International trade with sequential production

Abstract

We develop a tractable general equilibrium model of international trade with firm heterogeneity and sequential production, based on the partial equilibrium model of Antràs and Chor (Econometrica 81:2127–2204, 2013). The length of supply chains is endogenous, and in the autarky equilibrium more productive final-good producers are served by longer supply chains. International trade in final goods magnifies the differences in the length of supply chains between firms, with exporting firms being served by longer supply chains, and non-exporting firms being served by shorter supply chains than in autarky. We show that, for given model parameters, the gains from trade are larger than in the canonical Melitz model with Pareto-distributed productivities, since production along a sequential supply chain in the Antràs–Chor model is subject to external increasing returns to scale.

Appendices

Appendix A

In this Appendix, we show how to compute average profits $\overline{\pi }$ in the open economy. We can write

$$\begin{aligned} \overline{\pi }=\overline{\pi }_{d}+\chi \overline{\pi }_{x} \end{aligned}$$

where $\overline{\pi }_{d}$ are average profits made by all firms in their home market, and $\overline{\pi }_{x}$ are average profits made by exporting firms in their export market, which can be calculated as

$$\begin{aligned} \overline{\pi }_{x}&=\int _{\varphi _{x}^{*}}^{\infty }\pi _{x}\left( \varphi \right) \frac{g\left( \varphi \right) }{1-G\left( \varphi _{x}^{*}\right) }\hbox {d}\varphi \\&=\varTheta \theta \beta \left( 1-\beta \right) ^{\frac{\rho }{1-\rho }}\varOmega A \left[ M_{x}\left( \varphi _{x}^{*}\right) \right] ^{\frac{s\rho }{ 1-\rho }}\left( \varphi _{x}^{*}\right) ^{\frac{\rho }{1-\rho }}-f_{x} \end{aligned}$$

Using Eqs. (27), (28), and the exporter indifference condition

$$\begin{aligned} \beta \left( 1-\beta \right) ^{\frac{\rho }{1-\rho }}\varOmega A(\varphi _{x}^{*})^{\frac{\rho }{1-\rho }}\left[ M_{d}\left( \varphi _{x}^{*}\right) \right] ^{\frac{s\rho }{1-\rho }}\left[ (1+\theta )^{\frac{ \varepsilon (1-\rho )}{\rho }}-1\right] =f_{x}, \end{aligned}$$

we can rewrite this expression as

$$\begin{aligned} \overline{\pi }_{x}&=\left[ \frac{\varTheta \theta (1+\theta )^{s\varepsilon } }{(1+\theta )^{\frac{\varepsilon (1-\rho )}{\rho }}-1}-1\right] f_{x} \\&=\varTheta \theta (1+\theta )^{s\varepsilon }\xi ^{\varepsilon }f-f_{x} \end{aligned}$$

