Unilateral Climate Policy and Foreign Direct Investment with Firm and Country Heterogeneity

Abstract

We analyse the effects of unilateral climate policy in a two-country two-firm model, with endogenous plant location and heterogeneity in both country size and firm’s characteristics. The effectiveness of unilateral climate policy is shown to depend on the joint effect of country and firm heterogeneity, and on their impact on equilibrium location choice. For being effective and not leading to production relocation in the long-run, unilateral climate policy should be moderate, implemented by a sufficiently larger area and complemented by mechanisms promoting the international transfer of clean technologies. The model indicates that the smaller area cannot take the lead in global climate mitigation for a protracted time period. Finally, when the local community is not environmentally concerned, the unilateral policy unambiguously makes the society worse off.

Appendix 1

Considering for lack of space only firm 1’s objective functions, we have:

$$\begin{aligned} \Pi _{1}^{NR}(q_{i,K},w_{1,K})= & {} [a_{I}-b_{I}(q_{1,I}\!+\!q_{2,I} )]q_{1,I}\!+\![a_{II}-b_{II}(q_{1,II}+q_{2,II})]q_{1,II}-cq_{1,I}+\\&\quad -(c+s)q_{1,II} -\frac{w_{1,I}^{2}}{2}-t_{I}[\phi _{1}(q_{1,I}+q_{1,II})-w_{1,I}]-F-G\\ \Pi _{1}^{PR}(q_{i,K},w_{1,K})= & {} [a_{I}-b_{I}(q_{1,I}+q_{2,I} )]q_{1,I}+[a_{II}-b_{II}(q_{1,II}+q_{2,II})]q_{1,II}+\\&\quad -c(q_{1,I}+q_{1,II} )-\frac{w_{1,I}^{2}}{2}-\frac{w_{1,II}^{2}}{2}-t_{I}(\phi _{1}q_{1,I} -w_{1,I})+\\&\quad -t_{II}(\phi _{1}q_{1,II}-w_{1,II})-F-2G\\ \Pi _{1}^{TR}(q_{i,K},w_{1,K})= & {} [a_{I}-b_{I}(q_{1,I}+q_{2,I} )]q_{1,I}+[a_{II}-b_{II}(q_{1,II}+q_{2,II})]q_{1,II}+\\&\quad -cq_{1,II}-(c+s)q_{1,I} -\frac{w_{1,II}^{2}}{2}+\\&\quad -t_{II}[\phi _{1}(q_{1,I}+q_{1,II})-w_{1,II}]-F-G. \end{aligned}$$

1.1 Appendix 2

Let us remind that $A_{I}=[ a_{I}-c-\phi _{1}t_{I}+(\phi _{2}-\phi _{1})t_{II}]$ and $A_{II}=[ a_{II}-c-\phi _{1}t_{I} +(\phi _{2}-\phi _{1})t_{II}]$, with $A_{I}>0$ because of $\hat{q}_{1,I}^{NR}+\hat{q}_{1,I}^{TR}>0$, and $(A_{II}-s)>0$, due to $\hat{q}_{1,II}^{NR}+\hat{q}_{1,II}^{TR}>0$. Also, notice that Eq. (5) may be written as:

$$\begin{aligned} \hat{\Pi }_{1}^{NR}-\hat{\Pi }_{1}^{TR}=\frac{4}{9}\left\{ \left[ s-\phi _{1}(t_{I}-t_{II})\right] \frac{A_{I}}{b_{I}}-\left[ s+\phi _{1}(t_{I} -t_{II})\right] \frac{(A_{II}-s)}{b_{II}}\right\} +\frac{t_{I}^{2} -t_{II}^{2}}{2}. \end{aligned}$$

(17)

It immediately follows that, in the low transport cost case, namely in the case where $s<\phi _{1}(t_{I}-t_{II})$, then $\hat{\Pi }_{1}^{NR} -\hat{\Pi }_{1}^{TR}<0$ for a sufficiently low taxation gap $(t_{I}-t_{II}),$ namely when the centripetal effect of abatement is weak. Numerical simulations show that $\hat{\Pi }_{1}^{NR}-\hat{\Pi }_{1}^{TR}<0$ for most admissible parameters values (see Fig. 1).

