Appendix
1.1 A.1 Productivity thresholds
The free-entry condition can be written as \(\ \int _{\varphi _{D}}^{\infty } \left[ W^{1-\sigma }\varphi ^{\sigma -1}B-WF_{D}\right] dH\left( \varphi \right) +\int _{\varphi _{X}}^{\infty }\left[ W^{1-\sigma }\varphi ^{\sigma -1}B^{*}\tau ^{1-\sigma }-WF_{X}\right] dH\left( \varphi \right) =W\delta F_{E}.\) Next insert the productivity threshold conditions, \(W^{1-\sigma }\left( \varphi _{D}\right) ^{\sigma -1}B=WF_{D}\) and \(W^{1-\sigma }\left( \varphi _{X}\right) ^{\sigma -1}B^{*}\tau ^{1-\sigma }=WF_{X}\) to obtain
$$\begin{aligned} \frac{F_{D}}{\delta F_{E}}\int _{\varphi _{D}}^{\infty }\left[ \left( \frac{ \varphi }{\varphi _{D}}\right) ^{\sigma -1}-1\right] dH\left( \varphi \right) +\frac{F_{X}}{\delta F_{E}}\int _{\varphi _{X}}^{\infty }\left[ \left( \frac{\varphi }{\varphi _{X}}\right) ^{\sigma -1}-1\right] dH\left( \varphi \right) =1 \end{aligned}$$
which by use of the Pareto distribution of productivities can be written
$$\begin{aligned} \frac{F_{D}}{\delta F_{E}}\frac{\left( \sigma -1\right) }{k-\left( \sigma -1\right) }\left( \frac{\varphi _{D}}{\varphi _{\min }}\right) ^{-k}+\frac{ F_{X}}{\delta F_{E}}\frac{\left( \sigma -1\right) }{k-\left( \sigma -1\right) }\left( \frac{\varphi _{X}}{\varphi _{\min }}\right) ^{-k}=1 \end{aligned}$$
From the export threshold conditions it follows that \(\left( \frac{\varphi _{D}}{\varphi _{X}}\right) ^{\sigma -1}=\frac{F_{D}}{F_{X}}\frac{B^{*}}{B }\tau ^{1-\sigma }\) and we thus obtain
$$\begin{aligned} \varphi _{D} \; = \; & \varphi _{{\min }} \left( {\frac{{\sigma - 1}}{{k - \left( {\sigma - 1} \right)}}\frac{{F_{D} }}{{\delta F_{E} }}} \right)^{{\frac{1}{k}}} \left( {1 + \left( {\frac{{F_{X} }}{{F_{D} }}} \right)^{{1 - \frac{k}{{\sigma - 1}}}} \tau ^{{ - k}} \left( {\frac{B}{{B^{*} }}} \right)^{{ - \frac{k}{{\sigma - 1}}}} } \right)^{{\frac{1}{k}}} \\ & = \varphi _{{aut}} \left( {1 + \Phi \left( {\frac{B}{{B^{*} }}} \right)^{{ - \frac{k}{{\sigma - 1}}}} } \right)^{{\frac{1}{k}}} \\ \end{aligned}$$
and \(\varphi _{X}=\varphi _{D}\tau \left( \frac{F_{X}}{F_{D}}\frac{B}{ B^{*}}\right) ^{\frac{1}{\sigma -1}}=\varphi _{aut}\left( 1+\Phi ^{-1}\left( \frac{B}{B^{*}}\right) ^{\frac{k}{\sigma -1}}\right) ^{\frac{ 1}{k}}\left( \frac{F_{D}}{F_{X}}\right) ^{-\frac{1}{k}}\), where \(\varphi _{aut}\equiv \varphi _{\min }\left( \frac{\sigma -1}{k-\left( \sigma -1\right) }\frac{F_{D}}{\delta F_{E}}\right) ^{\frac{1}{k}}\) and \(\Phi \equiv \left( \frac{F_{X}}{F_{D}}\right) ^{1-\frac{k}{\sigma -1}}\tau ^{-k}\) .
1.2 A.2 General equilibrium
Balanced trade
Balanced trade requires that aggregate imports (I) equal aggregate exports (X), where \(I=N^{*}\int _{\varphi _{X}^{*}}^{\infty }E\left( P\right) ^{\sigma -1}\left( \frac{\sigma }{\sigma -1}\frac{W^{*}\tau ^{*}}{\varphi }\right) ^{1-\sigma }\frac{dH^{*}\left( \varphi \right) }{1-H^{*}\left( \varphi _{D}^{*}\right) }\) and \(X=N\int _{\varphi _{X}}^{\infty }E^{*}\left( P^{*}\right) ^{\sigma -1}\left( \frac{\sigma }{\sigma -1}\frac{W\tau }{\varphi }\right) ^{1-\sigma }\frac{dH\left( \varphi \right) }{1-H\left( \varphi _{D}\right) }\). Using the demand components, i.e., \(B\equiv EP^{\sigma -1}\left( \frac{\sigma }{ \sigma -1}\right) ^{-\sigma }\frac{1}{\sigma -1}\) and \(B^{*}\equiv E^{*}\left( P^{*}\right) ^{\sigma -1}\left( \frac{\sigma }{\sigma -1} \right) ^{-\sigma }\frac{1}{\sigma -1}\), the balanced trade condition reads \(N^{*}B\left( W^{*}\tau ^{*}\right) ^{1-\sigma }\int _{\varphi _{X}^{*}}^{\infty }\left( \varphi \right) ^{\sigma -1}\frac{dH^{*}\left( \varphi \right) }{1-H^{*}\left( \varphi _{D}^{*}\right) } =NB^{*}\left( W\tau \right) ^{1-\sigma }\int _{\varphi _{X}}^{\infty }\left( \varphi \right) ^{\sigma -1}\frac{dH\left( \varphi \right) }{ 1-H\left( \varphi _{D}\right) }\). Imposing the Pareto distributions, this expression can be written
$$\begin{aligned} \left( \frac{W}{W^{*}}\frac{\tau }{\tau ^{*}}\right) ^{\sigma -1}= \frac{k^{*}-\left( \sigma -1\right) }{k-\left( \sigma -1\right) }\frac{k }{k^{*}}\frac{N}{N^{*}}\frac{B^{*}}{B}\frac{\left( \varphi _{D}\right) ^{k}}{\left( \varphi _{D}^{*}\right) ^{k^{*}}}\frac{ \left( \varphi _{X}\right) ^{\sigma -1-k}}{\left( \varphi _{X}^{*}\right) ^{\sigma -1-k^{*}}} \end{aligned}$$
(7)
The metric of trade openness is given as
$$\begin{aligned} O\equiv & {} \frac{I}{E}=\frac{N^{*}\int _{\varphi _{X}^{*}}^{\infty }E\left( P\right) ^{\sigma -1}\left( \frac{\sigma }{\sigma -1}\frac{W^{*}\tau ^{*}}{\varphi }\right) ^{1-\sigma }\frac{dH^{*}\left( \varphi \right) }{1-H^{*}\left( \varphi _{D}^{*}\right) }}{N\int _{\varphi _{D}}^{\infty }E\left( P\right) ^{\sigma -1}\left( \frac{\sigma }{\sigma -1} \frac{W}{\varphi }\right) ^{1-\sigma }\frac{dH\left( \varphi \right) }{ 1-H\left( \varphi _{D}\right) }+N^{*}\int _{\varphi _{X}^{*}}^{\infty }E\left( P\right) ^{\sigma -1}\left( \frac{\sigma }{\sigma -1}\frac{W^{*}\tau ^{*}}{\varphi }\right) ^{1-\sigma }\frac{dH^{*}\left( \varphi \right) }{1-H^{*}\left( \varphi _{D}^{*}\right) }} \\= & {} \left( 1+\frac{N}{N^{*}}\left( \frac{W}{W^{*}}\right) ^{1-\sigma }\left( \tau ^{*}\right) ^{\sigma -1}\frac{\frac{k}{k-\left( \sigma -1\right) }\left( \varphi _{D}\right) ^{\sigma -1}}{\left( \varphi _{D}^{*}\right) ^{k^{*}}\frac{k^{*}}{k^{*}-\left( \sigma -1\right) }\left( \varphi _{X}^{*}\right) ^{\sigma -1-k^{*}}}\right) ^{-1} \end{aligned}$$
Imposing the balanced trade condition (7), we have
$$\begin{aligned} O= & {} \left( 1+\frac{B}{B^{*}}\left( \tau \right) ^{\sigma -1}\left( \frac{\varphi _{D}}{\varphi _{X}}\right) ^{\sigma -1-k}\right) ^{-1}=\left( 1+\left( \frac{B}{B^{*}}\right) ^{\frac{k}{\sigma -1}}\tau ^{k}\left( \frac{F_{X}}{F_{D}}\right) ^{\frac{k}{\sigma -1}-1}\right) ^{-1} \\= & {} \left( 1+\left( \frac{B}{B^{*}}\right) ^{\frac{k}{\sigma -1}}\Phi ^{-1}\right) ^{-1}\in \left( 0,1\right) \end{aligned}$$
where \(\frac{\partial O}{\partial \frac{B}{B^{*}}}<0\). Note that \(\lambda =\left( 1+\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{\sigma -1}}\right) ^{-1}=1-O.\)
Labor demand and labor market equilibrium
Aggregate labor demand is the sum of labor demand across firms , and it is given as
$$\begin{aligned} L^{d}= & {} \frac{NF_{E}}{1-H\left( \varphi _{D}\right) }+NF_{D}+NF_{X}\frac{ 1-H\left( \varphi _{X}\right) }{1-H\left( \varphi _{D}\right) } \\&+N\int _{\varphi _{D}}^{\infty }\frac{1}{\varphi }E\left( P\right) ^{\sigma -1}\left( p_{D}\left( \varphi \right) \right) ^{-\sigma }\frac{dH\left( \varphi \right) }{1-H\left( \varphi _{D}\right) } \\&+N\int _{\varphi _{X}}^{\infty }\frac{\tau }{\varphi }E^{*}\left( P^{*}\right) ^{\sigma -1}\left( p_{X}\left( \varphi \right) \right) ^{-\sigma }\frac{dH\left( \varphi \right) }{1-H\left( \varphi _{D}\right) } \\= & {} N\frac{k\sigma }{k-\left( \sigma -1\right) }F_{D}\left( 1+\left( \frac{ F_{D}}{F_{X}}\right) ^{\frac{k}{\sigma -1}-1}\tau ^{-k}\left( \frac{B}{ B^{*}}\right) ^{-\frac{k}{\sigma -1}}\right) \end{aligned}$$
where it has been used that
$$\begin{aligned}&N\int _{\varphi _{D}}^{\infty }\frac{1}{\varphi }E\left( P\right) ^{\sigma -1}\left( p_{D}\left( \varphi \right) \right) ^{-\sigma }\frac{dH\left( \varphi \right) }{1-H\left( \varphi _{D}\right) }\\&\qquad +N\int _{\varphi _{X}}^{\infty }\frac{\tau }{\varphi }E^{*}\left( P^{*}\right) ^{\sigma -1}\left( p_{X}\left( \varphi \right) \right) ^{-\sigma }\frac{ dH\left( \varphi \right) }{1-H\left( \varphi _{D}\right) } \\&\quad =NB(\sigma -1)W^{-\sigma }\left( \varphi _{D}\right) ^{k}k\int _{\varphi _{D}}^{\infty }\varphi ^{\sigma -1}\varphi ^{-k-1}d\varphi \\&\qquad +NB^{*}(\sigma -1)\tau ^{1-\sigma }W^{*-\sigma }\left( \varphi _{D}\right) ^{k}k\int _{\varphi _{X}}^{\infty }\varphi ^{\sigma -1}\varphi ^{-k-1}d\varphi \end{aligned}$$
and
$$\begin{aligned} \frac{NF_{E}}{1-H\left( \varphi _{D}\right) }=N\frac{F_{E}}{(\frac{\varphi _{D}}{\varphi _{\min }})^{-k}}=N\left( 1+\Phi \left( \frac{B}{B^{*}} \right) ^{-\frac{k}{\sigma -1}}\right) F_{D}\frac{\sigma -1}{k-(\sigma -1)} \end{aligned}$$
The labor market equilibrium condition becomes