The average domestic profits $\overline{\pi }_{d}$ can be calculated as

$$\begin{aligned} \overline{\pi }_{d}&=\int _{\varphi _{d}^{*}}^{\infty }\pi \left( \varphi \right) \frac{g\left( \varphi \right) }{1-G\left( \varphi _{d}^{*}\right) }\hbox {d}\varphi \\&=\left( \varphi _{d}^{*}\right) ^{k}\beta \left( 1-\beta \right) ^{ \frac{\rho }{1-\rho }}\varOmega A\\&\quad \times \left\{ \int _{\varphi _{d}^{*}}^{\varphi _{x}^{*}}\left[ M_{d}\left( \varphi \right) \right] ^{\frac{s\rho }{ 1-\rho }}\varphi ^{\frac{\rho }{1-\rho }}g\left( \varphi \right) \hbox {d}\varphi +\int _{\varphi _{x}^{*}}^{\infty }\left[ M_{x}(\varphi )\right] ^{\frac{ s\rho }{1-\rho }}\varphi ^{\frac{\rho }{1-\rho }}g\left( \varphi \right) \hbox {d}\varphi \right\} -f\\&=\left( \varphi _{d}^{*}\right) ^{k}\beta \left( 1-\beta \right) ^{ \frac{\rho }{1-\rho }}\varOmega A\\&\quad \times \left\{ \begin{array}{c} \int _{\varphi _{d}^{*}}^{\infty }\left[ M_{d}\left( \varphi \right) \right] ^{\frac{s\rho }{1-\rho }}\varphi ^{\frac{\rho }{1-\rho }}g\left( \varphi \right) \hbox {d}\varphi \\ +\left[ (1+\theta )^{s\varepsilon }-1\right] \int _{\varphi _{x}^{*}}^{\infty }\left[ M_{d}(\varphi )\right] ^{\frac{s\rho }{1-\rho }}\varphi ^{ \frac{\rho }{1-\rho }}g\left( \varphi \right) \hbox {d}\varphi \end{array} \right\} -f \\&=\left( \varphi _{d}^{*}\right) ^{k}\beta \left( 1-\beta \right) ^{ \frac{\rho }{1-\rho }}\varOmega A\int _{\varphi _{d}^{*}}^{\infty }\left[ M_{d}\left( \varphi \right) \right] ^{\frac{s\rho }{1-\rho }}\varphi ^{\frac{ \rho }{1-\rho }}g\left( \varphi \right) \hbox {d}\varphi \\&\quad +\left[ (1+\theta )^{s\varepsilon }-1\right] \xi ^{-k}\left( \varphi _{x}^{*}\right) ^{k}\\&\quad \times \left\{ \beta \left( 1-\beta \right) ^{\frac{ \rho }{1-\rho }}\varOmega A\int _{\varphi _{x}^{*}}^{\infty }\left[ M_{d}(\varphi )\right] ^{\frac{s\rho }{1-\rho }}\varphi ^{\frac{\rho }{ 1-\rho }}g\left( \varphi \right) \hbox {d}\varphi \right\} -f \\&=\varTheta f+\left[ (1+\theta )^{s\varepsilon }-1\right] \xi ^{-k}\varTheta \frac{ f_{x}}{\left[ (1+\theta )^{\frac{\varepsilon (1-\rho )}{\rho }}-1\right] }-f\\&=\varTheta f+\left[ (1+\theta )^{s\varepsilon }-1\right] \xi ^{-k}\varTheta \xi ^{\varepsilon }f-f \end{aligned}$$

Substituting for $\overline{\pi }_{d}$, $\overline{\pi }_{x}$ and $\chi$, we finally get

$$\begin{aligned} \overline{\pi }&=\overline{\pi }_{d}+\chi \overline{\pi }_{x} \\&=\varTheta f+\left[ (1+\theta )^{s\varepsilon }-1\right] \varTheta \xi ^{\varepsilon -k}f-f+\xi ^{-k}\left[ \varTheta \theta (1+\theta )^{s\varepsilon }\xi ^{\varepsilon }f-f_{x}\right] \\&=\left( \varTheta -1\right) f+\left[ (1+\theta )^{s\varepsilon }-1+\theta (1+\theta )^{s\varepsilon }\right] \varTheta \xi ^{\varepsilon -k}f-\xi ^{-k}f_{x} \\&=\left( \varTheta -1\right) f +\left[ (1+\theta )^{s\varepsilon +1}-1\right] \varTheta \xi ^{\varepsilon -k}f-\xi ^{-k}f_{x} \\&=\left( \varTheta -1\right) f +\left[ (1+\theta )^{\frac{\varepsilon (1-\rho ) }{\rho }}-1\right] f\varTheta \xi ^{\varepsilon -k}-\xi ^{-k}f_{x} \\&=\left( \varTheta -1\right) f +f_{x}\xi ^{-\varepsilon }\varTheta \xi ^{\varepsilon -k}-\xi ^{-k}f_{x} \\&=\left( \varTheta -1\right) f +\varTheta \xi ^{-k}f_{x}-\xi ^{-k}f_{x} \\&=\left( \varTheta -1\right) \left( \chi f_{x}+f\right) \\&=\overline{\pi }_{a}\left( 1+\chi \frac{f_{x}}{f}\right) , \end{aligned}$$

where the last line is Eq. (29).

