Carbon border adjustment: a unilateral solution to the multilateral problem?

Abstract

This study investigates whether the rationales for introducing carbon border adjustments (CBA) are consistent with theoretical predictions. Specifically, it addresses the following questions. When CBA is theoretically modelled for a level playing field in international trade, does it encourage climate efforts in developing countries, and does it improve global climate effectiveness? What are the implications for international environmental cooperation? The study highlights how the strategic interdependence between countries changes with the introduction of CBA by using a model that incorporates CBA into the two-stage reciprocal-markets model originated by Brander and Spencer (1984). Recognising that CBA is unidirectional, whereby only a country with a more stringent climate policy can impose CBA on imports, the results demonstrate that introducing CBA creates an incentive discrepancy between developed and developing countries; the former (the latter) may adopt a more aggressive (defensive) domestic climate policy.

Appendix

1.1 Proof of Proposition 2

Proposition 2

For each $i\in \{N,S\}$ , there exists the $i$ -lead equilibrium $({\overline{{\varvec{\uptau}}}}_{i},{\underset{\_}{{\varvec{\uptau}}}}_{j})$ if ${\alpha }_{i}{d}_{i}>\frac{1}{9-8{\beta }_{ij}{\lambda }_{ij}+4{\lambda }_{ij}}$ .

Proof.

Since ${\overline{\tau }}_{i}(\cdot )$ is strictly increasing in ${\tau }_{j}$ and ${\overline{\tau }}_{i}({\tau }_{j})>{\tau }_{j}$ for any ${\tau }_{j}\in \left[0,\frac{9{\alpha }_{i}{d}_{i}-1}{8{\alpha }_{j}}\right]$, ${\underset{\_}{\tau }}_{j}(\cdot )$ is decreasing in ${\tau }_{i}$ and ${\underset{\_}{\tau }}_{j}({\tau }_{i})<{\tau }_{i}$ for any ${\tau }_{i}>\frac{2{\alpha }_{j}-{\alpha }_{i}}{2{\alpha }_{i}}{d}_{j}$, ${\overline{\tau }}_{i}(\cdot )$ intersects ${\underset{\_}{\tau }}_{j}(\cdot )$ only once if $\frac{9{\alpha }_{i}{d}_{i}-1}{8{\alpha }_{i}}>\frac{2{\alpha }_{j}-{\alpha }_{i}}{2{\alpha }_{i}}{d}_{j}$, which is $(9-8{\beta }_{ij}{\lambda }_{ij}+4{\lambda }_{ij}){\alpha }_{i}{d}_{i}>1$.

1.2 Proof of Proposition 4

Proposition 4

For a given $(\beta ,\lambda )$ , there exists $\widehat{\alpha d}$ such that (i) $\Omega ({\overline{{\varvec{\uptau}}}}_{N},{\underset{\_}{{\varvec{\uptau}}}}_{S})<\stackrel{\sim }{\Omega }({\stackrel{\sim }{{\varvec{\uptau}}}}_{N},{\stackrel{\sim }{{\varvec{\uptau}}}}_{S})$ if and only if ${\alpha }_{N}{d}_{N}<\widehat{\alpha d}$ .

Proof.

When the equilibrium is an interior solution, $\Omega ({\overline{{\varvec{\uptau}}}}_{N},{\underset{\_}{{\varvec{\uptau}}}}_{S})<\stackrel{\sim }{\Omega }({\stackrel{\sim }{{\varvec{\uptau}}}}_{N},{\stackrel{\sim }{{\varvec{\uptau}}}}_{S})\iff {\alpha }_{N}{d}_{N}<\widehat{\alpha d}=\frac{8+14\beta }{63-108\beta +45{\beta }^{2}+\left(23-71\beta +50{\beta }^{2}\right)\lambda }$. Since $\widehat{\alpha d}$ $>\frac{1}{9-8\beta \lambda +4\lambda },\frac{1}{9\beta \lambda +4\beta -8}$ for any $\beta$ and $\lambda$, there exists such $\widehat{\alpha d}$ whenever the North-lead equilibrium exists. When the equilibrium is a corner solution, the threshold is ${\widehat{\alpha d}}_{0}=\frac{14+20\beta }{96-165\beta +45{\beta }^{2}+15\left(3-11\beta +10{\beta }^{2}\right)\lambda }$. One may check that $\widehat{\alpha d}>{\widehat{\alpha d}}_{0}$ and ${\widehat{\alpha d}}_{0}<\frac{1}{9+10\lambda -20\beta \lambda }$. Thus, once the North-lead equilibrium exists, $\Omega ({\overline{{\varvec{\uptau}}}}_{N},{\underset{\_}{{\varvec{\uptau}}}}_{S})\le \stackrel{\sim }{\Omega }({\stackrel{\sim }{{\varvec{\uptau}}}}_{N},{\stackrel{\sim }{{\varvec{\uptau}}}}_{S})$ in the case of a corner solution.