$$\begin{aligned} L=N\frac{k\sigma }{k-\left( \sigma -1\right) }F_{D}\left( 1+\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{\sigma -1}}\right) =N\left( \frac{ \varphi _{D}}{\varphi _{\min }}\right) ^{k}\frac{k\sigma }{\sigma -1}\delta F_{E} \end{aligned}$$
The real wage
Inserting the demand component, \(B\equiv EP^{\sigma -1}\left( \frac{\sigma }{\sigma -1}\right) ^{-\sigma }\frac{1}{\sigma -1}\), into the exit threshold condition, \(W^{1-\sigma }\left( \varphi _{D}\right) ^{\sigma -1}B\equiv WF_{D}\), implies
$$\begin{aligned} \left( W\right) ^{1-\sigma }\left( \varphi _{D}\right) ^{\sigma -1}E\left( P\right) ^{\sigma -1}\left( \frac{\sigma }{\sigma -1}\right) ^{-\sigma } \frac{1}{\sigma -1}=WF_{D} \end{aligned}$$
Using that \(E=WL\), it follows that
$$\begin{aligned} \frac{W}{P}=\varphi _{D}\left( L\left( \frac{\sigma }{\sigma -1}\right) ^{-\sigma }\frac{1}{\sigma -1}\frac{1}{F_{D}}\right) ^{\frac{1}{\sigma -1}} \end{aligned}$$
Deriving the equilibrium condition
The relative number of firms follows from the labor market equilibrium and is given by
$$\begin{aligned} \frac{N}{N^{*}}=\frac{L}{L^{*}}\frac{\left( \frac{\varphi _{D}^{*}}{\varphi _{\min }^{*}}\right) ^{k^{*}}\frac{k^{*}\sigma }{ \left( \sigma -1\right) }\delta ^{*}F_{E}^{*}}{\left( \frac{\varphi _{D}}{\varphi _{\min }}\right) ^{k}\frac{k\sigma }{\left( \sigma -1\right) } \delta F_{E}}=\frac{L}{L^{*}}\frac{\left( \frac{\varphi _{D}^{*}}{ \varphi _{\min }^{*}}\right) ^{k^{*}}k^{*}\delta ^{*}F_{E}^{*}}{\left( \frac{\varphi _{D}}{\varphi _{\min }}\right) ^{k}k\delta F_{E}} \end{aligned}$$
From the exit threshold conditions for Home and Foreign, we have
$$\begin{aligned} \frac{W}{W^{*}}=\left( \left( \frac{\varphi _{D}^{*}}{\varphi _{D}} \right) ^{\sigma -1}\frac{F_{D}}{F_{D}^{*}}\frac{B^{*}}{B}\right) ^{- \frac{1}{\sigma }} \end{aligned}$$
Inserting this into the balanced trade condition, we obtain
$$\begin{aligned} \frac{N}{N^{*}}=\left( \left( \frac{\varphi _{D}^{*}}{\varphi _{D}} \right) ^{\sigma -1}\frac{F_{D}}{F_{D}^{*}}\frac{B^{*}}{B}\right) ^{- \frac{\sigma -1}{\sigma }}\left( \frac{\tau }{\tau ^{*}}\right) ^{\sigma -1}\frac{k-\left( \sigma -1\right) }{k^{*}-\left( \sigma -1\right) } \frac{k^{*}}{k}\frac{B}{B^{*}}\frac{\left( \varphi _{X}^{*}\right) ^{\sigma -1-k^{*}}}{\left( \varphi _{X}\right) ^{\sigma -1-k}} \frac{\left( \varphi _{D}^{*}\right) ^{k^{*}}}{\left( \varphi _{D}\right) ^{k}} \end{aligned}$$
Equating the two terms for \(\frac{N}{N^{*}}\), we get
$$\begin{aligned} \frac{L}{L^{*}}\frac{\left( \frac{\varphi _{D}^{*}}{\varphi _{\min }^{*}}\right) ^{k^{*}}\frac{k^{*}\sigma }{\left( \sigma -1\right) }\delta ^{*}F_{E}^{*}}{\left( \frac{\varphi _{D}}{\varphi _{\min }}\right) ^{k}\frac{k\sigma }{\left( \sigma -1\right) }\delta F_{E}}= & {} \left( \left( \frac{\varphi _{D}^{*}}{\varphi _{D}}\right) ^{\sigma -1} \frac{F_{D}}{F_{D}^{*}}\frac{B^{*}}{B}\right) ^{-\frac{\sigma -1}{ \sigma }}\left( \frac{\tau }{\tau ^{*}}\right) ^{\sigma -1}\\&\frac{ k-\left( \sigma -1\right) }{k^{*}-\left( \sigma -1\right) }\frac{k^{*}}{k}\frac{B}{B^{*}}\frac{\left( \varphi _{X}^{*}\right) ^{\sigma -1-k^{*}}}{\left( \varphi _{X}\right) ^{\sigma -1-k}}\frac{\left( \varphi _{D}^{*}\right) ^{k^{*}}}{\left( \varphi _{D}\right) ^{k}} \end{aligned}$$
and insert the thresholds as functions of \(\frac{B}{B^{*}}\) to obtain
$$\begin{aligned}&\frac{L}{L^{*}}\frac{\left( \varphi _{\min }\right) ^{k}}{\left( \varphi _{\min }^{*}\right) ^{k^{*}}}\frac{\delta ^{*}F_{E}^{*}}{\delta F_{E}}\frac{k^{*}-\left( \sigma -1\right) }{ k-\left( \sigma -1\right) }\frac{\left( \varphi _{aut}\right) ^{\frac{\sigma -1}{\sigma }-k}}{\left( \varphi _{aut}^{*}\right) ^{\frac{\sigma -1}{ \sigma }-k^{*}}}\left( \frac{F_{D}}{F_{D}^{*}}\right) ^{\frac{\sigma -1}{\sigma }} \\&\quad =\left( \frac{B}{B^{*}}\right) ^{-\frac{1}{\sigma }}\frac{\left( 1+\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{\sigma -1}}\right) ^{1- \frac{1}{k}\frac{\sigma -1}{\sigma }}}{\Phi \left( \frac{B}{B^{*}} \right) ^{-\frac{k}{\sigma -1}}}\frac{\Phi ^{*}\left( \frac{B}{B^{*}} \right) ^{\frac{k^{*}}{\sigma -1}}}{\left( 1+\Phi ^{*}\left( \frac{B }{B^{*}}\right) ^{\frac{k^{*}}{\sigma -1}}\right) ^{1-\frac{1}{ k^{*}}\frac{\sigma -1}{\sigma }}} \end{aligned}$$
Next insert the autarky thresholds, \(\varphi _{aut}\), to get
$$\begin{aligned}&\frac{L}{L^{*}}\left( \frac{\varphi _{\min }}{\varphi _{\min }^{*}} \right) ^{\frac{\sigma -1}{\sigma }}\frac{\left( \frac{\sigma -1}{k-\left( \sigma -1\right) }\frac{F_{D}}{\delta F_{E}}\right) ^{\frac{1}{k}\frac{ \sigma -1}{\sigma }}}{\left( \frac{\sigma -1}{k^{*}-\left( \sigma -1\right) }\frac{F_{D}^{*}}{\delta ^{*}F_{E}^{*}}\right) ^{\frac{ 1}{k^{*}}\frac{\sigma -1}{\sigma }}}\left( \frac{F_{D}}{F_{D}^{*}} \right) ^{-\frac{1}{\sigma }} \\&\quad =\frac{\left( 1+\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{\sigma -1}}\right) ^{1-\frac{1}{k}\frac{\sigma -1}{\sigma }}}{\Phi \left( \frac{B}{ B^{*}}\right) ^{-\frac{k}{\sigma -1}}}\frac{\Phi ^{*}\left( \frac{B}{ B^{*}}\right) ^{\frac{k^{*}}{\sigma -1}}}{\left( 1+\Phi ^{*}\left( \frac{B}{B^{*}}\right) ^{\frac{k^{*}}{\sigma -1}}\right) ^{1- \frac{1}{k^{*}}\frac{\sigma -1}{\sigma }}}\left( \frac{B}{B^{*}} \right) ^{-\frac{1}{\sigma }} \end{aligned}$$
To ease notation in the following, we impose the definitions below, and rewrite the equilibrium condition as
$$\begin{aligned} \Gamma \left( \frac{B}{B^{*}},\Phi ,\Phi ^{*}\right)= & {} \digamma (Z,Z^{*})\nonumber \\ \digamma (Z,Z^{*})\equiv & {} \frac{L}{L^{*}}\left( \frac{\varphi _{\min }}{\varphi _{\min }^{*}}\right) ^{\frac{\sigma -1}{\sigma }}\frac{ \left( \frac{\sigma -1}{k-\left( \sigma -1\right) }\frac{F_{D}}{\delta F_{E}} \right) ^{\frac{1}{k}\frac{\sigma -1}{\sigma }}}{\left( \frac{\sigma -1}{ k^{*}-\left( \sigma -1\right) }\frac{F_{D}^{*}}{\delta ^{*}F_{E}^{*}}\right) ^{\frac{1}{k^{*}}\frac{\sigma -1}{\sigma }}} \left( \frac{F_{D}}{F_{D}^{*}}\right) ^{-\frac{1}{\sigma }} \end{aligned}$$
(8)
$$\begin{aligned} \Gamma \left( \frac{B}{B^{*}},\Phi ,\Phi ^{*}\right)\equiv & {} \frac{\left( 1+\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{\sigma -1}}\right) ^{1- \frac{1}{k}\frac{\sigma -1}{\sigma }}}{\Phi \left( \frac{B}{B^{*}} \right) ^{-\frac{k}{\sigma -1}}}\frac{\Phi ^{*}\left( \frac{B}{B^{*}} \right) ^{\frac{k^{*}}{\sigma -1}}}{\left( 1+\Phi ^{*}\left( \frac{B }{B^{*}}\right) ^{\frac{k^{*}}{\sigma -1}}\right) ^{1-\frac{1}{ k^{*}}\frac{\sigma -1}{\sigma }}}\left( \frac{B}{B^{*}}\right) ^{- \frac{1}{\sigma }} \end{aligned}$$
(9)
Note that
$$\begin{aligned}&\frac{\partial \Gamma }{\partial \frac{B}{B^{*}}}\frac{\frac{B}{ B^{*}}}{\Gamma } \\&\quad =\frac{k}{\sigma -1}\left( 1-\left( 1-\frac{1}{k}\frac{\sigma -1}{\sigma } \right) \frac{\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{\sigma -1}} }{1+\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{\sigma -1}}}\right) \\&\qquad + \frac{k^{*}}{\sigma -1}\left( 1-\left( 1-\frac{1}{k^{*}}\frac{\sigma -1}{\sigma }\right) \frac{\Phi ^{*}\left( \frac{B}{B^{*}}\right) ^{ \frac{k^{*}}{\sigma -1}}}{1+\Phi ^{*}\left( \frac{B}{B^{*}} \right) ^{\frac{k^{*}}{\sigma -1}}}\right) -\frac{1}{\sigma } \\&\quad =\frac{k}{\sigma -1}\left( 1-\left( 1-\frac{1}{k}\frac{\sigma -1}{\sigma } \right) O\right) +\frac{k^{*}}{\sigma -1}\left( 1-\left( 1-\frac{1}{ k^{*}}\frac{\sigma -1}{\sigma }\right) O^{*}\right) -\frac{1}{\sigma } \\&\quad =\left( \frac{k}{\sigma -1}-\frac{1}{\sigma }\right) \left( 1-O\right) +\left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma }\right) \left( 1-O^{*}\right) +\frac{1}{\sigma }>0 \end{aligned}$$
Hence, \(\Gamma\) is continuous and strictly increasing in \(\frac{B}{B^{*} }\). As \(\lim _{\frac{B}{B^{*}}\rightarrow \infty }\Gamma =\infty\) and \(\lim _{\frac{B}{B^{*}}\rightarrow 0}\Gamma =0\), it follows that a unique value of \(\frac{B}{B^{*}}\) solves the equilibrium condition. We define \(\Psi ^{-1}\equiv \left( \frac{k}{\sigma -1}-\frac{1}{\sigma }\right) \left( 1-O\right) +\left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma } \right) \left( 1-O^{*}\right) +\frac{1}{\sigma }\) implying that \(\frac{ \partial \Gamma }{\partial \frac{B}{B^{*}}}\frac{\frac{B}{B^{*}}}{ \Gamma }=\Psi ^{-1}\).