Appendix B

In this Appendix, we provide the proof of Proposition 1. Combining Eqs. (30) and (32) gives

$$\begin{aligned} \frac{M_{x}(\varphi )}{M_{a}(\varphi )}=(1+\theta )^{\frac{\varepsilon (1-\rho )}{\rho }}\left( 1+\chi \frac{f_{x}}{f}\right) ^{-\frac{\varepsilon }{k}} \end{aligned}$$

Rewriting this expression and substituting for $\chi$ gives

$$\begin{aligned} \frac{M_{x}(\varphi )}{M_{a}(\varphi )}=\left[ B(\theta ,f_x,f)\right] ^{-\frac{ \varepsilon }{k}} \end{aligned}$$

where

$$\begin{aligned} B (\theta ,f_{x},f )\equiv \left( 1+{\normalsize \theta }\right) ^{-\frac{\left( 1-\rho \right) k}{\rho }}\left[ 1+\left( \left( 1+\theta \right) ^{^{\frac{ \varepsilon (1-\rho )}{\rho }}}-1\right) ^{\frac{k}{\varepsilon }}\left( \frac{f_{x}}{f}\right) ^{1-\frac{k}{\varepsilon }}\right] \end{aligned}$$

From Eq. (28), we have

$$\begin{aligned} \frac{f_{x}}{f}>\left( 1+\theta \right) ^{\frac{ \varepsilon (1-\rho )}{\rho }}-1, \end{aligned}$$

and therefore

$$\begin{aligned} B (\theta ,f_{x},f )&<\left( 1+{\normalsize \theta }\right) ^{-\frac{\left( 1-\rho \right) k}{\rho }}\left[ 1+\left( \left( 1+\theta \right) ^{^{\frac{ \varepsilon (1-\rho )}{\rho }}}-1\right) ^{\frac{k}{\varepsilon }+1-\frac{k}{\varepsilon }}\right] \\&=\left( 1+{\normalsize \theta }\right) ^{\frac{\left( 1-\rho \right) }{\rho }\left( \varepsilon -k\right) } \end{aligned}$$

Since $k>\varepsilon >0$, this implies $B (\theta ,f_{x},f )<1$, and therefore $M_{x}(\varphi )/M_{a}(\varphi )>1$, which proves Proposition 1.

Appendix C

In this Appendix, we show how to derive for our model the share of expenditure on domestically produced goods. Using the solution for $\overline{\pi }_{x}$ from Appendix A, as well the symmetry of countries, we know that the average revenue earned by an importer—and therefore the expenditure falling on the average importer—is $\overline{r}_{x}=\frac{1}{\beta }\varTheta \theta (1+\theta )^{s\varepsilon }\xi ^{\varepsilon }f$. Use $\lambda _{x}$ to denote the share of expenditure going to imported products. Then we have

$$\begin{aligned} \lambda _{x} =\frac{\overline{r}_{x}\chi N}{L} =\frac{\varTheta \theta (1+\theta )^{s\varepsilon }\xi ^{\varepsilon }f\chi \beta L}{\beta L \left( 1+\chi \frac{f_x}{f}\right) \varTheta f} =\frac{\theta (1+\theta )^{s\varepsilon }\xi ^{\varepsilon }\chi }{1+\chi \frac{f_{x}}{f}} \end{aligned}$$

The share of domestic expenditure follows as

$$\begin{aligned} \lambda _{d} =1-\lambda _{x} =\frac{1+\chi \left[ \frac{f_{x}}{f}-\theta (1+\theta )^{s\varepsilon }\xi ^{\varepsilon }\right] }{1+\chi \frac{f_{x}}{f }}, \end{aligned}$$

and we can show that if $s>0$,

$$\begin{aligned} \frac{f_{x}}{f}-\theta (1+\theta )^{s\varepsilon }\xi ^{\varepsilon } =\frac{ f_{x}}{f}-\frac{f_{x}\theta (1+\theta )^{s\varepsilon }}{f\left[ (1+\theta )^{\frac{\varepsilon (1-\rho )}{\rho }}-1\right] } =\frac{f_{x}}{f}\left[ \frac{(1+\theta )^{s\varepsilon }-1}{(1+\theta )^{s\varepsilon +1}-1}\right] >0 \end{aligned}$$

Therefore

$$\begin{aligned} \lambda _{d}>\frac{1}{1+\chi \frac{f_{x}}{f}} \end{aligned}$$

The gains from the trade would be underestimated if we use the share of domestic expenditure as a sufficient statistic, conditional on observed trade data.