We prove now that: (i) $\breve{a}_{II}(s)$ is an increasing function of s; (ii) there exists a value of $G_{\min }$ s.t., for any $G\eqslantgtr G_{\min }=\max \{ \frac{t_{I}^{2}}{2},\frac{4}{9} \frac{[ s+\phi _{1}( t_{I}-t_{II})] ^{2}}{b_{II} }+\frac{t_{II}^{2}}{2}\} ,$ $\bar{a}_{II}(s)$ is a decreasing function of s ; (iii) $\bar{a}_{II}\mid _{s=0}>\breve{a}_{II}\mid _{s=0}.$

Finally, we derive the value of s, say $s^{*}$, such that $\breve{a}_{II}(s^{*})$ $=$ $\bar{a}_{II}(s^{*})$.

To this aim, let us first consider that

$$\begin{aligned} \breve{a}_{II}(s)=\frac{b_{II}}{b_{I}}\frac{\left[ (s-\phi _{1}\left( t_{I}-t_{II}\right) )A_{I}\right] }{[s+\phi _{1}\left( t_{I}-t_{II}\right) ]}+\frac{9}{8}\frac{b_{II}(t_{I}-t_{II})(t_{I}+t_{II})}{[s+\phi _{1}\left( t_{I}-t_{II}\right) ]}+c+s+\phi _{1}(t_{I}+t_{II})-\phi _{2}t_{II} \end{aligned}$$

(18)

and

$$\begin{aligned} \bar{a}_{II}(s)=\frac{9}{4}\left( \frac{b_{II}}{s+\phi _{1}\left( t_{I}-t_{II}\right) }\right) \left( G-\frac{t_{II}^{2}}{2}\right) +[c+s+t_{I}\phi _{1}-t_{II}\left( \phi _{2}-\phi _{1}\right) ]. \end{aligned}$$

(19)

• As far as (i), the derivative of $\breve{a}_{II}(s)$ w.r.t. s writes as:

  $$\begin{aligned} \frac{2b_{II}A_{I}\phi _{1}\left( t_{I}-t_{II}\right) }{b_{I}\left( s+\phi _{1}\left( t_{I}-t_{II}\right) \right) ^{2}}+1-\frac{9b_{II} (t_{I}-t_{II})(t_{I}+t_{II})}{8\left( s+\phi _{1}\left( t_{I}-t_{II}\right) \right) ^{2}}. \end{aligned}$$

  In order to evaluate the sign of this expression, we proceed as follows. As to the denominator, it is immediate to see that it is strictly positive. As to the numerator, it may be rewritten as a univariate quadratic function of the form: $as^{2}+bs+c$, with $a>0,$ $b>0$. The sign of the discriminant is thus crucial to assess whether $\frac{\partial \breve{a}_{II}(s)}{\partial s}>0,$ being this discriminant given by: $\Delta =$ $2(t_{I}-t_{II})b_{I}b_{II} [9b_{I}(t_{I}+t_{II})-16\phi _{1}A_{I}].$ To ensure that $\frac{\partial \breve{a}_{II}(s)}{\partial s}>0$ it has to hold that $\Delta <0,$ namely $16\phi _{1}A_{I}>9b_{I}(t_{I}+t_{II}).$ First, consider that $[9b_{I}(t_{I}+t_{II})-16\phi _{1}A_{I}]=9(t_{I}+t_{II})-24\phi _{1}(\hat{q}_{1,I}^{NR}+\hat{q}_{1,I}^{TR}).$ Since total net emissions by firm 1 in NR (resp. TR) location equilibrium are such that $\phi _{1}(\hat{q} _{1,I}^{NR}+\hat{q}_{1,II}^{NR})> t_{I},$ (resp. $\phi _{1}(\hat{q}_{1,I} ^{TR}+\hat{q}_{1,II}^{TR})> t_{II}$) it has to hold that $2\phi _{1}\hat{q}_{1,I}^{NR}>t_{I}$ and $2\phi _{1}\hat{q}_{1,I}^{TR}>t_{II}.$ Therefore, from these inequalities, we obtain that $2\phi _{1}(\hat{q}_{1,I}^{NR}+\phi _{1}\hat{q}_{1,I}^{TR})>t_{I}+t_{II}.$ Hence, $24\phi _{1}(\hat{q}_{1,I} ^{NR}+\phi _{1}\hat{q}_{1,I}^{TR})>12(t_{I}+t_{II})>9(t_{I}+t_{II}).$ Q.E.D.

