where the terms (1), (2) and (3) on the RHS of the above equation respectively correspond to the origin-specific, the pair-specific, and the destination-specific components of the gravity equation. As before, $\varepsilon_y^X$ is the elasticity of aggregate expenditure with respect to individual income and $\varepsilon_L^X$ is the elasticity of aggregate expenditure with respect to population size, which are respectively given by:

$$ \left\{ \begin{aligned} \varepsilon_y^X &= \frac{\partial \log X}{\partial \log y} = \chi_d \varepsilon_y^{X_d} + (1 - \chi_d) \varepsilon_y^{X_x} > 0 \\ \varepsilon_L^X &= \frac{\partial \log X}{\partial \log L} = \chi_d \varepsilon_L^{X_d} + (1 - \chi_d) \varepsilon_L^{X_x} > 0 \end{aligned} \right. \qquad (62) $$

Added flexibility in our generalized Gorman-Pollak demand system allows the price elasticity of demand $\varepsilon_D(e)$ to vary across firms, which thus implies that $(\varepsilon_y^{X_x} \neq \varepsilon_y^{X_d})$ and $(\varepsilon_L^{X_x} \neq \varepsilon_L^{X_d})$. Again, homothetic CES demand stands as a special case which imposes a constant demand elasticity, thus yielding: $(\varepsilon_y^{X_x} = \varepsilon_y^{X_d})$, $(\varepsilon_L^{X_x} = \varepsilon_L^{X_d})$. Now by simply rearranging equation (61), I obtain the following structural gravity equation:

$$ X_x = \underbrace{M_e}_{(1)} \underbrace{\tau^{-\theta} (f_x)^{\varepsilon_{f_x}^{X_x}}}_{(2)} \underbrace{(y)^{(1-\varepsilon_y^X + \varepsilon_y^{X_x})} (L)^{(1-\varepsilon_L^X + \varepsilon_L^{X_x})}}_{(3)} \Psi \qquad (63) $$

where $\Psi = [M_e [T(N, \tau, f_x)]^{(1-\chi_d)} [(f)^{\varepsilon_{f_x}^{X_d}}]^{\chi_d}]^{-1}$ 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 deterring domestic trade, higher fixed domestic costs foster bilateral trade.

### A.4 Final expression of $\zeta(S)$

In subsection 4.1.1 of the main text, I established that the productivity elasticity of the elasticity of firm-level operating profits with respect to firm productivity boils down to:

$$ \frac{\partial \log \varepsilon_{\varphi}^{\pi^{\circ}}(e)}{\partial \log \varphi} = \zeta(S) = (-S) a(e, S) $$

where $a(e, S)$ is strictly positive, and $\varepsilon_{\varphi}^{\pi^{\circ}}(e) = \frac{\partial \log \pi^{\circ}(e)}{\partial \log \varphi} = (1 - \rho(S))(\varepsilon_D(e) - 1) + \eta \rho(S) > 0$. To prove this result, I proceed in three steps. First, I recall the definition of $\zeta(S)$. Second, I separately simplify the expression of each of its terms. Finally, I obtain the above result with a simple rearrangement while providing an explicit expression of the strictly positive term $a(e, S)$.
Step 1. Definition of $\zeta(S)$

By definition, $\zeta(S)$ is given by:

$$ \zeta(S) = \frac{\partial \log \varepsilon_{\varphi}^{\pi^{o}}(e)}{\partial \log \varphi} = \underbrace{\frac{\partial \log \varepsilon_{\varphi}^{\pi^{o}}(e)}{\partial \log \varepsilon_{D}(z)}}_{(1)} \underbrace{\frac{\partial \log \varepsilon_{D}(z)}{\partial \log z}}_{(2)} \underbrace{\frac{\partial \log z}{\partial \log e}}_{(3)} \underbrace{\frac{\partial \log e}{\partial \log \varphi}}_{(4)} $$

where $z = \frac{\mu(e)}{e}$ is the argument of the demand function, and $e = \frac{y}{cF(\Lambda)}$ is a measure of firm-level efficiency.

Step 2. Simplification of terms (1), (2), (3), and (4)

$$ \left\{ \begin{array}{l} (1) : \frac{\partial \log \varepsilon_{\varphi}^{\pi^{o}}(e)}{\partial \log \varepsilon_{D}(z)} = \frac{(1-\rho(S)) \varepsilon_{D}(e)}{\varepsilon_{\varphi}^{\pi^{o}}(e)} > 0 \\ (2) : \frac{\partial \log \varepsilon_{D}(z)}{\partial \log z} = S \le 0 \\ (3) : \frac{\partial \log z}{\partial \log e} = (\rho(S) - 1) < 0 \\ (4) : \frac{\partial \log e}{\partial \log \varphi} = \frac{\partial \log e}{\partial \log c} \frac{\partial \log c}{\partial \log \varphi} = 1 > 0 \end{array} \right. $$

Step 3. Rearrangement

Now by simply rearranging, I obtain the following result:

$$ \zeta(S) = (-S) \underbrace{\frac{(1 - \rho(S))^2 \varepsilon_D(e)}{\varepsilon_{\varphi}^{\pi^o}(e)}}_{a(e,S)>0} \le 0 $$

A.5 Final expression of welfare per worker W

In section 5.1.2 of the main text, I justified the choice of the price aggregator $\Lambda$ as a sufficient statistic for welfare per worker, $W = \Lambda$, with the fact that it is the single industry-wide indicator that captures the intensity of competition. I also established that by using the budget constraint and invoking the unbounded Pareto assumption A3, welfare per capita $W$ can be written as a function of solely the total mass of available varieties $M$ and the domestic productivity cutoff $\varphi_d^*$. To show this in a simple way, I proceed in two steps. First, by using the budget constraint and the Pareto assumption and rearranging, I express welfare per worker $W$ as a function of the total mass of available varieties $M$, the domestic cutoff $\varphi_d^*$ and the export cutoff $\varphi_x^*$. Then, using general equilibrium expressions of productivity cutoffs, I obtain a simple expression for their ratio $(\varphi_x^*) / (\varphi_d^*)$. This allows me to further simplify the expression of welfare per worker $W$ and write it as a function of solely the total mass of available varieties and the domestic cutoff: $W(M, \varphi_d^*)$.
Step 1. Solving for welfare per capita (W = Λ) using the budget constraint

As before, the budget constraint can be rewritten as:

$$
\int_0^{\bar{M}} \frac{p_i}{y} \frac{D_i(F(\Lambda) \frac{p_i}{y})}{H(\Lambda)} di = 1 \qquad (64)
$$

Using the definition of the elasticities of functions Dᵢ, H, and F in their determinants from subsec-
tion 2.1 in the main text, the above equation boils down to:

$$
M(\bar{p})^{[1-\epsilon_D(\bar{e})]} (y)^{[\epsilon_D(\bar{e})-1]} (\Lambda)^{-[\epsilon_F \epsilon_D(\bar{e})+\epsilon_H]} = 1 \quad (65)
$$

where $M = M_d + M_m = M_e [1 - G(\varphi_d^*)] + (N - 1) M_e [1 - G(\varphi_x^*)]$ is the total mass of available varieties in any country, and $\bar{p} = \bar{c} \mu(\bar{e})$ is the weighted average price at the industry level, with:

$$
\begin{equation}
\begin{cases}
\bar{c} = \left(\frac{\theta-1}{\theta}\right) \left[ \left(\frac{Y}{Y_W}\right) (\varphi_d^*)^{-1} + \left(\frac{Y_{RoW}}{Y_W}\right) \tau (\varphi_x^*)^{-1} \right] \\
\mu(\bar{e}) = \left[ \left(\frac{\theta}{\theta-1}\right) \frac{Y}{F(\Lambda)} \right]^{\rho(S)} \left[ \left(\frac{Y}{Y_W}\right) (\varphi_d^*)^{\rho(S)} + \left(\frac{Y_{RoW}}{Y_W}\right) (\tau)^{-\rho(S)} (\varphi_x^*)^{\rho(S)} \right]
\end{cases}
\tag{66}
\end{equation}
$$

