Pollution Control with Time-Varying Model Mistrust of the Stock Dynamics

Abstract

I develop a framework to analyze the robust emissions tax policy for a stock pollutant when the environmental authority is not fully confident about its estimated model of pollution dynamics and, in contrast to previous research, the degree of model mistrust may change over time. I characterize the effect of time-varying model mistrust on emission taxes, pollution stocks and welfare. The general results of this paper show that introducing the possibility of a time-varying degree of model mistrust produces different emission taxes, abatement and welfare compared to the traditional assumption of a time-fixed model mistrust. This result holds even if the probability that the model mistrust may change in the future is small. If the environmental authority expects that the model uncertainty may decrease in the future then current emissions taxes should also decrease. Conversely, if the model mistrust may increase in the future then an active approach compatible with the Precautionary Principle is optimal and current emissions taxes should also increase.

Appendices

Appendix 1: Choice of $\theta _1$ and $\theta _2$

As mentioned in the main text, the environmental authority’s degree of mistrust about its own model has to come from outside the model in each regime. In Eq. (11) I narrow down the possible values of $\theta _{t+1, 1}$ and $\theta _{t+1, 2}$. The goal of this Appendix is to satisfy Eq. (11) by simultaneously finding numerical values of $\theta _{t+1, 1}$ and $\theta _{t+1, 2}$ that represent an environmental authority that is optimistic about its estimated model of pollution concentration dynamics in regime 1 and pessimistic in regime 2.

I follow with some differences the procedure outlined in Gonzalez and Rodriguez (2013) that adapts Hansen and Sargent’s (2008) detection error probability approach to include Markov switching and simultaneously choose $\theta _{t+1, 1}$ and $\theta _{t+1, 2}$.¹³ I consider two models: the original non-robust model without model uncertainty (given by Eqs. 6–7) and the time-varying model uncertainty model at the end of Sect. 2.4. The procedure consists of finding values of $\theta _{{t+1, 1}}$ and $\theta _{{t+1, 2}}$ for which it is statistically difficult to distinguish between the original and the time-varying model in regime 1 and that also satisfy Eq. (11).

First, I obtain the solution for the pollution stock in the original non-robust model:

$$\begin{aligned} \ddot{S}_{t+1} = \ddot{k} \ddot{S}_{t} + \ddot{H} +\ddot{\varepsilon }_{t+1} \end{aligned}$$

(35)

where $\ddot{k}=(\alpha + \beta \ddot{F})$ and $\ddot{H}= \beta \ddot{f} + \bar{x}$. Similarly, I obtain the solution for the pollution stock in the time-varying model:

$$\begin{aligned} S_{t+1, j} = k_{i} S_{t, i} + H_i + \varepsilon _{t+1} \qquad i,j=1,2 \end{aligned}$$

(36)

where $k_i=(\alpha + { \widetilde{\mathbf{B}}} \mathbf{F}_i)$ and $H_i=({ \widetilde{\mathbf{B}} }f _i + \bar{x})$. Next, I define the time-varying model in regime 1 as follows:

$$\begin{aligned} \breve{S_t} = (1-p) ( k_{1} S_{t, {1}} + H_{1}) + p ( k_{2} S_{t, {2}} + H_{2}) + \breve{\varepsilon }_{t+1} \end{aligned}$$

(37)

The worst-case shock assuming that the pollution stock was generated by the original model is given by:

$$\begin{aligned} \ddot{w} = F_{w} \ddot{S_t} + f_{w} \end{aligned}$$

(38)

The worst-case shock assuming that the pollution stock was generated by the time-varying model is then:

$$\begin{aligned} \breve{w}=F_{w} \breve{S_t} + f_{w} \end{aligned}$$

(39)

The log-likelihood ratio under the original model is then:

$$\begin{aligned} \ddot{r} =\frac{1}{T}\sum ^{T-1}_{t=0} 0.5 \{ \ddot{w}_{t+1} \ddot{w}_{t+1} - \ddot{w}_{t+1} \ddot{\varepsilon }_{t+1}\} \end{aligned}$$

(40)

The log-likelihood ratio under the time-varying model is then:

$$\begin{aligned} \breve{r} =\frac{1}{T}\sum ^{T-1}_{t=0} 0.5 \{ \breve{w}_{t+1} \breve{w}_{t+1} + \breve{w}_{t+1} \breve{\epsilon }_{t+1} \} \end{aligned}$$

(41)

I produce 200 random draws over $\ddot{\varepsilon }_{t+1}$ and $\breve{\varepsilon }_{t+1}$ and use Eqns. (35) and (37) above to simulate 200 years ($T=200$) of the pollution stock, which is roughly the amount of recorded CO$_2$ emissions data. Next, using Eqns. (38) and (41), I compute $\ddot{w}$, $\breve{w}$, $\ddot{r}$, and $\breve{r}$.