• As far as (ii), given that

  $$\begin{aligned} \frac{\partial \bar{a}_{II}(s)}{\partial s}=1-\frac{9}{4}\frac{b_{II} [G-t_{II}^{2}/2]}{\left[ s+\phi _{1}\left( t_{I}-t_{II}\right) \right] ^{2}}, \end{aligned}$$

  (20)

  we obtain that $\bar{a}_{II}(s)$ is a decreasing function of s iff $G\eqslantgtr \frac{4}{9}\frac{\left[ s+\phi _{1}\left( t_{I}-t_{II}\right) \right] ^{2}}{b_{II}}+\frac{t_{II}^{2}}{2}.$ Q.E.D.

• Finally, concerning (iii), the difference between $\bar{a}_{II}\mid _{s=0}$ and $\breve{a}_{II}\mid _{s=0}$ can be written as

  $$\begin{aligned} \frac{1}{8}b_{II}\frac{9b_{I}(2G-t_{I}^{2})+8A_{I}\phi _{1}(t_{I}-t_{II} )}{b_{I}\phi _{1}\left( t_{I}-t_{II}\right) } \end{aligned}$$

  (21)

  If $G\eqslantgtr G_{\min }=\max \left\{ \frac{t_{I}^{2}}{2},\frac{4}{9} \frac{\left[ s+\phi _{1}\left( t_{I}-t_{II}\right) \right] ^{2}}{b_{II} }+\frac{t_{II}^{2}}{2}\right\}$, we can conclude that $\bar{a}_{II} \mid _{s=0}>\breve{a}_{II}\mid _{s=0}$. Q.E.D.

• By simple algebra it is found that $\exists s :\breve{a}_{II}(s) =\bar{a}_{II}(s)$. Denote this value of s as $s^{*}$, where

  $$\begin{aligned} s^{*}=\frac{9}{4A_{I}}b_{I}\left( G-\frac{t_{I}^{2}}{2}\right) +\phi _{1}(t_{I}-t_{II}). \end{aligned}$$

  Since $s^{*}$ is such that $\hat{\Pi }_{1}^{NR}(s^{*})=\hat{\Pi }_{1} ^{TR}(s^{*})$ and $\hat{\Pi }_{1}^{NR}(s^{*})=\hat{\Pi }_{1}^{PR}(s^{*}),$ it follows that also $\hat{\Pi }_{1}^{PR}(s^{*})=\hat{\Pi }_{1} ^{TR}(s^{*})$, with $\hat{\Pi }_{1}^{PR}-\hat{\Pi }_{1}^{TR}>(<)$ 0 for $s>(<)$ $s^{*}$.

1.2 Appendix 3

When the equilibrium location choice is NR, we find:

$$\begin{aligned} \widehat{Q}_{1}^{NR}-\widetilde{Q}_{1}=-2\phi _{1}(t_{I}-t_{II})\left[ \frac{1}{3b_{I}}+\frac{1}{3b_{II}}\right] <0 \end{aligned}$$

(22)

and

$$\begin{aligned} \widehat{Q}_{2}^{NR}-\widetilde{Q}_{2}=\phi _{1}(t_{I}-t_{II})\left[ \frac{1}{3b_{I}}+\frac{1}{3b_{II}}\right] >0. \end{aligned}$$

(23)

When the equilibrium location choice is TR, we find:

$$\begin{aligned} \widehat{Q}_{1}^{TR}-\widetilde{Q}_{1}=-2s\left[ \frac{1}{3b_{I}}-\frac{1}{3b_{II}}\right] <0 \end{aligned}$$

(24)

and

$$\begin{aligned} \widehat{Q}_{2}^{TR}-\widetilde{Q}_{2}=s\left[ \frac{1}{3b_{I}}-\frac{1}{3b_{II}}\right] >0. \end{aligned}$$

(25)