where, $\frac{Y}{Y_W} = \frac{1}{N}$ 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. By plugging the above expressions of $\bar{c}$ and $\mu(\bar{e})$ in equation (65), and rearranging, we get:⁴

$$
W = \left[ \left( \frac{\theta}{\theta-1} \right) y \right]^{\frac{(1-\rho(S))(\epsilon_D(\bar{e})-1)}{\psi(.)}} (M)^{\frac{1}{\psi(.)}} \underbrace{\left[ \left( \frac{Y}{Y_W} \right) (\varphi_d^*)^{-1} + \left( \frac{Y_{RoW}}{Y_W} \right) \tau (\varphi_x^*)^{-1} \right]}_{(1) \bar{c}(\varphi_d^*, \varphi_x^*)}^{\frac{(1-\epsilon_D(\bar{e}))}{\psi(.)}} \underbrace{\left[ \left( \frac{Y}{Y_W} \right) (\varphi_d^*)^{\rho(S)} + \left( \frac{Y_{RoW}}{Y_W} \right) (\tau)^{-\rho(S)} (\varphi_x^*)^{\rho(S)} \right]}_{(2) \bar{\mu}(\varphi_d^*, \varphi_x^*)}^{\frac{(1-\epsilon_D(\bar{e}))}{\psi(.)}} \quad (67)
$$

where $\psi(.) = \psi(\epsilon_F, \epsilon_H, \epsilon_D(\bar{e}), \rho(S))$ is a strictly positive composite elasticity given by:

$$
\psi(\epsilon_F, \epsilon_H, \epsilon_D(\bar{e}), \rho(S)) = \epsilon_F \left[ (1 - \rho(S)) \epsilon_D(\bar{e}) + \rho(S) \right] + \epsilon_H \geq 1
$$

where $\epsilon_D(\bar{e})$ is the weighted average price elasticity of demand, $\rho(S)$ is the markup elasticity with respect to firm efficiency, and $\epsilon_F$ and $\epsilon_H$ are both dummies capturing the type of separability.⁵

Step 2. Expressing welfare per capita as a function of solely M and φ<sup>*</sup><sub>d</sub>: W(M, φ<sup>*</sup><sub>d</sub>)

Using general equilibrium expressions of domestic and export cutoffs, respectively from equations
(56) and (58), the ratio (φ\*_x / φ\*_d) can be written as:

$$
\frac{\varphi_x^*}{\varphi_d^*} = \tau \left[ \frac{(f_x)^{\epsilon_{f_x}^*}}{(f)^{\epsilon_f^*}} \right] (y)^{\Delta_{x,d}^y} (L)^{\Delta_{x,d}^L} (M_e)^{\Delta_{x,d}^{M_e}} [T(N, \tau, f_x)]^{\Delta_{x,d}^T} \quad (68)
$$

<footnote>$^4$Nominal wage, $w$, has been normalized to unity by choice of labor as numéraire. Also notice that the terms (1) and (2) on the RHS respectively capture endogenous components of the weighted average marginal cost $\bar{c}$ and the weighted average markup $\bar{\mu}$.</footnote>

<footnote>$^5$The weighted average price elasticity of demand is given by $\epsilon_D(\bar{e}) = (\frac{Y}{Y_W}) \epsilon_D(\bar{e}_d) + (\frac{Y_{RoW}}{Y_W}) \epsilon_D(\bar{e}_x)$.</footnote>
where $\Delta_{x,d}^y = \epsilon_y^{\varphi_x^*} - \epsilon_y^{\varphi_d^*}$, $\Delta_{x,d}^L = \epsilon_L^{\varphi_x^*} - \epsilon_L^{\varphi_d^*}$, $\Delta_{x,d}^{M_e} = \epsilon_{M_e}^{\varphi_x^*} - \epsilon_{M_e}^{\varphi_d^*}$, and $\Delta_{x,d}^T = \epsilon_{T(N,\tau,f_x)}^{\varphi_x^*} - \epsilon_{T(N,\tau,f_x)}^{\varphi_d^*}$. Now using the above relationship between the export and the domestic cutoff and rearranging, equation (67) can be rewritten as:

$$ W = \Lambda = \Omega(.) \underbrace{(M)^{\frac{1}{\psi(\varepsilon_F, \varepsilon_H, \varepsilon_D(\tilde{\theta}), \rho(S))}}_{(1) Variety \ effect} \underbrace{(\varphi_d^*)^{\frac{(\varepsilon_D(\tilde{\theta})-1)}{\psi(\varepsilon_F, \varepsilon_H, \varepsilon_D(\tilde{\theta}), \rho(S))}}_{(2)Selection \ effect} \underbrace{(\varphi_d^*)^{-\rho(S)} \frac{(\varepsilon_D(\tilde{\theta})-1)}{\psi(\varepsilon_F, \varepsilon_H, \varepsilon_D(\tilde{\theta}), \rho(S))}}_{(3)Markup \ effect} \quad (69) $$

where $\Omega(.) = [ (\frac{\theta}{\theta-1}) y ]^{\frac{(1-\rho(S))(\varepsilon_D(\tilde{\theta})-1)}{\psi(\varepsilon_F, \varepsilon_H, \varepsilon_D(\tilde{\theta}), \rho(S))}} \Omega_1(.) \ \Omega_2(.)$, with $\Omega_1(.)$ and $\Omega_2(.)$ are respectively given by:

$$ \left\{ \begin{aligned} \Omega_1(.) &= \left(\frac{Y}{Y_W}\right) + \left(\frac{Y_{RoW}}{Y_W}\right) \left[ \frac{(f_x)^{\varepsilon_x^{\varphi_x^*}}}{(f)^{\varepsilon_f^{\varphi_d^*}}} \right]^{-1} (y)^{-\Delta_{x,d}^y} (L)^{-\Delta_{x,d}^L} (M_e)^{-\Delta_{x,d}^{M_e}} [T(N, \tau, f_x)]^{-\Delta_{x,d}^T} \\ \Omega_2(.) &= \left(\frac{Y}{Y_W}\right) + \left(\frac{Y_{RoW}}{Y_W}\right) \left[ \frac{(f_x)^{\varepsilon_x^{\varphi_x^*}}}{(f)^{\varepsilon_f^{\varphi_d^*}}} \right]^{\rho(S)} (y)^{\rho(S)} \Delta_{x,d}^y (L)^{\rho(S)} \Delta_{x,d}^L (M_e)^{\rho(S)} \Delta_{x,d}^{M_e} [T(N, \tau, f_x)]^{\rho(S)} \Delta_{x,d}^T \end{aligned} \right. $$

## A.6 Proof of Theorem 1

**Theorem 1** establishes that in any market (domestic or export), super-convex demand provides an upper bound for the productivity cutoff in general equilibrium, while sub-convex demand provides a lower bound. Within these bounds, the CES delivers an intermediate result. This can be shown using the general equilibrium expression of the domestic cutoff in equation (56) as follows.<fnref n="6" /> Let $\varphi_d^*[super]$ and $\varphi_d^*[sub]$ respectively denote the domestic productivity cutoff under super-convex demand, and sub-convex demand. Their ratio $\frac{\varphi_d^*[super]}{\varphi_d^*[sub]}$ can be written as:

$$ \frac{\varphi_d^*[super]}{\varphi_d^*[sub]} = (f)^{\Delta_1} (M_e)^{\Delta_2} [T(N, \tau, f_x)]^{\Delta_3} (y)^{\Delta_4} (L)^{\Delta_5} \quad (70) $$

where $\Delta_1, \Delta_2, \Delta_3, \Delta_4$, and $\Delta_5$ are respectively given by:

$$ \left\{ \begin{aligned} \Delta_1 &= \varepsilon_f^{\varphi_d^*[super]} - \varepsilon_f^{\varphi_d^*[sub]}; \quad \Delta_2 = \varepsilon_{M_e}^{\varphi_d^*[super]} - \varepsilon_{M_e}^{\varphi_d^*[sub]} \\ \Delta_3 &= \varepsilon_{T(N, \tau, f_x)}^{\varphi_d^*[super]} - \varepsilon_{T(N, \tau, f_x)}^{\varphi_d^*[sub]}; \quad \Delta_4 = \varepsilon_y^{\varphi_d^*[super]} - \varepsilon_y^{\varphi_d^*[sub]} \\ \Delta_5 &= \varepsilon_L^{\varphi_d^*[super]} - \varepsilon_L^{\varphi_d^*[sub]} \end{aligned} \right. $$

where explicit expressions of the above elasticities are provided in equation (57) and imply that: $\varepsilon_y^{\pi^0}(e_d^*) = (-\varepsilon_{F(\Lambda)}^{\pi^0}(e_d^*)) = 1 + \varepsilon_q^{\pi^0}(e_d^*)$. By recalling that while the cutoff domestic producer faces the lowest demand elasticity under super-convex demand, it faces the highest demand elasticity under sub-convex demand ($\varepsilon_D(e_d^*)$ [super-convex] $\ll \varepsilon_D(e_d^*)$ [sub-convex]) , I prove that $\frac{\varphi_d^*[super]}{\varphi_d^*[sub]} > 1$.

