Environmental Policy in a Dynamic Model with Heterogeneous Agents and Voting

Abstract

We consider a population of infinitely-lived households split into two: some agents have a high discount factor (the patients), and some others have a low one (the impatients). Polluting emissions due to economic activity harm environmental quality. The governmental policy consists in proposing households to vote for a tax to maintain environmental quality. By studying the voting equilibrium at steady states we show that the equilibrium maintenance level is the one of the median voter. We also show that (i) an increase in total factor productivity may produce effects described by the Environmental Kuznets Curve, (ii) an increase in the patience of impatient households may foster environmental quality if the median voter is impatient and maintenance positive, finally (iii) a decrease in inequality among the patient households leads to an increase in environmental quality if the median voter is patient and maintenance is positive. We show that, when the median income of the median voter is lower than the mean (which is empirically founded), our model with heterogeneous agents predicts a lower level of environmental quality than what the representative agent model would predict, and that increasing the public debt decreases the level of environmental quality.

Appendix

1.1 8.1 Proof of Proposition 1

It is sufficient to notice that since in a steady-state equilibrium we have $\mu\varPhi(Q^{*})=\mu(Q^{*}-\kappa P^{*})+L\bar{m}$ and for each i, the sequence $(\tilde{s}_{t-1}^{i}, \tilde {c}_{t}^{i})_{t=0}^{\infty}$ given by $\tilde{s}_{t-1}^{i}=s^{i*}$, $\tilde{c}_{t}^{i}=c^{i*}$ is a solution to

$$\begin{aligned} &\max\sum_{t=0}^{\infty}\beta_{i}^{t}u(c_{t}), \quad c_{t}+s_{t}\leq \bigl(w^{*}-\bar{m} \bigr)+ \bigl(1+r^{*} \bigr)s_{t-1},\ s_{-1}^{i}=s^{i*},\\ &\quad c_{t}\geq0, \ s_{t}\geq0 \end{aligned}$$

and to refer to Becker (1980, 2006).

1.2 8.2 Proof of Lemma 1

We have:

$$\frac{\partial V_{i,0}(s^{i*},Q^{*},\mathbf{m}^{*})}{\partial m^{*}_{0}} =\frac{\partial\varLambda_{i,0}(Q^{*},\mathbf {m}^{*})}{\partial m^{*}_{0}} +\frac{\partial\varGamma _{i,t}(s^{i*},\mathbf{m}^{*})}{\partial m^{*}_{0}}, $$

where the functions Λ _i,0 and Γ _i,0 are defined as follows:

$$\begin{aligned} &\varLambda_{i,0} \bigl(Q_{0},\mathbf{m}^{*} \bigr) = \max_{(Q_{t})_{t=1}^{\infty}} \Biggl\{ \sum_{t=0}^{\infty} \beta_{i}^{t}v(Q_{t}) \mid\mu\varPhi(Q_{t+1})\leq\mu \bigl(Q_{t}-\kappa P^{*} \bigr)+Lm^{*}_{t},\\ &\phantom{\varLambda_{i,0} \bigl(Q_{0},\mathbf{m}^{*} \bigr) = \max_{(Q_{t})_{t=1}^{\infty}} \Biggl\{ } Q_{t+1}\geq0, \ t=0,1,\ldots \Biggr\} ,\\ &\varGamma_{i,0} \bigl(s_{-1},\mathbf{m}^{*} \bigr)= \max_{(c_{t})_{t=0}^{\infty},(s_{t})_{t=0}^{\infty}} \Biggl\{ \sum_{t=0}^{\infty} \beta_{i}^{t} u(c_{t}) \mid c_{t}+s_{t}+m^{*}_{t}\leq w^{*}+ \bigl(1+r^{*} \bigr)s_{t-1},\\ &\phantom{\varGamma_{i,0} \bigl(s_{-1},\mathbf{m}^{*} \bigr)= \max_{(c_{t})_{t=0}^{\infty},(s_{t})_{t=0}^{\infty}} \Biggl\{ } c_{t}\geq0, \ s_{t}\geq0, \ t=0,1,\ldots \Biggr\} . \end{aligned}$$

It is not difficult to check that

$$\begin{aligned} &\frac{\partial\varLambda_{i,0}(Q^{*},\mathbf{m}_{t}^{*})}{\partial m^{*}_{t}} =\beta_{i}\frac{Lv'(Q^{*})}{\mu(\varPhi'(Q^{*})-\beta_{i})},\\ &\frac{\partial\varGamma_{i,0}(s^{i*},\mathbf{m}_{t}^{*})}{\partial m^{*}_{t}} =-u' \bigl(c^{*} \bigr). \end{aligned}$$

Therefore,

$$\frac{\partial V_{i,0}(s^{i*},Q^{*},\mathbf{m}^{*})}{\partial m^{*}_{t}} =\beta_{i}\frac{Lv'(Q^{*})}{\mu(\varPhi'(Q^{*})-\beta_{i})}-u' \bigl(c^{*} \bigr), $$

which implies (21).

1.3 8.3 Proof of Lemma 2

Using a traditional argument (see e.g. McKenzie 1986) we can prove that a sequence $(\tilde{s}_{t-1},\tilde{c}_{t},\tilde {m}_{t},\tilde{Q}_{t})_{t=0}^{\infty}$ given by (22) is a steady-state solution to problem $\mathcal{P}_{2}$ if and only if there exist q and p such that for $p_{t}=\beta_{i} p_{t-1}=\cdots=\beta _{i}^{t}p$ and $q_{t+1}=\beta_{i} q_{t}=\cdots=\beta_{i}^{t+1}q$ the following relationships hold:

$$\begin{aligned} &\beta_{i}^{t}u'(\tilde{c}_{t})=p_{t},\\ &\beta_{i}^{t}v'(\tilde{Q}_{t})+q_{t+1} \mu-q_{t}\mu\varPhi'(\tilde{Q}_{t})=0,\\ &\bigl(1+r^{*} \bigr)p_{t}\leq p_{t-1}\ (= \mathrm{if}\ \tilde{s}_{t-1}>0),\\ &q_{t+1}L-p_{t}\geq0 \ (= \mathrm{if}\ \tilde{m}_{t}>0),\\ &q_{t+1}\tilde{Q}_{t}+p_{t}\tilde{s}_{t-1} \rightarrow_{t\rightarrow\infty} 0, \end{aligned}$$

or, equivalently,

$$\begin{aligned} &u'(\tilde{c})=p,\\ &v'(\tilde{Q})=\mu q \bigl(\varPhi'(\tilde{Q})- \beta_{i} \bigr),\\ &\beta_{i}\leq\frac{1}{1+r^{*}} \ (= \mathrm{if}\ \tilde{s}>0),\\ &\beta_{i} Lq-p\geq0 \ (= \mathrm{if}\ \tilde{m}>0). \end{aligned}$$

The existence of such q and p is equivalent to conditions (23)–(24).