1.3 Appendix 4

As far as consumer surplus variation under tighter environmental measures, we find that $\widehat{CS}_{I}^{NR}-\widetilde{CS}_{I}$ $=\frac{b_{I}}{2}[(\hat{Q}_{I}^{NR}+\tilde{Q}_{I})(\hat{Q}_{I}^{NR}-\tilde{Q}_{I})].$ So, the sign of $(\widehat{CS}_{I}^{NR}-\widetilde{CS}_{I})$ is determined by sign $(\hat{Q}_{I}^{NR}-\tilde{Q}_{I}^{NR})=-\frac{1}{3b_{I}}( t_{I}-t_{II}) \phi _{1}<0.$ In the NR case, the effect on government revenue from a unilateral tax in country I is given by$:\hat{T}_{I} ^{NR}-\tilde{T}_{I}=t_{I}[\phi _{1}(\hat{q}_{1,I}^{NR}+\hat{q}_{1,II} ^{NR})-t_{I}]-t_{II}[\phi _{1}(\tilde{q}_{1,I}^{NR}+\tilde{q}_{1,II} ^{NR})-t_{II}]$ namely $(t_{I}-t_{II})[ \phi _{1}\left( \hat{q} _{1,I}^{NR}-\frac{2t_{II}\phi _{1}}{3b_{II}}\right) +\phi _{1}( \hat{q}_{1,II}^{NR}-\frac{2t_{II}\phi _{1}}{3b_{I}}) -(t_{I}+t_{II})]$. The sign of this expression is ambiguous. However, assuming that $t_{II}=0$ implies that $\hat{T}_{I}^{NR}-\tilde{T}_{I}>0$ holds, since $[ \phi _{1}(\hat{q}_{1,I}^{NR}+\hat{q}_{1,II}^{NR}) -t_{I}]$ are net emissions in country I. In general, we find that $\hat{T}_{I}^{NR}-\tilde{T}_{I}>0$ whenever

$$\begin{aligned}&\frac{\left( a_{II}-c-2s-2t_{I}\phi _{I}-t_{II}(2\phi _{1}-\phi _{2})\right) }{3b_{II}}+\frac{\left( a_{I}-c+s-2t_{I}\phi _{1}-t_{II}(2\phi _{1}-\phi _{2})\right) }{3b_{I}}\\&\quad >\frac{\left( t_{I}+t_{II}\right) }{\phi _{I}}. \end{aligned}$$

As shown in Sect. 5, a higher unilateral carbon tax reduces global emissions ($\hat{E}_{W}^{NR}$) with respect to the baseline scenario ($\tilde{E}_{W}$) whenever $\phi _{1}\ge \frac{\phi _{2}}{2}$. The same condition applies when one examines the effect on damage.

Then, let us consider the aggregate consumers’ welfare. Focusing on the scenario with $\phi _{1}\ge \frac{\phi _{2}}{2}$, in order to conclude whether aggregate consumers’ welfare raises under NR, we evaluate the sign of the following expression: $\frac{b_{I}}{2}[(\hat{Q}_{I}^{NR}+\tilde{Q}_{I} )(\hat{Q}_{I}^{NR}-\tilde{Q}_{I})]+\hat{T}_{I}^{NR}-\tilde{T}_{I}$. It emerges that $[(\hat{Q}_{I}^{NR}+\tilde{Q}_{I})(\hat{Q}_{I}^{NR}-\tilde{Q}_{I})]<0$ always holds while $\hat{T}_{I}^{NR}-\tilde{T}_{I}>0$ may occur. From standard computations, we find that aggregate consumers’ welfare under a unilateral policy in the NR equilibrium is higher than in the baseline, namely $\frac{b_{I}}{2}[(\hat{Q}_{I}^{NR}+\tilde{Q}_{I})(\hat{Q}_{I}^{NR}-\tilde{Q}_{I})]+\hat{T}_{I}^{NR}-\tilde{T}_{I}>0$ whenever:

$$\begin{aligned} \frac{b_{I}}{2(t_{I}+t_{II})}[(\hat{Q}_{I}^{NR}+\tilde{Q}_{I})(\hat{Q} _{I}^{NR}-\tilde{Q}_{I})]-\frac{\phi _{1}}{(t_{I}+t_{II})}(\hat{Q}_{I} ^{NR}t_{I}-\tilde{Q}_{I}t_{II})>(t_{I}-t_{II}). \end{aligned}$$

