Foreign competition and optimal privatization with excess burden of taxation

Abstract

We examine in a mixed oligopoly setting how foreign competition and the excess burden of taxation will affect privatization policy in the presence of strategic tax/subsidy policies. We show that in the presence of excess burden of taxation with foreign competitors, output subsidy coupled with import tariff and partial privatization is adopted to improve the social welfare. However, if the excess burden of taxation is relatively large, the government may switch to use production tax coupled with tariff policy and partial privatization to improve the social welfare.

Appendices

Appendix I

(i)

$$\begin{aligned} \frac{{\partial q}_{di}^{*}}{\partial t}= & {} \frac{m(n+(2+n)\theta )+2m(1+n)(1-\theta )\lambda }{2(4+2m+n+(2+n)\theta )+4(3+m+n)(1-\theta )\lambda }>0 \\ \frac{\partial q_{0}^{*}}{\partial t}= & {} \frac{-m(n(1-\theta )+2\theta \lambda -2(\theta +\lambda ))}{2(4+2m+n+(2+n)\theta )+4(3+m+n)(1-\theta )\lambda }>0, \\ \textit{if}\, \theta> & {} \frac{n-2\lambda }{2+n-2\lambda }. \\ \frac{\partial {(q}_{0}^{*}+nq_{di}^{*})}{\partial t}= & {} \frac{m(1+(1-\theta )\lambda )}{4+2m+n+2\theta +n\theta +2(3+m+n)(1-\theta )\lambda }>0 \\ \frac{\partial q_{fj}^{*}}{\partial t}= & {} -\frac{4+n+\left( 2+n \right) \theta +2(3+n)(1-\theta )\lambda }{2(4+2m+n+(2+n)\theta )+4(3+m+n)(1-\theta )\lambda }<0 \\ \frac{\partial Q^{*}}{\partial t}= & {} -\frac{2m(1+\lambda -\theta \lambda )}{4+2m+n+2\theta +n\theta +2(3+m+n)(1-\theta )\lambda }<0. \\ \frac{{\partial q}_{di}^{*}}{\partial s}= & {} \frac{(2+m)(2\theta +n(1+\theta )+2n(1-\theta )\lambda )}{2(4+2m+n+(2+n)\theta )+4(3+m+n)(1-\theta )\lambda }>0 \\ \frac{\partial q_{0}^{*}}{\partial s}= & {} \frac{2\left( 2+m \right) \theta -n(1-\theta )(2+m+2\lambda )}{2(4+2m+n+(2+n)\theta )+4(3+m+n)(1-\theta )\lambda }>0, \\ \textit{if} \,\theta> & {} \frac{n(2+m+2\lambda )}{(2+m)(2+n)+2n\lambda }. \\ \frac{\partial {(q}_{0}^{*}+nq_{di}^{*})}{\partial s}= & {} \frac{2+m+(3+m)(1-\theta )\lambda }{4+2m+n+2\theta +n\theta +2(3+m+n)(1-\theta )\lambda }>0 \\ \frac{\partial q_{fj}^{*}}{\partial s}= & {} -\frac{n+\left( 2+n \right) \theta +2n(1-\theta )\lambda }{2(4+2m+n+(2+n)\theta )+4(3+m+n)(1-\theta )\lambda }<0 \\ \frac{\partial Q^{*}}{\partial s}= & {} \frac{2\theta +n(1+\theta )+2n(1-\theta )\lambda }{4+2m+n+2\theta +n\theta +2(3+m+n)(1-\theta )\lambda }>0 \end{aligned}$$

(ii)