<footnote n="6">⁶This result holds for both markets and can be alternatively proved using the general equilibrium expression of the export productivity cutoff in equation (58).</footnote>
CES demand is a special case where the cutoff productivity firm faces the same constant price elasticity of demand ($\epsilon_D(e_d^*)$ [CES] = $\epsilon_D \forall e$) and thus delivers an intermediate outcome.

$$ \varphi_d^*[sub - convex] < \varphi_d^*[CES] < \varphi_d^*[super - convex] $$

## A.7 Proof of Theorem 2

**Theorem 2** establishes that 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, the CES delivers an intermediate result. This can be shown using the general equilibrium expression of the ratio $\frac{\varphi_x^*}{\varphi_d^*}$ in equation (68), as follows:

$$ \frac{\varphi_x^*}{\varphi_d^*} = \underbrace{\left[ \frac{(f_x)^{\epsilon_{f_x}^*}}{(f)^{\epsilon_d^*}} \right]}_{(1)} \tau (y)^{\Delta_{x,d}^y} (L)^{\Delta_{x,d}^L} (M_e)^{\Delta_{x,d}^{M_e}} [T(N, \tau, f_x)]^{\Delta_{x,d}^T} \quad (71) $$

As highlighted in the main text, countries are symmetric and thus share the same characteristics, including individual income $y$, population size $L$, mass of entrants $M_e$, and the degree of exposure to trade $T(N, \tau, f_x)$. This, in turn, implies that the partitioning of firms by export status is solely driven by the presence of variable and fixed trade costs. Therefore, I restrict my focus on the first ratio (1) where explicit expressions of the elasticities it incorporates are provided in equations (57) and (59). Now by recalling that under super-convex demand, the firm at the domestic cutoff faces a lower demand elasticity than the firm at the export cutoff ($\epsilon_D(e_d^*) < \epsilon_D(e_x^*)$) and that this order is reversed under sub-convex demand ($\epsilon_D(e_d^*) > \epsilon_D(e_x^*)$), I prove that:

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

Finally, recalling that under CES demands both cutoff productivity firms face the same demand elasticity ($\epsilon_D(e_d^*) = \epsilon_D(e_x^*) = \epsilon_D$) ensures that the CES delivers an intermediate outcome, and thus completes the proof.

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

## A.8 Proof of Theorem 3

**Theorem 3** establishes that in our theoretical environment, the magnitude of the competitive pressure that higher exposure to trade exerts on domestic firms forcing the least productive among
them to exit (CPD) takes the following simple form:<fnref n="7" />

$$CPD = \frac{\partial \log \varphi_d^*}{\partial \log T(N, \tau, f_x)} = \frac{(1 - \chi_d)}{\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))} > 0$$

This can be shown with a four-step procedure.

### Step 1:

Let us recall that from equations (56) and (57), the elasticity of the domestic cutoff with respect to the trade exposure index is given by:

$$CPD = \frac{\partial \log \varphi_d^*}{\partial \log T(N, \tau, f_x)} = \frac{[\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^o}(e_d^*)]}{\varepsilon_{\varphi}^{\pi^o}(e_d^*)} \frac{(1 - \chi_d)}{(-\varepsilon_{\Lambda}^{X})} > 0$$

### Step 2:

Let us recall that $(-\varepsilon_{\Lambda}^{X})$ is the absolute value of the elasticity of aggregate expenditure with respect to the price aggregator $\Lambda$. Using equation (52), $(-\varepsilon_{\Lambda}^{X})$ boils down to:

$$(-\varepsilon_{\Lambda}^{X}) = -\frac{\partial \log X}{\partial \log \Lambda} = \chi_d (-\varepsilon_{\Lambda}^{X_d}) + (1 - \chi_d) (-\varepsilon_{\Lambda}^{X_x}) > 0 \quad (72)$$

Now using the expressions of $\varepsilon_{\Lambda}^{X_x}$ and $\varepsilon_{\Lambda}^{X_d}$, respectively from equations (41) and (43), I can simplify the above expression of $(-\varepsilon_{\Lambda}^{X})$ with successive rearrangement as follows:

$$\begin{align*}
(-\varepsilon_{\Lambda}^{X}) &= \chi_d (-\varepsilon_{\Lambda}^{X_d}) + (1 - \chi_d) (-\varepsilon_{\Lambda}^{X_x}) \\
&= \chi_d \left[ [\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^r(\bar{e}_d)] + [\theta - \varepsilon_{\varphi}^r(\bar{e}_d)] \frac{[\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^o}(e_d^*)]}{\varepsilon_{\varphi}^{\pi^o}(e_d^*)} \right] \\
&\quad + (1 - \chi_d) \left[ [\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^r(\bar{e}_x)] + [\theta - \varepsilon_{\varphi}^r(\bar{e}_x)] \frac{[\varepsilon_H - \varepsilon_F \varepsilon_{F(\Lambda)}^{\pi^o}(e_x^*)]}{\varepsilon_{\varphi}^{\pi^o}(e_x^*)} \right] \\
&= \varepsilon_H \left[ \chi_d \left(1 + \frac{[\theta - \varepsilon_{\varphi}^r(\bar{e}_d)]}{\varepsilon_{\varphi}^{\pi^o}(e_d^*)}\right) + (1 - \chi_d) \left(1 + \frac{[\theta - \varepsilon_{\varphi}^r(\bar{e}_x)]}{\varepsilon_{\varphi}^{\pi^o}(e_x^*)}\right) \right] \\
&\quad + \varepsilon_F \chi_d \left[ (-\varepsilon_{F(\Lambda)}^r(\bar{e}_d)) + (-\varepsilon_{F(\Lambda)}^{\pi^o}(e_d^*)) \frac{[\theta - \varepsilon_{\varphi}^r(\bar{e}_d)]}{\varepsilon_{\varphi}^{\pi^o}(e_d^*)} \right] \\
&\quad + \varepsilon_F (1 - \chi_d) \left[ (-\varepsilon_{F(\Lambda)}^r(\bar{e}_x)) + (-\varepsilon_{F(\Lambda)}^{\pi^o}(e_x^*)) \frac{[\theta - \varepsilon_{\varphi}^r(\bar{e}_x)]}{\varepsilon_{\varphi}^{\pi^o}(e_x^*)} \right] \\
&\Rightarrow (-\varepsilon_{\Lambda}^{X}) = \varepsilon_H \left[ \frac{1}{\varepsilon_{\varphi}^{\pi^o}(e_d^*)} \chi_d (\theta + \varepsilon_{\varphi}^{\pi^o}(e_d^*) - \varepsilon_{\varphi}^r(\bar{e}_d)) + \frac{1}{\varepsilon_{\varphi}^{\pi^o}(e_x^*)} (1 - \chi_d) (\theta + \varepsilon_{\varphi}^{\pi^o}(e_x^*) - \varepsilon_{\varphi}^r(\bar{e}_x)) \right]
\end{align*}$$