Finally, when moving to producers’surplus—evaluated in terms of global profits—we have that:

$$\begin{aligned} \hat{\Pi }_{1}^{NR}-\tilde{\Pi }_{1}&=\left\{ b_{I}[(\hat{q}_{1,I} ^{NR}+\tilde{q}_{1,I}^{NR})(\hat{q}_{1,I}^{NR}-\tilde{q}_{1,I}^{NR} )]+b_{II}[(\hat{q}_{1,II}^{NR}+\tilde{q}_{1,II}^{NR})(\hat{q}_{1,II} ^{NR}-\tilde{q}_{1,II}^{NR})]\right\} +\\&\quad +\frac{\left( t_{I}+t_{II}\right) \left( t_{I}-t_{II}\right) }{2}. \end{aligned}$$

The sign of $\hat{\Pi }_{1}^{NR}-\tilde{\Pi }_{1}^{NR}$ depends on one hand on the terms in curly brackets: this component is negative since $(\hat{q} _{1,I}^{NR}-\tilde{q}_{1,I}^{NR})=-\frac{2}{3b_{I}}\phi _{1}( t_{I}-t_{II}) <0$ and $(\hat{q}_{1,II}^{NR}-\tilde{q}_{1,II} ^{NR})=-\frac{1}{3b_{II}}2( t_{I}-t_{II}) \phi _{1}<0$. On the other hand, the abatement effort exerted in NR under unilateral climate policy boosts the corresponding profits. It is reasonable to conclude that the first negative component in curly brackets prevails over the second positive one, namely $\frac{( t_{I}+t_{II})}{2}.$ Therefore, $\hat{\Pi }_{1}^{NR}-\tilde{\Pi }_{1}^{NR}<0.$ Q.E.D.

1.4 Appendix 5

Consider the function

$$\begin{aligned} \widehat{W}_{I}^{NR}=\widehat{CS}_{I}^{NR}+\widehat{\Pi }_{1}^{NR}+\widehat{T}_{I}^{NR}-\widehat{D}_{I}^{NR}. \end{aligned}$$

In order to evaluate the sign of $\frac{\partial \widehat{W}_{I}^{NR}}{\partial t_{I}},$ we first remind that

$$\begin{aligned}&\widehat{CS}_{I}^{NR}=\frac{b_{I}}{2}(\hat{q}_{1,I}^{NR}+\hat{q}_{2,I} ^{NR})^{2}\\&\widehat{\Pi }_{1}^{NR}=b_{I}(\hat{q}_{1,I}^{NR})^{2}+b_{II}(\hat{q} _{1,II}^{NR})^{2}+\frac{t_{I}^{2}}{2}-F-G\\&\widehat{T}_{I}^{NR}=t_{I}[\phi _{1}(\hat{q}_{1,I}^{NR}+\hat{q}_{1,II} ^{NR})-t_{I}]\\&\widehat{D}_{I}^{NR}=\frac{\gamma _{I}}{2}\left[ \phi _{1}(\hat{q} _{1,I}^{NR}+\hat{q}_{1,II}^{NR})+\phi _{2}(\hat{q}_{2,I}^{NR}+\hat{q} _{2,II}^{NR})-(t_{I}+t_{II})\right] ^{2}. \end{aligned}$$

Thus we find that:

$$\begin{aligned} \frac{\partial \widehat{CS}_{I}^{NR}}{\partial t_{I}}+\frac{\partial \widehat{\Pi }_{1}^{NR}}{\partial t_{I}}+\frac{\partial \widehat{T}_{I}^{NR} }{\partial t_{I}}= & {} -\frac{\phi _{1}}{3}(5\hat{q}_{1,I}^{NR}+\hat{q}_{2,I} ^{NR}+4\hat{q}_{1,II}^{NR})+\\&-\frac{2}{3b_{I}b_{II}}\phi _{1} ^{2}\left( b_{I}+b_{II}\right) (\hat{q}_{1,I}^{NR}+\hat{q}_{1,II} ^{NR})-t_{I}<0. \end{aligned}$$