Analyzing the equilibrium condition
Properties of the equilibrium condition are crucial for effects of changing exogenous parameters of the model. We therefore explore the equilibrium condition. We find for \(\digamma\) that
$$\begin{aligned} \frac{\partial \digamma }{\partial L}\frac{L}{\digamma }= & {} -\frac{\partial \digamma }{\partial L^{*}}\frac{L^{*}}{\digamma }=1>0 \\ \frac{\partial LHS}{\partial \varphi _{\min }}\frac{\varphi _{\min }}{LHS}= & {} -\frac{\partial \digamma }{\partial \varphi _{\min }^{*}}\frac{ \varphi _{\min }^{*}}{\digamma }=\frac{\sigma -1}{\sigma }>0 \\ \frac{\partial \digamma }{\partial F_{D}}\frac{F_{D}}{\digamma }= & {} \frac{1}{ k}\frac{\sigma -1}{\sigma }-\frac{1}{\sigma }=\frac{1}{\sigma }\left( \frac{ \sigma -1}{k}-1\right)<0 \\ \frac{\partial \digamma }{\partial F_{D}^{*}}\frac{F_{D}^{*}}{ \digamma }= & {} \frac{1}{\sigma }\left( 1-\frac{\sigma -1}{k^{*}}\right)>0 \\ \frac{\partial \digamma }{\partial \delta F_{E}}\frac{\delta F_{E}}{\digamma }= & {} -\frac{1}{k}\frac{\sigma -1}{\sigma }<0 \\ \frac{\partial \digamma }{\partial \delta ^{*}F_{E}^{*}}\frac{\delta ^{*}F_{E}^{*}}{\digamma }= & {} \frac{1}{k^{*}}\frac{\sigma -1}{ \sigma }>0 \end{aligned}$$
and for \(\Gamma\) that
$$\begin{aligned} \frac{\partial \Gamma }{\partial \Phi }\frac{\Phi }{\Gamma }= & {} -1+\left( 1- \frac{1}{k}\frac{\sigma -1}{\sigma }\right) \frac{\Phi \left( \frac{B}{ B^{*}}\right) ^{-\frac{k}{\sigma -1}}}{1+\Phi \left( \frac{B}{B^{*}} \right) ^{-\frac{k}{\sigma -1}}}=-1+\left( 1-\frac{1}{k}\frac{\sigma -1}{ \sigma }\right) O<0 \\ \frac{\partial \Gamma }{\partial \Phi ^{*}}\frac{\Phi ^{*}}{\Gamma }= & {} 1-\left( 1-\frac{1}{k^{*}}\frac{\sigma -1}{\sigma }\right) O^{*}>0. \end{aligned}$$
This in turn implies that \(\frac{B}{B^{*}}\left( \underset{+}{\Phi }, \underset{-}{\Phi }^{*},\underset{+}{L},\underset{-}{L}^{*},\underset{+}{\varphi _{\min }},\underset{-}{\varphi _{\min }^{*}},\underset{-}{ F_{D}},\underset{+}{F_{D}^{*}},\underset{-}{\delta F_{E}},\underset{+}{ \delta ^{*}F_{E}^{*}}\right)\), where the signs denote the signs of the partial derivative of the relative demand component.
1.3 A.3 Static comparative analysis
Reduced barriers to trade
Consider the effects of changes in fixed and iceberg trade costs. From the properties of the equilibrium condition it follows that
$$\begin{aligned} \frac{d\frac{B}{B^{*}}}{d\Phi }\frac{\Phi }{\frac{B}{B^{*}}}= & {} - \frac{\frac{\partial \Gamma }{\partial \Phi }\frac{\Phi }{\Gamma }}{\frac{ \partial \Gamma }{\partial \frac{B}{B^{*}}}\frac{\frac{B}{B^{*}}}{ \Gamma }}=\left( 1-\left( 1-\frac{1}{k}\frac{\sigma -1}{\sigma }\right) O\right) \Psi >0 \\ \frac{d\frac{B}{B^{*}}}{d\Phi ^{*}}\frac{\Phi ^{*}}{\frac{B}{ B^{*}}}= & {} -\frac{\frac{\partial \Gamma }{\partial \Phi ^{*}}\frac{ \Phi ^{*}}{\Gamma }}{\frac{\partial \Gamma }{\partial \frac{B}{B^{*}} }\frac{\frac{B}{B^{*}}}{\Gamma }}=-\left( 1-\left( 1-\frac{1}{k^{*}} \frac{\sigma -1}{\sigma }\right) O^{*}\right) \Psi <0. \end{aligned}$$
For the productivity thresholds this implies
$$\begin{aligned} \frac{d\varphi _{D}}{d\Phi }\frac{\Phi }{\varphi _{D}}= & {} \frac{\partial \varphi _{D}}{\partial \Phi }\frac{\Phi }{\varphi _{D}}+\frac{\partial \varphi _{D}}{\partial \frac{B}{B^{*}}}\frac{\frac{B}{B^{*}}}{ \varphi _{D}}\frac{d\frac{B}{B^{*}}}{d\Phi }\frac{\Phi }{\frac{B}{ B^{*}}} \\= & {} \frac{1}{k}\frac{\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{ \sigma -1}}}{1+\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{\sigma -1}} }\\&+\frac{1}{k}\left( -\frac{k}{\sigma -1}\right) \frac{\Phi \left( \frac{B}{ B^{*}}\right) ^{-\frac{k}{\sigma -1}}}{1+\Phi \left( \frac{B}{B^{*}} \right) ^{-\frac{k}{\sigma -1}}}\left( 1-\left( 1-\frac{1}{k}\frac{\sigma -1 }{\sigma }\right) O\right) \Psi \\= & {} O\frac{1}{k}\left( 1-\frac{k}{\sigma -1}\left( 1-\left( 1-\frac{1}{k} \frac{\sigma -1}{\sigma }\right) O\right) \Psi \right) \\= & {} \frac{O\frac{1}{k}\left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma } \right) \left( 1-O^{*}\right) }{\left( \frac{k}{\sigma -1}-\frac{1}{ \sigma }\right) \left( 1-O\right) +\left( \frac{k^{*}}{\sigma -1}-\frac{1 }{\sigma }\right) \left( 1-O^{*}\right) +\frac{1}{\sigma }} \\= & {} O\frac{1}{k}\left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma }\right) \left( 1-O^{*}\right) \Psi>0\\ \frac{d\varphi _{D}}{d\Phi ^{*}}\frac{\Phi ^{*}}{\varphi _{D}}= & {} \frac{ \partial \varphi _{D}}{\partial \frac{B}{B^{*}}}\frac{\frac{B}{B^{*}} }{\varphi _{D}}\frac{d\frac{B}{B^{*}}}{d\Phi ^{*}}\frac{\Phi ^{*} }{\frac{B}{B^{*}}}=\left( 1-\left( 1-\frac{1}{k^{*}}\frac{\sigma -1}{ \sigma }\right) O^{*}\right) \Psi \frac{1}{\sigma -1}O>0. \end{aligned}$$
Hence, reduced trade barriers in any country increase the exit thresholds in both countries. The welfare effects follow directly from the effect on the exit thresholds as \({\mathcal {W}}=\frac{W}{P}=\varphi _{D}\left( \left( \frac{ \sigma }{\sigma -1}\right) ^{-\sigma }\frac{1}{\sigma -1}\frac{L}{F_{D}} \right) ^{\frac{1}{\sigma -1}}\) and reduced trade barriers in any country increase welfare in both countries. Turning to the export thresholds we find
$$\begin{aligned} \frac{d\varphi _{X}}{d\Phi ^{*}}\frac{\Phi ^{*}}{\varphi _{X}}= & {} -\left( 1-\left( 1-\frac{1}{k^{*}}\frac{\sigma -1}{\sigma }\right) O^{*}\right) \Psi \frac{1}{\sigma -1}\left( 1-O\right) <0\\ \frac{d\varphi _{X}}{dF_{X}}\frac{F_{X}}{\varphi _{X}}= & {} \frac{1}{k}+\frac{1 }{k}\frac{\Phi ^{-1}\left( \frac{B}{B^{*}}\right) ^{\frac{k}{\sigma -1}} }{1+\Phi ^{-1}\left( \frac{B}{B^{*}}\right) ^{\frac{k}{\sigma -1}}} \left( \frac{k}{\sigma -1}\frac{d\frac{B}{B^{*}}}{dF_{X}}\frac{F_{X}}{ \frac{B}{B^{*}}}-\frac{d\Phi }{dF_{X}}\frac{F_{X}}{\Phi }\right) \\= & {} \frac{1}{k}+\frac{1}{k}\frac{\Phi ^{-1}\left( \frac{B}{B^{*}}\right) ^{\frac{k}{\sigma -1}}}{1+\Phi ^{-1}\left( \frac{B}{B^{*}}\right) ^{ \frac{k}{\sigma -1}}}\left( \frac{k}{\sigma -1}\frac{d\frac{B}{B^{*}}}{ d\Phi }\frac{\Phi }{\frac{B}{B^{*}}}\frac{d\Phi }{dF_{X}}\frac{F_{X}}{ \Phi }-\frac{d\Phi }{dF_{X}}\frac{F_{X}}{\Phi }\right) \\= & {} \frac{1}{k}-\frac{1}{k}\left( 1-O\right) \left( \frac{k}{\sigma -1}\left( 1-\left( 1-\frac{1}{k}\frac{\sigma -1}{\sigma }\right) O\right) \Psi -1\right) \left( \frac{k}{\sigma -1}-1\right) \\= & {} \frac{1}{k}\left( 1+\left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma } \right) \left( 1-O^{*}\right) \left( 1-O\right) \left( \frac{k}{\sigma -1 }-1\right) \Psi \right)>0\\ \frac{d\varphi _{X}}{d\tau }\frac{\tau }{\varphi _{X}}= & {} \frac{1}{k}\frac{ \Phi ^{-1}\left( \frac{B}{B^{*}}\right) ^{\frac{k}{\sigma -1}}}{1+\Phi ^{-1}\left( \frac{B}{B^{*}}\right) ^{\frac{k}{\sigma -1}}}\left( \frac{k }{\sigma -1}\frac{d\frac{B}{B^{*}}}{d\Phi }\frac{\Phi }{\frac{B}{B^{*}}}\frac{d\Phi }{d\tau }\frac{\tau }{\Phi }-\frac{d\Phi }{d\tau }\frac{\tau }{\Phi }\right) \\= & {} \left( 1-O\right) \left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma } \right) \left( 1-O^{*}\right) \Psi >0. \end{aligned}$$
Hence, reduced trade barriers in any country reduce the export thresholds in both countries. For the relative wage we have
$$\begin{aligned} \frac{d\frac{W}{W^{*}}}{d\Phi }\frac{\Phi }{\frac{W}{W^{*}}}= & {} \frac{1}{\sigma }\frac{d\frac{B}{B^{*}}}{d\Phi }\frac{\Phi }{\frac{B}{ B^{*}}}+\frac{\sigma -1}{\sigma }\left( \frac{d\varphi _{D}}{d\Phi } \frac{\Phi }{\varphi _{D}}-\frac{d\varphi _{D}^{*}}{d\Phi }\frac{\Phi }{ \varphi _{D}^{*}}\right) \\= & {} \frac{1}{\sigma }\left( 1-O+O\frac{k^{*}}{k}\right) \left( 1-O^{*}\right) \Psi >0. \end{aligned}$$
Reduced barriers to entry
Reduced barriers to entry—a reduction in \(\delta F_{E}\) - may either be through lower sunk entry costs or reduced probability of exogenous exit. We have that
$$\begin{aligned} \frac{d\frac{B}{B^{*}}}{d\delta F_{E}}\frac{\delta F_{E}}{\frac{B}{ B^{*}}}= & {} \frac{\frac{\partial \digamma }{\partial \delta F_{E}}\frac{ \delta F_{E}}{\digamma }}{\frac{\partial \Gamma }{\partial \frac{B}{B^{*} }}\frac{\frac{B}{B^{*}}}{\Gamma }}=-\frac{1}{k}\frac{\sigma -1}{\sigma } \Psi <0 \\ \frac{d\frac{B}{B^{*}}}{d\delta ^{*}F_{E}^{*}}\frac{\delta ^{*}F_{E}^{*}}{\frac{B}{B^{*}}}= & {} \frac{\frac{\partial \digamma }{\partial \delta ^{*}F_{E}^{*}}\frac{\delta ^{*}F_{E}^{*}}{ \digamma }}{\frac{\partial \Gamma }{\partial \frac{B}{B^{*}}}\frac{\frac{ B}{B^{*}}}{\Gamma }}=\Psi \frac{1}{k^{*}}\frac{\sigma -1}{\sigma }>0 \end{aligned}$$
and it follows that
$$\begin{aligned} \frac{d\varphi _{D}}{d\delta F_{E}}\frac{\delta F_{E}}{\varphi _{D}}= & {} - \frac{1}{k}+\frac{1}{k}\frac{\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{ k}{\sigma -1}}}{1+\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{\sigma -1}}}\left( -\frac{k}{\sigma -1}\right) \frac{d\frac{B}{B^{*}}}{d\delta F_{E}}\frac{\delta F_{E}}{\frac{B}{B^{*}}} \\= & {} -\frac{1}{k}+\frac{1}{k}O\left( -\frac{k}{\sigma -1}\right) \left( -\frac{ 1}{k}\frac{\sigma -1}{\sigma }\Psi \right) =\frac{1}{k}\left( -1+O\frac{1}{ \sigma }\Psi \right) \\= & {} -\frac{1}{k}\Psi \left( \frac{k}{\sigma -1}\left( 1-O\right) +\left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma }\right) \left( 1-O^{*}\right) \right)<0\\ \frac{d\varphi _{D}}{d\delta ^{*}F_{E}^{*}}\frac{\delta ^{*}F_{E}^{*}}{\varphi _{D}}= & {} \frac{1}{k}\frac{\Phi \left( \frac{B}{ B^{*}}\right) ^{-\frac{k}{\sigma -1}}}{1+\Phi \left( \frac{B}{B^{*}} \right) ^{-\frac{k}{\sigma -1}}}\left( -\frac{k}{\sigma -1}\right) \frac{d \frac{B}{B^{*}}}{d\delta ^{*}F_{E}^{*}}\frac{\delta ^{*}F_{E}^{*}}{\frac{B}{B^{*}}} \\= & {} -O\Psi \frac{1}{k^{*}}\frac{1}{\sigma }<0 \end{aligned}$$
