Greenhouse-Gas Emission Controls and Firm Locations in North–South Trade

Abstract

This paper studies greenhouse-gas emission (GHG) controls in the presence of international carbon leakage through international firm relocation. In a new economic geography model with two countries (‘North’ and ‘South’), only North sets a target for GHG emissions. We compare the consequences of emission quotas and emission taxes under trade liberalization on location of two manufacturing sectors with different emission intensities and degrees of carbon leakage. With low trade costs, further trade liberalization increases global emissions by facilitating carbon leakage. Regulation by quotas leads to spatial sorting, resulting in less carbon leakage and less global emissions than regulation by taxes.

Appendices

Appendix 1

In this appendix, we prove Proposition 2. First, note that the relocation of firms from North to South under trade liberalization means $\frac{dn}{d\phi }<0$ and $\frac{dm}{d\phi }<0$.

Differentiating global emissions in sector C with respect to $\phi$, we can derive

$$\begin{aligned} \frac{d\chi _C ^{W}}{d\phi }= & {} \frac{d\chi _C ^{W}}{dn}\frac{dn}{d\phi }+\frac{d\chi _C ^{W}}{d\phi }+\frac{d\chi _C ^{W}}{dB_C }\frac{dB_C }{d\Delta _C }\left( {\frac{d\Delta _C }{d\phi }+\frac{d\Delta _C }{dn}\frac{dn}{d\phi }} \right) \\&+\,\frac{d\chi _C ^{W}}{dB_C ^{*}}\frac{dB_C ^{*}}{d\Delta _C ^{*}}\left( {\frac{d\Delta _C ^{*}}{d\phi }+\frac{d\Delta _C ^{*}}{dn}\frac{dn}{d\phi }} \right) \end{aligned}$$

where $B_C \equiv \frac{s}{\Delta _C }$ and $B_C ^{*}\equiv \frac{1-s}{\Delta _C ^{*}}$. We now focus on emissions by C-sector firms. D-sector is isomorphic. Here we omit subscript “C” for simplicity.

The first term can be rewritten as

$$\begin{aligned} \frac{d\chi ^{W}}{dn}\frac{dn}{d\phi }=\left( {(B+\phi B^{*})T_C -(\phi B+B^{*})} \right) \frac{dn}{d\phi }=\left( {(T_C -\phi )B+(\phi T_C -1)B^{*}} \right) \frac{dn}{d\phi } \end{aligned}$$

where $T_C \equiv (1+t)^{-\sigma }$. The second term can be given as $\frac{d\chi ^{W}}{d\phi }=nB^{*}T_C +(1-n)B$. The third and fourth terms are

$$\begin{aligned} \frac{d\chi ^{W}}{dB}\frac{dB}{d\Delta }\left( {\frac{d\Delta }{d\phi }+\frac{d\Delta }{dn}\frac{dn}{d\phi }} \right)= & {} -B\frac{nT_C +(1-n)\phi }{n\tilde{T}+(1-n)\phi }\left( {(1-n)+(\tilde{T}-\phi )\frac{dn}{d\phi }} \right) \frac{d\chi ^{W}}{dB^{*}}\frac{dB^{*}}{d\Delta ^{*}}\\ \quad \left( {\frac{d\Delta ^{*}}{d\phi }+\frac{d\Delta ^{*}}{dn}\frac{dn}{d\phi }} \right)= & {} -B^{*}\frac{nT_C \phi +(1-n)}{n\tilde{T}\phi +1-n}\left( {n\tilde{T}+(\phi \tilde{T}-1)\frac{dn}{d\phi }} \right) \end{aligned}$$

where $\tilde{T}\equiv (1+t)^{1-\sigma }>T_C$.

Then we can reorganize the above four terms into the following three parts. The first and second parts are given as

$$\begin{aligned} ((T_C -\phi )-\alpha (\tilde{T}-\phi ))B\frac{dn}{d\phi }>0\quad \hbox {and}\quad ((T_C \phi -1)-\beta (\tilde{T}\phi -1))B^{*}\frac{dn}{d\phi }>0 \end{aligned}$$

