Trade and the emissions trading system in a small open economy

Abstract

To control greenhouse gas emissions efficiently, we should regulate all pollution emission sources in the economy equally. However, in the real world, environmental regulations differ by sector. Using a small open economy model, this paper examines how the enforcement of an economy-wide emissions trading system (ETS) (i.e., uniform environmental regulation) affects welfare under a trade policy, where the government initially implements different environmental regulations across sectors. It is important to consider this relation between environmental regulations and trade policy because countries tend to impose weak environmental regulations in sectors facing global competition often protected by import tariffs. We find that an economy-wide ETS may harm a small open economy even under a low import tariff rate as long as the environmental regulation in the exporting industry is weaker than in the importing industry. To avoid welfare deterioration, the government should then decrease the import tariff rate in line with mitigation of the sectoral differences in environmental regulation.

Appendices

Appendix 1: Derivation of Eq. (5)

Using $Z_{2} = Z - Z_{1}$ and totally differentiating Eqs. (1–3), we obtain

$$ {\text{d}}L_{1} = p_{1} X_{Z}^{1} \left( {p_{1} X_{LZ}^{1} + p_{2} X_{LZ}^{2} } \right)\frac{{{\text{d}}\theta }}{\Delta } = Ap_{1} X_{Z}^{1} \frac{{{\text{d}}\theta }}{\Delta } > 0, $$

(7)

$$ dZ_{1} = - p_{1} X_{Z}^{1} \left( {p_{1} X_{LL}^{1} + p_{2} X_{LL}^{2} } \right)\frac{d\theta }{\Delta } = Bp_{1} X_{Z}^{1} \frac{d\theta }{\Delta } > 0 . $$

(8)

$\Delta$ represents the Jacobian determinant of the system given by Eqs. (2) and (3).

Using the properties of homogeneous functions, $X_{LZ}^{i} = - e_{i} X_{ZZ}^{i} = - X_{LL}^{i} /e_{i} = X_{ZL}^{i}$, we can rewrite the Jacobian determinant as $\Delta = p_{1} X_{LL}^{1} p_{2} X_{LL}^{2} \left( {e_{1} } \right)^{2} \left( {e_{1} - e_{2} } \right)\left( {e_{1} - \theta e_{2} } \right)$.

From Eq. (4), using Eqs. (1–3) and $p_{1} = \left( {1 + t} \right)p_{1}^{*}$, we have

$$ \begin{aligned} \left( {E_{\text{u}} - tp_{1}^{*} E_{{p_{1} {\text{u}}}} } \right){\text{d}}u &= \,\left( {p_{1} X_{L}^{1} - p_{2} X_{L}^{2} - \frac{t}{t + 1}p_{1} X_{L}^{1} } \right){\text{d}}L_{1} + \left( {p_{1} X_{Z}^{1} - p_{2} X_{Z}^{2} - \frac{t}{t + 1}p_{1} X_{Z}^{1} } \right){\text{d}}Z_{1} \\& = \,\left( {w - w - \frac{t}{t + 1}p_{1} X_{L}^{1} } \right){\text{d}}L_{1} + \left( {1 - \frac{{r_{2} }}{{r_{1} }} - \frac{t}{t + 1}} \right)p_{1} X_{Z}^{1} {\text{d}}Z_{1} \\ & = \, - \frac{t}{t + 1}p_{1} X_{L}^{1} {\text{d}}L_{1} - \left( {\theta - \frac{1}{t + 1}} \right)p_{1} X_{Z}^{1} {\text{d}}Z_{1} . \\ \end{aligned} $$

Substituting Eqs. (7) and (8) for the above equation, we have

$$ \left( {E_{\text{u}} - tp_{1}^{*} E_{{p_{1} {\text{u}}}} } \right){\text{d}}u = - \frac{t}{t + 1}p_{1} X_{L}^{1} Ap_{1} X_{Z}^{1} \frac{{{\text{d}}\theta }}{\Delta } - \left( {\theta - \frac{1}{t + 1}} \right)p_{1} X_{Z}^{1} Bp_{1} X_{Z}^{1} \frac{{{\text{d}}\theta }}{\Delta }, $$

which is equal to Eq. (5).