and welfare effects follow trivially as \({\mathcal {W}}=\frac{W}{P}=\varphi _{D}\left( \left( \frac{\sigma }{\sigma -1}\right) ^{-\sigma }\frac{1}{ \sigma -1}\frac{L}{F_{D}}\right) ^{\frac{1}{\sigma -1}}\). Turning to export thresholds we have
$$\begin{aligned} \frac{d\varphi _{X}}{d\delta F_{E}}\frac{\delta F_{E}}{\varphi _{X}}= & {} \frac{d\varphi _{D}}{d\delta F_{E}}\frac{\delta F_{E}}{\varphi _{D}}+\frac{1 }{\sigma -1}\frac{d\frac{B}{B^{*}}}{d\delta F_{E}}\frac{\delta F_{E}}{ \frac{B}{B^{*}}} \\= & {} -\frac{1}{k}\left( 1+\frac{1}{\sigma }\Psi \left( 1-O\right) \right) <0\\ \frac{d\varphi _{X}}{d\delta ^{*}F_{E}^{*}}\frac{\delta ^{*}F_{E}^{*}}{\varphi _{X}}= & {} \frac{d\varphi _{D}}{d\delta ^{*}F_{E}^{*}}\frac{\delta ^{*}F_{E}^{*}}{\varphi _{D}}+\frac{1}{ \sigma -1}\frac{d\frac{B}{B^{*}}}{d\delta ^{*}F_{E}^{*}}\frac{ \delta ^{*}F_{E}^{*}}{\frac{B}{B^{*}}} \\= & {} \left( 1-O\right) \Psi \frac{1}{k^{*}}\frac{1}{\sigma }>0. \end{aligned}$$
For the relative wage
$$\begin{aligned} \frac{d\frac{W}{W^{*}}}{d\delta F_{E}}\frac{\delta F_{E}}{\frac{W}{ W^{*}}}= & {} \frac{1}{\sigma }\frac{d\frac{B}{B^{*}}}{d\delta F_{E}} \frac{\delta F_{E}}{\frac{B}{B^{*}}}+\frac{\sigma -1}{\sigma }\left( \frac{d\varphi _{D}}{d\delta F_{E}}\frac{\delta F_{E}}{\varphi _{D}}-\frac{ d\varphi _{D}^{*}}{d\delta F_{E}}\frac{\delta F_{E}}{\varphi _{D}^{*} }\right) \\= & {} -\frac{\sigma -1}{\sigma }\frac{1}{k}\left[ \frac{k}{\sigma -1}\left( 1-O\right) +\frac{k^{*}}{\sigma -1}\left( 1-O^{*}\right) \right] \Psi <0. \end{aligned}$$
Productivity levels
Consider changes in the location parameter of the Pareto distribution. We have that
$$\begin{aligned} \frac{d\frac{B}{B^{*}}}{d\frac{\varphi _{\min }}{\varphi _{\min }^{*} }}\frac{\frac{\varphi _{\min }}{\varphi _{\min }^{*}}}{\frac{B}{B^{*} }}=\frac{\frac{\partial \digamma }{\partial \left( \frac{\varphi _{\min }}{ \varphi _{\min }^{*}}\right) }\frac{\frac{\varphi _{\min }}{\varphi _{\min }^{*}}}{\digamma }}{\frac{\partial \Gamma }{\partial \frac{B}{ B^{*}}}\frac{\frac{B}{B^{*}}}{\Gamma }}=\frac{\sigma -1}{\sigma } \Psi >0 \end{aligned}$$
and it follows that
$$\begin{aligned} \frac{d\varphi _{D}}{d\varphi _{\min }}\frac{\varphi _{\min }}{\varphi _{D}}= & {} 1+\frac{1}{k}\frac{\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{ \sigma -1}}}{1+\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{\sigma -1}} }\left( -\frac{k}{\sigma -1}\right) \frac{d\frac{B}{B^{*}}}{d\varphi _{\min }}\frac{\varphi _{\min }}{\frac{B}{B^{*}}} \\= & {} 1-O\frac{1}{\sigma }\Psi =\Psi \left[ \frac{k}{\sigma -1}\left( 1-O\right) +\left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma }\right) \left( 1-O^{*}\right) \right]>0\\ \frac{d\varphi _{D}}{d\varphi _{\min }^{*}}\frac{\varphi _{\min }^{*} }{\varphi _{D}}= & {} \frac{1}{k}\frac{\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{\sigma -1}}}{1+\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k }{\sigma -1}}}\left( -\frac{k}{\sigma -1}\right) \frac{d\frac{B}{B^{*}}}{ d\varphi _{\min }^{*}}\frac{\varphi _{\min }^{*}}{\frac{B}{B^{*}} } \\= & {} O\frac{1}{\sigma }\Psi >0. \end{aligned}$$
Hence, a better productivity distribution in any country improves welfare in both countries as \({\mathcal {W}}=\frac{W}{P}=\varphi _{D}\left( \left( \frac{\sigma }{\sigma -1}\right) ^{-\sigma }\frac{1}{\sigma -1}\frac{L }{F_{D}}\right) ^{\frac{1}{\sigma -1}}\). Turning to the export thresholds we have
$$\begin{aligned} \frac{d\varphi _{X}}{d\varphi _{\min }}\frac{\varphi _{\min }}{\varphi _{X}}= & {} \frac{d\varphi _{D}}{d\varphi _{\min }}\frac{\varphi _{\min }}{\varphi _{D}}+\frac{1}{\sigma -1}\frac{d\frac{B}{B^{*}}}{d\varphi _{\min }}\frac{ \varphi _{\min }}{\frac{B}{B^{*}}} \\= & {} 1+\frac{1}{\sigma }\Psi \left( 1-O\right) >0\\ \frac{d\varphi _{X}}{d\varphi _{\min }^{*}}\frac{\varphi _{\min }^{*} }{\varphi _{X}}= & {} \frac{d\varphi _{D}}{d\varphi _{\min }^{*}}\frac{ \varphi _{\min }^{*}}{\varphi _{D}}+\frac{1}{\sigma -1}\frac{d\frac{B}{ B^{*}}}{d\varphi _{\min }^{*}}\frac{\varphi _{\min }^{*}}{\frac{B }{B^{*}}} \\= & {} -\left( 1-O\right) \frac{1}{\sigma }\Psi <0. \end{aligned}$$
Changes in the relative wage are given by
$$\begin{aligned} \frac{d\frac{W}{W^{*}}}{d\varphi _{\min }}\frac{\varphi _{\min }}{\frac{W }{W^{*}}}= & {} \frac{1}{\sigma }\frac{d\frac{B}{B^{*}}}{d\varphi _{\min }}\frac{\varphi _{\min }}{\frac{B}{B^{*}}}+\frac{\sigma -1}{ \sigma }\left[ \frac{d\varphi _{D}}{d\varphi _{\min }}\frac{\varphi _{\min } }{\varphi _{D}}-\frac{d\varphi _{D}^{*}}{d\varphi _{\min }}\frac{\varphi _{\min }}{\varphi _{D}^{*}}\right] \\= & {} \frac{\sigma -1}{\sigma }\Psi \left( \frac{k}{\sigma -1}\left( 1-O\right) +\frac{k^{*}}{\sigma -1}\left( 1-O^{*}\right) \right) >0. \end{aligned}$$
Population size
For population size we have
$$\begin{aligned} \frac{d\frac{B}{B^{*}}}{d\frac{L}{L^{*}}}\frac{\frac{L}{L^{*}}}{ \frac{B}{B^{*}}}=\frac{\frac{\partial \digamma }{\partial \left( \frac{L }{L^{*}}\right) }\frac{\frac{L}{L^{*}}}{\digamma }}{\frac{\partial \Gamma }{\partial \frac{B}{B^{*}}}\frac{\frac{B}{B^{*}}}{\Gamma }} =\Psi >0 \end{aligned}$$
and it follows that
$$\begin{aligned} \frac{d\varphi _{D}}{dL}\frac{L}{\varphi _{D}}= & {} -\frac{1}{\sigma -1}O\Psi <0 \\ \frac{d\varphi _{D}}{dL^{*}}\frac{L^{*}}{\varphi _{D}}= & {} \frac{1}{ \sigma -1}O\Psi >0. \end{aligned}$$
For welfare we have
$$\begin{aligned} \frac{d{\mathcal {W}}}{dL}\frac{L}{{\mathcal {W}}}= & {} \frac{d\varphi _{D}}{dL} \frac{L}{\varphi _{D}}+\frac{1}{\sigma -1} \\= & {} \frac{1}{\sigma -1}\left( 1-O\Psi \right) \\= & {} \frac{\Psi }{\sigma -1}\left[ \left( \frac{k}{\sigma -1}-\frac{1}{\sigma }\right) \left( 1-O\right) +\left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma }\right) \left( 1-O^{*}\right) +\frac{1}{\sigma }-O\right] \\= & {} \frac{\Psi }{\sigma -1}\left[ \left( \frac{k}{\sigma -1}\right) \left( 1-O\right) \right. \\&\left. +\left( \frac{k^{*}}{\sigma -1}-1\right) \left( 1-O^{*}\right) +\frac{\sigma -1}{\sigma }\left( 1-O-O^{*}\right) \right] >0 \end{aligned}$$
as \(1-O-O^{*}>0\), which follows from \(1-O-O^{*}>0\Leftrightarrow 1>\Phi ^{*}\Phi\). We further find
$$\begin{aligned} \frac{d{\mathcal {W}}}{dL^{*}}\frac{L^{*}}{{\mathcal {W}}}=\frac{d\varphi _{D}}{dL^{*}}\frac{L^{*}}{\varphi _{D}}=\frac{1}{\sigma -1}O\Psi >0. \end{aligned}$$
Hence, an increase in population size in either country increases welfare in both countries. For the export thresholds we find
$$\begin{aligned} \frac{d\varphi _{X}}{dL}\frac{L}{\varphi _{X}}= & {} \frac{d\varphi _{D}}{dL} \frac{L}{\varphi _{D}}+\frac{1}{\sigma -1}\frac{d\frac{B}{B^{*}}}{dL} \frac{L}{\frac{B}{B^{*}}} \\= & {} \frac{1-O}{\sigma -1}\Psi >0\\ \frac{d\varphi _{X}}{dL^{*}}\frac{L^{*}}{\varphi _{X}}= & {} \frac{ d\varphi _{D}}{dL^{*}}\frac{L^{*}}{\varphi _{D}}+\frac{1}{\sigma -1} \frac{d\frac{B}{B^{*}}}{dL^{*}}\frac{L^{*}}{\frac{B}{B^{*}}} \\= & {} -\frac{1-O}{\sigma -1}\Psi <0. \end{aligned}$$
Changes in the relative wage are given by
$$\begin{aligned} \frac{d\frac{W}{W^{*}}}{dL}\frac{L}{\frac{W}{W^{*}}}= & {} \frac{1}{ \sigma }\frac{d\frac{B}{B^{*}}}{dL}\frac{L}{\frac{B}{B^{*}}}+\frac{ \sigma -1}{\sigma }\left[ \frac{d\varphi _{D}}{dL}\frac{L}{\varphi _{D}}- \frac{d\varphi _{D}^{*}}{dL}\frac{L}{\varphi _{D}^{*}}\right] \\= & {} \Psi \frac{1}{\sigma }\left( 1-O-O^{*}\right) >0. \end{aligned}$$
Fixed costs of serving the domestic market
Here we have that
$$\begin{aligned} \frac{d\frac{B}{B^{*}}}{dF_{D}}\frac{F_{D}}{\frac{B}{B^{*}}}= & {} \frac{\frac{\partial \digamma }{\partial F_{D}}\frac{F_{D}}{\digamma }-\frac{ \partial \Gamma }{\partial \Phi }\frac{\Phi }{\Gamma }\frac{\partial \Phi }{ \partial F_{D}}\frac{F_{D}}{\Phi }}{\frac{\partial \Gamma }{\partial \frac{B }{B^{*}}}\frac{\frac{B}{B^{*}}}{\Gamma }} \\= & {} \Psi \left[ \frac{1}{\sigma }\left( \frac{\sigma -1}{k}-1\right) -\left( -1+\left( 1-\frac{1}{k}\frac{\sigma -1}{\sigma }\right) O\right) \left( \frac{k}{\sigma -1}-1\right) \right] \\= & {} \Psi \left( 1-O\right) \left( \frac{k}{\sigma -1}-1\right) \left( 1-\frac{ \sigma -1}{\sigma }\frac{1}{k}\right) >0\\ \frac{d\frac{B}{B^{*}}}{dF_{D}^{*}}\frac{F_{D}^{*}}{\frac{B}{ B^{*}}}= & {} \frac{\left( \frac{\partial \digamma }{\partial F_{D}^{*}} \frac{F_{D}^{*}}{\digamma }-\frac{\partial \Gamma }{\partial \Phi ^{*}}\frac{\Phi ^{*}}{\Gamma }\frac{\partial \Phi ^{*}}{\partial F_{D}^{*}}\frac{F_{D}^{*}}{\Phi ^{*}}\right) }{\frac{\partial \Gamma }{\partial \frac{B}{B^{*}}}\frac{\frac{B}{B^{*}}}{\Gamma }} \\= & {} \Psi \left[ \frac{1}{\sigma }\left( 1-\frac{\sigma -1}{k^{*}}\right) -\left( 1-\left( 1-\frac{1}{k^{*}}\frac{\sigma -1}{\sigma }\right) O^{*}\right) \left( \frac{k^{*}}{\sigma -1}-1\right) \right] \\= & {} \Psi \left( 1-O^{*}\right) \left( \frac{k^{*}}{\sigma -1} -1\right) \left( \frac{1}{k^{*}}\frac{\sigma -1}{\sigma }-1\right) <0 \end{aligned}$$
and it follows that