Appendix D

In this Appendix, we provide derivations for the results presented in Sect. 3. Solving program (34), we obtain the optimal choice of labour employment:

$$\begin{aligned} l\left[ m,\beta \left( m\right) \right] =\left[ \left( 1-\beta \left( m\right) \right) \rho \left( A^{1-\rho }\varphi ^{\rho }M^{\left( s+1-1/\alpha \right) \rho }\right) ^{\frac{\alpha }{\rho }}r\left( M,m,\varphi \right) ^{\frac{\rho -\alpha }{\rho }}\right] ^{\frac{1}{ 1-\alpha }} \end{aligned}$$

Plugging this expression into $r^{\prime }\left( M,m,\varphi \right)$ delivers the differential equation

$$\begin{aligned} r^{\prime }\left( M,m,\varphi \right)&=\frac{\rho }{\alpha }\left( A^{1-\rho }\varphi ^{\rho }M^{\left( s+1-1/\alpha \right) \rho }\right) ^{\frac{\alpha }{\rho \left( 1-\alpha \right) }}\\&\quad \times r\left( M,m,\varphi \right) ^{\frac{\rho -\alpha }{\rho \left( 1-\alpha \right) }}\left[ \rho \left( 1-\beta \left( m\right) \right) \right] ^{\frac{\alpha }{1-\alpha }}, \end{aligned}$$

and we can verify that the solution to this differential equation is given by:

$$\begin{aligned} r\left( M,m,\varphi \right)&=A\left( \varphi M^{s+1-1/\alpha }\right) ^{ \frac{\rho }{1-\rho }}\left( \frac{1-\rho }{1-\alpha }\right) ^{\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }}\\&\quad \times \rho ^{\frac{\rho }{ 1-\rho }}\left[ \int _{0}^{m}\left( 1-\beta \left( i\right) \right) ^{\frac{ \alpha }{1-\alpha }}{\text{ d }}i\right] ^{\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }} \\&=\varOmega A\left( \varphi M^{s+1-1/\alpha }\right) ^{\frac{\rho }{1-\rho }} \left[ \int _{0}^{m}\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{ 1-\alpha }}{\text{ d }}i\right] ^{\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }} \end{aligned}$$

Differentiating $r\left( M,m,\varphi \right)$ and substituting into $\int _{0}^{M}\beta \left( i\right) r^{\prime }\left( M,i,\varphi \right) di$, we can compute operating profits as:

$$\begin{aligned} \pi _{op}=&\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) } \varOmega A\left( \varphi M^{s+1-1/\alpha }\right) ^{\frac{\rho }{1-\rho }}\\&\quad \times \int _{0}^{M}\beta \left( i\right) \left( 1-\beta \left( i\right) \right) ^{ \frac{\alpha }{1-\alpha }}\left[ \int _{0}^{i}\left( 1-\beta \left( u\right) \right) ^{\frac{\alpha }{1-\alpha }}du\right] ^{\frac{\rho -\alpha }{\alpha \left( 1-\rho \right) }}di \end{aligned}$$

Following the analysis in Antràs and Chor (2013) and Alfaro et al. (2018), we know that for a final-good producer $\varphi$ served by a supply chain of length M, there exist thresholds $m^*_c \in \left( 0,M\right]$ and $m^*_s\in \left( 0,M\right]$ separating the stages that are outsourced from those that are integrated in the complements case and the substitutes case, respectively. With sequential complements, all production stages before $m^*_c$ are outsourced, with the remainder being integrated, while in the case of sequential substitutes all stages before $m^*_s$ are integrated, with the remainder being outsourced.

Consider first the complements case. Here, we can write the operating profit $\pi _{op}$ as

$$\begin{aligned} \pi _{op}=\varOmega A\left( \varphi M^{s+1-1/\alpha }\right) ^{\frac{\rho }{ 1-\rho }}\int _{0}^{M}\beta \left( i\right) \frac{\partial \left( \left[ \int _{0}^{i}\left( 1-\beta \left( u\right) \right) ^{\frac{\alpha }{1-\alpha }}du\right] ^{\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }}\right) }{\partial i}di, \end{aligned}$$