<footnote n="7">⁷CPD: Competitive Pressure on Domestic firms.</footnote>
$$ + \varepsilon_F \frac{1}{\varepsilon_\varphi^{\pi^o}(e_d^*)} \chi_d [ \varepsilon_\varphi^{\pi^o}(e_d^*) (-\varepsilon_{F(\Lambda)}^r(\bar{e}_d)) + (-\varepsilon_{F(\Lambda)}^{\pi^o}(e_d^*)) [\theta - \varepsilon_\varphi^r(\bar{e}_d)] ] \\ + \varepsilon_F \frac{1}{\varepsilon_\varphi^{\pi^o}(e_x^*)} (1 - \chi_d) [ \varepsilon_\varphi^{\pi^o}(e_x^*) (-\varepsilon_{F(\Lambda)}^r(\bar{e}_x)) + (-\varepsilon_{F(\Lambda)}^{\pi^o}(e_x^*)) [\theta - \varepsilon_\varphi^r(\bar{e}_x)] ] \quad (73) $$

where $\chi_d$ is the domestic trade share, and $\varepsilon_H$ and $\varepsilon_F$ are both preference dummies capturing the type of separability.<fnref n="8" /> Inspection of the expressions of firm-level elasticities in equations (25), (27), (29), and (31) reveals the following relationships:

$$ \begin{cases} \forall e, (-\varepsilon_{F(\Lambda)}^r(e)) = 1 + \varepsilon_\varphi^r(e) > 0 \\ \forall e, (-\varepsilon_{F(\Lambda)}^{\pi^o}(e)) = 1 + \varepsilon_\varphi^{\pi^o}(e) > 0 \end{cases} \quad (74) $$

Now using the above relationships, the penultimate line of the expression of $(-\varepsilon_\Lambda^X)$ in equation (73) further simplifies to:

$$ \begin{aligned} & \varepsilon_F \frac{1}{\varepsilon_\varphi^{\pi^o}(e_d^*)} \chi_d [ \varepsilon_\varphi^{\pi^o}(e_d^*) (-\varepsilon_{F(\Lambda)}^r(\bar{e}_d)) + (-\varepsilon_{F(\Lambda)}^{\pi^o}(e_d^*)) [\theta - \varepsilon_\varphi^r(\bar{e}_d)] ] \\ &= \varepsilon_F \frac{\chi_d}{\varepsilon_\varphi^{\pi^o}(e_d^*)} (-\varepsilon_{F(\Lambda)}^r(e_d^*)) [ (1 - \frac{1}{-\varepsilon_{F(\Lambda)}^{\pi^o}(e_d^*)}) (1 + \varepsilon_\varphi^r(\bar{e}_d)) + [\theta - \varepsilon_\varphi^r(\bar{e}_d)] ] \\ &= \varepsilon_F \frac{\chi_d}{\varepsilon_\varphi^{\pi^o}(e_d^*)} (1 + \varepsilon_\varphi^{\pi^o}(e_d^*)) [ \theta + (1 - \frac{1}{(1 + \varepsilon_\varphi^{\pi^o}(e_d^*))}) - \frac{\varepsilon_\varphi^r(\bar{e}_d)}{(1 + \varepsilon_\varphi^{\pi^o}(e_d^*))} ] \end{aligned} \quad (75) $$

Using the same relationships, the ultimate line of the expression of $(-\varepsilon_\Lambda^X)$ in equation (73) further simplifies to:

$$ \begin{aligned} & \varepsilon_F \frac{1}{\varepsilon_\varphi^{\pi^o}(e_x^*)} (1 - \chi_d) [ \varepsilon_\varphi^{\pi^o}(e_x^*) (-\varepsilon_{F(\Lambda)}^r(\bar{e}_x)) + (-\varepsilon_{F(\Lambda)}^{\pi^o}(e_x^*)) [\theta - \varepsilon_\varphi^r(\bar{e}_x)] ] \\ &= \varepsilon_F \frac{(1 - \chi_d)}{\varepsilon_\varphi^{\pi^o}(e_x^*)} (1 + \varepsilon_\varphi^{\pi^o}(e_x^*)) [ \theta + (1 - \frac{1}{(1 + \varepsilon_\varphi^{\pi^o}(e_x^*))}) - \frac{\varepsilon_\varphi^r(\bar{e}_x)}{(1 + \varepsilon_\varphi^{\pi^o}(e_x^*))} ] \end{aligned} \quad (76) $$

Hence, using equations (75) and (76), the expression of $(-\varepsilon_\Lambda^X)$ in equation (73) boils down to:

$$ (-\varepsilon_\Lambda^X) = \varepsilon_H \left[ \frac{1}{\varepsilon_\varphi^{\pi^o}(e_d^*)} \chi_d C_d^H + \frac{1}{\varepsilon_\varphi^{\pi^o}(e_x^*)} (1 - \chi_d) C_x^H \right] + \varepsilon_F \left[ \frac{1 + \varepsilon_\varphi^{\pi^o}(e_d^*)}{\varepsilon_\varphi^{\pi^o}(e_d^*)} \chi_d C_d^F + \frac{1 + \varepsilon_\varphi^{\pi^o}(e_x^*)}{\varepsilon_\varphi^{\pi^o}(e_x^*)} (1 - \chi_d) C_x^F \right] \quad (77) $$

where $C_d^H, C_x^H, C_d^F$, and $C_x^F$ are respectively given by:

$$ \begin{cases} C_d^H = [\theta + \varepsilon_\varphi^{\pi^o}(e_d^*) - \varepsilon_\varphi^r(\bar{e}_d)]; \quad C_x^H = [\theta + \varepsilon_\varphi^{\pi^o}(e_x^*) - \varepsilon_\varphi^r(\bar{e}_x)] \\ C_d^F = [\theta + 1 - \frac{1+\varepsilon_\varphi^r(\bar{e}_d)}{1+\varepsilon_\varphi^{\pi^o}(e_d^*)}]; \quad C_x^F = [\theta + 1 - \frac{1+\varepsilon_\varphi^r(\bar{e}_x)}{1+\varepsilon_\varphi^{\pi^o}(e_x^*)}] \end{cases} \quad (78) $$

<footnote n="8">⁸Consumer preferences are directly separable if $(\varepsilon_H = 0; \varepsilon_F = 1)$, and indirectly separable if $(\varepsilon_H = 1; \varepsilon_F = 0)$.</footnote>
**Step 3:**

Now using the final expression of $(-\epsilon_{\Lambda}^{X})$ from equation (77), the expression of the magnitude of the competitive pressure of trade exposure on domestic firms (CPD) initially defined in Step 1 can be simplified as follows:

$$ 
\begin{aligned}
CPD &= \frac{\partial \log \varphi_d^*}{\partial \log T(N, \tau, f_x)} = \frac{[\epsilon_H - \epsilon_F \epsilon_{F(\Lambda)}^{\pi^0}(e_d^*)]}{\epsilon_{\varphi}^{\pi^0}(e_d^*)} \frac{(1 - \chi_d)}{(-\epsilon_{\Lambda}^{X})} \\
&= \frac{(1 - \chi_d)}{\left( \frac{\epsilon_{\varphi}^{\pi^0}(e_d^*)}{[\epsilon_H - \epsilon_F \epsilon_{F(\Lambda)}^{\pi^0}(e_d^*)]} \right) (-\epsilon_{\Lambda}^{X})} = \frac{(1 - \chi_d)}{\Gamma(\theta, \epsilon_F, \epsilon_H, \epsilon_D(e), \rho(S))} > 0
\end{aligned}
$$

where $\Gamma(\theta, \epsilon_F, \epsilon_H, \epsilon_D(e), \rho(S))$ is a composite function given by:

$$ 
\begin{aligned}
\Gamma(\theta, \epsilon_F, \epsilon_H, \epsilon_D(e), \rho(S)) = \epsilon_H \left[ \frac{\chi_d}{[\epsilon_H - \epsilon_F \epsilon_{F(\Lambda)}^{\pi^0}(e_d^*)]} C_d^H + \frac{(1 - \chi_d)}{[\epsilon_H - \epsilon_F \epsilon_{F(\Lambda)}^{\pi^0}(e_d^*)]} \frac{\epsilon_{\varphi}^{\pi^0}(e_d^*)}{\epsilon_{\varphi}^{\pi^0}(e_x^*)} C_x^H \right] \\
+ \epsilon_F \left[ \chi_d \frac{1 + \epsilon_{\varphi}^{\pi^0}(e_d^*)}{[\epsilon_H - \epsilon_F \epsilon_{F(\Lambda)}^{\pi^0}(e_d^*)]} C_d^F + (1 - \chi_d) \frac{1 + \epsilon_{\varphi}^{\pi^0}(e_x^*)}{[\epsilon_H - \epsilon_F \epsilon_{F(\Lambda)}^{\pi^0}(e_d^*)]} \frac{\epsilon_{\varphi}^{\pi^0}(e_d^*)}{\epsilon_{\varphi}^{\pi^0}(e_x^*)} C_x^F \right]
\end{aligned}
\quad (79) $$

where the explicit expressions of $C_d^H$, $C_x^H$, $C_d^F$, and $C_x^F$ are provided by equation (78). We can notice that the first line on the RHS of equation (79) captures the case of indirectly-separable preferences ($\epsilon_H = 1$; $\epsilon_F = 0$) and its second line captures the case of directly-separable preferences ($\epsilon_F = 1$; $\epsilon_H = 0$). Now by applying this nesting of both types of separability and using the relationships in equation (74), the composite function $\Gamma(.)$ further simplifies to:

$$ \Gamma(.) = \epsilon_H [\chi_d C_d^H + (1 - \chi_d) \frac{\epsilon_{\varphi}^{\pi^0}(e_d^*)}{\epsilon_{\varphi}^{\pi^0}(e_x^*)} C_x^H] + \epsilon_F [\chi_d C_d^F + (1 - \chi_d) \frac{(1 + \epsilon_{\varphi}^{\pi^0}(e_x^*))}{(1 + \epsilon_{\varphi}^{\pi^0}(e_d^*))} \frac{\epsilon_{\varphi}^{\pi^0}(e_d^*)}{\epsilon_{\varphi}^{\pi^0}(e_x^*)} C_x^F] \quad (80) $$

Finally, by defining $R_{xd}^* = \frac{\epsilon_{\varphi}^{\pi^0}(e_x^*)}{\epsilon_{\varphi}^{\pi^0}(e_d^*)}$, the above expression of $\Gamma(.)$ can be rewritten as:

$$ \Gamma(.) = \epsilon_H [\chi_d C_d^H + (1 - \chi_d) C_x^H \Phi^H(R_{xd}^*)] + \epsilon_F [\chi_d C_d^F + (1 - \chi_d) C_x^F \Phi^F(R_{xd}^*)] \quad (81) $$

where $\Phi^H(R_{xd}^*)$ and $\Phi^F(R_{xd}^*)$ are both strictly decreasing functions in the ratio $R_{xd}^*$, and are respectively given by:

$$ 
\left\{ 
\begin{aligned}
\Phi^H(R_{xd}^*) &= (R_{xd}^*)^{-1}, \quad \text{with } R_{xd}^* = \frac{\epsilon_{\varphi}^{\pi^0}(e_x^*)}{\epsilon_{\varphi}^{\pi^0}(e_d^*)} = \frac{(1-\rho(S))(\epsilon_D(e_x^*)-1)+\eta\rho(S)}{(1-\rho(S))(\epsilon_D(e_d^*)-1)+\eta\rho(S)} \\
\Phi^F(R_{xd}^*) &= \frac{[\frac{\epsilon_{\varphi}^{\pi^0}(e_d^*)}{\epsilon_{\varphi}^{\pi^0}(e_d^*)} + R_{xd}^*]}{[\frac{1}{\epsilon_{\varphi}^{\pi^0}(e_d^*)} + 1]} (R_{xd}^*)^{-1}, \quad \text{with } R_{xd}^* = \frac{\epsilon_{\varphi}^{\pi^0}(e_x^*)}{\epsilon_{\varphi}^{\pi^0}(e_d^*)} = \frac{(1-\rho(S))(\epsilon_D(e_x^*)-1)+\eta\rho(S)}{(1-\rho(S))(\epsilon_D(e_d^*)-1)+\eta\rho(S)}
\end{aligned}
\right. 
\quad (82) $$
**Step 4:**

Now the final step is to provide a more explicit expression of each element ($C_d^H, C_x^H, C_d^F, C_x^F$) of the composite function $\Gamma(\cdot)$ using firm-level elasticities. In deed, using the elasticities of firm-level variables with respect to their determinants from equations (29), (31), (34) and (36), the expression of each of these elements can be written more explicitly as follows:

$$ \left\{ \begin{aligned} C_d^H &= [\theta + (1 - \rho(S)) (\varepsilon_D(e_d^*) - \varepsilon_D(\bar{e}_d)) + \eta \rho(S)] \\ C_x^H &= [\theta + (1 - \rho(S)) (\varepsilon_D(e_x^*) - \varepsilon_D(\bar{e}_x)) + \eta \rho(S)] \end{aligned} \right. \qquad (83) $$

$$ \left\{ \begin{aligned} C_d^F &= [\theta + 1 - \frac{1 + (1 - \rho(S)) (\varepsilon_D(\bar{e}_d) - 1)}{[(1 - \rho(S)) \varepsilon_D(e_d^*) + (1 + \eta) \rho(S)]}] \\ C_x^F &= [\theta + 1 - \frac{1 + (1 - \rho(S)) (\varepsilon_D(\bar{e}_x) - 1)}{[(1 - \rho(S)) \varepsilon_D(e_x^*) + (1 + \eta) \rho(S)]}] \end{aligned} \right. \qquad (84) $$

Therefore, as established in **Theorem 3**, in our theoretical environment, the magnitude of the competitive pressure that exposure to trade exerts on domestic firms forcing the least productive among them to exit the market (CPD), takes a simple form:

$$ CPD = \frac{\partial \log \varphi_d^*}{\partial \log T(N, \tau, f_x)} = \frac{(1 - \chi_d)}{\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))} > 0 $$

where $\chi_d$ is the domestic trade share in any country, the final expression of the composite function $\Gamma(\theta, \varepsilon_F, \varepsilon_H, \varepsilon_D(e), \rho(S))$ is provided by equation (81), and the explicit expressions of its components are provided by equations (82), (83), and (84).

## A.9 Proof of Proposition 2

**Proposition 2** establishes that 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. This can be shown as follows. As stated in the main text, 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 $\tau$. Accordingly, upon plugging the general equilibrium price aggregator $\Lambda$ from equation (55) in the partial equilibrium expression of firm-level operating profits from export sales in equation (35), the
absolute value of this total elasticity evaluates to:

$$ 
\begin{aligned}
\left(-\frac{d\log \pi^o(e_x)}{d\log \tau}\right) &= \underbrace{\left(-\frac{\partial \log \pi^o(e_x)}{\partial \log \tau}\right)}_{(I^+)>0} + \underbrace{\frac{\partial \log \pi^o(e_x)}{\partial \log \Lambda} \frac{\partial \log \Lambda}{\partial \log T(N, \tau, f_x)} \frac{\partial \log T(N, \tau, f_x)}{\partial \log \tau}}_{(C^-)<0} \\
&= \underbrace{\left(-\varepsilon_{\tau}^{\pi^o}(e_x)\right)}_{(I^+)>0} + \underbrace{\varepsilon_{\Lambda}^{\pi^o}(e_x) \left[ \frac{(1-\chi_d)\theta}{(-\varepsilon_{\Lambda}^X)} \right]}_{(C^-)<0} \quad (85)
\end{aligned}
$$

where $\theta$ is the shape parameter of the Pareto, $\chi_d$ is the domestic trade share in any country, the final expression of $(-\varepsilon_{\Lambda}^X)$ and those of its components are provided by equations (77) and (78), and the explicit expressions of $\varepsilon_{\Lambda}^{\pi^o}(e_x)$ and $(-\varepsilon_{\tau}^{\pi^o}(e_x))$ are respectively provided by equations (27) and (36) and given by:

$$ 
\left\{
\begin{array}{l}
\varepsilon_{\Lambda}^{\pi^o}(e_x) = -[\varepsilon_H + \varepsilon_F(-\varepsilon_{F(\Lambda)}^{\pi^o}(e_x))] = -[\varepsilon_H + \varepsilon_F[(1-\rho(S))\varepsilon_D(e_x) + (1+\eta)\rho(S)]]] < 0 \\
(-\varepsilon_{\tau}^{\pi^o}(e_x)) = \varepsilon_{\phi}^{\pi^o}(e_x) = (1-\rho(S))(\varepsilon_D(e_x) - 1) + \eta\rho(S) > 0
\end{array}
\right.
\quad (86)
$$

Inspection of equation (85) shows that 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. To determine the sign of the net outcome, I proceed in two steps. First, I examine the sign of the net outcome under homothetic CES demand. Then, I examine its sign beyond the CES.