$$\begin{aligned} \frac{d\varphi _{D}}{dF_{D}}\frac{F_{D}}{\varphi _{D}}= & {} \frac{1}{k}+\frac{1 }{k}\frac{\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{\sigma -1}}}{ 1+\Phi \left( \frac{B}{B^{*}}\right) ^{-\frac{k}{\sigma -1}}}\left( \frac{\partial \Phi }{\partial F_{D}}\frac{F_{D}}{\Phi }-\frac{k}{\sigma -1} \frac{d\frac{B}{B^{*}}}{dF_{D}}\frac{F_{D}}{\frac{B}{B^{*}}}\right) \\= & {} \frac{1}{k}+\frac{1}{k}O\left( \left( \frac{k}{\sigma -1}-1\right) -\frac{ k}{\sigma -1}\Psi \left( 1-O\right) \left( \frac{k}{\sigma -1}-1\right) \left( 1-\frac{\sigma -1}{\sigma }\frac{1}{k}\right) \right) \\= & {} \frac{1}{k}+\frac{1}{k}O\left( \frac{k}{\sigma -1}-1\right) \left( 1- \frac{k}{\sigma -1}\Psi \left( 1-O\right) \left( 1-\frac{\sigma -1}{\sigma } \frac{1}{k}\right) \right) \\= & {} \frac{1}{k}+\frac{1}{k}O\left( \frac{k}{\sigma -1}-1\right) \left( \left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma }\right) \left( 1-O^{*}\right) +\frac{1}{\sigma }\right) \Psi>0\\ \frac{d\varphi _{D}}{dF_{D}^{*}}\frac{F_{D}^{*}}{\varphi _{D}}= & {} - \frac{1}{\sigma -1}O\frac{d\frac{B}{B^{*}}}{dF_{D}^{*}}\frac{ F_{D}^{*}}{\frac{B}{B^{*}}} \\= & {} O\Psi \left( 1-O^{*}\right) \frac{1}{k^{*}}\left( \frac{k^{*} }{\sigma -1}-1\right) \left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma } \right) >0 \end{aligned}$$
and for welfare
$$\begin{aligned} \frac{d{\mathcal {W}}}{dF_{D}}\frac{F_{D}}{{\mathcal {W}}}= & {} -\frac{1}{\sigma -1}+ \frac{d\varphi _{D}}{dF_{D}}\frac{F_{D}}{\varphi _{D}} \\= & {} -\frac{1}{\sigma -1}+\frac{1}{k}+\frac{1}{k}O\left( \frac{k}{\sigma -1} -1\right) \Psi \left( \left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma } \right) \left( 1-O^{*}\right) +\frac{1}{\sigma }\right) \\= & {} -\left( \frac{k}{\sigma -1}-1\right) \left( 1-O\right) \frac{1}{k}\left( \frac{k}{\sigma -1}+\left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma } \right) \left( 1-O^{*}\right) \right) \Psi <0\\ \frac{d{\mathcal {W}}}{dF_{D}^{*}}\frac{F_{D}^{*}}{{\mathcal {W}}}= & {} \frac{ d\varphi _{D}}{dF_{D}^{*}}\frac{F_{D}^{*}}{\varphi _{D}}>0. \end{aligned}$$
For the export thresholds
$$\begin{aligned} \frac{d\varphi _{X}}{dF_{D}}\frac{F_{D}}{\varphi _{X}}= & {} \frac{d\varphi _{D} }{dF_{D}}\frac{F_{D}}{\varphi _{D}}-\frac{1}{\sigma -1}+\frac{1}{\sigma -1} \frac{d\frac{B}{B^{*}}}{dF_{D}}\frac{F_{D}}{\frac{B}{B^{*}}} \\= & {} \frac{1}{k}\left( \frac{k}{\sigma -1}-1\right) \left[ -1+O\Psi \left( \left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma }\right) \left( 1-O^{*}\right) +\frac{1}{\sigma }\right) \right. \\&\left. +\Psi \left( 1-O\right) \left( \frac{k}{ \sigma -1}-\frac{1}{\sigma }\right) \right] \\= & {} \frac{1}{k}\left( \frac{k}{\sigma -1}-1\right) \left[ -1+\Psi \left[ \left( \frac{k^{*}}{\sigma -1}\right. \right. \right. \\&\left. \left. \left. -\frac{1}{\sigma }\right) \left( 1-O^{*}\right) O+\frac{1}{\sigma }O+\left( 1-O\right) \left( \frac{k}{\sigma -1}- \frac{1}{\sigma }\right) \right] \right] \\= & {} -\frac{1}{k}\left( \frac{k}{\sigma -1}-1\right) \left( 1-O\right) \\&\frac{ \left( \frac{k^{*}}{\sigma -1}-\frac{1}{\sigma }\right) \left( 1-O^{*}\right) +\frac{1}{\sigma }}{\left( \frac{k}{\sigma -1}-\frac{1}{\sigma } \right) \left( 1-O\right) +\left( \frac{k^{*}}{\sigma -1}-\frac{1}{ \sigma }\right) \left( 1-O^{*}\right) +\frac{1}{\sigma }}<0\\ \frac{d\varphi _{X}}{dF_{D}^{*}}\frac{F_{D}^{*}}{\varphi _{X}}= & {} \frac{d\varphi _{D}}{dF_{D}^{*}}\frac{F_{D}^{*}}{\varphi _{D}}+\frac{ 1}{\sigma -1}\frac{d\frac{B}{B^{*}}}{dF_{D}^{*}}\frac{F_{D}^{*}}{ \frac{B}{B^{*}}} \\= & {} \frac{1}{\sigma -1}\left( 1-O\right) \frac{d\frac{B}{B^{*}}}{ dF_{D}^{*}}\frac{F_{D}^{*}}{\frac{B}{B^{*}}}<0. \end{aligned}$$
Changes in the relative wage are given by
$$\begin{aligned} \frac{d\frac{W}{W^{*}}}{dF_{D}}\frac{F_{D}}{\frac{W}{W^{*}}}= & {} - \frac{1}{\sigma }+\frac{1}{\sigma }\frac{d\frac{B}{B^{*}}}{dF_{D}}\frac{ F_{D}}{\frac{B}{B^{*}}}+\frac{\sigma -1}{\sigma }\left[ \frac{d\varphi _{D}}{dF_{D}}\frac{F_{D}}{\varphi _{D}}-\frac{d\varphi _{D}^{*}}{dF_{D}} \frac{F_{D}}{\varphi _{D}^{*}}\right] \\= & {} -\frac{1-O}{\sigma }\left( 1-\frac{\sigma -1}{k}\right) \left( \frac{k}{ \sigma -1}O^{*}+\frac{k^{*}}{\sigma -1}\left( 1-O^{*}\right) \right) \Psi <0. \end{aligned}$$
1.4 A.4 No selection case
Consider the case with no selection, i.e., where \(\varphi _{\min }>\varphi _{X}\) for both countries. Let \({\hat{\varphi }}\equiv \int _{\varphi _{\min }}^{\infty }\varphi ^{\sigma -1}dH\left( \varphi \right)\) denote “average” productivity. In that case, the free-entry conditions read
$$\begin{aligned} W^{1-\sigma }\left( B+B^{*}\tau ^{1-\sigma }\right) {\hat{\varphi }}= & {} W\delta F_{E}+WF_{D}+WF_{X} \\ \left( W^{*}\right) ^{1-\sigma }\left( B^{*}+B\left( \tau ^{*}\right) ^{1-\sigma }\right) {\hat{\varphi }}^{*}= & {} W^{*}\delta ^{*}F_{E}^{*}+W^{*}F_{D}^{*}+W^{*}F_{X}^{*} \end{aligned}$$
which imply a relative wage of
$$\begin{aligned} \frac{W}{W^{*}}=\left( \frac{B+B^{*}\tau ^{1-\sigma }}{B^{*}+B\left( \tau ^{*}\right) ^{1-\sigma }}\frac{{\hat{\varphi }}}{{\hat{\varphi }}^{*}}\frac{\delta ^{*}F_{E}^{*}+F_{D}^{*}+F_{X}^{*}}{ \delta F_{E}+F_{D}+F_{X}}\right) ^{\frac{1}{\sigma }} \end{aligned}$$
and the balanced trade condition becomes
$$\begin{aligned} N^{*}B\left( W^{*}\tau ^{*}\right) ^{1-\sigma }{\hat{\varphi }} ^{*}=NB^{*}\left( W\tau \right) ^{1-\sigma }{\hat{\varphi }}. \end{aligned}$$
Openness reads
$$\begin{aligned} O=\frac{N^{*}B\left( W^{*}\tau ^{*}\right) ^{1-\sigma }{\hat{\varphi }}^{*}}{N^{*}B\left( W^{*}\tau ^{*}\right) ^{1-\sigma } {\hat{\varphi }}^{*}+NB\left( W\right) ^{1-\sigma }{\hat{\varphi }}}=\frac{ \left( \frac{B}{B^{*}}\right) ^{-1}\tau ^{1-\sigma }}{1+\left( \frac{B}{ B^{*}}\right) ^{-1}\tau ^{1-\sigma }} \end{aligned}$$
and the labor market equilibrium condition reads
$$\begin{aligned} L= & {} N\delta F_{E}+NF_{D}+NF_{X} \\&+N\int _{\varphi _{\min }}^{\infty }\frac{1}{\varphi }E\left( P\right) ^{\sigma -1}\left( p_{D}\left( \varphi \right) \right) ^{-\sigma }dH\left( \varphi \right) \\&+N\int _{\varphi _{\min }}^{\infty }\frac{\tau }{\varphi }E^{*}\left( P^{*}\right) ^{\sigma -1}\left( p_{X}\left( \varphi \right) \right) ^{-\sigma }dH\left( \varphi \right) \\= & {} N\delta F_{E}+NF_{D}+NF_{X} \\&+NB\left( W\right) ^{-\sigma }\int _{\varphi _{\min }}^{\infty }\varphi ^{\sigma -1}dH\left( \varphi \right) \\&+NB^{*}\left( W\right) ^{-\sigma }\left( \tau \right) ^{1-\sigma }\int _{\varphi _{\min }}^{\infty }\varphi ^{\sigma -1}dH\left( \varphi \right) \\= & {} N\left( \delta F_{E}+F_{D}+F_{X}+\left( B+B^{*}\left( \tau \right) ^{1-\sigma }\right) \left( W\right) ^{-\sigma }{\hat{\varphi }}\right) . \end{aligned}$$
From the labor market equilibria in the two countries it follows that
$$\begin{aligned} \frac{N}{N^{*}}=\frac{\delta ^{*}F_{E}^{*}+F_{D}^{*}+F_{X}^{*}+\left( B^{*}+B\left( \tau ^{*}\right) ^{1-\sigma }\right) \left( W^{*}\right) ^{-\sigma }{\hat{\varphi }}^{*}}{\delta F_{E}+F_{D}+F_{X}+\left( B+B^{*}\left( \tau \right) ^{1-\sigma }\right) \left( W\right) ^{-\sigma }{\hat{\varphi }}}\frac{L}{L^{*}} \end{aligned}$$
and inserting this into the balanced trade condition yields
$$\begin{aligned} \frac{B\left( W^{*}\tau ^{*}\right) ^{1-\sigma }}{B^{*}\left( W\tau \right) ^{1-\sigma }}=\frac{\delta ^{*}F_{E}^{*}+F_{D}^{*}+F_{X}^{*}+\left( B^{*}+B\left( \tau ^{*}\right) ^{1-\sigma }\right) \left( W^{*}\right) ^{-\sigma }{\hat{\varphi }}^{*}}{\delta F_{E}+F_{D}+F_{X}+\left( B+B^{*}\left( \tau \right) ^{1-\sigma }\right) \left( W\right) ^{-\sigma }{\hat{\varphi }}}\frac{{\hat{\varphi }}}{{\hat{\varphi }} ^{*}}\frac{L}{L^{*}}. \end{aligned}$$
Next insert the free-entry condition to obtain
$$\begin{aligned} \frac{B\left( \tau ^{*}\right) ^{1-\sigma }}{B^{*}\left( \tau \right) ^{1-\sigma }}=\frac{1+\frac{B}{B^{*}}\left( \tau ^{*}\right) ^{1-\sigma }}{\frac{B}{B^{*}}+\tau ^{1-\sigma }}\frac{W}{W^{*}}\frac{ L}{L^{*}} \end{aligned}$$
and finally insert the relative wage from above to obtain
$$\begin{aligned} \frac{B}{B^{*}}\left( \frac{\tau ^{*}}{\tau }\right) ^{1-\sigma }\left( \frac{\frac{B}{B^{*}}+\tau ^{1-\sigma }}{1+\frac{B}{B^{*}} \left( \tau ^{*}\right) ^{1-\sigma }}\right) ^{\frac{\sigma -1}{\sigma } }=\left( \frac{{\hat{\varphi }}}{{\hat{\varphi }}^{*}}\frac{\delta ^{*}F_{E}^{*}+F_{D}^{*}+F_{X}^{*}}{\delta F_{E}+F_{D}+F_{X}}\right) ^{\frac{1}{\sigma }}\frac{L}{L^{*}} \end{aligned}$$
which we can write as
$$\begin{aligned} {\hat{\Gamma }}\left( \frac{B}{B^{*}},\tau ,\tau ^{*}\right) ={\hat{\digamma }}\left( Z,Z^{*}\right) , \end{aligned}$$
where \({\hat{\Gamma }}\left( \frac{B}{B^{*}},\tau ,\tau ^{*}\right) \equiv \frac{B}{B^{*}}\left( \frac{\tau ^{*}}{\tau }\right) ^{1-\sigma }\left( \frac{\frac{B}{B^{*}}+\tau ^{1-\sigma }}{1+\frac{B}{ B^{*}}\left( \tau ^{*}\right) ^{1-\sigma }}\right) ^{\frac{\sigma -1 }{\sigma }}\) and \({\hat{\digamma }}\left( Z,Z^{*}\right) \equiv \left( \frac{{\hat{\varphi }}}{{\hat{\varphi }}^{*}}\frac{\delta ^{*}F_{E}^{*}+F_{D}^{*}+F_{X}^{*}}{\delta F_{E}+F_{D}+F_{X}}\right) ^{\frac{1}{ \sigma }}\frac{L}{L^{*}}\). Note that \({\hat{\Gamma }}\) is continuous and increasing in \(\frac{B}{B^{*}}\) and that \(\lim _{\frac{B}{B^{*}} \rightarrow 0}{\hat{\Gamma }}=0\) and \(\lim _{\frac{B}{B^{*}}\rightarrow \infty }{\hat{\Gamma }}=\infty\). Hence, there exists a unique equilibrium for the relative demand component.