Appendix 2: Stability condition

We consider the dynamic system consisting of Eqs. (2) and (3) as follows:

$$ \dot{L}_{1} = p_{1} X_{L}^{1} \left( {L_{1} , Z_{1} } \right) - p_{2} X_{L}^{2} \left( {L - L_{1} , Z - Z_{1} } \right), $$

$$ \dot{Z}_{1} = \theta p_{1} X_{Z}^{1} \left( {L_{1} , Z_{1} } \right) - p_{2} X_{Z}^{2} \left( {L - L_{1} , Z - Z_{1} } \right). $$

Linearizing the system at the equilibrium values of the variables, $L_{1}^{*}$ and $Z_{1}^{*}$, we derive

$$ \left[ {\begin{array}{*{20}c} {\dot{L}_{1} } \\ {\dot{Z}_{1} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - B} & A \\ {\theta p_{1} X_{ZL}^{1} + p_{2} X_{ZL}^{2} } & {\theta p_{1} X_{ZZ}^{1} + p_{2} X_{ZZ}^{2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\text{d}}L_{1} } \\ {{\text{d}}Z_{1} } \\ \end{array} } \right], $$

where ${\text{d}}L_{1} = L_{1} - L_{1}^{*}$ and ${\text{d}}Z_{1} = Z_{1} - Z_{1}^{*}$.

From the Routh–Hurwitz theorem, the equilibrium is locally stable if $- B + \theta p_{1} X_{ZZ}^{1} + p_{2} X_{ZZ}^{2} < 0$ and the Jacobian determinant $\Delta$ is positive.

Appendix 3: Approach of the piecemeal reform

From Eqs. (1–3), we have

$$ p_{1} X_{L}^{1} \left( {L_{1} , Z_{1} } \right) = p_{2} X_{L}^{2} \left( {L - L_{1} , Z_{2} } \right), $$

$$ \theta p_{1} X_{Z}^{1} \left( {L_{1} , Z_{1} } \right) = p_{2} X_{Z}^{2} \left( {L - L_{1} , Z_{2} } \right). $$

Totally differentiating these equations, we obtain

$$ {\text{d}}L_{1} = \frac{{p_{1} }}{\Delta }\left[ {\left( {A\theta X_{Z}^{1} + DX_{L}^{1} } \right)\frac{{{\text{d}}t}}{1 + t} + AX_{Z}^{1} {\text{d}}\theta } \right], $$

(9)

$$ {\text{d}}Z_{1} = \frac{{p_{1} }}{\Delta }\left[ {\left( {B\theta X_{Z}^{1} + CX_{L}^{1} } \right)\frac{{{\text{d}}t}}{1 + t} + BX_{Z}^{1} {\text{d}}\theta } \right], $$

(10)

where $C = \theta p_{1} X_{ZL}^{1} + p_{2} X_{ZL}^{2} > 0$ and $D = - \left( {\theta p_{1} X_{ZZ}^{1} + p_{2} X_{ZZ}^{2} } \right) > 0$.

Totally differentiating the expenditure function $E\left( {p_{1} ,p_{2} ,u,Z} \right)$, we derive

$$ \left( {E_{\text{u}} - tp_{1}^{*} E_{{p_{ 1} {\text{u}}}} } \right){\text{d}}u = - \frac{t}{t + 1}p_{1} X_{L}^{1} {\text{d}}L_{1} - \left( {\theta - \frac{1}{t + 1}} \right)p_{1} X_{Z}^{1} {\text{d}}Z_{1} + \left( {\frac{{p_{1} }}{1 + t}} \right)^{2} tE_{{p_{1} p_{1} }} {\text{d}}t. $$

Substituting Eqs. (9) and (10) for the above equation, we obtain

$$ \begin{aligned} \left( {E_{\text{u}} - tp_{1}^{*} E_{{p_{ 1} {\text{u}}}} } \right){\text{d}}u = & - \underbrace {{\left[ {Ap_{1} X_{L}^{1} \frac{t}{1 + t} + Bp_{1} X_{Z}^{1} \left( {\theta - \frac{1}{1 + t}} \right)} \right]\frac{{p_{1} X_{Z}^{1} }}{\Delta }}}_{\alpha }{\text{d}}\theta \\ & - \frac{{p_{1} }}{1 + t}\left\{ {\underbrace {{\frac{1}{\Delta }\left[ {\frac{t}{1 + t}p_{1} X_{L}^{1} \left( {A\theta X_{Z}^{1} + DX_{L}^{1} } \right) + \left( {\theta - \frac{1}{1 + t}} \right)p_{1} X_{Z}^{1} \left( {B\theta X_{Z}^{1} + CX_{L}^{1} } \right)} \right]}}_{\beta }\underbrace {{ - \left( {\frac{{p_{1} }}{1 + t}} \right)tE_{{p_{1} p_{1} }} }}_{\gamma }} \right\}{\text{d}}t \\ \end{aligned} $$

(11)

We know that $\gamma > 0$ always holds. From Eq. (11), we can characterize the path for ${\text{d}}u \ge 0$ as

$$ - \alpha {\text{d}}\theta \ge \left[ {p_{1} /(1 + t)} \right]\left( {\beta + \gamma } \right){\text{d}}t. $$