where $\alpha \equiv \frac{nT_C +(1-n)\phi }{n\tilde{T}+(1-n)\phi }<1$ and $\beta \equiv \frac{n\phi T_C +(1-n)}{n\phi \tilde{T}+(1-n)}<1$. This is because $\tilde{T}>T_C \quad T_C -\alpha \tilde{T}=\frac{(1-n)\phi (T_C -\tilde{T})}{n\tilde{T}+(1-n)\phi }<0$ and $T_C -\beta \tilde{T}=\frac{(1-n)(T_C -\tilde{T})}{n\phi \tilde{T}+(1-n)}<0$. The third part is given by

$$\begin{aligned}&-\alpha B(1-n)-\beta B^{*}n\tilde{T}+B(1-n)+B^{*}nT_C =(1-\alpha )B(1-n)+(T_C -\beta \tilde{T})nB^{*} \\&\quad =\frac{n(1-n)(\tilde{T}-T_C )}{\Delta }B-\frac{n(1-n)(\tilde{T}-T_C )}{\Delta ^{*}}B^{*}>0 \end{aligned}$$

Note that B$>$B* and $\Delta <\Delta ^{*}$. Since all of these three parts are positive and the same is true for D-sector, $\frac{d\chi ^{W}}{d\phi }>0$ always holds when $\frac{dn}{d\phi }<0$.

Appendix 2

In this appendix, we prove Proposition 4. South emissions are given by$\chi ^{*}=(1-n)(\phi B_C +B_C ^{*})+(1-m)(\phi B_D +B_D ^{*})$. Differentiating South emissions with respect to $\phi$, we get

$$\begin{aligned} \frac{d\chi ^{*}}{d\phi }= & {} \frac{d\chi ^{*}}{dn}\left( {\frac{dn}{d\phi }+\frac{dn}{dq}\frac{dq}{d\phi }} \right) +\frac{d\chi ^{*}}{dB_C }\left( {\frac{dB_C }{d\Delta _C }\left( {\frac{d\Delta _C }{d\phi }+\frac{d\Delta _C }{dn}\frac{dn}{d\phi }+\frac{d\Delta _C }{dn}\frac{dn}{dq}\frac{dq}{d\phi }} \right) } \right) \\&+\,\frac{d\chi ^{*}}{dB_C ^{*}}\left( {\frac{dB_C ^{*}}{d\Delta _C ^{*}}\left( {\frac{d\Delta _C ^{*}}{d\phi }+\frac{d\Delta _C ^{*}}{dn}\frac{dn}{d\phi }+\frac{d\Delta _C ^{*}}{dn}\frac{dn}{dq}\frac{dq}{d\phi }} \right) } \right) +\frac{d\chi ^{*}}{dm}\left( {\frac{dm}{d\phi }+\frac{dm}{dq}\frac{dq}{d\phi }} \right) \\&+\,\frac{d\chi ^{*}}{dB_D }\left( {\frac{dB_D }{d\Delta _D }\left( {\frac{d\Delta _D }{d\phi }+\frac{d\Delta _D }{dm}\frac{dm}{d\phi }+\frac{d\Delta _D }{dm}\frac{dm}{dq}\frac{dq}{d\phi }} \right) } \right) \\&+\, \frac{d\chi ^{*}}{dB_D ^{*}}\left( {\frac{dB_D ^{*}}{d\Delta _D ^{*}}\left( {\frac{d\Delta _D ^{*}}{d\phi }+\frac{d\Delta _D ^{*}}{dm}\frac{dm}{d\phi }+\frac{d\Delta _D ^{*}}{dm}\frac{dm}{dq}\frac{dq}{d\phi }} \right) } \right) \\&+\, (1-n)B_C +(1-m)B_D \end{aligned}$$

Note that $\frac{d\chi ^{*}}{dn}<0$ and $\frac{d\chi ^{*}}{dm}<0$.

Now we consider two extreme cases, (i) the bifurcation point and (ii) free trade.

1.1 At the Bifurcation Point

Note that in the following, we consider D-sector only, because C-sector is in isomorphic expression. First of all, we prove $\frac{dm}{dq}<0$. We can derive

$$\begin{aligned} \frac{dm}{dq}=\left( {\frac{(s(1-\phi ^{2})+\phi ^{2})}{(1-\phi Q)(Q-\phi )}-\frac{(Q(s(1-\phi ^{2})+\phi ^{2})-\phi )(\phi ^{2}-2\phi Q+1)}{(1-\phi Q)^{2}(Q-\phi )^{2}}} \right) \frac{dQ}{dq} \end{aligned}$$