in which

$$\begin{aligned}&\int _{0}^{M}\beta \left( i\right) \frac{\partial \left( \left[ \int _{0}^{i}\left( 1-\beta \left( u\right) \right) ^{\frac{\alpha }{1-\alpha }}du\right] ^{\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }}\right) }{\partial i}di \\&\quad =\beta _{o}\left( 1-\beta _{o}\right) ^{\frac{\rho }{1-\rho }}m_{c}^{\frac{ \rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }}\\&\qquad +\beta _{v}\left( \begin{array}{c} \left[ m_{c}\left( 1-\beta _{o}\right) ^{\frac{\alpha }{1-\alpha }}+\left( M-m_{c}\right) \left( 1-\beta _{v}\right) ^{\frac{\alpha }{1-\alpha }}\right] ^{\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }} \\ -\left[ m_{c}\left( 1-\beta _{o}\right) ^{\frac{\alpha }{1-\alpha }}\right] ^{\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }} \end{array} \right) \end{aligned}$$

Taking the first-order condition with respect to $m_{c}$, we find:

$$\begin{aligned} \left( \beta _{v}-\beta _{o}\right) \left( 1-\beta _{o}\right) ^{\frac{\rho }{1-\rho }}&=\beta _{v}\left[ \left( 1-\beta _{o}\right) ^{\frac{\alpha }{ 1-\alpha }}-\left( 1-\beta _{v}\right) ^{\frac{\alpha }{1-\alpha }}\right] \\&\quad \times \left( \left( 1-\beta _{o}\right) ^{\frac{\alpha }{1-\alpha }}+\frac{ M-m_{c}^{*}}{m_{c}^{*}}\left( 1-\beta _{v}\right) ^{\frac{\alpha }{ 1-\alpha }}\right) ^{\frac{\rho -\alpha }{\alpha \left( 1-\rho \right) }} \end{aligned}$$

From this, we obtain

$$\begin{aligned} \frac{m_{c}^{*}}{M}=h_{c}^{*}=\left\{ 1+\left( \frac{1-\beta _{o}}{ 1-\beta _{v}}\right) ^{\frac{\alpha }{1-\alpha }}\left[ \left( \frac{1-\frac{ \beta _{o}}{\beta _{v}}}{1-\left( \frac{1-\beta _{o}}{1-\beta _{v}}\right) ^{-\frac{\alpha }{1-\alpha }}}\right) ^{\frac{\alpha \left( 1-\rho \right) }{ \rho -\alpha }}-1\right] \right\} ^{-1}, \end{aligned}$$

The threshold in the substitutes case can be derived in an analogous way, and we get

$$\begin{aligned} \frac{m_{s}^{*}}{M}=h_{s}^{*}=\left\{ 1+\left( \frac{1-\beta _{v}}{ 1-\beta _o}\right) ^{\frac{\alpha }{1-\alpha }}\left[ \left( \frac{\frac{ \beta _{v}}{\beta _o}-1}{\left( \frac{1-\beta _{v}}{1-\beta _o}\right) ^{- \frac{\alpha }{1-\alpha }}-1}\right) ^{\frac{\alpha \left( 1-\rho \right) }{ \rho -\alpha }}-1\right] \right\} ^{-1} \end{aligned}$$

Substituting for $m_{k}^{*}$ with $k=c,s$ in Eq. (35), we can write total revenue along a supply chain as

$$\begin{aligned} r\left( M,\varphi \right)&=\varOmega A\left( M^{\left( s+1-1/\alpha \right) }\varphi \right) ^{\frac{\rho }{1-\rho }}\\&\quad \times \left[ \int _{0}^{h_{k}^{*}M}\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{1-\alpha } }di+\int _{h_{k}^{*}M}^{M}\left( 1-\beta \left( j\right) \right) ^{\frac{ \alpha }{1-\alpha }}dj\right] ^{\frac{\rho \left( 1-\alpha \right) }{\left( 1-\rho \right) \alpha }} \\&=\varOmega A\left( M^{\left( s+1-1/\alpha \right) }\varphi \right) ^{\frac{ \rho }{1-\rho }}\\&\quad \times \left[ h_{k}^{*}M\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{1-\alpha }}+\left( M-h_{k}^{*}M\right) \left( 1-\beta \left( j\right) \right) ^{\frac{\alpha }{1-\alpha }}\right] ^{\frac{\rho \left( 1-\alpha \right) }{\left( 1-\rho \right) \alpha }} \\&=\varOmega A\left( M^{s}\varphi \right) ^{\frac{\rho }{1-\rho }}\left[ h_{k}^{*}\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{ 1-\alpha }}+\left( 1-h_{k}^{*}\right) \left( 1-\beta \left( j\right) \right) ^{\frac{\alpha }{1-\alpha }}\right] ^{\frac{\rho \left( 1-\alpha \right) }{\left( 1-\rho \right) \alpha }} \end{aligned}$$