Combining the free entry condition and the definition of the demand component, B, and inserting that \(E=WL\) implies that
$$\begin{aligned} {\mathcal {W}}= & {} \frac{W}{P} \\= & {} \frac{\sigma -1}{\sigma }\left( \frac{{\hat{\varphi }}}{\sigma }\frac{L}{ \delta F_{E}+F_{D}+F_{X}}\right) ^{\frac{1}{\sigma -1}}\left( 1+\left( \frac{ B}{B^{*}}\right) ^{-1}\tau ^{1-\sigma }\right) ^{\frac{1}{\sigma -1}} \\= & {} \frac{\sigma -1}{\sigma }\left( \frac{{\hat{\varphi }}}{\sigma }\frac{L}{ \delta F_{E}+F_{D}+F_{X}}\right) ^{\frac{1}{\sigma -1}}\left( 1-O\right) ^{- \frac{1}{\sigma -1}}\\= & {} \frac{\sigma -1}{\sigma }\left( \frac{{\hat{\varphi }}}{ \sigma }\frac{L}{\delta F_{E}+F_{D}+F_{X}}\right) ^{\frac{1}{\sigma -1} }\lambda ^{-\frac{1}{\sigma -1}} \end{aligned}$$
where
$$\begin{aligned} \frac{\partial {\mathcal {W}}}{\partial \frac{B}{B^{*}}}\frac{\frac{B}{ B^{*}}}{{\mathcal {W}}}=-\frac{1}{\sigma -1}O<0. \end{aligned}$$
Properties of the equilibrium condition
Starting with \({\hat{\Gamma }}\left( \frac{B}{B^{*}},\tau ,\tau ^{*}\right)\) we have that
$$\begin{aligned} \frac{\partial {\hat{\Gamma }}\left( \frac{B}{B^{*}},\tau ,\tau ^{*}\right) }{\partial \frac{B}{B^{*}}}\frac{\frac{B}{B^{*}}}{{\hat{\Gamma }}}= & {} 1+\frac{\sigma -1}{\sigma }\left( \frac{\frac{B}{B^{*}}}{ \frac{B}{B^{*}}+\tau ^{1-\sigma }}-\frac{\frac{B}{B^{*}}\left( \tau ^{*}\right) ^{1-\sigma }}{1+\frac{B}{B^{*}}\left( \tau ^{*}\right) ^{1-\sigma }}\right) \\= & {} 1+\frac{\sigma -1}{\sigma }\frac{B}{B^{*}}\frac{1-\tau ^{1-\sigma }\left( \tau ^{*}\right) ^{1-\sigma }}{\left( \frac{B}{B^{*}}+\tau ^{1-\sigma }\right) \left( 1+\frac{B}{B^{*}}\left( \tau ^{*}\right) ^{1-\sigma }\right) } \\= & {} 1+\frac{\sigma -1}{\sigma }\left( 1-O-O^{*}\right)>0\\ \frac{\partial {\hat{\Gamma }}\left( \frac{B}{B^{*}},\tau ,\tau ^{*}\right) }{\partial \tau }\frac{\tau }{{\hat{\Gamma }}}= & {} \left( \sigma -1\right) \left( 1-\frac{\sigma -1}{\sigma }O\right) >0 \\ \frac{\partial {\hat{\Gamma }}\left( \frac{B}{B^{*}},\tau ,\tau ^{*}\right) }{\partial \tau ^{*}}\frac{\tau ^{*}}{{\hat{\Gamma }}}= & {} -\left( \sigma -1\right) \left( 1-\frac{\sigma -1}{\sigma }O^{*}\right) <0 \end{aligned}$$
and turning to \({\hat{\digamma }}\left( Z,Z^{*}\right)\) where
$$\begin{aligned} \frac{\partial {\hat{\digamma }}\left( Z,Z^{*}\right) }{\partial L}\frac{L}{ {\hat{\digamma }}}= & {} -\frac{\partial {\hat{\digamma }}\left( Z,Z^{*}\right) }{ \partial L^{*}}\frac{L^{*}}{{\hat{\digamma }}}=1 \\ \frac{\partial {\hat{\digamma }}\left( Z,Z^{*}\right) }{\partial \left( \delta F_{E}+F_{D}+F_{X}\right) }\frac{\delta F_{E}+F_{D}+F_{X}}{{\hat{\digamma }}}= & {} -\frac{\partial {\hat{\digamma }}\left( Z,Z^{*}\right) }{ \partial \left( \delta ^{*}F_{E}^{*}+F_{D}^{*}+F_{X}^{*}\right) }\frac{\delta ^{*}F_{E}^{*}+F_{D}^{*}+F_{X}^{*}}{ {\hat{\digamma }}}\\= & {} -\frac{1}{\sigma } \\ \frac{\partial {\hat{\digamma }}\left( Z,Z^{*}\right) }{\partial {\hat{\varphi }}}\frac{{\hat{\varphi }}}{{\hat{\digamma }}}= & {} -\frac{\partial {\hat{\digamma }}\left( Z,Z^{*}\right) }{\partial {\hat{\varphi }}^{*}}\frac{ {\hat{\varphi }}^{*}}{{\hat{\digamma }}}=\frac{1}{\sigma }>0 \end{aligned}$$
and we thus have \(\frac{B}{B^{*}}\left( \underset{-}{\tau },\underset{+}{ \tau ^{*}},\underset{+}{L},\underset{-}{L^{*}},\underset{+}{{\hat{\varphi }}},\underset{-}{{\hat{\varphi }}^{*}},\underset{-}{\left( \delta F_{E}+F_{D}+F_{X}\right) },\underset{+}{\left( \delta ^{*}F_{E}^{*}+F_{D}^{*}+F_{X}^{*}\right) }\right)\), where the signs denote the signs of the partial derivatives of the relative demand component.
Welfare
Recall that
$$\begin{aligned} {\mathcal {W}}=\frac{\sigma -1}{\sigma }\left( \frac{1}{\sigma }\frac{L}{\delta F_{E}+F_{D}+F_{X}}\right) ^{\frac{1}{\sigma -1}}\left( {\hat{\varphi }}\right) ^{\frac{1}{\sigma -1}}\left( 1+\left( \frac{B}{B^{*}}\right) ^{-1}\tau ^{1-\sigma }\right) ^{\frac{1}{\sigma -1}}, \end{aligned}$$
where
$$\begin{aligned} \frac{\partial {\mathcal {W}}}{\partial \frac{B}{B^{*}}}\frac{\frac{B}{ B^{*}}}{{\mathcal {W}}}=-\frac{1}{\sigma -1}O<0. \end{aligned}$$
It follows that
$$\begin{aligned} \frac{\partial {\mathcal {W}}}{\partial \left( \delta F_{E}+F_{D}+F_{X}\right) } \frac{\left( \delta F_{E}+F_{D}+F_{X}\right) }{{\mathcal {W}}}= & {} -\frac{1}{ \sigma -1} -\frac{1}{\sigma -1}O\frac{\partial \frac{B}{B^{*}}}{\partial \left( \delta F_{E}+F_{D}+F_{X}\right) }\\&\frac{\left( \delta F_{E}+F_{D}+F_{X}\right) }{\frac{B}{B^{*}}} \\= & {} -\frac{1}{\sigma -1}\frac{1-\frac{1}{\sigma }O+\frac{\sigma -1}{\sigma } \left( 1-O-O^{*}\right) }{1+\frac{\sigma -1}{\sigma }\left( 1-O-O^{*}\right) }<0\\ \frac{\partial {\mathcal {W}}}{\partial \left( \delta ^{*}F_{E}^{*}+F_{D}^{*}+F_{X}^{*}\right) }\frac{\left( \delta ^{*}F_{E}^{*}+F_{D}^{*}+F_{X}^{*}\right) }{{\mathcal {W}}}= & {} -\frac{1}{ \sigma -1}O\frac{\partial \frac{B}{B^{*}}}{\partial \left( \delta ^{*}F_{E}^{*}+F_{D}^{*}+F_{X}^{*}\right) }\\&\frac{\left( \delta ^{*}F_{E}^{*}+F_{D}^{*}+F_{X}^{*}\right) }{\frac{B}{B^{*} }}<0\\ \frac{\partial {\mathcal {W}}}{\partial {\hat{\varphi }}}\frac{{\hat{\varphi }}}{ {\mathcal {W}}}= & {} \frac{1}{\sigma -1}-\frac{1}{\sigma -1}O\frac{\partial \frac{ B}{B^{*}}}{\partial {\hat{\varphi }}}\frac{{\hat{\varphi }}}{\frac{B}{B^{*} }} \\= & {} \frac{1}{\sigma -1}\frac{1-O\frac{1}{\sigma }+\frac{\sigma -1}{\sigma } \left( 1-O-O^{*}\right) }{1+\frac{\sigma -1}{\sigma }\left( 1-O-O^{*}\right) }>0\\ \frac{\partial {\mathcal {W}}}{\partial {\hat{\varphi }}^{*}}\frac{{\hat{\varphi }}^{*}}{{\mathcal {W}}}= & {} -\frac{1}{\sigma -1}O\frac{\partial \frac{B}{B^{*}}}{\partial {\hat{\varphi }}^{*}}\frac{{\hat{\varphi }}^{*}}{\frac{B}{ B^{*}}}>0\\ \frac{\partial {\mathcal {W}}}{\partial \tau }\frac{\tau }{{\mathcal {W}}}= & {} \frac{1}{\sigma -1}O\left( \left( 1-\sigma \right) -\frac{\partial \frac{B}{ B^{*}}}{\partial \tau }\frac{\tau }{\frac{B}{B^{*}}}\right) \\= & {} -O\frac{\frac{\sigma -1}{\sigma }\left( 1-O^{*}\right) }{1+\frac{ \sigma -1}{\sigma }\left( 1-O-O^{*}\right) }<0\\ \frac{\partial {\mathcal {W}}}{\partial \tau ^{*}}\frac{\tau ^{*}}{ {\mathcal {W}}}= & {} -\frac{1}{\sigma -1}O\frac{\partial \frac{B}{B^{*}}}{ \partial \tau ^{*}}\frac{\tau ^{*}}{\frac{B}{B^{*}}}<0\\ \frac{\partial {\mathcal {W}}}{\partial L}\frac{L}{{\mathcal {W}}}= & {} \frac{1}{ \sigma -1}+\frac{1}{\sigma -1}O\left( -\frac{\partial \frac{B}{B^{*}}}{ \partial L}\frac{L}{\frac{B}{B^{*}}}\right) \\= & {} \frac{1}{\sigma -1}\frac{1-O+\frac{\sigma -1}{\sigma }\left( 1-O-O^{*}\right) }{1+\frac{\sigma -1}{\sigma }\left( 1-O-O^{*}\right) }>0\\ \frac{\partial {\mathcal {W}}}{\partial L^{*}}\frac{L^{*}}{{\mathcal {W}}}= & {} -\frac{1}{\sigma -1}O\frac{\partial \frac{B}{B^{*}}}{\partial L^{*}} \frac{L^{*}}{\frac{B}{B^{*}}}>0. \end{aligned}$$
In sum we have that \({\mathcal {W}}\left( \underset{-}{\tau },\underset{-}{\tau ^{*}},\underset{+}{L},\underset{+}{L^{*}},\underset{+}{{\hat{\varphi }}} ,\underset{+}{{\hat{\varphi }}^{*}},\underset{-}{\left( \delta F_{E}+F_{D}+F_{X}\right) },\underset{-}{\left( \delta ^{*}F_{E}^{*}+F_{D}^{*}+F_{X}^{*}\right) }\right)\), where the signs denote the signs of the partial derivatives of domestic welfare (real income). From the expression for the relative wage
$$\begin{aligned} \frac{W}{W^{*}}=\left( \frac{\frac{B}{B^{*}}+\tau ^{1-\sigma }}{1+ \frac{B}{B^{*}}\left( \tau ^{*}\right) ^{1-\sigma }}\frac{{\hat{\varphi }}}{{\hat{\varphi }}^{*}}\frac{\delta ^{*}F_{E}^{*}+F_{D}^{*}+F_{X}^{*}}{\delta F_{E}+F_{D}+F_{X}}\right) ^{\frac{1}{ \sigma }}, \end{aligned}$$
it follows that any improvement in domestic business conditions increases the relative wage, whereas the opposite applies for improvements in foreign business conditions.