Moreover,

$$\begin{aligned} \frac{\partial \widehat{D}_{I}^{NR}}{\partial t_{I}}= & {} \gamma _{I}\left[ -(2\phi _{1}-\phi _{2})\frac{\phi _{1}}{3}\left( \frac{1}{b_{I}}+\frac{1}{b_{II}}\right) -1\right] \\&\left[ \phi _{1}(\hat{q}_{1,I}^{NR}+\hat{q} _{1,II}^{NR})+\phi _{2}(\hat{q}_{2,I}^{NR}+\hat{q}_{2,II}^{NR})-(t_{I} +t_{II})\right] . \end{aligned}$$

Denoting by $\Gamma =[ -(2\phi _{1}-\phi _{2})\frac{\phi _{1}}{3}\left( \frac{1}{b_{I}}+\frac{1}{b_{II}}\right) ]$ and considering that global net emissions under the NR equilibrium are positive, or $\phi _{1}(\hat{q}_{1,I}^{NR}+\hat{q}_{1,II}^{NR})+\phi _{2}(\hat{q}_{2,I} ^{NR}+\hat{q}_{2,II}^{NR})-(t_{I}+t_{II}) >0$, it follows that

$$\begin{aligned} sign\left( \frac{\partial \widehat{D}_{I}^{NR}}{\partial t_{I}}\right) =sign(\Gamma -1), \end{aligned}$$

with $\Gamma$ concave in $\phi _{1}$ and positive for any $\phi _{1}\in ]0,\frac{\phi _{2}}{2}[.$ Therefore, we obtain that the condition $\phi _{1}<\frac{\phi _{2}}{2}$ is necessary for $\frac{\partial \widehat{D}_{I}^{NR} }{\partial t_{I}}>0$ to occur and thus for $\frac{\partial \widehat{W}_{I} ^{NR}}{\partial t_{I}}<0.$ Viceversa, for any $\phi _{1}\ge \frac{\phi _{2}}{2}$, the function $\Gamma$ is negative or null and thus damage unambigously falls. In this scenario, it may happen that $\frac{\partial \widehat{W}_{I}^{NR}}{\partial t_{I}}>0.$ Q.E.D.

1.5 Appendix 6

As to the comparison between the welfare properties of equilibria under NR and under TR,  let us consider first that $\widehat{CS}_{I}^{NR}-\widehat{CS}_{I}^{TR}$ $=\frac{b_{I}}{2}[(\hat{Q}_{I}^{NR}+\hat{Q}_{I}^{TR})(\hat{Q}_{I}^{NR}-\hat{Q}_{I}^{TR})].$ Since it is most likely to observe either NR or TR equilibrium location choice in the high transport cost scenario, we focus on this case thereby stating that $\widehat{CS}_{I}^{NR}-\widehat{CS}_{I}^{TR}>0$, due to $(\hat{Q}_{I}^{NR}-\hat{Q}_{I}^{TR})=\left[ \frac{s-\phi _{1}(t_{I}-t_{II})}{3b_{I}}\right] >0.$ Further, it is straightforward that $\widehat{T}_{I}^{NR}>\widehat{T}_{I}^{TR}$, being $\widehat{T}_{I}^{TR}=0$ and $\widehat{T}_{I}^{NR}>0.$ As to producer surplus, if one considers only firm $1^{\prime }$s domestic profits, it comes out that $\widehat{\pi }_{1I}^{NR}>\widehat{\pi }_{1I}^{TR}$ as $\widehat{\pi }_{1I}^{TR}$ is nil. As to the difference in damage under the two equilibrium location choices, this is given by $\frac{\gamma _{_{I}}}{2}[(\hat{E} _{W}^{NR})^{2}-(\hat{E}_{W}^{TR})^{2}].$ Thus the sign of this difference is determined by $sign(\hat{E}_{W}^{NR}-\hat{E}_{W}^{TR})$. It comes out that

$$\begin{aligned} (\hat{E}_{W}^{NR}-\hat{E}_{W}^{TR})=\frac{1}{3}(\phi _{2}-2\phi _{1})\left[ \phi _{1}(t_{I}-t_{II})\left( \frac{1}{b_{I}}+\frac{1}{b_{II}}\right) -s\left( \frac{1}{b_{I}}-\frac{1}{b_{II}}\right) \right] -(t_{I}-t_{II}). \end{aligned}$$