where $Q\equiv (1+\gamma q)^{1-\sigma }$ and $\frac{dQ}{dq}<0$. At the bifurcation point, as discussed in main text, the permit price is zero, i.e. Q=1, $\frac{dm}{dq}=\left( {\frac{\phi }{(1-\phi )^{2}}} \right) \frac{dQ}{dq}<0$. Since the bifurcation point is assumed to be small (i.e., $\phi ^{B}$ is assumed to be small), $\frac{dm}{dq}$ is close to zero. In the same manner, we can derive $\frac{dn}{dq}<0$ which is close to zero. Next, we prove $\frac{dm}{d\phi }>0$. With Q=1, we have

$$\begin{aligned} \frac{dm}{d\phi }= & {} \frac{2Q(1-s)\phi -1}{(1-\phi Q)(Q-\phi )}+\frac{(Q(s(1-\phi ^{2})+\phi ^{2})-\phi )(Q^{2}-2\phi Q+1)}{(1-\phi Q)^{2}(Q-\phi )^{2}} \\= & {} \frac{2(1-s)\phi -1}{(1-\phi )^{2}}+\frac{2(s(1+\phi )-\phi )}{(1-\phi )^{2}}=\frac{2s-1}{(1-\phi )^{2}}>0 \end{aligned}$$

Thus, $\frac{dm}{d\phi }>0$ always holds. $\frac{dn}{d\phi }>0$ for sector C can be derived in the same manner. Using these results, we have

$$\begin{aligned} \frac{d\chi ^{*}}{dm}\left( {\frac{dm}{d\phi }+\frac{dm}{dq}\frac{dq}{d\phi }} \right) =\frac{d\chi ^{*}}{dm}\left( {\frac{dm}{d\phi }} \right) <0. \end{aligned}$$

Then the other three terms for D-sector are given as$(1-m)B_D$,

$$\begin{aligned}&\frac{d\chi ^{*}}{dB_D }\left( {\frac{dB_D }{d\Delta _D }\left( {\frac{d\Delta _D }{d\phi }+\frac{d\Delta _D }{dm}\frac{dm}{d\phi }+\frac{d\Delta _D }{dm}\frac{dm}{dq}\frac{dq}{d\phi }} \right) } \right) \\&\quad \cong \frac{d\chi ^{*}}{dB_D }\left( {\frac{dB_D }{d\Delta _D }\left( {\frac{d\Delta _D }{d\phi }+\frac{d\Delta _D }{dm}\frac{dm}{d\phi }} \right) } \right) <0 \end{aligned}$$

and $\frac{d\chi ^{*}}{dB_D ^{*}}\left( {\frac{dB_D ^{*}}{d\Delta _D ^{*}}\left( {\frac{d\Delta _D ^{*}}{d\phi }+\frac{d\Delta _D ^{*}}{dm}\frac{dm}{d\phi }+\frac{d\Delta _D ^{*}}{dm}\frac{dm}{dq}\frac{dq}{d\phi }} \right) } \right) \cong \frac{d\chi ^{*}}{dB_D ^{*}}\left( {\frac{dB_D ^{*}}{d\Delta _D ^{*}}\left( {\frac{d\Delta _D ^{*}}{d\phi }+\frac{d\Delta _D ^{*}}{dm}\frac{dm}{d\phi }} \right) } \right)$

The last term can be decomposed as $\frac{d\chi ^{*}}{dB_D ^{*}}\left( {\frac{dB_D ^{*}}{d\Delta _D ^{*}}\frac{d\Delta _D ^{*}}{d\phi }} \right) <0$ and $\frac{d\chi ^{*}}{dB_D ^{*}}\left( {\frac{dB_D ^{*}}{d\Delta _D ^{*}}\frac{d\Delta _D ^{*}}{dm}\frac{dm}{d\phi }} \right) =(1-m)\frac{B^{*}}{\Delta ^{*}}(1-Q\phi )\frac{dm}{d\phi }>0$. Here the subscript “D” is omitted for simplicity.