1.5 A.5 Three-country analysis
Consider a three-country extension of the framework, which accounts for third country-effects, i.e., that a change in business conditions in say, country 1 affects country 3 directly through trade between the two countries, but also indirectly via trade with country 2. Below, the model is written up for \(J\ge 2\) countries and eventually a numerical analysis is conducted for the three-country case.
Firm-level profits of a firm located in country i with productivity \(\varphi\) read
$$\begin{aligned} \pi _{i}\left( \varphi \right) =\max \left[ W_{i}^{1-\sigma }\varphi ^{\sigma -1}\sum _{j}B_{j}I_{j}\tau _{ij}^{1-\sigma }-W_{i}\sum _{j}I_{j}F_{ij},0\right] , \end{aligned}$$
where \(B_{j}\equiv E_{j}P_{j}^{\sigma -1}\left( \frac{\sigma }{\sigma -1} \right) ^{-\sigma }\frac{1}{\sigma -1}\) is the demand component in country \(j,W_{i}\) is the wage in country i, \(\tau _{ij}\) is the iceberg trade costs when exporting from country i to country j (assume that \(\tau _{ii}=1\), i.e., no domestic iceberg trade costs), \(F_{ij}\) is the fixed costs (in country i labor units) of serving country j for a firm in country i, and \(\ I_{j}\) is a dummy variable taking value 1 if serving country j and zero otherwise. To ease notation, let \(\Phi _{ij}\equiv \left( F_{ij}\right) ^{1-\frac{k_{i}}{\sigma -1}}\left( \tau _{ij}\right) ^{-k_{i}}\), which is a trade costs aggregate for exports from country i to country j. The zero-profit conditions read \(W_{i}^{1-\sigma }\left( \varphi _{ij}\right) ^{\sigma -1}B_{j}\tau _{ij}^{1-\sigma }\equiv W_{i}F_{ij}\) for all \(\left( i,j\right)\), while the free-entry conditions read \(\int _{\varphi _{\min ,i}}^{\infty }\pi _{i}\left( \varphi \right) dH_{i}\left( \varphi \right) =W_{i}\delta _{i}F_{\mathop {\mathrm{Ei}}\nolimits }\) for all i, where \(F_{\mathop {\mathrm{Ei}}\nolimits }\) is the entry costs (in local labor units) in country i, and \(H_{i}\left( \varphi \right)\) is the underlying productivity distribution in country i, and \(\varphi _{ij}\) is the productivity threshold for a firm in country i to serve country j. It follows from the zero-profit condition that \(B_{j}\tau _{ij}^{1-\sigma }\equiv \frac{W_{i}F_{ij}}{W_{i}^{1-\sigma }\left( \varphi _{ij}\right) ^{\sigma -1}},\) and taking the ratio for country i and country j yields \(\varphi _{ij}\equiv \varphi _{ii}\left( \frac{B_{i}}{ B_{j}}\frac{F_{ij}}{F_{ii}}\right) ^{\frac{1}{\sigma -1}}\frac{\tau _{ij}}{ \tau _{ii}}\), which in turn implies that the free entry condition may be rewritten as
$$\begin{aligned} \sum _{j}F_{ij}\left( \left[ \int _{\varphi _{ij}}^{\infty }\left( \frac{ \varphi }{\varphi _{ij}}\right) ^{\sigma -1}-1\right] dH_{i}\left( \varphi \right) \right) =\frac{\left( \sigma -1\right) }{k_{i}-\left( \sigma -1\right) }\sum _{j}F_{ij}\left( \frac{\varphi _{ij}}{\varphi _{\min ,i}} \right) ^{-k_{i}}=\delta _{i}F_{\mathop {\mathrm{Ei}}\nolimits }, \end{aligned}$$
where \(k_{i}\) is the shape parameter of the Pareto distribution for productivities in country i. Using \(\varphi _{ij}\equiv \varphi _{ii}\left( \frac{B_{i}}{B_{j}}\frac{F_{ij}}{F_{ii}}\right) ^{\frac{1}{ \sigma -1}}\frac{\tau _{ij}}{\tau _{ii}}\), it follows that
$$\begin{aligned} \varphi _{ii}= & {} \varphi _{\min ,i}\left( \frac{\left( \sigma -1\right) }{ k_{i}-\left( \sigma -1\right) }\sum _{j}\frac{F_{ij}}{\delta _{i}F_{\mathop {\mathrm{Ei}}\nolimits } }\left( \frac{B_{i}}{B_{j}}\frac{F_{ij}}{F_{ii}}\right) ^{-\frac{k_{i}}{ \sigma -1}}\left( \frac{\tau _{ij}}{\tau _{ii}}\right) ^{-k_{i}}\right) ^{ \frac{1}{k_{i}}} \\= & {} \varphi _{\min ,i}\left( \frac{\left( \sigma -1\right) }{k_{i}-\left( \sigma -1\right) }\frac{F_{ii}}{\delta _{i}F_{\mathop {\mathrm{Ei}}\nolimits }}\sum _{j}\frac{\Phi _{ij}}{\Phi _{ii}}\left( \frac{B_{i}}{B_{j}}\right) ^{-\frac{k_{i}}{\sigma -1 }}\right) ^{\frac{1}{k_{i}}} \\= & {} \varphi _{aut,i}\left( \sum _{j}\frac{\Phi _{ij}}{\Phi _{ii}}\left( \frac{ B_{i}}{B_{j}}\right) ^{-\frac{k_{i}}{\sigma -1}}\right) ^{\frac{1}{k_{i}}}, \end{aligned}$$
where \(\varphi _{aut,i}=\varphi _{\min ,i}\left( \frac{\left( \sigma -1\right) }{k_{i}-\left( \sigma -1\right) }\frac{F_{ii}}{\delta _{i}F_{\mathop {\mathrm{ Ei}}\nolimits }}\right) ^{\frac{1}{k_{i}}}\) is the autarky threshold of country i.
Imports and exports read
$$\begin{aligned} I_{i}=\sum _{j\ne i}N_{j}\int _{\varphi _{ji}}^{\infty }E_{i}\left( P_{i}\right) ^{\sigma -1}\left( \frac{\sigma }{\sigma -1}\frac{W_{j}\tau _{ji}}{\varphi }\right) ^{1-\sigma }\frac{dH_{j}\left( \varphi \right) }{ 1-H_{j}\left( \varphi _{jj}\right) } \end{aligned}$$
and
$$\begin{aligned} X_{i}=N_{i}\sum _{j\ne i}\int _{\varphi _{ij}}^{\infty }E_{j}\left( P_{j}\right) ^{\sigma -1}\left( \frac{\sigma }{\sigma -1}\frac{W_{i}\tau _{ij}}{\varphi }\right) ^{1-\sigma }\frac{dH_{i}\left( \varphi \right) }{ 1-H_{i}\left( \varphi _{ii}\right) }, \end{aligned}$$
where \(N_{i}\) is the number of active firms (exporters and non-exporters) in country i. Trade balance implies
$$\begin{aligned} N_{i}\sum _{j\ne i}\int _{\varphi _{ij}}^{\infty }B_{j}\left( \frac{W_{i}\tau _{ij}}{\varphi }\right) ^{1-\sigma }\frac{dH_{i}\left( \varphi \right) }{ 1-H_{i}\left( \varphi _{ii}\right) }=\sum _{j\ne i}N_{j}\int _{\varphi _{ji}}^{\infty }B_{i}\left( \frac{W_{j}\tau _{ji}}{\varphi }\right) ^{1-\sigma }\frac{dH_{j}\left( \varphi \right) }{1-H_{j}\left( \varphi _{jj}\right) } \end{aligned}$$
which can be rewritten as
$$\begin{aligned} W_{i}N_{i}\frac{k_{i}}{k_{i}-\left( \sigma -1\right) }\sum _{j\ne i}F_{ij}\left( \frac{\varphi _{ij}}{\varphi _{ii}}\right) ^{-k_{i}}=\sum _{j\ne i}W_{j}N_{j}F_{ji}\frac{k_{j}}{k_{j}-\left( \sigma -1\right) }\left( \frac{\varphi _{ji}}{\varphi _{jj}}\right) ^{-k_{j}} \end{aligned}$$
i.e., one equation per country (one is redundant due to Walras’ law). The domestic expenditure share reads
$$\begin{aligned} \lambda _{i}= & {} 1-\frac{I_{i}}{E_{i}} \\= & {} \frac{N_{i}\int _{\varphi _{ii}}^{\infty }E_{i}\left( P_{i}\right) ^{\sigma -1}\left( \frac{\sigma }{\sigma -1}\frac{W_{i}\tau _{ii}}{\varphi } \right) ^{1-\sigma }\frac{dH_{i}\left( \varphi \right) }{1-H_{i}\left( \varphi _{ii}\right) }}{\sum _{j\ne i}N_{j}\int _{\varphi _{ji}}^{\infty }E_{i}\left( P_{i}\right) ^{\sigma -1}\left( \frac{\sigma }{\sigma -1}\frac{ W_{j}\tau _{ji}}{\varphi }\right) ^{1-\sigma }\frac{dH_{j}\left( \varphi \right) }{1-H_{j}\left( \varphi _{jj}\right) }+N_{i}\int _{\varphi _{ii}}^{\infty }E_{i}\left( P_{i}\right) ^{\sigma -1}\left( \frac{\sigma }{ \sigma -1}\frac{W_{i}\tau _{ii}}{\varphi }\right) ^{1-\sigma }\frac{ dH_{i}\left( \varphi \right) }{1-H_{i}\left( \varphi _{ii}\right) }} \\= & {} \frac{N_{i}\int _{\varphi _{ii}}^{\infty }\left( \frac{W_{i}\tau _{ii}}{ \varphi }\right) ^{1-\sigma }\frac{dH_{i}\left( \varphi \right) }{ 1-H_{i}\left( \varphi _{ii}\right) }}{N_{i}\sum _{j\ne i}\int _{\varphi _{ij}}^{\infty }\frac{B_{j}}{B_{i}}\left( \frac{W_{i}\tau _{ij}}{\varphi } \right) ^{1-\sigma }\frac{dH_{i}\left( \varphi \right) }{1-H_{i}\left( \varphi _{ii}\right) }+N_{i}\int _{\varphi _{ii}}^{\infty }\left( \frac{ W_{i}\tau _{ii}}{\varphi }\right) ^{1-\sigma }\frac{dH_{i}\left( \varphi \right) }{1-H_{i}\left( \varphi _{ii}\right) }} \\= & {} \frac{1}{\sum _{j\ne i}\left( \frac{\tau _{ij}}{\tau _{ii}}\right) ^{1-\sigma }\left( \frac{\varphi _{ij}}{\varphi _{ii}}\right) ^{-\left( k_{i}-\left( \sigma -1\right) \right) }\frac{B_{j}}{B_{i}}+1} \\= & {} \frac{1}{\sum _{j\ne i}\left( \frac{\tau _{ij}}{\tau _{ii}}\right) ^{1-\sigma }\left( \frac{B_{i}}{B_{j}}\right) ^{-\frac{k_{i}}{\sigma -1} }\left( \frac{F_{ij}}{F_{ii}}\right) ^{1-\frac{k_{i}}{\sigma -1}}\left( \frac{\tau _{ij}}{\tau _{ii}}\right) ^{-\left( k_{i}-\left( \sigma -1\right) \right) }+1}=\frac{\sum _{j\ne i}\Phi _{ii}B_{i}^{\frac{k_{i}}{\sigma -1}}}{ \sum _{j}\Phi _{ij}B_{j}^{\frac{k_{i}}{\sigma -1}}} \\= & {} \left( \frac{\varphi _{ii}}{\varphi _{\min ,i}}\right) ^{-k_{i}}\frac{ \sigma -1}{k_{i}-\left( \sigma -1\right) }\frac{F_{ii}}{\delta _{i}F_{\mathop {\mathrm{ Ei}}\nolimits }}=\left( \frac{\varphi _{ii}}{\varphi _{aut,i}}\right) ^{-k_{i}}. \end{aligned}$$
The exit threshold may thus be written as \(\varphi _{ii}=\varphi _{aut,i}\lambda _{i}^{-\frac{1}{k_{i}}}\).
