\Lambda^{X_x}} = yL \\ \Rightarrow M_e(w)^{\varepsilon_w^{X_d}} (f)^{\varepsilon_f^{X_d}} (y)^{\varepsilon_y^{X_d}} (L)^{\varepsilon_L^{X_d}} (\Lambda)^{\varepsilon_\Lambda^{X_d}} + \left(\frac{Y_{RoW}}{Y_W}\right) N M_e(w)^{\varepsilon_w^{X_x}} (\tau)^{\varepsilon_\tau^{X_x}} (f_x)^{\varepsilon_{f_x}^{X_x}} (y)^{\varepsilon_y^{X_x}} (L)^{\varepsilon_L^{X_x}} (\Lambda)^{\varepsilon_\Lambda^{X_x}} = yL \end{gathered} \quad (51) $$

where, as before, $\frac{Y_{RoW}}{Y_W} = \frac{N-1}{N}$ is the GDP share of the rest of the world. Now by applying the EEM, the above expression of aggregate expenditure $X(M_e, w, f, N, \tau, f_x, y, L, \Lambda)$ on the LHS of equation (51) further simplifies to:

$$ X(M_e, w, f, N, \tau, f_x, y, L, \Lambda) \equiv M_e(w)^{\varepsilon_w^X} (f)^{\varepsilon_f^X} (N)^{\varepsilon_N^X} (\tau)^{\varepsilon_\tau^X} (f_x)^{\varepsilon_{f_x}^X} (y)^{\varepsilon_y^X} (L)^{\varepsilon_L^X} (\Lambda)^{\varepsilon_\Lambda^X} \quad (52) $$

where the elasticities of aggregate expenditure $X(M_e, w, f, N, \tau, f_x, y, L, \Lambda)$ with respect to the cost of labor $w$, the fixed cost of producing for the domestic market $f$, the number of trading countries at the world level $N$, the variable trade cost $\tau$, the fixed cost of exporting $f_x$, individual income $y$, population size $L$, and the price aggregator $\Lambda$ are respectively given by

$$ \left\{ \begin{aligned} \varepsilon_w^X &= \frac{\partial \log X}{\partial \log w} = \chi_d \varepsilon_w^{X_d} + (1 - \chi_d) \varepsilon_w^{X_x} < 0; & \varepsilon_f^X &= \frac{\partial \log X}{\partial \log f} = \chi_d \varepsilon_f^{X_d} < 0 \\ \varepsilon_N^X &= \frac{\partial \log X}{\partial \log N} = (1 - \chi_d) > 0; & \varepsilon_\tau^X &= \frac{\partial \log X}{\partial \log \tau} = (1 - \chi_d) \varepsilon_\tau^{X_x} = (1 - \chi_d) (-\theta) < 0 \\ \varepsilon_{f_x}^X &= \frac{\partial \log X}{\partial \log f_x} = (1 - \chi_d) \varepsilon_{f_x}^{X_x} < 0; & \varepsilon_y^X &= \frac{\partial \log X}{\partial \log y} = \chi_d \varepsilon_y^{X_d} + (1 - \chi_d) \varepsilon_y^{X_x} > 0 \\ \varepsilon_L^X &= \frac{\partial \log X}{\partial \log L} = \chi_d \varepsilon_L^{X_d} + (1 - \chi_d) \varepsilon_L^{X_x} > 0; & \varepsilon_\Lambda^X &= \frac{\partial \log X}{\partial \log \Lambda} = \chi_d \varepsilon_\Lambda^{X_d} + (1 - \chi_d) \varepsilon_\Lambda^{X_x} < 0 \end{aligned} \right. $$

where $\chi_d = \frac{X_d}{X}$ is the domestic trade share in any country, and $(1 - \chi_d) = \frac{(N-1)X_x}{X}$ corresponds to the import penetration rate. Let $T(N, \tau, f_x) = N \tau^{-\theta} (f_x)^{\varepsilon_{f_x}^{X_x}} > 0$ denote the trade exposure index. That is, a reduction in trade costs or an increase in the number of trading countries yields a higher level of the trade exposure index, thus reflecting that an economy is more exposed to trade. Conversely, an increase in trade costs or a decrease in the number of trading countries lowers the level of this
index, thus showing that an economy is less exposed to trade. Since labor is chosen as numéraire, I hereafter normalize wage to one and rewrite the expression of aggregate expenditure in equation (52) using the trade exposure index:

$$X(M_e, f, T(N, \tau, f_x), y, L, \Lambda) \equiv M_e (f)^{\chi_d} \epsilon_f^{\chi_d} T(N, \tau, f_x)^{(1-\chi_d)} (y)^{\epsilon_y} (L)^{\epsilon_L} (\Lambda)^{\epsilon_\Lambda} \quad (53)$$

Now using the above expression of aggregate expenditure, the budget constraint in equation (51) can be rewritten as:

$$X(M_e, f, T(N, \tau, f_x), y, L, \Lambda) \equiv M_e (f)^{\chi_d} \epsilon_f^{\chi_d} T(N, \tau, f_x)^{(1-\chi_d)} (y)^{\epsilon_y} (L)^{\epsilon_L} (\Lambda)^{\epsilon_\Lambda} = yL \quad (54)$$