with $\beta \left( i\right) =\beta _{o}$ and $\beta \left( j\right) =\beta _{v}$ in the complements case, and with $\beta \left( i\right) =\beta _{v}$ and $\beta \left( j\right) =\beta _{o}$ in the substitutes case.

From the expression of $r\left( M,m,\varphi \right)$, we have

$$\begin{aligned} r^{\prime }\left( M,m,\varphi \right)&=\frac{\rho \left( 1-\alpha \right) }{ \alpha \left( 1-\rho \right) }\varOmega A\left( \varphi M^{s+1-1/\alpha }\right) ^{\frac{\rho }{1-\rho }}\\&\quad \times \left[ \int _{0}^{m}\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{1-\alpha }}{\text{ d }}i\right] ^{\frac{\rho -\alpha }{\alpha \left( 1-\rho \right) }}\left( 1-\beta \left( m\right) \right) ^{ \frac{\alpha }{1-\alpha }}, \end{aligned}$$

and rewriting this expression in terms of $h=m/M$ gives

$$\begin{aligned} r^{\prime }\left( h,M,\varphi \right) =&\frac{\rho \left( 1-\alpha \right) }{ \alpha \left( 1-\rho \right) }\varOmega A\varphi ^{\frac{\rho }{1-\rho }}M^{ \frac{1-\left( s+1\right) \rho }{\rho -1}}\left( 1-\beta \left( j\right) \right) ^{\frac{\alpha }{1-\alpha }}\\&\quad \times \left[ h_{k}^{*}\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{1-\alpha }}+\left( h-h_{k}^{*}\right) \left( 1-\beta \left( j\right) \right) ^{\frac{\alpha }{1-\alpha } }\right] ^{\frac{\rho -\alpha }{\alpha \left( 1-\rho \right) }} \end{aligned}$$

if $h>h_{k}^{*}$ and

$$\begin{aligned} r^{\prime }\left( h,M,\varphi \right)&= \frac{\rho \left( 1-\alpha \right) }{ \alpha \left( 1-\rho \right) }\varOmega \varphi ^{\frac{\rho }{1-\rho }}M^{ \frac{1-\left( s+1\right) \rho }{\rho -1}}\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{1-\alpha }}\\&\quad \times \left[ h_{k}^{*}\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{1-\alpha }}\right] ^{\frac{\rho -\alpha }{\alpha \left( 1-\rho \right) }} \end{aligned}$$

if $h<h_{k}^{*}$. Since both suppliers’ profits and wage costs are proportional to $r^{\prime }\left( h,M,\varphi \right)$, we have

$$\begin{aligned} \frac{M\left( \varphi _{1}\right) }{M\left( \varphi _{2}\right) }=\left( \frac{\varphi _{1}}{\varphi _{2}}\right) ^{\varepsilon }, \end{aligned}$$

and therefore the length of supply chains is increasing in $\varphi$ with the same constant elasticity as in the benchmark model. Plugging the expression of $m_{k}^{*}$ with $k=c,s$ into the final-good producer’s operating profits $\pi _{op}$, we can show that $\pi _{op}=\varGamma _{k}\left( \beta _{v},\beta _{o},\alpha ,\rho \right) \varOmega A\left( M^{s}\varphi \right) ^{\frac{\rho }{ 1-\rho }}$ with $k=c,s$, in which $\varGamma _{k}\left( \beta _{v},\beta _{o},\alpha ,\rho \right)$ collects $\beta _{v}$ and $\beta _{o}$. The profit of a final-good producer with productivity $\varphi$ is given by

$$\begin{aligned} \pi \left( M,\varphi \right) =\varGamma _{k}\left( \beta _{v},\beta _{o},\alpha ,\rho \right) \varOmega A\left( M^{s}\varphi \right) ^{\frac{\rho }{1-\rho }} -f \end{aligned}$$