Labor demand in country i reads
$$\begin{aligned} L_{i}^{d}= & {} \frac{N_{i}\delta _{i}F_{\mathop {\mathrm{Ei}}\nolimits }}{1-H_{i}\left( \varphi _{ii}\right) }+N_{i}\sum _{j}F_{ij}\frac{1-H_{i}\left( \varphi _{ij}\right) }{ 1-H_{i}\left( \varphi _{ii}\right) } \\&+N_{i}\sum _{j}\int _{\varphi _{ij}}^{\infty }\frac{\tau _{ij}}{\varphi } E_{j}\left( P_{j}\right) ^{\sigma -1}\left( \tau _{ij}p_{i}\left( \varphi \right) \right) ^{-\sigma }\frac{dH_{i}\left( \varphi \right) }{ 1-H_{i}\left( \varphi _{ii}\right) } \\= & {} N_{i}\frac{k_{i}\sigma }{k_{i}-\left( \sigma -1\right) } \sum _{j}F_{ij}\left( \frac{\varphi _{ij}}{\varphi _{ii}}\right) ^{-k_{i}} \\= & {} N_{i}\frac{k_{i}\sigma }{k_{i}-\left( \sigma -1\right) } \sum _{j}F_{ij}\left( \frac{B_{i}}{B_{j}}\frac{F_{ij}}{F_{ii}}\right) ^{- \frac{k_{i}}{\sigma -1}}\left( \frac{\tau _{ij}}{\tau _{ii}}\right) ^{-k_{i}} \end{aligned}$$
and labor market equilibrium implies for country i that
$$\begin{aligned} L_{i}= & {} N_{i}\frac{k_{i}\sigma }{k_{i}-\left( \sigma -1\right) } \sum _{j}F_{ij}\left( \frac{\varphi _{ij}}{\varphi _{ii}}\right) ^{-k_{i}}=N_{i}\frac{k_{i}\sigma }{\sigma -1}\left( \frac{\varphi _{ii}}{ \varphi _{\min ,i}}\right) ^{k_{i}}\delta _{i}F_{\mathop {\mathrm{Ei}}\nolimits } \\= & {} N_{i}\frac{k_{i}\sigma }{k_{i}-\left( \sigma -1\right) }\frac{F_{ii}}{ \lambda _{i}}. \end{aligned}$$
It follows that the number of firms become
$$\begin{aligned} N_{i}=\frac{k_{i}-\left( \sigma -1\right) }{k_{i}\sigma }\frac{L_{i}}{F_{ii}} \lambda _{i}. \end{aligned}$$
From the domestic threshold condition and the definition of the demand component it follows that
$$\begin{aligned} {\mathcal {W}}_{i}= & {} \frac{W_{i}}{P_{i}}=\left( \frac{L_{i}}{F_{ii}}\left( \frac{\sigma }{\sigma -1}\right) ^{-\sigma }\frac{1}{\sigma -1}\right) ^{ \frac{1}{\sigma -1}}\frac{\varphi _{ii}}{\tau _{ii}} \\= & {} \left( \lambda _{i}\right) ^{-\frac{1}{k_{i}}}\varphi _{aut,i}\left( \frac{L_{i}}{F_{ii}}\left( \frac{\sigma }{\sigma -1}\right) ^{-\sigma }\frac{ 1}{\sigma -1}\tau _{ii}^{1-\sigma }\right) ^{\frac{1}{\sigma -1}} \\= & {} {\mathcal {W}}_{aut,i}\left( \lambda _{i}\right) ^{-\frac{1}{k_{i}}}. \end{aligned}$$
Now revisit the balanced trade conditions
$$\begin{aligned} W_{i}N_{i}\frac{k_{i}}{k_{i}-\left( \sigma -1\right) }\sum _{j\ne i}F_{ij}\left( \frac{\varphi _{ij}}{\varphi _{ii}}\right) ^{-k_{i}}=\sum _{j\ne i}W_{j}N_{j}F_{ji}\frac{k_{j}}{k_{j}-\left( \sigma -1\right) }\left( \frac{\varphi _{ji}}{\varphi _{jj}}\right) ^{-k_{j}}. \end{aligned}$$
Insert the number of firms to obtain
$$\begin{aligned} W_{i}L_{i}\frac{\sum _{j\ne i}F_{ij}\left( \frac{\varphi _{ij}}{\varphi _{ii} }\right) ^{-k_{i}}}{\sum _{j}F_{ij}\left( \frac{\varphi _{ij}}{\varphi _{ii}} \right) ^{-k_{i}}}=\frac{\sum _{j\ne i}W_{j}L_{j}F_{ji}\left( \frac{\varphi _{ji}}{\varphi _{jj}}\right) ^{-k_{j}}}{\sum _{i}F_{ji}\left( \frac{\varphi _{ji}}{\varphi _{jj}}\right) ^{-k_{j}}}. \end{aligned}$$
Next, use the definition of \(\lambda\) and insert wages to obtain
$$\begin{aligned} L_{i}\left( 1-\lambda _{i}\right) =\sum _{j\ne i}\lambda _{j}\left( \frac{ \varphi _{jj}}{\varphi _{ii}}\frac{\tau _{ii}}{\tau _{jj}}\right) ^{\frac{ \sigma -1}{\sigma }}\left( \frac{F_{ii}}{F_{jj}}\right) ^{\frac{1}{\sigma } }L_{j}\left( \frac{B_{j}}{B_{i}}\right) ^{-\frac{k_{j}}{\sigma -1}}\frac{ \Phi _{ji}}{\Phi _{jj}}. \end{aligned}$$
Three-country case: numerical analysis
Normalize \(B_{1}=1\) and let \(b_{21}\equiv \frac{B_{2}}{B_{1}}=B_{2}\) and \(b_{31}\equiv \frac{B_{3}}{B_{1}}=B_{3}\). It follows that \(\frac{B_{2}}{B_{3}} =\frac{b_{21}}{b_{31}}\). Assume that \(\tau _{ii}=1\) and \(k_{i}=k\) for all i . It follows that
$$\begin{aligned} \varphi _{11}= & {} \varphi _{aut,1}\left( 1+\frac{\Phi _{12}}{\Phi _{11}} \left( b_{21}\right) ^{\frac{k}{\sigma -1}}+\frac{\Phi _{13}}{\Phi _{11}} \left( b_{31}\right) ^{\frac{k}{\sigma -1}}\right) ^{\frac{1}{k}} \\ \varphi _{22}= & {} \varphi _{aut,2}\left( \frac{\Phi _{21}}{\Phi _{22}}\left( b_{21}\right) ^{-\frac{k}{\sigma -1}}+1+\frac{\Phi _{23}}{\Phi _{22}}\left( \frac{b_{21}}{b_{31}}\right) ^{-\frac{k}{\sigma -1}}\right) ^{\frac{1}{k}} \\ \varphi _{33}= & {} \varphi _{aut,3}\left( \frac{\Phi _{31}}{\Phi _{33}}\left( b_{31}\right) ^{-\frac{k}{\sigma -1}}+\frac{\Phi _{32}}{\Phi _{33}}\left( \frac{b_{31}}{b_{21}}\right) ^{-\frac{k}{\sigma -1}}+1\right) ^{\frac{1}{k}}. \end{aligned}$$
In the numerical analysis we assume that \(L_{1}=L_{2}=L_{3}\) and \(F_{11}=F_{22}=F_{33}\), which implies that the balanced trade conditions become
$$\begin{aligned} 1= & {} \left( \frac{\varphi _{22}}{\varphi _{aut,2}}\right) ^{-k}\left( \frac{ \varphi _{22}}{\varphi _{11}}\right) ^{\frac{\sigma -1}{\sigma }}\left( b_{21}\right) ^{-\frac{k}{\sigma -1}}\frac{\Phi _{21}}{\Phi _{22}}\\&+\left( \frac{\varphi _{33}}{\varphi _{aut,3}}\right) ^{-k}\left( \frac{\varphi _{33} }{\varphi _{11}}\right) ^{\frac{\sigma -1}{\sigma }}\left( b_{31}\right) ^{- \frac{k}{\sigma -1}}\frac{\Phi _{31}}{\Phi _{33}}+\left( \frac{\varphi _{11} }{\varphi _{aut,1}}\right) ^{-k} \\ 1= & {} \left( \frac{\varphi _{11}}{\varphi _{aut,1}}\right) ^{-k}\left( \frac{ \varphi _{11}}{\varphi _{22}}\right) ^{\frac{\sigma -1}{\sigma }}\left( b_{21}\right) ^{\frac{k}{\sigma -1}}\frac{\Phi _{12}}{\Phi _{11}}\\&+\left( \frac{\varphi _{33}}{\varphi _{aut,3}}\right) ^{-k}\left( \frac{\varphi _{33} }{\varphi _{22}}\right) ^{\frac{\sigma -1}{\sigma }}\left( \frac{b_{31}}{ b_{21}}\right) ^{-\frac{k}{\sigma -1}}\frac{\Phi _{32}}{\Phi _{33}}+\left( \frac{\varphi _{22}}{\varphi _{aut,2}}\right) ^{-k}, \end{aligned}$$
which can be solved for \(b_{21}\) and \(b_{31}\), which in turn pin down all other endogenous variables.Footnote 12 In line with Melitz and Redding (2015), set \(\sigma =4\) and \(k=4.25\). Assume that trade barriers are symmetric at the bilateral level and that
$$\begin{aligned} \Phi =\left[ \begin{array}{ccc} \Phi _{11} &{}\quad \Phi _{12} &{}\quad \Phi _{13} \\ \Phi _{21} &{}\quad \Phi _{22} &{}\quad \Phi _{23} \\ \Phi _{31} &{}\quad \Phi _{32} &{}\quad \Phi _{33} \end{array} \right] =\left[ \begin{array}{ccc} 1 &{}\quad 0.8 &{}\quad \Phi _{13} \\ 0.8 &{}\quad 1 &{}\quad 0.8 \\ \Phi _{31} &{}\quad 0.8 &{}\quad 1 \end{array} \right] . \end{aligned}$$
Hence, country pairs {1,2} and {2,3} have identical trade barriers, while trade barriers for country pair {1,3} is allowed to vary. The numerical analysis considers the welfare response in country 2 and country 3 as business conditions in country 1 improves as a function of \(\Phi _{13}\) (\(\Phi _{13}=\Phi _{31}\)). Assume that \(\varphi _{aut,1}=\varphi _{aut,2}=\varphi _{aut,3}=1\) in the initial equilibrium and that \(\varphi _{aut,1}\) increases to \(\varphi _{aut,1}=1.1\) due to improved business conditions in country 1 (e.g., due to a reduction in \(\delta _{1}\), a reduction in \(F_{E1}\) or an increase in \(\varphi _{\min ,1}\)). The figure below plots welfare in country 2 and country 3 relative to the initial equilibrium against \(\Phi _{31}\). Recall that trade barriers between country 1 and country 3 increase as \(\Phi _{13}\) decreases and that \(\Phi _{12}=\Phi _{21}=\Phi _{23}=\Phi _{32}=0.8\).
The figure reveals a symmetric positive spillover in the case of symmetric countries, i.e., for \(\Phi _{13}=0.8\). Further, it shows that the more important trade partner, country 2 for \(\Phi _{13}<0.8\), gains more than the less important trade partner, country 3. As \(\Phi _{13}\) decreases, the welfare gain in country 3 decreases (less direct impact from improved business conditions in country 1), while the welfare gain increases for country 2 (export supply of country 1 is increasingly directed toward country 2). Importantly, when \(\Phi _{13}\) becomes sufficiently low, welfare in country 3 is adversely affected, i.e., relative welfare is below unity. The reason being that the direct trade-creation channel (export expansion of country 1 toward country 3) is dominated by the indirect trade-diversion channel (shift in country 2 export toward country 1 at the expense of country 3) driven by general equilibrium mechanisms. The key take-away is that a pro-competitive improvement in business conditions may hurt some trade partners in settings with more than two (asymmetric) countries.