Finally, by rearranging the above expression of the budget constraint, I can immediately solve for the general equilibrium price aggregator $\Lambda$:

$$\Lambda \equiv (M_e)^{\frac{1}{(-\epsilon_\Lambda^X)}} (y)^{\frac{(\epsilon_y^X - 1)}{(-\epsilon_\Lambda^X)}} (L)^{\frac{(\epsilon_L^X - 1)}{(-\epsilon_\Lambda^X)}} (f)^{\chi_d} \frac{\epsilon_f^{\chi_d}}{(-\epsilon_\Lambda^X)} [T(N, \tau, f_x)]^{\frac{(1-\chi_d)}{(-\epsilon_\Lambda^X)}} \quad (55)$$

Since the domestic trade share in any country is strictly lower than one ($\chi_d < 1$), and $(-\epsilon_\Lambda^X)$ is strictly positive, the general equilibrium price aggregator $\Lambda$ is thus monotonically increasing in the trade exposure index $T(N, \tau, f_x)$. Therefore, an increase in an economy's exposure to trade always intensifies competition on its domestic market.

## Step 4. Solving for the Domestic and Export Productivity Cutoffs in General Equilibrium

Now the final step is to plug the general equilibrium price aggregator $\Lambda$ from equation (55) in the partial equilibrium expressions of the domestic and the export productivity cutoffs in equations (32) and (37). By simply rearranging, I obtain their respective explicit expressions in general equilibrium:

$$\varphi_d^* \equiv (f)^{\varphi_d^*} (y)^{\varphi_d^*} (L)^{\varphi_d^*} (M_e)^{\varphi_d^*} [T(N, \tau, f_x)]^{\varphi_d^*} \quad (56)$$

where the above exponents are respectively given by:

$$\left\{
\begin{aligned}
\varepsilon_f^{\varphi_d^*} &= \frac{d \log \varphi_d^*}{d \log f} = \frac{1}{\varepsilon_{\varphi}^{\pi^0}(e_d^*)} + \frac{[\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^0}(e_d^*)]}{\varepsilon_{\varphi}^{\pi^0}(e_d^*)} \frac{\chi_d \varepsilon_f^{\chi_d}}{(-\varepsilon_\Lambda^X)} > 0; & \varepsilon_y^{\varphi_d^*} &= \frac{d \log \varphi_d^*}{d \log y} = -\frac{\varepsilon_y^{\pi^0}(e_d^*)}{\varepsilon_{\varphi}^{\pi^0}(e_d^*)} + \frac{[\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^0}(e_d^*)]}{\varepsilon_{\varphi}^{\pi^0}(e_d^*)} \frac{(\varepsilon_y^X - 1)}{(-\varepsilon_\Lambda^X)} \le 0 \\
\varepsilon_L^{\varphi_d^*} &= \frac{d \log \varphi_d^*}{d \log L} = -\frac{1}{\varepsilon_{\varphi}^{\pi^0}(e_d^*)} + \frac{[\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^0}(e_d^*)]}{\varepsilon_{\varphi}^{\pi^0}(e_d^*)} \frac{(\varepsilon_L^X - 1)}{(-\varepsilon_\Lambda^X)} \ge 0; & \varepsilon_{M_e}^{\varphi_d^*} &= \frac{d \log \varphi_d^*}{d \log M_e} = \frac{[\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^0}(e_d^*)]}{\varepsilon_{\varphi}^{\pi^0}(e_d^*)} \frac{1}{(-\varepsilon_\Lambda^X)} > 0 \\
\varepsilon_T^{\varphi_d^*} &= \frac{d \log \varphi_d^*}{d \log T(N, \tau, f_x)} = \frac{[\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^0}(e_d^*)]}{\varepsilon_{\varphi}^{\pi^0}(e_d^*)} \frac{(1 - \chi_d)}{(-\varepsilon_\Lambda^X)} > 0
\end{aligned}
\right.
\quad (57)$$

and correspond to the total elasticities of the domestic cutoff with respect to each of its determinants, and the explicit expressions of the firm-level elasticities they embody $\{\varepsilon_{\varphi}^{\pi^0}(e_d^*), \varepsilon_{\varphi}^{\pi^0}(e_d^*), \varepsilon_{F(\Lambda)}^{\pi^0}(e_d^*)\}$
can be immediately obtained from equation (31) by simply considering the case of the firm at the domestic cutoff: $e_d = e_d^*$. We can also notice that $\epsilon_f^{\phi_d^*}$ is always strictly positive since the domestic trade share is always strictly lower than one ($\chi_d < 1$) at the trade equilibrium.<fnref n="3" /> Now using the same procedure, I obtain the following expression of the general equilibrium export productivity cutoff $\varphi_x^*$:

$$ \varphi_x^* \equiv \tau (f_x)^{\epsilon_f^{\phi_x^*}} (y)^{\epsilon_y^{\phi_x^*}} (L)^{\epsilon_L^{\phi_x^*}} (M_e)^{\epsilon_{M_e}^{\phi_x^*}} [T(N, \tau, f_x)]^{\epsilon_T^{\phi_x^*}} \quad (58) $$

where $\epsilon_{f_x}^{\phi_x^*}$ is the partial elasticity of the export cutoff with respect to the fixed export cost $f_x$, and the rest of the above exponents embody the total elasticities of the export cutoff with respect to its other determinants, and are respectively given by:

$$ \left\{ \begin{aligned} \epsilon_{f_x}^{\phi_x^*} &= \frac{\partial \log \varphi_x^*}{\partial \log f_x} = \frac{1}{\epsilon_{\phi}^{\pi^0}(e_x^*)} > 0; \quad \epsilon_y^{\phi_x^*} = \frac{d \log \varphi_x^*}{d \log y} = -\frac{\epsilon_y^{\pi^0}(e_x^*)}{\epsilon_{\phi}^{\pi^0}(e_x^*)} + \frac{[\epsilon_H - \epsilon_F \epsilon_{F(\Lambda)}^{\pi^0}(e_x^*)]}{\epsilon_{\phi}^{\pi^0}(e_x^*)} \frac{(\epsilon_y^X - 1)}{(-\epsilon_{\Lambda}^X)} \le 0 \\ \epsilon_{L}^{\phi_x^*} &= \frac{d \log \varphi_x^*}{d \log L} = -\frac{1}{\epsilon_{\phi}^{\pi^0}(e_x^*)} + \frac{[\epsilon_H - \epsilon_F \epsilon_{F(\Lambda)}^{\pi^0}(e_x^*)]}{\epsilon_{\phi}^{\pi^0}(e_x^*)} \frac{(\epsilon_L^X - 1)}{(-\epsilon_{\Lambda}^X)} \ge 0; \quad \epsilon_{M_e}^{\phi_x^*} = \frac{d \log \varphi_x^*}{d \log M_e} = \frac{[\epsilon_H - \epsilon_F \epsilon_{F(\Lambda)}^{\pi^0}(e_x^*)]}{\epsilon_{\phi}^{\pi^0}(e_x^*)} \frac{1}{(-\epsilon_{\Lambda}^X)} > 0 \\ \epsilon_{T}^{\phi_x^*} &= \frac{d \log \varphi_x^*}{d \log T(N, \tau, f_x)} = \frac{[\epsilon_H - \epsilon_F \epsilon_{F(\Lambda)}^{\pi^0}(e_x^*)]}{\epsilon_{\phi}^{\pi^0}(e_x^*)} \frac{(1 - \chi_d)}{(-\epsilon_{\Lambda}^X)} > 0 \end{aligned} \right. \quad (59) $$

where the explicit expressions of the firm-level elasticities $\{\epsilon_{\phi}^{\pi^0}(e_x^*), \epsilon_{y}^{\pi^0}(e_x^*), \epsilon_{F(\Lambda)}^{\pi^0}(e_x^*)\}$ present in the above exponents can be immediately obtained from equation (36) by simply considering the case of the firm at the export cutoff: $e_x = e_x^*$.

### A.3 Structural Gravity

Upon normalizing wage to unity by choice of labor as numéraire, the value of bilateral trade flows in partial equilibrium evaluates to:

$$ X_x(M_e, \tau, f_x, y, L, \Lambda) \equiv M_e \tau^{-\theta} (f_x)^{\epsilon_{f_x}^{\chi_x}} (y)^{\epsilon_y^{\chi_x}} (L)^{\epsilon_L^{\chi_x}} (\Lambda)^{\epsilon_{\Lambda}^{\chi_x}} \quad (60) $$

where explicit expressions of the above exponents are provided in equation (41). Now by plugging the general equilibrium price aggregator $\Lambda$ from equation (55) in the above expression and rearranging, I solve for bilateral trade flows in general equilibrium:

$$ X_x(M_e, \tau, f_x, y, L, \Lambda) \equiv \underbrace{M_e}_{(1)} \underbrace{\tau^{-\theta} (f_x)^{\epsilon_{f_x}^{\chi_x}}}_{(2)} \underbrace{(y)^{(1-\epsilon_y^X+\epsilon_y^{\chi_x})} (L)^{(1-\epsilon_L^X+\epsilon_L^{\chi_x})} [(M_e)^{-1} [T(N, \tau, f_x)]^{(1-\chi_d)}]^{-1} (f)^{\chi_d} (-\epsilon_f^{\chi_d})]}_{(3)} \quad (61) $$

<footnote n="3">³We can first notice the presence of a negative general equilibrium effect in the second term on the RHS of $\epsilon_f^{\phi_d^*}$. All components of this term are strictly positive, except the elasticity of domestic trade with respect to the fixed production cost, which is strictly negative: $\epsilon_f^{\chi_d} < 0$. Despite that, $\epsilon_f^{\phi_d^*}$ remains strictly positive since the domestic trade share is always strictly lower than one ($\chi_d < 1$) at the trade equilibrium.</footnote>