Effects on growth of environmental policy in a small open economy

Abstract

This paper examines the effect of environmental policy on economic growth in a small open economy in a neoclassical framework with pollution as an argument in both the utility and production functions. We show that environmental policy imposes a drag on long-run growth in both the open and closed economy cases. The effect of environmental policy on growth is stronger in the open economy case relative to the closed economy model if the country has strong aversion to pollution and thus serves as a net exporter of capital in the international capital market. On the other hand, if the agents in the economy have low aversion to pollution and thus import capital, the effect of environmental care on growth is stronger in the closed economy relative to the open economy. Thus, from our setup, environmental policy is harmful to growth but environmental sustainability need not be incompatible with continued economic growth.

Appendices

Appendix 1: decentralized solution

Then the objective of the household is to maximize

$$ u = \int\limits_{0}^{\infty } {u(C/L,Z){\text{e}}^{ - (\rho - n)t} } {\text{d}}t = \int\limits_{0}^{\infty } {\left[ {\frac{{(C/L)^{1 - \theta } - 1}}{1 - \theta } - \phi \frac{{Z^{1 + \eta } }}{1 + \eta }} \right]} {\text{e}}^{ - (\rho - n)t} {\text{d}}t $$

(48)

with respect to C, subject to

$$ \dot{A} = wL + rA + \varphi - C, $$

(49)

The current value Hamiltonian for the above dynamic optimization problem is

$$ H(C/L,Z,K,\lambda ) = \frac{{(C/L)^{1 - \theta } }}{1 - \theta } - \phi \frac{{Z^{1 + \eta } }}{1 + \eta } + \lambda \left[ {wL + rA + \varphi - C} \right] $$

(50)

Note that the household cannot influence Z. It is therefore not a direct choice variable here. The associated optimality conditions for the representative household are

$$ \frac{\partial H}{\partial C} = 0 \Leftrightarrow C^{ - \theta } L^{\theta - 1} - \lambda = 0 \Leftrightarrow C^{ - \theta } = \lambda L^{1 - \theta } $$

(51)

$$ \frac{\partial H}{\partial A} = - \left[ {\dot{\lambda } - (\rho - n)\lambda } \right] \Leftrightarrow r\lambda = \dot{\lambda } - (\rho - n)\lambda \Rightarrow \dot{\lambda } = (\rho + \delta - n - \alpha /v)\lambda $$

(52)

$$ \lim_{t \to \infty } \left[ {\lambda (t)A(t)} \right]{\text{e}}^{ - (\rho - n)t} = 0 $$

(53)

Note that (6) has been used in Eq. (52). It can conveniently be rewritten as

$$ \frac{{\dot{\lambda }}}{\lambda } = \rho + \delta - n - \alpha /v $$

(54)

Taking logs of Eq. (51), differentiating both sides with respect to time and using condition (54), we obtain the familiar consumption Euler equation.

$$ \frac{{\dot{C}}}{C} = \frac{1}{\theta }\left[ {(\alpha /v) + \theta n - (\rho + \delta )} \right] $$

(55)

Appendix 2: equivalence between command and decentralized solutions

2.1 Optimal tax rule

From the optimality condition in (4), we obtain the following relationship among the growth rates of pollution tax, aggregate production and aggregate emissions as follows.

$$ \frac{{\dot{\tau }}}{\tau } = \frac{{\dot{Y}}}{Y} - \frac{{\dot{Z}}}{Z} $$

Imposing the planners steady state growth rate of aggregate production and pollution into the above condition for the growth rate of the pollution tax, we derived the socially optimal growth rate of the tax as follows.

$$ \frac{{\dot{Y}}}{Y} = \frac{(1 + \eta )(1 - \alpha - \beta )}{(\beta (\theta - 1) + (1 - \alpha )(1 + \eta ))}x + \frac{(\beta (\theta - 1) + (1 - \alpha - \beta )(1 + \eta ))}{(\beta (\theta - 1) + (1 - \alpha )(1 + \eta ))}n $$

$$ \frac{{\dot{Z}}}{Z} = - \left( {\frac{(\theta - 1)(1 - \alpha - \beta )x - \beta (\theta - 1)n}{\beta (\theta - 1) + (1 - \alpha )(1 + \eta )}} \right) $$

$$ \begin{aligned} \frac{{\dot{\tau }}}{\tau } &= \frac{(1 + \eta )(1 - \alpha - \beta )}{(\beta (\theta - 1) + (1 - \alpha )(1 + \eta ))}x + \frac{(\beta (\theta - 1) + (1 - \alpha - \beta )(1 + \eta ))}{(\beta (\theta - 1) + (1 - \alpha )(1 + \eta ))}n \hfill \\ &\quad+ \left( {\frac{(\theta - 1)(1 - \alpha - \beta )x - \beta (\theta - 1)n}{\beta (\theta - 1) + (1 - \alpha )(1 + \eta )}} \right) \hfill \\ \end{aligned} $$

$$ \begin{aligned} \frac{{\dot{\tau }}}{\tau } &= \left( {\frac{(1 + \eta )(1 - \alpha - \beta )}{(\beta (\theta - 1) + (1 - \alpha )(1 + \eta ))} + \frac{(\theta - 1)(1 - \alpha - \beta )}{\beta (\theta - 1) + (1 - \alpha )(1 + \eta )}} \right)x \hfill \\ &\quad+ \left( {\frac{(\beta (\theta - 1) + (1 - \alpha - \beta )(1 + \eta ))}{(\beta (\theta - 1) + (1 - \alpha )(1 + \eta ))} - \frac{\beta (\theta - 1)}{\beta (\theta - 1) + (1 - \alpha )(1 + \eta )}} \right)n \hfill \\ \end{aligned} $$

$$ \frac{{\dot{\tau }}}{\tau } = \frac{(\theta + \eta )(1 - \alpha - \beta )x + (1 - \alpha - \beta )(1 + \eta )n}{{\left( {\beta (\theta - 1) + (1 - \alpha )(1 + \eta )} \right)}} $$

(56)

2.2 Steady state growth rates of aggregate variables

Imposing the steady state condition that $\dot{K}/K = \dot{Y}/Y$ on Eq. (9) we obtain the steady state growth rate of aggregate output as

$$ \frac{{\dot{Y}}}{Y} = n + x - \xi \frac{{\dot{\tau }}}{\tau }. $$

Substituting Eq. (56) into this together with some simplifications, we have

$$ \frac{{\dot{Y}}}{Y} = \frac{(1 - \alpha - \beta )(1 + \eta )x + (\beta (\theta - 1) + (1 - \alpha - \beta )(1 + \eta ))n}{(\beta (\theta - 1) + (1 - \alpha )(1 + \eta ))} $$

Note that we have the same results as the one obtained under command allocation. Thus, we can conclude that with the Pigouvian tax optimally set, the command allocation and decentralized allocation are equivalent.