Then, if the condition $\phi _{1}\ge \frac{\phi _{2}}{2}$ holds, then $\hat{E}_{W}^{TR}>\hat{E}_{W}^{NR}$ provided $b_{II}<\breve{b}_{II} =\frac{( b_{I}s+b_{I}\phi _{1}(t_{I}-t_{II})) }{s-\phi _{1} (t_{I}-t_{II}){}_{}}.$

Moving to the PR equilibrium location choice, first notice that the sign of the difference $\widehat{CS}_{I}^{NR}-\widehat{CS}_{I}^{PR}$ depends on $sign(\hat{Q}_{I}^{NR}-\hat{Q}_{I}^{PR}).$ Since $\hat{Q}_{I}^{NR}=\hat{Q} _{I}^{PR}$, it immediately follows that consumer surplus in country I in the NR scenario coincides with that observed in the PR one. As far as the government revenue in the PR case, we find that $\hat{T}_{I}^{PR}=t_{I} (\phi _{1}\hat{q}_{1,I}^{PR}-t_{I}),$ while $\hat{T}_{I}^{NR}=$ $t_{I}(\phi _{1}(\hat{q}_{1,I}^{NR}+\hat{q}_{1,II}^{NR})-t_{I}).$ Since $\hat{q} _{1,I}^{PR}=\hat{q}_{1,I}^{NR},$ it immediately follows that $\hat{T}_{I} ^{NR}>$ $\hat{T}_{I}^{PR}.$ Moreover, as to producer surplus, it results that $\widehat{\pi }_{1I}^{NR}>\widehat{\pi }_{1I}^{PR}$, since (i) $\hat{q}_{1,II}^{PR}$ is not included in firm 1’s domestic profits, while $\hat{q}_{1,I}^{PR}=\hat{q}_{1,I}^{NR}$ , (ii) the abatement at equilibrium is the same for firm 1 in country I both in NR and PR. Finally, when considering global emissions, it emerges that

$$\begin{aligned} (\hat{E}_{W}^{PR}-\hat{E}_{W}^{NR})=(2\phi _{1}-\phi _{2})\left[ \frac{s+\phi _{1}(t_{I}-t_{II})}{3b_{II}}\right] -t_{II}. \end{aligned}$$

Assuming that $\phi _{1}>\frac{\phi _{2}}{2}$ holds—which is a sufficient condition for global emissions to decrease both in the NR case and in the PR one–, then $(\hat{E}_{W}^{PR}-\hat{E}_{W}^{NR})>0$ for $b_{II} <\widetilde{b_{II}},$ with $\widetilde{b_{II}}=\frac{(s+\phi _{1}(t_{I} -t_{II}))\left( 2\phi _{1}-\phi _{2}\right) }{3t_{II}}.$ Q.E.D.

1.6 Appendix 7

In line with the findings emerged when country I is larger than country II, we get that under reverse asymmetry in the low transport cost case TR always dominates PR (see Eq. 7). As to the high transport cost case, the benefits of partial over total relocation are lower, the smaller the size of country I (see Eq. 7). Indeed, for a given size of the world market, the parameter $A_{I}$ (resp. $b_{I})$ is lower (resp. higher) under reverse market asymmetry than under the traditional assumption of market asymmetry. Also, reminding that $s^{*}$ is the value of s satisfying $\hat{\Pi }_{1}^{PR}(s^{*})=\hat{\Pi }_{1}^{TR}(s^{*})\$ under market asymmetry, and defining by $s^{**}$ the value of unit transport cost s.t. $\hat{\Pi }_{1}^{PR}(s^{**})=\hat{\Pi }_{1}^{TR}(s^{**})$ under reverse market asymmetry, we can borrow from the analysis developed in “Appendix 2”. In particular, we can state that the incentive to total relocation dominates the one to partial relocation whenever $s<s^{**},$ while it is dominated otherwise, namely when $s\ge s^{**}$. Finally, since both $s^{*}$ and $s^{**}$ are decreasing (resp. increasing) in $a_{I}$ (resp. $b_{I})$, as a by-product, $s^{**}>s^{*}$ . Accordingly, there exists a set of parameters such that, under market asymmetry, firm 1 does not totally relocate (thereby choosing PR), while preferring total relocation under reverse market asymmetry.