Fiscal and Environmental Sustainability: Is Public Debt Environmentally Friendly?

Abstract

This article assesses the dilemma that most governments face when seeking to ensure the sustainability of their public finances through economic growth while simultaneously protecting the environment. We propose a growth model in which the government finances abatement-spending through taxation or public debt and which follows a fiscal rule that targets the long-run debt-to-GDP ratio. We show that there is a threshold for the debt ratio below which debt and environmental sustainability are secured. In steady state, the debt ratio exerts a nonlinear effect on environmental quality in the form of an inverted U-shaped curve, and the environmental tax is good for the environment when public debt is not. A fiscal rule authorizing a small but strictly positive debt ratio could help the government to implement adaptation policies for environmental protection while supporting long-run economic growth.

Appendices

Appendix A. Steady State

Proof of Proposition 3. The long-run environmental quality is given by

$$\begin{aligned} E^{*}(\theta )= \bar{E} -\frac{\beta }{\varepsilon \tau _p g^*(\theta )} \end{aligned}$$

(A.1)

where

$$\begin{aligned}g^*(\theta )=\beta +\frac{As \theta }{1+ A\theta }-r\theta , \text { and }\gamma ^*(\theta )=\frac{As}{1+A\theta }.\end{aligned}$$

First: $\gamma ^*(\theta )-1>0 \Leftrightarrow \theta < \bar{\theta }:=(As-1)/A$. $\bar{\theta }$ is positive as we assume $As>1$.

Second: $E^*(\theta )>0 \Leftrightarrow g^*(\theta )-\frac{\beta }{ \bar{E} \varepsilon \tau _p}=:h(\theta )>0$, where

$$\begin{aligned} h(\theta ):=\beta + \frac{As \theta }{1+A\theta }- r\theta -\frac{\beta }{ \bar{E} \varepsilon \tau _p}. \end{aligned}$$

Let us suppose that $\bar{E} \varepsilon \tau _p>1$. Then h is a continuous mapping on $\mathbb {R}^+$, with the following properties: $h(0)=\beta -\beta /\bar{E} \varepsilon \tau _p >0$, $h(+\infty )=-\infty$, and

$$\begin{aligned} h'(\theta )=\frac{As}{(1+A\theta )^2}-r \ge 0 \Leftrightarrow \ \theta \le \hat{\theta }=:\frac{1}{A}\left[ \sqrt{\frac{As}{r}}-1\right] . \end{aligned}$$

The threshold $\hat{\theta }$—which is positive as $As>1>r$—is the level of the long-run debt target that maximizes the abatement-spending ratio. Hence, h describes an inverted U-shaped curve with a maximum at $\theta =\hat{\theta }$, as described in Fig. 1.

Additionally, we have $h(\bar{\theta })=\beta + (s-1)- r(s-1)-\frac{A\beta }{ \bar{E} \varepsilon \tau _p}=\beta +\frac{(As-1)(1-r)}{A} -\frac{\beta }{ \bar{E} \varepsilon \tau _p}>\beta -\frac{\beta }{ \bar{E} \varepsilon \tau _p}>0$. Hence, if $\beta >\beta /\bar{E} \varepsilon \tau _p$, then $h(\theta )>0$, for any $\theta \in (0,\bar{\theta })$.

Consequently, if $\theta <\bar{\theta }$ (i.e., $\gamma ^*-1>0$, debt sustainability), the environmental sustainability (i.e., $E^*(\theta )>0$) is ensured. $\square$

Proof of Proposition 4. We first prove the uniqueness of the steady state then we focus on the local stability.

Uniqueness. From Eq. (23.b), it is clear that the law of motion of $b_t$ does not depend on $E_t$; such that $b_t$ monotonically converges to its target $b^*=\theta$ since $\phi \in (0,1)$. This explains the vertical line in the diagram phase (see Fig. 5). Let $b_{t+1}=b_t=b$ and $E_{t+1}=E_t=E$ in system (23). From Eq. (23.a), the stationary locus of the environmental quality is depicted by the following link between E and b

$$\begin{aligned} E=E(b)=\bar{E}-\frac{\beta }{\varepsilon \tau _p}\left( \frac{1}{\frac{As b}{1+Ab}-rb+\beta }\right) . \end{aligned}$$

Let us suppose that $\theta< \bar{\theta }\Leftrightarrow b<\bar{b}$, namely $E(b)>0$, as stated in the proof of Proposition 3. It is clear that E(b) is a continuous mapping on $[0,\bar{b}$].

