Appendix
Appendix A: maximum value of \(c\):
We see from (14) and (24) that \( q_{p}^{x}\) and \(q_{p}^{R}\) are positive respectively for:
$$\begin{aligned} c&< \bar{c}(\alpha ,s)\equiv \frac{1}{2\alpha +4}\left( s+\alpha +s\alpha +3\right) , \end{aligned}$$
(39)
$$\begin{aligned} c&< \hat{c}(\alpha )\equiv \frac{3-\alpha }{\alpha +\alpha ^{2}+3}. \end{aligned}$$
(40)
We further see that:
$$\begin{aligned} \frac{\partial \bar{c}(\alpha ,s)}{\partial \alpha }<0\quad \text { and }\quad \frac{d \hat{c}(\alpha )}{d\alpha }<0. \end{aligned}$$
(41)
Following (41), the least possible values of (39) and (40) are respectively:
$$\begin{aligned} c&< \bar{c}_{\min }\equiv \bar{c}(1,s)=\frac{s+2}{3} \end{aligned}$$
(42)
$$\begin{aligned} c&< \hat{c}_{\min }\equiv \hat{c}(1)=\frac{2}{5}. \end{aligned}$$
(43)
Comparing (42) and (43), we see that \(\hat{c}(1)<\bar{c} (1,s).\) Hence, the relevant constraint is (3).
Appendix B: \(W_{h}^{R}\) as a function of \(\alpha \):
Differentiating domestic welfare \(W_{h}^{R}\ \)under FDI in (35) partially with respect to \(\alpha ,\) we see that there are two solutions to \(\partial W_{h}^{R}/\partial \alpha =0,\) which we call \(\underline{\alpha } \) and \(\alpha _{R}^{*}\):
$$\begin{aligned} \underline{\alpha }=c-3,\, \, \, \,\, \, \, \alpha _{R}^{*}=\frac{c}{2-c}. \end{aligned}$$
(44)
We see that \(0<\alpha _{R}^{*}<1\) and \(\underline{\alpha }\) is negative.
In order to determine whether these two stationary points are maxima or minima, we differentiate partially with respect to \(\alpha \) again:
$$\begin{aligned} \left. \frac{d^{2}\left( W_{h}^{R}\right) }{d\alpha ^{2}}\right| _{\alpha =\underline{\alpha }}&= \frac{1}{-4c+c^{2}+6}>0, \\ \left. \frac{d^{2}\left( W_{h}^{R}\right) }{d\alpha ^{2}}\right| _{\alpha =\alpha _{R}^{*}}&= -\frac{1}{4}\frac{\left( c-2\right) ^{4}}{ c^{2}-4c+6}<0. \end{aligned}$$
Hence, domestic welfare reaches a global maximum for \(\alpha \in \left[ 0,1 \right] \) at \(\alpha =\alpha _{R}^{*}\) as given by (44) .
Appendix C: \(W_{h}^{R}\) is not higher than \(W_{h}^{x}\) for \(\alpha =1\):
Setting \(\alpha =1\) in (35) we get the domestic welfare under FDI at \(\alpha =1\) as:
$$\begin{aligned} W_{h\left| \alpha =1\right| }^{R}=\frac{7c^{2}-8c+4}{12}. \end{aligned}$$
(45)
Similarly setting \(\alpha =1\) in (33), we get domestic welfare under export at \(\alpha =1\) as:
$$\begin{aligned} W_{h_{|\alpha =1|}}^{x\max }=\frac{\left( 7c^{2}-8c+4\right) -s\left( 3\right) \left( 2c-s\right) }{8}. \end{aligned}$$
(46)
From (45) and (46), we see that:
$$\begin{aligned} \frac{d\left( W_{h}^{x}-W_{h}^{R}\right) _{\left| \alpha =1\right| }}{dc}=-\frac{1}{12}\left( 7c+9s-4\right) <0. \end{aligned}$$
The minimum value of \(\left( W_{h}^{x}-W_{h}^{R}\right) _{\left| \alpha =1\right| }\) is at the maximum value that \(c\) can take. From (45), (46) and the maximum value of \(c\) given by (3), we see that:
$$\begin{aligned} \left( W_{h}^{x}-W_{h}^{R}\right) _{\left| \alpha =1\right| _{c_{\max }}}=\frac{1}{200}\left( -60s+75s^{2}+16\right) >0. \end{aligned}$$
Thus, we see that
$$\begin{aligned} W_{h\left| \alpha =1\right| }^{x}>W_{h\left| \alpha =1\right| }^{R} \end{aligned}$$
at \(\alpha =1.\) Also, setting \(\alpha =0,\)in (33) and (35) we see that:
$$\begin{aligned} \left[ W_{h}^{R}-W_{h}^{x}\right] _{\alpha =0}=\frac{1}{6}s\left( 2c-s\right) >0. \end{aligned}$$
Appendix D: \(\hat{F}\) could be negative at \(\alpha =0\):
Substituting \(\alpha =0\) into (31), we see that \(\hat{F}>0\) at \( \alpha =0\) for:
$$\begin{aligned} c>\check{c}\equiv 2s. \end{aligned}$$
(47)
Thus, we see that when \(c<\check{c}\), \(\hat{F}\) is negative at \(\alpha =0\).
Appendix E: \(\alpha _{R}^{*}<a_{F1}\):
If \(\alpha _{R}^{*}>a_{F1},\) it implies from (29) that \(\hat{F}\) should be positive at \(\alpha =\alpha _{R}^{*}\equiv \frac{c}{2-c}.\)
We see from (29) that \(\hat{F}_{\alpha =\alpha _{R}^{*}}>0\) for:
$$\begin{aligned} c>\mathring{c}\equiv \frac{1}{2}\sqrt{16s+1}-\frac{1}{2}. \end{aligned}$$
(48)
However, for this to be consistent with the model setting, \(\mathring{c}\) should be less than \(\hat{c}\) in (3). On comparison from (48) and (3), we see that \(\mathring{c}<\hat{c}\) for \(s<0.14.\) Thus, for cases where \(s<0.14,\) we see that \(\alpha _{R}^{*}>a_{F1}.\)