I perform 1000 simulations of this procedure and obtain the frequency, over all the 1000 simulations, for which $\ddot{r}$ and $\breve{r}$ are less than zero:

$$\begin{aligned} \ddot{pr} = freq(\ddot{r}< 0) \qquad \breve{pr} = freq(\breve{r} < 0) \end{aligned}$$

(42)

The error detection probability is then defined as follows:

$$\begin{aligned} prob(\theta _1, \theta _2, p, q) = 0.5 \{ \ddot{pr} + \breve{pr} \} \end{aligned}$$

(43)

In this paper, I use the error detection probability as the measure of model uncertainty and set $\pi = prob(\theta _1, \theta _2, p, q)$. The environmental authority will pick the correct model with probability of one when $\pi =0$ and it is equally likely to pick either model when $\pi =0.5$. Therefore, when $\pi \rightarrow 0.5$ overall time-varying model uncertainty is small and consequently it is difficult for the environmental to distinguish between the two models. When $\pi \rightarrow 0$ the overall time-varying model uncertainty is large and it is easy for the environmental authority to distinguish between the two model. Following Hansen and Sargent (2008), I choose values of $\theta _{{t+1, 1}}$ and $\theta _{{t+1, 2}}$ associated with $\pi$ of 0.1 and 0.2 because they correspond to commonly used confidence intervals of 95 and 90 %, respectively.

A drawback of the error detection probability algorithm in this context is that it does not generate a unique set of values of $\theta _{{t+1, 1}}$ and $\theta _{{t+1, 2}}$ for each $\pi$. Therefore, in each of the simulations, I include additional restrictions on the possible parameters of $\theta _i$. In particular, I select a combination of $\theta _{1}$ and $\theta _2$ for each pair (p, q) that satisfies Eq. (11) and produces the desired value of $\pi$ (0.1 or 0.2) for which: (i) $\theta _{1}$ is large enough that the results in regime 1 are close to those of the original non-robust model, (ii) increases of $\theta _2$ produce almost no changes in the results of regime 2 and (iii) $\theta _1>> \theta _2$.

One last clarification is pertinent. I do not claim that the error detection probability is the only measure of overall model uncertainty. Rather, I use the error detection probability associated with the time-varying model along with a set of restrictions outlined above to find values of $(\theta _1, \theta _2)$ that generate reasonable levels of robustness.

Appendix 2: Worst-Case Shock Numerical Results

In this Appendix, I present the solution for the worst-case shock. To keep the exposition succinct I only show the contour graphs.

Fig. 4

Regime 1 and 2 worst-case emissions shocks for $\pi$ = 10, 20 %

The two graphs in the first row of Fig. 4 show the worst-case shock ($\omega$) for all the different combinations of the transition probabilities for regime 1 and regime 2 when $\pi =10\,\%$. The second row show the results when $\pi =20\,\%$.

Appendix 3: Emission Taxes for Uncompensated Changes in p and q

In this Appendix, I show the effect of uncompensated changes in the transition probabilities. That is, changes in p and q without the corresponding change in $\theta _2$ to keep the time-varying model uncertainty the same. Figure 5 shows the contour graph of emission taxes in each regime. The degree of overall model uncertainty varies across the graph but it is $10\,\%$ at $p=q=0.1$. Uncompensated increases in p produce a higher overall model uncertainty and generate lower emissions taxes in both regimes, although emission taxes are mostly insensitive to changes in p in regime 1. Uncompensated increases in q produce a lower overall model uncertainty and generate higher emissions taxes in both regimes, although emission taxes are mostly insensitive to changes in q in regime 2. These results along with those of Sects. 5.1 and 5.2 show that increases in the overall model uncertainty due to either higher p, lower q or higher $\theta _i$ lead to higher emission taxes.

Fig. 5

Emission taxes under uncompensated changes in p and q