$$\begin{aligned}&\frac{{\partial q}_{di}^{*}}{\partial \theta }=\frac{2a(1+m)-(2+n)((2+m)s+mt)-2(3+m+n)s\lambda }{{(4+2m+n+2\theta +n\theta +2(3+m+n)(1-\theta )\lambda )}^{2}}>0, \textit{if}\, \,{\lambda<\lambda }^{*}.\\&\frac{\partial q_{0}^{*}}{\partial \theta }=\frac{-(2+m+n)(2a(1+m)-(2+n)((2+m)s+mt)-2(3+m+n)s\lambda )}{{(4+2m+n+2\theta +n\theta +2(3+m+n)(1-\theta )\lambda )}^{2}}<0,\\&\quad \textit{if}\, \,{\lambda<\lambda }^{*}. \\&\frac{\partial {(q}_{0}^{*}+nq_{fi}^{*})}{\partial \theta }=\frac{-(2+m)(2a(1+m)-(2+n)((2+m)s+mt)-2(3+m+n)s\lambda )}{{(4+2m+n+2\theta +n\theta +2(3+m+n)(1-\theta )\lambda )}^{2}}<0, \\&\quad \textit{if}\,\, {\lambda<\lambda }^{*}. \\&\frac{\partial q_{fj}^{*}}{\partial \theta }=\frac{2a(1+m)-(2+n)((2+m)s+mt)-2(3+m+n)s\lambda }{{(4+2m+n+2\theta +n\theta +2(3+m+n)(1-\theta )\lambda )}^{2}}>0, \textit{if}\, \,{\lambda<\lambda }^{*}.\\&\frac{\partial Q^{*}}{\partial \theta }=\frac{-2(2a(1+m)-(2+n)((2+m)s+mt)-2(3+m+n)s\lambda )}{{(4+2m+n+2\theta +n\theta +2(3+m+n)(1-\theta )\lambda )}^{2}}<0, \textit{if }\,\,{\lambda <\lambda }^{*},\\ \end{aligned}$$

where $\lambda ^{*}\equiv \frac{2a+2am-4s-2ms-2ns-mns-2mt-mnt}{2(3+m+n)s}$. When a is sufficiently large, this condition is no longer needed.

Appendix II: The second-order conditions for optimal policies

We check the second-order conditions for the welfare maximization on ($s,t,\theta )$ at the neighborhood of the equilibrium ($s^{*} t^{*} \theta ^{*})$.

