2.1 Production

We consider an AK model of economic growth in which TFP decreases with pollution stock $P_{t}$ .Footnote ⁵ Considering that pollution or climate change is detrimental to production is particularly relevant in addressing debt and environmental issues, as it allows to focus on funding adaptation efforts, extending beyond mere mitigation. The need for adaptation strategies will increase with the intensification of climate change impacts. These adaptations come with associated costs, such as building infrastructure to protect against rising sea levels, creating drought-resistant agriculture or developing air conditioning systems to cope with heat waves. Public action can provide the necessary financial resources to implement these adaptation measures, reducing the vulnerability of countries to environmental shocks. We thus assume that the capacity for adaptation reduces the incremental damage caused by pollution stock. This ability to adapt is ensured by the public authorities, who devote an amount $G_{1t}$ , specifically for this purpose.Footnote ⁶

Therefore, the production $Y_t$ is given by:

(1) \begin{equation} Y_t=A\left (\frac {G_{1t}}{P_t}\right )K_t^\alpha (\bar {K}_tL_t)^{1- \alpha } \end{equation}

with $K_t$ the capital, $\bar {K}_t$ the aggregate level of capital, $L_t$ the labor, $\alpha \in (0,1)$ and:

This last assumption implies an elasticity of production to the adaptation to pollution ratio lower than one. When the adaptation to pollution ratio goes up it causes a less-than-proportional increase in productivity.

Note that $A(X)$ may capture the fact that climate change destroys a part of aggregate output at each period (Golosov et al. Reference Golosov, Hassler, Krusell and Tsyvinski2014; Dietz and Stern, Reference Dietz and Stern2015). It can also represent the health effects of global pollution stock or the impacts of a change in temperature, which results in reduced aggregate productivity (Dasgupta et al. Reference Dasgupta, van Maanen, Gosling, Piontek, Otto and Schleussner2021; Burke et al. Reference Burke, Hsiang and Miguel2015).

Example: we can consider the following specifications for $A(X)$ :

(2) \begin{equation} A(X)=\frac {A_1 X}{1+X} \end{equation}

with $A(0)=0$ . This function is increasing and concave, with:

(3) \begin{equation} \frac {A^{\prime }(X)X}{A(X)}=\frac {1}{1+X}\in (0, 1) \end{equation}

Let $r_t$ be the interest rate and $w_t$ the wage. At equilibrium, we have $ \bar {K}_t=K_t$ and assuming that the population in this economy is equal to one, labor input is $L_t=1$ . Therefore, profit maximization gives:

(4) \begin{align} r_t&= \alpha A\left (\frac {G_{1t}}{P_t}\right ) \end{align}

(5) \begin{align} w_t &= (1-\alpha ) A\left (\frac {G_{1t}}{P_t}\right )K_t \\[6pt]\nonumber\end{align}

Returns of factor being a positive function of productivity, they decrease with pollution.

2.2 Government

We consider public actions to tackle environmental issues. Public spending $G_t$ linearly increases with GDP:

(6) \begin{equation} G_t=g Y_t \end{equation}

and are divided into public spending which attenuates the effect of pollution on production $G_{1t}$ , i.e. adaptation to climate effect, and mitigation $G_{2t}$ :

(7) \begin{eqnarray} G_{it}=g_i Y_t \end{eqnarray}

for $i=1,2$ , with $g_1+g_2=g$ .

Environmental policy instruments consist of public spending on both mitigation and adaptation. While most of the literature has focused on their potential substitutability, these two strategies are now seen as simultaneously needed in the face of climate emergencies. This need is reflected in international ambitions to balance climate finance spending between the two strategies (Sadler et al. Reference Sadler, Ranger, Fankhauser, Marotta and O’Callaghan2024).

Since we are in an endogenous growth framework, we assume that the government determines its public spending for adaptation and mitigation by fixing their amount per GDP unit. This means that the policy of adaptation will be determined by $g_1$ and the policy of mitigation by $g_2$ .

To finance these spending, the government collects taxes on labor and capital incomes, $\tau _L$ and $\tau _K$ , and issues debt $B_t$ . Therefore, its expenditures include repayment of debt and interest payments. The government faces the following budget constraint at each period:

(8) \begin{equation} R^b_tB_t+G_t=B_{t+1}+\tau _L w_t+\tau _K r_tK_t \end{equation}

with $R^b_t$ the interest factor of debt and $B_0\gt 0$ the initial stock of debt. We are in an economy with a positive initial stock of public debt. The different policy parameters as well as the interest factor of debt and the income will determine how public debt evolves through time. We will precisely study the interplay between debt accumulation, dynamics of pollution stock, and growth.