Now we combine two parts $\frac{d\chi ^{*}}{dB^{*}}\left( {\frac{dB^{*}}{d\Delta ^{*}}\frac{d\Delta ^{*}}{dm}\frac{dm}{d\phi }} \right) >0$ and $\frac{d\chi ^{*}}{dm}\left( {\frac{dm}{d\phi }} \right) <0$. Then we finally get$\left( {\frac{d\chi ^{*}}{dB^{*}}\frac{dB^{*}}{d\Delta ^{*}}\frac{d\Delta ^{*}}{dm}+\frac{d\chi ^{*}}{dm}} \right) \left( {\frac{dm}{d\phi }} \right) =\left( {(1-m)\frac{B^{*}}{\Delta ^{*}}(1-Q\phi )-(\phi B+B^{*})} \right) \frac{dm}{d\phi }<0$

because $\frac{(1-m)(1-Q\phi )}{\Delta ^{*}}<1$.

Next, the remaining parts are combined as

$$\begin{aligned}&\frac{d\chi ^{*}}{dB}\left( {\frac{dB}{d\Delta }\left( {\frac{d\Delta }{d\phi }} \right) } \right) +\frac{d\chi ^{*}}{dB^{*}}\left( {\frac{dB^{*}}{d\Delta ^{*}}\left( {\frac{d\Delta ^{*}}{d\phi }} \right) } \right) +(1-m)B\\&\quad =-\frac{1-m}{\Delta }B(1-m)-\frac{1-m}{\Delta ^{*}}B^{*}mQ+(1-m)B\\&\quad =\left( {\frac{\Delta -(1-m)}{\Delta }} \right) B(1-m)-\frac{1-m}{\Delta ^{*}}B^{*}mQ\\&\quad =\left( {\frac{(1-m)(\phi -1)+mQ}{\Delta }} \right) B(1-m)-\frac{1-m}{\Delta ^{*}}B^{*}mQ \\&\quad =\left( {\frac{(1-m)(\phi -1)}{\Delta }+\frac{m}{\Delta }-\frac{m}{\Delta ^{*}}} \right) B(1-m)<0 \end{aligned}$$

The same is true for C-sector. Thus, we can conclude that $\frac{d\chi ^{*}}{d\phi }<0$ always holds at the bifurcation point.

1.2 Free Trade

To examine the case of free trade, we first show that the permit price is hump-shaped in $\phi$. The hump-shaped permit price indicates that $\frac{dq}{d\phi }$ is first positive, then negative and finally goes to zero at $\phi =1$. It is obvious that $\frac{dq}{d\phi }>0$ when quota is binding at the bifurcation point. Thus, here we prove $\lim _{\phi \rightarrow 1} \frac{dq}{d\phi }=-0$. Defining the quota constraint (15) as a function F, we differentiate (15) with respect to q:

$$\begin{aligned} \frac{dF}{dq}= & {} -\frac{1}{(1+q)^{2}}-\frac{\sigma m\gamma }{1+\gamma q}\Omega \\&+\,\gamma m(1+\gamma q)^{-\sigma }\left( {\frac{dB}{d\Delta }\frac{d\Delta }{dq}+\phi \frac{dB^{*}}{d\Delta ^{*}}\frac{d\Delta ^{*}}{dq}} \right) \\&+\,\gamma m(1+\gamma q)^{-\sigma }\left( {\frac{dB}{d\Delta }\frac{d\Delta }{dm}\frac{dm}{dq}+\phi \frac{dB^{*}}{d\Delta ^{*}}\frac{d\Delta ^{*}}{dm}\frac{dm}{dq}} \right) \end{aligned}$$

where $\Omega \equiv (1+\gamma q)^{-\sigma }\left( {B+\phi B^{*}} \right)$. Note that the first term is from North emissions by C-firms, $\frac{1}{1+q}$.

C-firms already concentrate in North at the sustain point and trade cost reduction at$\phi$=1 does not influence C-firms’ location. The change of the permit price affects only total C-firm emissions. Setting m=0, we can get $\frac{dF}{dq}=-\frac{1}{(1+q)^{2}}<0$ and $\lim \limits _{m\rightarrow +0} \frac{dF}{dq}=-\frac{1}{(1+q)^{2}}<0$. Likewise we derive

$$\begin{aligned} \frac{dF}{d\phi }= & {} \gamma m(1+\gamma q)^{-\sigma }\left( {\frac{dB}{d\Delta }\frac{d\Delta }{d\phi }+\phi \frac{dB^{*}}{d\Delta ^{*}}\frac{d\Delta ^{*}}{d\phi }+B^{*}} \right) \\= & {} \gamma m(1+\gamma q)^{-\sigma }\left( {-\frac{B}{\Delta }(1-m)-\phi \frac{B^{*}}{\Delta ^{*}}mQ+B^{*}} \right) <0 \end{aligned}$$

and $\lim \limits _{m\rightarrow +0} \frac{dF}{d\phi }=0$. Using $\frac{dq}{d\phi }=-\frac{\frac{dF}{d\phi }}{\frac{dF}{dq}}$, we obtain $\frac{dq}{d\phi }<0$ and $\lim \limits _{\phi \rightarrow 1} \frac{dq}{d\phi }=-0$.