$$\begin{aligned} \frac{\partial ^{2}W}{\partial \theta ^{2}}= & {} -\frac{a^{2}\left( 1+\lambda \right) ^{2}\left( 8+8m+m^{2}+5n+2mn+n^{2}+\left( 2+m+n \right) \left( 6+m+n \right) \lambda \right) \left( 24\lambda +\left( 1+\lambda \right) \left( m+4\left( m+n \right) \lambda \right) \right) ^{4}}{{\begin{array}{l} 4\left( 8+m+4n+2\left( 22+3m+6n \right) \lambda +8\left( 6+m+n \right) \lambda ^{2} \right) ^{2} \\ {\times \left( m\left( 3+m+n \right) +\left( 5m^{2}+m\left( 39+9n \right) +4\left( 12+n\left( 6+n \right) \right) \right) \lambda +4\left( 3+m+n \right) \left( 6+m+n \right) \lambda ^{2} \right) }^{2} \\ \end{array}}}<0\\ \frac{\partial ^{2}W}{\partial t^{2}}= & {} \frac{{\begin{array}{l} -\{m(1+\lambda )(m^{3}(1+\lambda )\left( 1+4\lambda \right) ^{2}(5+n+9\lambda +4n\lambda )+64\lambda ^{2}\left( 12+6n+n^{2}+\left( 3+n \right) \left( 6+n \right) \lambda \right) ^{2} \\ +4m^{2}\left( 1+\lambda \right) \left( 1+4\lambda \right) \left( \left( 3+n \right) ^{2}+\left( 93+n\left( 46+7n \right) \right) \lambda +6\left( 24+n\left( 15+2n \right) \right) \lambda ^{2} \right) \\ +16m\lambda (n^{3}(1+\lambda )(2+3\lambda )(1+4\lambda )+36(2+3\lambda )(1+7\lambda (1+\lambda )) \\ +6n(10+\lambda (79+3\lambda (55+34\lambda )))+n^{2}(18+\lambda (134+\lambda (260+153\lambda )))))\} \\ \end{array}}}{(4{(48\lambda +m^{2}(1+\lambda )(1+4\lambda )+m(1+\lambda )(3+n+36\lambda +8n\lambda )+4(6+n)\lambda (n+(3+n)\lambda ))}^{2})}<0 \\ \frac{\partial ^{2}W}{\partial s^{2}}= & {} \frac{{\begin{array}{l} -4(8m^{2}+8m^{3}+m^{4}+12m^{2}n+12m^{3}n+m^{4}n+4m^{2}n^{2}+4m^{3}n^{2}+92m^{2}\lambda +88m^{3}\lambda \\ +11m^{4}\lambda +256mn\lambda +408m^{2}n\lambda +176m^{3}n\lambda +14m^{4}n\lambda +192mn^{2}\lambda +240m^{2}n^{2}\lambda \\ +52m^{3}n^{2}\lambda +32mn^{3}\lambda +32m^{2}n^{3}\lambda +384m^{2}\lambda ^{2}+344m^{3}\lambda ^{2}+43m^{4}\lambda ^{2} \\ +2304n\lambda ^{2}+4992mn\lambda ^{2}+3852m^{2}n\lambda ^{2}+1028m^{3}n\lambda ^{2}+73m^{4}n\lambda ^{2}+2048n^{2}\lambda ^{2} \\ +3744mn^{2}\lambda ^{2}+2004m^{2}n^{2}\lambda ^{2}+252m^{3}n^{2}\lambda ^{2}+576n^{3}\lambda ^{2}+832mn^{3}\lambda ^{2} \\ +240m^{2}n^{3}\lambda ^{2}+64n^{4}\lambda ^{2}+64mn^{4}\lambda ^{2}+716m^{2}\lambda ^{3}+584m^{3}\lambda ^{3}+73m^{4}\lambda ^{3} \\ +16128n\lambda ^{3}+24576mn\lambda ^{3}+13584m^{2}n\lambda ^{3}+2760m^{3}n\lambda ^{3}+172m^{4}n\lambda ^{3} \\ +11392n^{2}\lambda ^{3}+14880mn^{2}\lambda ^{3}+5608m^{2}n^{2}\lambda ^{3}+556m^{3}n^{2}\lambda ^{3}+2816n^{3}\lambda ^{3} \\ +2976mn^{3}\lambda ^{3}+576m^{2}n^{3}\lambda ^{3}+256n^{4}\lambda ^{3}+192mn^{4}\lambda ^{3}+608m^{2}\lambda ^{4} \\ +448m^{3}\lambda ^{4}+56m^{4}\lambda ^{4}+32832n\lambda ^{4}+39808mn\lambda ^{4}+17616m^{2}n\lambda ^{4}+3080m^{3}n\lambda ^{4} \\ +176m^{4}n\lambda ^{4}+19904n^{2}\lambda ^{4}+20736mn^{2}\lambda ^{4}+6288m^{2}n^{2}\lambda ^{4}+544m^{3}n^{2}\lambda ^{4} \\ +4288n^{3}\lambda ^{4}+3584mn^{3}\lambda ^{4}+560m^{2}n^{3}\lambda ^{4}+320n^{4}\lambda ^{4}+192mn^{4}\lambda ^{4}+192m^{2}\lambda ^{5} \\ +128m^{3}\lambda ^{5}+16m^{4}\lambda ^{5}+20736n\lambda ^{5}+19968mn\lambda ^{5}+7488m^{2}n\lambda ^{5} \\ +1184m^{3}n\lambda ^{5}+64m^{4}n\lambda ^{5}+11136n^{2}\lambda ^{5}+9408mn^{2}\lambda ^{5}+2448m^{2}n^{2}\lambda ^{5}+192m^{3}n^{2}\lambda ^{5} \\ +2048n^{3}\lambda ^{5}+1408mn^{3}\lambda ^{5}+192m^{2}n^{3}\lambda ^{5}+128n^{4}\lambda ^{5}+64mn^{4}\lambda ^{5}) \\ \end{array}}}{(16{(48\lambda +m^{2}(1+\lambda )(1+4\lambda )+m(1+\lambda )(3+n+36\lambda +8n\lambda )+4(6+n)\lambda (n+(3+n)\lambda ))}^{2})}<0 \\ \frac{\partial ^{2}W}{\partial \theta \partial t}= & {} \frac{\partial ^{2}W}{\partial t\partial \theta }= \frac{{\begin{array}{l} am\left( 1+\lambda \right) ^{2}[m\left( 10+m+4n \right) +\left( 6+m+n \right) \left( 5m+4\left( 4+n \right) \right) \lambda \\ +4\left( 6+m+n \right) ^{2}\lambda ^{2}{(24\lambda +(1+\lambda )(m+4(m+n)\lambda ))}^{2}] \\ \end{array}}}{{\begin{array}{l} 4\left( 8+m+4n+2\left( 22+3m+6n \right) \lambda +8\left( 6+m+n \right) \lambda ^{2} \right) \\ \times {(m(3+m+n)+(5m^{2}+m(39+9n)+4(12+n(6+n)))\lambda +4(3+m+n)(6+m+n)\lambda ^{2})}^{2} \\ \end{array}}} \\ \frac{\partial ^{2}W}{\partial \theta \partial s}= & {} \frac{\partial ^{2}W}{\partial s\partial \theta }=\frac{{\begin{array}{l} (a(1+\lambda )\left( 24\lambda +\left( 1+\lambda \right) \left( m+4\left( m+n \right) \lambda \right) \right) ^{2} \\ {(m}^{3}\left( 1{+}\lambda \right) ^{2}\left( 1{+}4\lambda \right) {+}m^{2}\left( 1+\lambda \right) \left( 8+4n+40\lambda +9n\lambda +8\left( 4+n \right) \lambda ^{2} \right) -8n\lambda \left( 2+n+\left( 11+3n \right) \lambda +2\left( 6+n \right) \lambda ^{2} \right) \\ +2m(1+\lambda )(4+2n+22\lambda +n(13+2n)\lambda +2(12+n(4+n))\lambda ^{2}))) \\ \end{array}}}{{\begin{array}{l} (4(8+m+4n+2(22+3m+6n)\lambda +8(6+m+n)\lambda ^{2}) \\ {(m(3+m+n)+(5m^{2}+m(39+9n)+4(12+n(6+n)))\lambda +4(3+m+n)(6+m+n)\lambda ^{2})}^{2}) \\ \end{array}}} \\ \frac{\partial ^{2}W}{\partial t\partial s}= & {} \frac{\partial ^{2}W}{\partial s\partial t}= \frac{{\begin{array}{l} (m(1+\lambda )(-m^{2}(1+n)(10+m+4n)-m(m^{2}(10+13n)+16(3+n)(2+n(7+2n))+6m(16+n(29+8n)))\lambda \\ -(m^{3}(33+60n)+16m(3+n)(13+n(62+13n))+6m^{2}(53+2n(70+17n))+32n(84+n(68+n(19+2n))))\lambda ^{2} \\ -8\left( 6+m+n \right) \left( m^{2}\left( 5+14n \right) +4n\left( 53+n\left( 27+4n \right) \right) +m\left( 23+2n\left( 64+15n \right) \right) \right) \lambda ^{3} \\ -16{(6+m+n)}^{2}(m+4mn+4n(4+n))\lambda ^{4})) \\ \end{array}}}{(4{(m(3+m+n)+(5m^{2}+m(39+9n)+4(12+n(6+n)))\lambda +4(3+m+n)(6+m+n)\lambda ^{2})}^{2})} \end{aligned}$$

