Foreign-owned firms and zoning under spatial price discrimination

Abstract

This paper shows that the nationality of firms influences the design of the optimal zoning by a regulator in a duopoly model of spatial price discrimination. For high enough values of the bias of the regulator towards firms, the size of the zone in which firms are allowed to locate is greater when firms are partially foreign-owned than when firms are fully domestic-owned.

Appendix

Proof of Proposition 1

Equilibrium price policies are described by $p_{1}^{*}(x_{1},x_{2},x)=p_{2}^{*}(x_{1},x_{2},x)$ = $max\{f_{1}\left( x_{1},x\right)$, $f_{2}\left( x_{2},x\right) \}.$ Given a pair of locations $\left( x_{1},x_{2}\right)$ such that $x_{1}\leqslant x_{2}$, firms’ total profits, $\pi =\pi _{1}+\pi _{2}$, are: $\pi =\int _{\frac{x_{2}-x_{1}}{2} }^{1-x_{1}}txdx+\int _{\frac{x_{2}-x_{1}}{2}}^{x_{2}}txdx- \int _{0}^{x_{1}}txdx-2\int _{0}^{\frac{x_{2}-x_{1}}{2}}txdx- \int _{0}^{1-x_{2}}txdx;$ the first two terms are firms’ revenues and the rest are total transportation costs. So, the weighted welfare is: $W=\alpha \beta (\pi _{1}+\pi _{2})+(1-\alpha )CS=$ $\alpha \beta t\left[ \frac{1}{2}(1-x_{1})^{2}-\frac{^{x_{1}2}}{2}-\frac{1}{2 }(1-x_{2})^{2}+\frac{^{x_{2}2}}{2}-\frac{1}{2}(x_{2}-x_{1})^{2}\right] +$ $(1-\alpha )\overline{s}-(1-\alpha )t\left[ \frac{1}{2}(1-x_{1})^{2}+\frac{ ^{x_{2}2}}{2}-\frac{1}{4}(x_{2}-x_{1})^{2}\right] .$ From the first order conditions these two equations are obtained: $x_{1}=\frac{2(1-\alpha (1+\beta ))-x_{2}(1-\alpha (1+2\beta ))}{1-\alpha (1-2\beta )},$ $x_{2}= \frac{2\alpha \beta -x_{1}(1-\alpha (1+2\beta ))}{1-\alpha (1-2\beta )}.$ They are valid when $\alpha >\frac{1}{1+2\beta }.$ The solution is thus: $x_{1}^{^{*}}=\frac{1}{4}+\frac{1-\alpha (1+\beta )}{4\alpha \beta },$ $x_{2}^{^{*}}=\frac{3}{4}-\frac{1-\alpha (1+\beta )}{4\alpha \beta },$ when $\alpha >\frac{1}{1+2\beta },$ and $x_{1}^{^{*}}=x_{2}{}^{^{*}}= \frac{1}{2}$ when $\alpha \le \frac{1}{1+2\beta }.$ The second order conditions are met. $\square$