**1. Sign of the net outcome under homothetic CES demand:** whether it is taken as a 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 has two immediate implications for both effects. The first is that the initial positive effect $(I^+)$ is fixed to $(\varepsilon_D - 1)$, as indicated by equation (86). The second follows by inspection of equations (85), (77) and (78) and consists in the fact that under CES demand, the negative competition effect $(C^-)$ is always equal to $[-(1 - \chi_d)(\varepsilon_D - 1)]$. Therefore, the net outcome is strictly positive and identical across exporters under this special case:

$$ \forall \varphi \ge \varphi_x^*, \left(-\frac{d\log \pi^o(e_x)}{d\log \tau}\right)[\text{CES}] = \chi_d(\varepsilon_D - 1) > 0 $$
## 2. Sign of the net outcome beyond the homothetic CES case: Here, I proceed in three steps.

**Step 1:** I start by recalling that the below relationship from equation (74) is valid for all exporters:

$$ \forall e_x, (-\varepsilon_{F(\Lambda)}^{\pi^0}(e_x)) = 1 + \varepsilon_{\phi}^{\pi^0}(e_x) > 0 \quad (87) $$

Accordingly, equation (85) can be rewritten as:

$$ \begin{aligned} \left(-\frac{d\log \pi^0(e_x)}{d\log \tau}\right) &= \underbrace{\varepsilon_{\phi}^{\pi^0}(e_x)}_{(I^+) > 0} - \underbrace{\left[\varepsilon_H + \varepsilon_F(1 + \varepsilon_{\phi}^{\pi^0}(e_x))\right]}_{(C^-) < 0} \left[\frac{(1 - \chi_d)\theta}{(-\varepsilon_{\Lambda}^X)}\right] \\ &= \underbrace{\varepsilon_{\phi}^{\pi^0}(e_x)}_{> 0} \left[1 - \frac{\left[\varepsilon_H + \varepsilon_F(1 + \varepsilon_{\phi}^{\pi^0}(e_x))\right]}{\varepsilon_{\phi}^{\pi^0}(e_x)}\right] \left[\frac{(1 - \chi_d)\theta}{(-\varepsilon_{\Lambda}^X)}\right] \end{aligned} \quad (88) $$

**Step 2:** Now by plugging the simplified expression of $(-\varepsilon_{\Lambda}^X)$ from equation (77) in equation (88) and rearranging, I obtain the following final expressions of the net outcome under each type of separability:

* Type 1: Direct Separability ($\varepsilon_H = 0$ ; $\varepsilon_F = 1$)

$$ \left(-\frac{d\log \pi^0(e_x)}{d\log \tau}\right) = \varepsilon_{\phi}^{\pi^0}(e_x) \left[ 1 - (1 - \chi_d) R_1^F(e_x, e_x^*) \underbrace{\frac{\theta}{[R_2^F(e_x^*, e_d^*) \chi_d C_d^F + (1 - \chi_d) C_x^F]}}_{R_3^F} \right] \quad (89) $$

where $R_1^F(e_x, e_x^*)$, $R_2^F(e_x^*, e_d^*)$, $C_d^F$, and $C_x^F$ are respectively given by:

$$ \left\{ \begin{aligned} R_1^F(e_x, e_x^*) &= \frac{\varepsilon_{\phi}^{\pi^0}(e_x^*) (1 + \varepsilon_{\phi}^{\pi^0}(e_x))}{\varepsilon_{\phi}^{\pi^0}(e_x) (1 + \varepsilon_{\phi}^{\pi^0}(e_x^*))}; & R_2^F(e_x^*, e_d^*) &= \frac{\varepsilon_{\phi}^{\pi^0}(e_x^*) (1 + \varepsilon_{\phi}^{\pi^0}(e_d^*))}{\varepsilon_{\phi}^{\pi^0}(e_d) (1 + \varepsilon_{\phi}^{\pi^0}(e_x^*))} \\ C_d^F &= [\theta + 1 - \frac{1 + \varepsilon_{\phi}^r(\bar{e}_d)}{1 + \varepsilon_{\phi}^{\pi^0}(e_d^*)}]; & C_x^F &= [\theta + 1 - \frac{1 + \varepsilon_{\phi}^r(\bar{e}_x)}{1 + \varepsilon_{\phi}^{\pi^0}(e_x^*)}] \end{aligned} \right. \quad (90) $$

* Type 2: Indirect separability ($\varepsilon_H = 1$ ; $\varepsilon_F = 0$)

$$ \left(-\frac{d\log \pi^0(e_x)}{d\log \tau}\right) = \varepsilon_{\phi}^{\pi^0}(e_x) \left[ 1 - (1 - \chi_d) R_1^H(e_x, e_x^*) \underbrace{\frac{\theta}{[R_2^H(e_x^*, e_d^*) \chi_d C_d^F + (1 - \chi_d) C_x^F]}}_{R_3^H} \right] \quad (91) $$

where $R_1^H(e_x, e_x^*)$, $R_2^H(e_x^*, e_d^*)$, $C_d^H$, and $C_x^H$ are respectively given by:

$$ \left\{ \begin{aligned} R_1^H(e_x, e_x^*) &= \frac{\varepsilon_{\phi}^{\pi^0}(e_x^*)}{\varepsilon_{\phi}^{\pi^0}(e_x)}; & R_2^H(e_x^*, e_d^*) &= \frac{\varepsilon_{\phi}^{\pi^0}(e_x^*)}{\varepsilon_{\phi}^{\pi^0}(e_d^*)} \\ C_d^H &= [\theta + \varepsilon_{\phi}^{\pi^0}(e_d^*) - \varepsilon_{\phi}^r(\bar{e}_d)]; & C_x^H &= [\theta + \varepsilon_{\phi}^{\pi^0}(e_x^*) - \varepsilon_{\phi}^r(\bar{e}_x)] \end{aligned} \right. \quad (92) $$

**Step 3:** Interpretation of the results and determination of the sign of the net outcome:

When demand is super convex ($S < 0$), the price elasticity of demand increases in firm productivity. This implies that the cutoff exporter faces a higher demand elasticity than the cutoff domestic
producer: $\varepsilon_{\phi}^{\pi^0}(e_x^*) > \varepsilon_{\phi}^{\pi^0}(e_d^*)$, for $\phi_x^* > \phi_d^*$. This also implies that any active exporter with productivity level exceeding the export cutoff ($\phi > \phi_x^*$) faces a higher demand elasticity than the firm at the export cutoff: $\forall \phi > \phi_x^*, \varepsilon_D(e_x) > \varepsilon_D(e_x^*)$, and so $\varepsilon_{\phi}^{\pi^0}(e_x) > \varepsilon_{\phi}^{\pi^0}(e_x^*)$. Inspection of equations (89), (90), (91) and (92) reveals then that under both types of separability, super-convex demand implies that: $\forall e_x > e_x^* > e_d^*, R_1^F(e_x, e_x^*) < 1, R_1^H(e_x, e_x^*) < 1, R_2^F(e_x^*, e_d^*) > 1, R_2^H(e_x^*, e_d^*) > 1$. It also leads to a negative deviation of $C_d^F, C_x^F$, and $C_d^H, C_x^H$ from their CES levels, which immediately implies that $R_3^F$ and $R_3^H$ are strictly higher than 1:

$$ \begin{cases} (C_d^F; C_x^F)[\text{super - convex}] < (C_d^F; C_x^F)[\text{CES}] = \theta \Rightarrow R_3^F[\text{super - convex}] > 1 \\ (C_d^H; C_x^H)[\text{super - convex}] < (C_d^H; C_x^H)[\text{CES}] = \theta \Rightarrow R_3^H[\text{super - convex}] > 1 \end{cases} $$