In analogy to the benchmark model, suppliers’ ex ante average profits along a supply chain are given by $\left( 1-\alpha \right) \left[ r\left( M,\varphi \right) -\pi _{op}\right] /M$, which are equalised across supply chains in equilibrium:

$$\begin{aligned}&\frac{\left( 1-\alpha \right) \left( \varLambda _{k}-\varGamma _{k}\right) \varOmega A \left[ M\left( \varphi _{1}\right) ^{s}\varphi _{1}\right] ^{\frac{\rho }{ 1-\rho }}}{M\left( \varphi _{1}\right) }\\&\quad =\frac{\left( 1-\alpha \right) \left( \varLambda _{k}-\varGamma _{k}\right) \varOmega A\left[ M\left( \varphi _{2}\right) ^{s}\varphi _{2}\right] ^{\frac{\rho }{1-\rho }}}{M\left( \varphi _{2}\right) }=f_{m} \end{aligned}$$

The zero cutoff profit condition $\pi _{op}=f$ follows as

$$\begin{aligned} \varOmega A\left[ M\left( \varphi ^{*}\right) ^{s}\varphi ^{*}\right] ^{ \frac{\rho }{1-\rho }}=\frac{f}{\varGamma _{k}} \end{aligned}$$

Immediately,

$$\begin{aligned} M\left( \varphi ^{*}\right) =\frac{f\left( 1-\alpha \right) \left( \varLambda _{k}-\varGamma _{k}\right) }{f_{m}\varGamma _{k}} \end{aligned}$$

and the average length of supply chains follows as $\overline{M}=\varTheta M\left( \varphi ^{*}\right)$. As in the benchmark model, the cutoff productivity $\varphi ^{*}$ is given by

$$\begin{aligned} \varphi ^{*}=\left[ \frac{\left( \varTheta -1\right) f}{f_{e}}\right] ^{ \frac{1}{k}} \end{aligned}$$

The average revenue of final-good producers equals $\varTheta f \varLambda _{k}/\varGamma _{k}$, the number of final-good producers is given by

$$\begin{aligned} N=\frac{\varGamma _{k}L}{\varLambda _{k}\varTheta f}, \end{aligned}$$

and the total number of suppliers follows as

$$\begin{aligned} \overline{M}N=\frac{\left( 1-\alpha \right) \left( \varLambda _{k}-\varGamma _{k}\right) }{\varLambda _{k}f_{m}}L. \end{aligned}$$

Using the zero profit condition for final-good producers, welfare per worker is given by

$$\begin{aligned} W=P^{-1}=\left( \frac{\varGamma _{k}\varOmega L}{f}\right) ^{\frac{1-\rho }{\rho }} \left[ M\left( \varphi ^{*}\right) \right] ^{s}\varphi ^{*} \end{aligned}$$

In the open economy, two additional costs are involved. An exporting firm has the total revenue that is a multiple $1+\theta$ of its domestic revenue. International trade has no effect on the threshold relative position $h_{k}^{*}$. In analogy to Eq. (26), we rewrite the indifference condition for the marginal exporting firm as:

$$\begin{aligned}&\left( 1+\theta \right) \varGamma _{k}\left( \beta _{v},\beta _{o},\alpha ,\rho \right) \varOmega A\left[ M_{x}\left( \varphi _{x}^{*}\right) ^{s}\varphi _{x}^{*}\right] ^{\frac{\rho }{1-\rho }}\\&\quad -\varGamma _{k}\left( \beta _{v},\beta _{o},\alpha ,\rho \right) \varOmega A\left[ M_{d}\left( \varphi _{x}^{*}\right) ^{s}\varphi _{x}^{*}\right] ^{\frac{\rho }{1-\rho } }=f_{x} \end{aligned}$$

Similarly, we can also show that the cross-regime SIC implies

$$\begin{aligned} \frac{M_{x}\left( \varphi \right) }{M_{d}\left( \varphi \right) }=\left( 1+\theta \right) ^{\frac{\varepsilon \left( 1-\rho \right) }{\rho }}, \end{aligned}$$

exactly as in the benchmark model. As a consequence, we find that many results from this model are still valid in the extension discussed here. In particular, the share of exporting firms $\chi$ is the same as before, and hence the gains from trade $W/W_{a}$ in this extension are the same as in our benchmark model.