First, we have

$$\begin{aligned}E(0)=\bar{E}-\frac{1}{\varepsilon \tau _p }>0,\text { and }E(\bar{b})=\bar{E} -\frac{\beta }{\varepsilon \tau _p[ (As-1)(1-r)/A+\beta ] }>0,\end{aligned}$$

as we assume $\bar{E} \varepsilon \tau _p>1$, and $As>1>r$.

Second, we compute

$$\begin{aligned} E'(b) \ge 0 \Leftrightarrow b \le \hat{b}:=\frac{1}{A}\left[ \sqrt{\frac{As}{r}}-1\right] . \end{aligned}$$

Hence, the stationary locus of the environmental quality depicts an inverted U-shaped curve on the phase portrait (b, E), with a maximum at $b=\hat{b}$, as depicted in Fig. 5. Consequently, for $b^* \in [0,\bar{b}]$, there is a unique crossing point between $b^*$ and E(b) that defines the unique steady state of the model $(b^*,E^*)$.

Fig. 5

Diagram phase

Local stability. From system (23), the Jacobian matrix evaluated at the steady state ($b^*,E^{*})$ is

$$\begin{aligned} \textbf{J} = \begin{bmatrix} 1-\varepsilon &{}K \\ 0 &{} 1-\phi \end{bmatrix}, \end{aligned}$$

where K is a (finite) scalar. Then, the determinant and trace are $\det (\textbf{J})=(1-\varepsilon )(1-\phi ),$ and $\text {tr}(\textbf{J})=2-\phi -\varepsilon$. For the steady state to be well determined, we must ensure that $\det (\textbf{J})<1$ and $\text {tr}(\textbf{J})<2$. As $\varepsilon <1$ and $\phi <1$, it follows the steady state is well determined. $\square$

Appendix B. Comparative Statics

Proof of Proposition 6. At steady state, the environmental quality is given Eq. (A.1), namely

$$\begin{aligned} E^{*}(\theta )= \bar{E} -\frac{\beta }{\varepsilon \tau _p \left[ h(\theta )+\frac{\beta }{\bar{E} \varepsilon \tau _p}\right] }. \end{aligned}$$

As proved in “Appendix A”, $h(\theta )$ describes an inverted U-shaped curve on $\theta \in [0,\bar{\theta }]$. Hence, $E(\theta )$ also describes an inverted U-shaped curve on $\theta \in [0,\bar{\theta }]$, with a threshold at $\theta =\hat{\theta }$. $\square$

Proof of Proposition 7. From Eq. (25), we define $E^*=E^{*}(\theta ,\tau _p)$, where

$$\begin{aligned} E^{*}(\theta ,\tau _p)= \bar{E} -\frac{\beta }{\varepsilon \tau _p g^*(\theta ,\tau _p)}, \end{aligned}$$

(B.1)

with, using $r=\alpha A(\tau _p)$,

$$\begin{aligned}g^*(\theta ,\tau _p)=\beta +\frac{A(\tau _p)s \theta }{1+ A(\tau _p)\theta }-\alpha A(\tau _p)\theta . \end{aligned}$$

We compute

$$\begin{aligned} \frac{\partial g^*(\theta ,\tau _p)}{\partial \tau _p}=A'(\tau _p)\theta \left[ \frac{s}{(1+A(\tau _p)\theta )^2}-\alpha \right] . \end{aligned}$$

The bracketed-term is negative if and only if

$$\begin{aligned}\theta \ge \ \frac{1}{A}\left[ \sqrt{\frac{s}{\alpha }}-1\right] =\frac{1}{A}\left[ \sqrt{\frac{As}{r}}-1\right] =\hat{\theta }. \end{aligned}$$

Consequently, as $A'(\tau _p) \le 0$, it follows that $\partial g^*(\theta ,\tau _p)/\partial \tau _p \ge 0 \Leftrightarrow \theta \ge \hat{\theta }$. As shown in “Appendix A”, this condition ($\theta \ge \hat{\theta }$) is precisely equivalent to $\partial g^*(\theta ,\tau _p)/\partial \theta \le 0$. In other words, while increasing public debt is detrimental to abatement spending, increasing taxes is not, and vice versa. Finally, from (B.1), if $\theta \ge \hat{\theta }$, we derive $\partial E^{*}(\theta ,\tau _p)/\partial \tau _p \ge 0$.