For instance, whether the CES case is taken as a homothetic exception under direct or indirect separability, the fact that it imposes the price elasticity of demand to be identical across firms and constant markups yields the following benchmark values: $R_1^F(e_x, e_x^*) = R_1^H(e_x, e_x^*) = R_2^F(e_x^*, e_D^*) = R_2^H(e_x^*, e_D^*) = 1$, $C_d^F = C_x^F = C_d^H = C_x^H = \theta > 0$. This is readily verified from equations (90) and (92) and now by inspection of equations (89) and (91), these CES benchmark values immediately imply that $R_3^F = R_3^H = 1$. Inspection of equations (89) and (91) reveals thus that regardless of the type of separability, super-convex demand yields a strictly positive net effect which is heterogeneous across exporters. As compared with the homothetic CES demand, super-convex magnifies the net positive effect for the most productive exporters (since $\lim_{e_x \to +\infty} R_1^H = \lim_{e_x \to +\infty} R_1^F \ll 1$), whereas it dampens it for the least productive ones (since $\lim_{e_x \to e_x^*} R_1^H = \lim_{e_x \to e_x^*} R_1^F = 1$). This result is reversed when demand is sub-convex. A summary of these different predictions is provided in Table 1 in the main text.

## Discussion: Disentangling the role of the curvature of demand from that of the type of separability in determining the net outcome:

When preferences are indirectly separable ($\varepsilon_H = 1$; $\varepsilon_F = 0$), firm-level markups are invariant to changes in the intensity of competition since the aggregate demand shifter $F(\Lambda)$ is fixed to unity ($\varepsilon_F = 0 \Rightarrow F(\Lambda) = 1$). Inspection of the first term [(1)] of $(C^-)$ in equation (88) clearly shows that under this type of separability, tougher competition leads only to a proportional decrease in the level of demand for all active exporters: [(1)] = $-\varepsilon_H = -1 < 0$. Hence, the competition effect $(C^-)$ solely corresponds to a quantity effect which is identical across active exporters. By contrast, when preferences are directly separable ($\varepsilon_H = 0$; $\varepsilon_F = 1$) and the CES case is excluded, firm-level markups are always responsive to changes in the intensity of competition since the aggregate demand shifter $F(\Lambda)$ is proportional to the price aggregator ($\varepsilon_F = 1 \Rightarrow F(\Lambda) = \Lambda$). Under direct separability, the first term [(1)] of $(C^-)$ in equation (88) boils down to: [(1)] = $-[(1 - \rho(S))\varepsilon_D(e_x)] + [-(1 + \eta)\rho(S)] < 0$.<fnref n="9" />

<footnote n="9">⁹This can be immediately obtained by recalling that $\varepsilon_{F(\Lambda)}^{\pi^0}(e_x) = \frac{\partial \log \pi^0(e_x)}{\partial \log F(\Lambda)} = -[(1 - \rho(S))\varepsilon_D(e_x) + (1 + \eta)\rho(S)] < 0$.</footnote>
This expression clearly indicates that the quantity effect is always negative and firm-specific, and the price effect may be positive or negative depending on demand curvature. That is, under sub-convex demand, the quantity effect is magnified for less productive exporters since they face higher demand elasticity, and the price effect is negative ($\rho(S) > 0$) as tougher competitive conditions make each single exporter relatively less efficient, and so forces all active exporters to reduce their markups. These patterns are reversed when demand is super-convex: now the quantity effect is magnified for more productive exporters as they face a higher demand elasticity, and the price effect is positive since firm-level markups are now increasing in the intensity of competition ($\rho(S) < 0$). As previously shown, irrespective of the type of separability, demand curvature critically determines whether the net effect is firm-specific or not and, if so, which category of exporters—either the most or the least productive—experiences a magnified impact. Since the type separability affects only the pattern of the competition effect, it can thus be regarded as playing a secondary role in driving the overall result.

## A.10 Analysis of Markup Response to Trade Exposure

### A.10.1 A Simple Measure for Average Markup Response

Using equations (43) and (44) from the main text and invoking the unbounded Pareto assumption A3, the weighted average markup at the industry level can be written as:

$$ \mu(\bar{e}) = \left[ \frac{y}{F(\Lambda)} \right]^{\rho(S)} \left[ \left( \frac{Y}{Y_W} \right) (\bar{\phi}_d)^{\rho(S)} + \left( \frac{Y_{RoW}}{Y_W} \right) \left( \frac{\theta}{\theta-1} \varphi_x^* \tau^{-1} \right)^{\rho(S)} \right] \quad (93) $$

Now under each of the aforementioned scenarios, the adjustment of the weighted average markup at the industry level $\mu(\bar{e})$ in response to a small change in trade exposure can be simply measured as follows:

$$ \left\{ \begin{array}{l} \Delta\mu(\bar{e})|(-\tau) = \left(-\frac{d\log \mu(\bar{e})}{d\log \tau}\right) = \underbrace{\alpha \rho(S) \left(1 - \frac{\partial \log \varphi_x^*}{\partial \log \tau}\right)}_{\text{direct markup effect}} + \underbrace{\varepsilon_F (-\rho(S)) \frac{\partial \log \Lambda}{\partial \log T(N, \tau, f_x)} \left(-\frac{\partial \log T(N, \tau, f_x)}{\partial \log \tau}\right)}_{\text{indirect markup effect}} \le 0 \\[1.5em] \Delta\mu(\bar{e})|(-f_x) = \left(-\frac{d\log \mu(\bar{e})}{d\log f_x}\right) = \underbrace{0}_{\text{direct markup effect}} + \underbrace{\varepsilon_F (-\rho(S)) \frac{\partial \log \Lambda}{\partial \log T(N, \tau, f_x)} \left(-\frac{\partial \log T(N, \tau, f_x)}{\partial \log f_x}\right)}_{\text{indirect markup effect}} \ge 0 \\[1.5em] \Delta\mu(\bar{e})|(+N) = \frac{d\log \mu(\bar{e})}{d\log N} = \underbrace{0}_{\text{direct markup effect}} + \underbrace{\varepsilon_F (-\rho(S)) \frac{\partial \log \Lambda}{\partial \log T(N, \tau, f_x)} \frac{\partial \log T(N, \tau, f_x)}{\partial \log N}}_{\text{indirect markup effect}} \le 0 \end{array} \right. \quad (94) $$

where $\alpha = \frac{(Y_{RoW}) \mu(\bar{e}_x)}{\mu(\bar{e})} \in ]0, 1[$, and as before, $\varepsilon_F$ is a dummy variable equal to 1 if preferences are directly-separable and zero otherwise, and $\rho(S) = \frac{\delta S}{1+\delta S} \le 0$ is the elasticity of firm-level markups with respect to firm-level efficiency. Let us first recall that under homothetic CES demand, the
markup is identical across firms and exogenously fixed to ($m = \frac{\epsilon_D}{\epsilon_D - 1}$). Thus, under this special case, the markup is unresponsive to changes in an economy's trade exposure, as reflected by the zero markup elasticity: $\rho(S) = 0$. By contrast, markups may be responsive beyond the CES and their adjustment is captured by the sum of two effects: (i) the direct markup effect (hereafter *DME*), and (ii) the indirect markup effect (hereafter *IME*), which can be explained in detail as follows.