1.3 Bertrand case

Define the consumer’s utility:

$$U_{i} = q_{i}^{i} + q_{j}^{i} - \frac{1}{2}\left[ {\left( {q_{i}^{i} } \right)^{2} + \left( {q_{j}^{i} } \right)^{2} + 2bq_{i}^{i} q_{j}^{i} } \right] + y,$$

where the parameter $b\in \mathrm{0,1}]$ is a measure of the degree of product differentiation and $b=1$ implies that the goods are homogenous. With fixed income, utility maximisation yields the following inverse demand functions:

$$p_{i}^{i} = 1 - q_{i}^{i} - bq_{j}^{i} {\text{ and }}p_{j}^{i} = 1 - q_{j}^{i} - bq_{i}^{i} .$$

From these, I derive the demand functions ${q}_{i}^{i}=\frac{\left(1-b\right)-{p}_{i}^{i}+b{p}_{j}^{i}}{1-{b}^{2}}$ and ${q}_{i}^{j}=\frac{\left(1-b\right)-{p}_{i}^{j}+b{p}_{j}^{j}}{1-{b}^{2}}$. For a given $(B,\tau )$, a firm chooses its price to maximise profit, yielding the following best responses:

$$p_{i}^{i} (p_{j}^{i} ) = \frac{{(1 - b) + \alpha_{i} \tau_{i} + bp_{j}^{i} }}{2},$$

$${\mathbf{p}}_{i}^{j} \left( {p_{j}^{j} } \right) = \frac{{\left( {1 - b} \right) + \left( {\alpha_{i} \tau_{i} + B_{j} } \right) + bp_{j}^{j} }}{2}.$$

At Nash equilibrium,

$${\mathbf{p}}_{i}^{i} = \frac{{\left( {2 + b} \right)\left( {1 - b} \right) + 2\alpha_{i} \tau_{i} + b\left( {\alpha_{j} \tau_{j} + B_{i} } \right)}}{{4 - b^{2} }}$$

$${\mathbf{p}}_{i}^{j} = \frac{{\left( {2 + b} \right)\left( {1 - b} \right) + b\alpha_{j} \tau_{j} + 2\left( {\alpha_{i} \tau_{i} + B_{j} } \right)}}{{4 - b^{2} }}.$$

Plugging in these equilibrium prices provides the equilibrium production level:

$${\mathbf{q}}_{i}^{i} = \frac{{\left( {2 - b - b^{2} } \right) - \left( {2 - b^{2} } \right)\alpha_{i} \tau_{i} + b\alpha_{j} \tau_{j} - b\left( {1 - b} \right)B_{i} }}{{\left( {4 - b^{2} } \right)\left( {1 - b^{2} } \right)}}$$

$${\mathbf{q}}_{i}^{j} = \frac{{\left( {2 - b - b^{2} } \right) - \left( {2 - b^{2} } \right)\alpha_{i} \tau_{i} + b\alpha_{j} \tau_{j} - b\left( {1 - b} \right)B_{j} }}{{\left( {4 - b^{2} } \right)\left( {1 - b^{2} } \right)}}.$$

Given a firm’s behaviour, each government chooses a domestic carbon tax to maximise social welfare. Following a similar progression for the Cournot case, the best response of carbon tax is

$${\overline{\mathbf{\tau }}}_{i} \left( {\tau_{j} } \right) = \frac{{\rho_{j} \tau_{j} + \rho_{d} d_{i} + \rho_{c} }}{\rho },$$

where $\rho ={\alpha }_{i}\left(12-4b-8{b}^{2}+{b}^{3}+{b}^{4}\right)>0$, ${\rho }_{j}={\alpha }_{j}\left(8-4b-6{b}^{2}+2{b}^{3}+{b}^{4}\right)>0$, ${\rho }_{d}={\alpha }_{j}\left(8-8b-6{b}^{2}+2{b}^{3}+{b}^{4}\right)+{\alpha }_{i}\left(16-4b-12{b}^{2}+{b}^{3}+2{b}^{4}\right)$ and ${\rho }_{c}=2{b}^{2}-{b}^{3}-{b}^{4}$.

Since $\rho >0$ and ${\rho }_{j}>0$ for any $b\in [\mathrm{0,1})$,

$$\frac{{\partial {\overline{\mathbf{\tau }}}_{i} \left( {\tau_{j} } \right)}}{{\partial \tau_{j} }} = \frac{{\rho_{j} }}{\rho } > 0,$$

which is a strategic complement to the leading case. Next, the best response in the lagging case is given by

$${\mathbf{\underline {\tau } }}_{i} \left( {\tau_{j} } \right) = \frac{{ - \left( {2 - b - b^{2} } \right)\tau_{j} + \theta d_{i} + 1 - b}}{{\alpha_{i} }},$$

where $\theta =(2-{b}^{2}){\alpha }_{i}-b{\alpha }_{j}$.

$$\frac{{\partial {\mathbf{\underline {\tau } }}_{i} \left( {\tau_{j} } \right)}}{{\partial \tau_{j} }} = - \left( {2 - b - b^{2} } \right) < 0,$$

since $b\in [\mathrm{0,1})$, which is a strategic substitute for a lagging country. These results are similar to the Cournot case, as the carbon tax of a leading country strategically complements that of its opponent, while that of the lagging country strategically substitutes it. This result shows that the strategic relationships between home and foreign carbon taxes depend on the relative stringency of carbon tax rates and the design of the CBA; hence, the issue of whether competition is about price or quantity is largely irrelevant.