Optimal Environmental Policy in the Presence of Multiple Equilibria and Reversible Hysteresis

Abstract

We study optimal environmental policy in an economy-ecology model featuring multiple stable steady-state ecological equilibria. The policy instruments consist of public abatement and a tax on the polluting production input, which we assume to be the stock of capital. The isocline for the stock of pollution features two stable branches, a low-pollution (good) and a high-pollution (bad) one. Assuming that the ecology is initially located on the bad branch of the isocline, the ecological equilibrium is reversibly hysteretic and a suitably designed environmental policy can be used to steer the environment from the bad to the good equilibrium. We study both first-best and second-best social optima. We show that, compared to capital taxation, abatement constitutes a very cheap instrument of environmental policy.

Appendix: Computational Details

First-Best (FB)

We use a continuation method to compute the first-best. Let X _s =(K,P) denote the state variables, X _c =(λ _K ,λ _P ) the costates, and X=(X _s ,X _c ). The controls and the Lagrange-multipliers are denoted by U=(C,G,η _I ,η _D ). From Pontryagin’s maximum principle, we get U=U ^∗(X): the state- and costate-variables determine consumption, abatement and the multipliers for the investment- and dirt constraints. Recall that the other first-order conditions can be written as follows:

$$\dot{X}(t)=H\bigl(X(t),U^{\ast}\bigl(X(t)\bigr)\bigr). $$

The optimal path is determined by constraints on X _s (0) and X(∞). In particular $X_{s}(0)=(\hat{K},\hat{P}_{B})$ and X(∞)=X ^∗, where X ^∗ is a root of H(⋅). The end condition is replaced by the requirement that at time T=200, the trajectory is orthogonal to the stable manifold of X ^∗. We approximate the first-order condition as follows. First, we discretize the time grid t∈{0,1,…,200} and at time t we replace the differential equation by a fourth-order Runge-Kutta approximation. This leads to a system of equations of which we have to find the root.

For the continuation method, we need a trivial solution. Note that X(t)≡X ^∗ is a solution for the initial condition $X_{s}(0)=X_{s}^{*}$. We slowly change this initial condition into the direction of the actual initial condition, using a simple predictor-correction algorithm. See Grass (2012) for details.

The time at which the investment constraint stops being binding is calculated in the following manner. Suppose that for t≤t ^∗, we have η _I (t)>0 (and η _I (t)=0 for t>t ^∗). This means that the investment constraint is binding until t _I ∈[t ^∗,t ^∗+1]. Using cubic extrapolation, we determine the value of t _I . It turns out that t _I =1.27. For the dirt constraint, we use a similar method and it turns out that t _D =27.01.

Time-Varying Tax (TVT)

In principle, in this case we should be able to use a similar algorithm as for the first-best. However, the continuation algorithm fails to terminate (the path “bends back” to the $X_{s}(0)=X_{s}^{\ast}$). We note that at some point the investment constraint becomes binding. Therefore, we postulate that the optimal path first goes through a regime where the investment constraint is binding. If the investment constraint is binding until t=t _I , then we can calculate the value of capital and pollution at t=t _I . We take these as the initial value for capital and pollution and solve for the optimal time-varying tax from that point onward. Then we choose t _I such that this is the point where the investment constraint stops being binding (i.e. λ _K (t _I )F(K(t _I ),1)=1). It turns out that this is the case for t _I =21.46.

Please note that in both FB and TVT the long-run tax rate is $\hat{\theta}=0.1066$.

Time-Invariant Tax (TIT)

In the long-run, we set the tax rate equal to $\hat{\theta}$, but we start with a higher tax rate to move the system towards a lower pollution level. It turns out that the initial tax rate θ ₀ is high enough to make the investment constraint binding. This means that we have to determine t _I (the time at which the investment constraint stops being binding) and t _E (the time at which the tax rate shifts from θ ₀ to $\hat{\theta}$). Since the consumption path cannot jump, we can only switch from θ ₀ to $\hat{\theta}$ if we are on the stable saddle path leading to the clean equilibrium. Hence, the free variables are θ ₀ and t _I . We somewhat crudely search for the lowest values that can force the system to the clean steady state by increasing θ ₀ with step size 0.05 and t _I with step size 1. We end up with θ ₀=0.85 and t _I =32. Since the EV under TIT is close to the EV under TIA (time-invariant abatement), we are confident that these values are close to the optimal tax of this form.

Time-Varying Abatement and Time-Invariant Abatement (TVA and TIA)

See Heijdra and Heijnen (2013). We have added a bit of accuracy for the case with TVA: full abatement until t _E =28.2, increases the EV to 40.5 %.

Calculation of Utility Levels

In the FB, we calculate utility level by calculating the Lagrangian at time zero and dividing this value by ρ. In all other cases, we use the following method to calculate the utility of the representative consumer. Given paths for consumption and pollution, this amounts to evaluating an integral of the form

$$W=\int_{0}^{\infty}u\bigl(C(s),P(s)\bigr) \mathrm{e}^{-\rho s}ds. $$

As inputs we have the levels of consumption and pollution at discrete points in time t∈{t ₀,t ₁,t ₂,…,t _n }, where t _n is sufficiently large for consumption and pollution to be close to the steady state values. Then, as is also noted by Heijnen and Wagener (2013), W is approximately equal to:

$$\begin{aligned} W& \approx\frac{1}{2}\sum_{i=1}^{n} \bigl[u\bigl(C(t_{i}),P(t_{i})\bigr)\mathrm{e}^{-\rho t_{i}}+u \bigl(C(t_{i-1}),P(t_{i-1})\bigr)\mathrm{e}^{-\rho t_{i-1}} \bigr](t_{i}-t_{i-1}) \\ & \quad{}+u\bigl(C(t_{n}),P(t_{n})\bigr)\frac{\mathrm{e}^{-\rho t_{n}}}{\rho}. \end{aligned}$$

Since our grid is not very dense, this gives a rather rough approximation, limiting the accuracy with which we can calculate the optimal policy.