(i) the direct markup effect (DME).– This effect is captured by the first term on the RHS of each measure of average markup response in equation (94). Visibly, this effect is operative only when an increase in trade exposure occurs through a small reduction in the variable trade cost, and thus directly affects the price competitiveness of exporters. The *DME* can be decomposed into two sub-effects. The first, denoted by *DME*_(1)_, occurs on active exporters and can be explained as follows: at the initial level of competition, a small reduction in the variable trade cost makes active exporters relatively more efficient. This leads either to an upward markup adjustment ($\rho(S) > 0$) if demand is sub-convex or to a downward adjustment ($\rho(S) < 0$) if demand is super-convex. By inspection of equation (94), we can see that this sub-effect is captured by ($\alpha \rho(S)$). The second sub-effect, denoted by *DME*_(2)_, occurs on new exporters and can be explained as follows: at the initial level of competition, a small reduction in the variable trade cost induces a proportional decrease in the export cutoff, $\frac{\partial \log \varphi_x^*}{\partial \log \tau} = 1$, and thus enables domestic firms with productivity $\varphi \to (\varphi_x^*)^{-}$ to start exporting. These new exporters are relatively less productive, and may set lower markups ($\rho(S) > 0$) if demand is sub-convex or higher markups ($\rho(S) < 0$) if demand is super-convex. From equation (94), we can see that this sub-effect is captured by [$\alpha(-\rho(S)) \frac{\partial \log \varphi_x^*}{\partial \log \tau}$].

(ii) the indirect markup effect (IME).– This effect is captured by the second term on the RHS of each measure of average markup response in equation (94). Visibly, this effect is always operative under the three different scenarios. The *IME* captures the adjustment of firm-level markups in response to an increase in the demand shifter $F(\Lambda)$, which reflects a more intense level of competition. Whether such a tougher competitive environment leads to a decrease ($if \rho(S) > 0$) or to an increase ($if \rho(S) < 0$) in domestic and foreign markups depends on demand curvature. Importantly, while the sign of the indirect markup effect hinges on demand curvature, its existence crucially depends on the type of separability. As previously mentioned, the aggregate demand shifter $F(\Lambda)$ can be either a constant equal to one when preferences are indirectly-separable ($\epsilon_F = 0$), or a monotonically increasing function in $\Lambda$ when preferences are directly-separable ($\epsilon_F = 1$). Hence, the indirect markup effect is absent when preferences are indirectly separable, and active only when preferences are directly separable.

## A.10.2 A Granular Analysis of Markups Response to Small Changes in Trade Exposure

In this subsection, I disentangle the role of demand curvature from that of the type of separability in determining the sign of the average markup response ($\Delta \mu(\bar{e})$) under each of the three scenarios.
(i). A small reduction in the variable trade cost ($-\tau$)

From equation (94), the adjustment of the weighted average markup at the industry level in response to a small reduction in the variable trade cost is given by:

$$ \Delta\mu(\bar{e})|(-\tau) = \underbrace{\alpha\rho(S)\left(1 - \frac{\partial\log\varphi_x^*}{\partial\log\tau}\right)}_{\text{direct markup effect}} + \underbrace{\varepsilon_F(-\rho(S))\frac{\partial\log\Lambda}{\partial\log T(N, \tau, f_x)}\left(-\frac{\partial\log T(N, \tau, f_x)}{\partial\log\tau}\right)}_{\text{indirect markup effect}} \le 0 \quad (95) $$

(2)-1) Direct markup effect (DME).– As clearly indicated in equation (30) from the main text, the export productivity cutoff always increases proportionally with the variable trade cost, $\frac{\partial\log\varphi_x^*}{\partial\log\tau} = 1$, at the initial level of competition. It follows then by inspection of the above equation that the direct markup effect (DME) always vanishes regardless of the curvature of demand and of the type of separability. This implies that the two opposite sub-effects, $DME_{(1)}$ and $DME_{(2)}$, exactly cancel out. That is, when demand is sub-convex (super-convex), the fact that active exporters set higher (lower) markups upon a small reduction in the variable trade cost exactly cancels out with the fact that new exporters set relatively lower (higher) markups.<fnref n="10" /> The underlying economics are driven by the presence of fixed trade costs in the current environment and the unbounded Pareto assumption. While the presence of fixed export costs ensures that the second sub-effect, $DME_{(2)}$, is non-negligible since cutoff-varieties are consumed in strictly positive amounts, unbounded Pareto immediately implies that these two sub-effects exactly cancel out. In this sense, the fact that the direct markup effect always vanishes in the current setting ($DME = 0$) can be seen as a straightforward implication of the presence of fixed trade costs and unbounded Pareto distribution.

(2)-2) Indirect markup effect (IME).– Inspection of equation (95) reveals that the existence of the indirect markup effect (IME) crucially depends on the type of separability. That is, when preferences are indirectly-separable, the indirect markup effect is absent since $\varepsilon_F = 0$ under this case. For instance, as described in section 2, under this class of preferences, firm-level markups vary with measures of absolute firm-level efficiency since the aggregate demand shifter $F(\Lambda)$ is a constant fixed to unity under this case. This implies then that firm-level markups are invariant to changes in the intensity of competition under indirect separability, as in Bertoletti et al. (2018).

By contrast, the indirect markup effect (IME) is always operative when preferences are directly separable (since $\varepsilon_F = 1$), and its sign crucially depends on demand curvature. A small reduction in the variable trade cost induces a more intense level of competition, which makes each active firm relatively less efficient. This may force domestic and foreign firms to reduce their markups when demand is sub-convex, as in Krugman (1979), or may lead to an increase in markups of all firms when demand is super-convex, as in (Zhelobodko et al., 2012; Mrázová and Neary, 2017).

<footnote n="10">¹⁰Note that "relatively" refers to an immediate comparison of new exporters with already active ones.</footnote>
(ii). A small reduction in the fixed trade cost ($-f_x$)

From equation (94), the adjustment of the weighted average markup at the industry level in response to a small reduction in the fixed trade cost is given by:

$$ \Delta\mu(\bar{e})|(-f_x) = \underbrace{(-\frac{d\log \mu(\bar{e})}{d\log f_x})}_{\text{direct markup effect}} + \underbrace{\varepsilon_F(-\rho(S)) \frac{\partial \log \Lambda}{\partial \log T(N, \tau, f_x)} (-\frac{\partial \log T(N, \tau, f_x)}{\partial \log f_x})}_{\text{indirect markup effect}} \le 0 \quad (96) $$

Inspection of the above equation clearly shows that while the direct markup effect is absent under this scenario, the indirect markup effect (IME) is operative only when preferences are directly separable ($\varepsilon_F = 1$). As before, the sign of the (IME) is pinned down by the curvature of demand. A small reduction in the fixed trade cost allows an additional fringe of purely domestic firms to start exporting. This induces a more intense level of competition, and thus makes each active firm relatively less efficient. This may force domestic and foreign firms to reduce their markups when demand is sub-convex, as in Krugman (1979), or may lead to an upward adjustment of markups of all firms when demand is super-convex, as in (Zhelobodko et al., 2012; Mrázová and Neary, 2017).

(iii). A small increase in the number of trading countries ($+N$)

From equation (94), the adjustment of the weighted average markup at the industry level in response to a small increase in the number of trading partners is given by:

$$ \Delta\mu(\bar{e})|(+N) = \underbrace{\frac{d\log \mu(\bar{e})}{d\log N}}_{\text{direct markup effect}} + \underbrace{\varepsilon_F(-\rho(S)) \frac{\partial \log \Lambda}{\partial \log T(N, \tau, f_x)} \frac{\partial \log T(N, \tau, f_x)}{\partial \log N}}_{\text{indirect markup effect}} \le 0 \quad (97) $$

The above equation clearly shows that while the direct markup effect is absent under this scenario, the indirect markup effect (IME) is operative when preferences are directly-separable ($\varepsilon_F = 1$). As before, the sign of the (IME) is pinned down by the curvature of demand. A small increase in the number of trading countries intensifies competition in any economy as its active firms compete henceforth with an additional mass of highly productive exporters. This, in turn, makes each active firm relatively less efficient and may force domestic and foreign firms in any economy to reduce their markups when demand is sub-convex, as in Krugman (1979), or may lead to an upward adjustment of markups of all firms when demand is super-convex, as in (Zhelobodko et al., 2012; Mrázová and Neary, 2017).
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