Since all C-sector firms already concentrate in North at the sustain point, the marginal change of trade costs at $\phi$=1 affects only D-sector firms. Thus, the differentiation in terms of$\phi$is only for D-sector firms:

$$\begin{aligned} \frac{d\chi ^{*}}{d\phi }= & {} \frac{d\chi ^{*}}{dm}\left( {\frac{dm}{d\phi }+\frac{dm}{dq}\frac{dq}{d\phi }} \right) +\frac{d\chi ^{*}}{dB}\left( {\frac{dB}{d\Delta }\left( {\frac{d\Delta }{d\phi }+\frac{d\Delta }{dm}\frac{dm}{d\phi }+\frac{d\Delta }{dm}\frac{dm}{dq}\frac{dq}{d\phi }} \right) } \right) \\&+\,\frac{d\chi ^{*}}{dB^{*}}\left( {\frac{dB^{*}}{d\Delta ^{*}}\left( {\frac{d\Delta ^{*}}{d\phi }+\frac{d\Delta ^{*}}{dm}\frac{dm}{d\phi }+\frac{d\Delta ^{*}}{dm}\frac{dm}{dq}\frac{dq}{d\phi }} \right) } \right) +(1-m)B \end{aligned}$$

where the subscript “D” is omitted. We now prove $\frac{dm}{d\phi }<0$.

$$\begin{aligned} \frac{dm}{d\phi }= & {} \frac{2Q(1-s)\phi -1}{(1-\phi Q)(Q-\phi )}+\frac{(Q(s(1-\phi ^{2})+\phi ^{2})-\phi )(Q^{2}-2\phi Q+1)}{(1-\phi Q)^{2}(Q-\phi )^{2}} \\= & {} \frac{(2Q(1-s)\phi -1)(1-\phi Q)(Q-\phi )+(Q(s+(1-s)\phi ^{2})-\phi )(Q^{2}-2\phi Q+1)}{(1-\phi Q)^{2}(Q-\phi )^{2}} \end{aligned}$$

Note that $2Q(1-s)\phi -1<0$. Keeping $Q>\phi$, as Q and $\phi$ converge to one, we can get $Q(s+(1-s)\phi ^{2})-\phi \cong \phi (s+(1-s)\phi ^{2})-\phi <0$. Thus, we have $\frac{dm}{d\phi }<0$. Using these results, we can derive $\frac{d\chi ^{*}}{dm}\left( {\frac{dm}{d\phi }+\frac{dm}{dq}\frac{dq}{d\phi }} \right) =\frac{d\chi ^{*}}{dm}\left( {\frac{dm}{d\phi }} \right) >0$.

Then, converging Q and $\phi$ to one, we can simplify other terms:

$$\begin{aligned}&\frac{d\chi ^{*}}{dB^{*}}\left( {\frac{dB^{*}}{d\Delta ^{*}}\left( {\frac{d\Delta ^{*}}{d\phi }+\frac{d\Delta ^{*}}{dm}\frac{dm}{d\phi }+\frac{d\Delta ^{*}}{dm}\frac{dm}{dq}\frac{dq}{d\phi }} \right) } \right) =\frac{d\chi ^{*}}{dB^{*}}\left( {\frac{dB^{*}}{d\Delta ^{*}}\left( {\frac{d\Delta ^{*}}{d\phi }+\frac{d\Delta ^{*}}{dm}\frac{dm}{d\phi }} \right) } \right) \\&\quad =\frac{d\chi ^{*}}{dB^{*}}\left( {\frac{dB^{*}}{d\Delta ^{*}}\left( {mQ+(\phi Q-1)\frac{dm}{d\phi }} \right) } \right) =0. \end{aligned}$$

The last term is (1-m)B = B$>$0. Based on all of these results, $\frac{d\chi ^{*}}{d\phi }>0$ always holds at $\phi$=1.