Table 2 Equilibrium abatement quantities...

Table 2

Equilibrium abatement quantities and emissions taxes with and without dimming

Pollutant/ region	Without dimming		With dimming
	Nash (⁠ ${\underline{\hat{q}}}_{n}^{k}$ ⁠)	Second-best (⁠ ${\underline{q}}_{n}^{k^{*}}$ ⁠)	Nash (⁠ ${\hat{q}}_{n}^{k}$ ⁠)	First-best (⁠ $q_{n}^{k^{*}}$ ⁠)
${SO}_{2} i$	$[{\underline{\hat{τ}}}_{i}^{L} = 32]$	$[{\underline{τ}}_{i}^{L^{*}} = 32]$	$[{\hat{τ}}_{i}^{L} = 32 - 31.11 α_{i}]$	$[τ_{i}^{L^{*}} = 5.33]$
${SO}_{2} j$	48 $[{\hat{τ}}_{j}^{L} = 48]$	48 $[{\underline{τ}}_{j}^{L^{*}} = 48]$	$48 - 31.11 α_{j} [{\hat{τ}}_{j}^{L} = 48 - 31.11 α_{j}]$	21.33 $[τ_{j}^{L^{*}} = 21.33]$
${CO}_{2} i$	$70 α_{i}$	70 $[{\underline{τ}}_{i}^{G^{*}} = 140]$	$44.44 α_{i} [{\hat{τ}}_{i}^{G} = 44.44 α_{i}]$	33.33 $[τ_{i}^{G^{*}} = 66.67]$
${CO}_{2} j$	$70 α_{j} [{\hat{τ}}_{j}^{G} = 140 α_{j}]$	70 $[{\underline{τ}}_{j}^{G^{*}} = 140]$	$44.44 α_{j} [{\hat{τ}}_{j}^{G} = 44.44 α_{j}]$	33.33 $[τ_{j}^{G^{*}} = 66.67]$

Pollutant/ region	Without dimming		With dimming
${SO}_{2} i$	$[{\underline{\hat{τ}}}_{i}^{L} = 32]$	$[{\underline{τ}}_{i}^{L^{*}} = 32]$	$[{\hat{τ}}_{i}^{L} = 32 - 31.11 α_{i}]$	$[τ_{i}^{L^{*}} = 5.33]$
${SO}_{2} j$	48 $[{\hat{τ}}_{j}^{L} = 48]$	48 $[{\underline{τ}}_{j}^{L^{*}} = 48]$	$48 - 31.11 α_{j} [{\hat{τ}}_{j}^{L} = 48 - 31.11 α_{j}]$	21.33 $[τ_{j}^{L^{*}} = 21.33]$
${CO}_{2} i$	$70 α_{i}$	70 $[{\underline{τ}}_{i}^{G^{*}} = 140]$	$44.44 α_{i} [{\hat{τ}}_{i}^{G} = 44.44 α_{i}]$	33.33 $[τ_{i}^{G^{*}} = 66.67]$
${CO}_{2} j$	$70 α_{j} [{\hat{τ}}_{j}^{G} = 140 α_{j}]$	70 $[{\underline{τ}}_{j}^{G^{*}} = 140]$	$44.44 α_{j} [{\hat{τ}}_{j}^{G} = 44.44 α_{j}]$	33.33 $[τ_{j}^{G^{*}} = 66.67]$

Notes: The outcomes are based on parameter values ${\overset{�}{L}}_{i} = 40, {\overset{�}{L}}_{j} = 60, {\overset{�}{G}}_{i} = 80, {\overset{�}{G}}_{j} = 120$ ⁠, $c^{L} = 1, c^{G} = 2, r = 4$ and $g = 140$ ⁠. These values ensure that conditions $R_{1}, R_{2}, R_{3}$ defined by (41) all hold, ensuring interior equilibrium solutions. Corresponding sulphur and carbon tax rates shown in squared brackets.

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Pollutant/ region	Without dimming		With dimming
	Nash ( ${\underline{\hat{q}}}_{n}^{k}$ )	Second-best ( ${\underline{q}}_{n}^{k^{*}}$ )	Nash ( ${\hat{q}}_{n}^{k}$ )	First-best ( $q_{n}^{k^{*}}$ )
${SO}_{2} i$	$[{\underline{\hat{τ}}}_{i}^{L} = 32]$	$[{\underline{τ}}_{i}^{L^{*}} = 32]$	$[{\hat{τ}}_{i}^{L} = 32 - 31.11 α_{i}]$	$[τ_{i}^{L^{*}} = 5.33]$
${SO}_{2} j$	48 $[{\hat{τ}}_{j}^{L} = 48]$	48 $[{\underline{τ}}_{j}^{L^{*}} = 48]$	$48 - 31.11 α_{j} [{\hat{τ}}_{j}^{L} = 48 - 31.11 α_{j}]$	21.33 $[τ_{j}^{L^{*}} = 21.33]$
${CO}_{2} i$	$70 α_{i}$	70 $[{\underline{τ}}_{i}^{G^{*}} = 140]$	$44.44 α_{i} [{\hat{τ}}_{i}^{G} = 44.44 α_{i}]$	33.33 $[τ_{i}^{G^{*}} = 66.67]$
${CO}_{2} j$	$70 α_{j} [{\hat{τ}}_{j}^{G} = 140 α_{j}]$	70 $[{\underline{τ}}_{j}^{G^{*}} = 140]$	$44.44 α_{j} [{\hat{τ}}_{j}^{G} = 44.44 α_{j}]$	33.33 $[τ_{j}^{G^{*}} = 66.67]$

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