We have

$$\begin{aligned}&\left| {\begin{array}{*{20}c} \frac{\partial ^{2}W}{\partial \theta ^{2}} &{} \frac{\partial ^{2}W}{\partial \theta \partial t} &{} \frac{\partial ^{2}W}{\partial \theta \partial s}\\ \frac{\partial ^{2}W}{\partial t\partial \theta } &{} \frac{\partial ^{2}W}{\partial t^{2}} &{} \frac{\partial ^{2}W}{\partial t\partial s}\\ \frac{\partial ^{2}W}{\partial s\partial \theta } &{} \frac{\partial ^{2}W}{\partial s\partial t} &{} \frac{\partial ^{2}W}{\partial s^{2}}\\ \end{array} } \right| \\&\quad =-\frac{a^{2}mn\left( 1+\lambda \right) ^{3}\left( 24\lambda +\left( 1+\lambda \right) \left( m+4\left( m+n \right) \lambda \right) \right) ^{4}}{{\begin{array}{l} 16\left( 8+m+4n+2\left( 22+3m+6n \right) \lambda +8\left( 6+m+n \right) \lambda ^{2} \right) \\ {\times \left( m\left( 3+m+n \right) +\left( 5m^{2}+m\left( 39+9n \right) +4\left( 12+n\left( 6+n \right) \right) \right) \lambda +4\left( 3+m+n \right) \left( 6+m+n \right) \lambda ^{2} \right) }^{2} \\ \end{array}}}<0 \end{aligned}$$

The second-order conditions for the optimal degree of privatization, output subsidy and import tariff hold.