2.3 Pollution

The stock of pollution increases with the emission flow and partly leaves the atmosphere through a natural process in a share $0\lt m\lt 1$ . Emission flow is assumed to be proportional to the stock of capital. Mitigation measures $G_{2t}$ are implemented by public authorities to further reduce pollution flows. Mitigation expenditure may include investment in carbon capture and sequestration or carbon dioxide removal solutions. The stock of pollution evolves according to:

(9) \begin{equation} P_{t+1}=(1-m)P_{t}-\psi G_{2t}+\mu K_{t} \end{equation}

The parameter $\psi \gt 0$ captures the efficiency of public mitigation and $\mu \gt 0$ the pollution flow resulting from capital stock, as in Heijdra et al. (Reference Heijdra, Kooiman and Ligthart2006) or Chiroleu-Assouline and Fodha (Reference Chiroleu-Assouline and Fodha2014). Capital is the source of pollution. Indeed, capital accumulation favors the production of pollution-intensive goods. This is consistent with the evidences (e.g. Cole et al. Reference Cole, Elliott and Shimamoto2005; Andersen, Reference Andersen2017). For example, Cole et al. (Reference Cole, Elliott and Shimamoto2005) find that industrial processes that tend to be physical capital intensive generate more emissions than less capital-intensive processes. Andersen (Reference Andersen2017) obtains that pollution emissions are higher for industries that use more intensively tangible assets, such as physical capital, than intangible assets, such as labor.

Note that since $G_{2t}$ will increase with income, public adaptation will also have a direct negative effect on the pollution stock by increasing TFP, income and therefore mitigation. In addition, polluting capital which will be derived from a portion of savings will increase with labor income and therefore with TFP and public adaptation.

This stock of pollution only affects the real side of the economy through its negative effect on the TFP. We will not consider a direct negative effect of pollution on households welfare.

2.4 Consumers

Consumers are in overlapping generations. The population size of each generation is constant and normalized to one. Each consumer lives for two periods, consumes in both periods, and saves through two assets, public debt and capital. Capital depreciates at rate $\delta \in (0,1)$ , meaning that return on capital is given by $1-\delta +(1-\tau _K) r_{t}$ .

The utility function of the generation born in $t$ is given by:

(10) \begin{equation} U(c_t, d_{t+1})= \ln c_t + \beta \ln d_{t+1} \end{equation}

with $c_t$ the consumption when young, $d_{t+1}$ the consumption when old, and $\beta \in (0,1)$ . The household maximizes her utility under the two budget constraints:

(11) \begin{align} c_t+K_{t+1}+B_{t+1}&= (1-\tau _L)w_t \end{align}

(12) \begin{align} d_{t+1}&= [1-\delta +(1-\tau _K) r_{t+1}]K_{t+1}+R^b_{t+1}B_{t+1} \\[9pt]\nonumber\end{align}

The household can solve her optimal behavior in two steps. Given her saving, she determines her portfolio choice between capital and debt holding. Given this choice, she chooses her optimal saving. Maximizing the utility with respect to $K_{t+1}$ and $B_{t+1}$ , we obtain the following equation:

(13) \begin{equation} 1-\delta +(1-\tau _K) r_{t+1}=R^b_{t+1}\equiv R_{t+1} \end{equation}

Bonds and capital assets provide the same return, which means that they are perfect substitutes.Footnote ⁷ Given this result, the utility maximization with respect to $K_{t+1}+B_{t+1}$ gives the saving function:Footnote ⁸

(14) \begin{equation} K_{t+1}+B_{t+1}=\frac {\beta }{1+\beta } (1-\tau _L)w_t \end{equation}

4.1 Existence and multiplicity of BGPs

Along a balanced growth path, capital, debt, and pollution grow at a constant rate $\gamma -1$ . A balanced growth path is thus characterized by $b_t=b_{t+1}=b$ and $\pi _t=\pi _{t+1}=\pi$ solving (24) and (25). Hence, it is a stationary solution $(b, \pi )$ to:

(26) \begin{align} b &= \frac {\frac {R(\pi )}{a(\pi )} b+g -(\tau _L (1-\alpha )+\tau _K \alpha ) }{\Sigma (1-\tau _L)(1-\alpha ) +(\tau _L (1-\alpha )+\tau _K \alpha )-g-\frac {R(\pi )}{a(\pi )} b} \end{align}

(27) \begin{align} \pi &=\frac {(1-m)\frac {\pi }{a(\pi )}-\psi g_2 +\frac {\mu }{a(\pi )} }{\Sigma (1-\tau _L)(1-\alpha ) +(\tau _L (1-\alpha )+\tau _K \alpha )-g-\frac {R(\pi )}{a(\pi )} b} \\[12pt]\nonumber \end{align}

Given such a solution, the growth factor corresponds to:

(28) \begin{equation} \gamma = a(\pi )\left [\Sigma (1-\tau _L)(1-\alpha ) +(\tau _L (1-\alpha )+\tau _K \alpha )-g-\frac {R(\pi )}{a(\pi )} b\right ] \end{equation}

The ratio of (26) and (27) gives:

(29) \begin{equation} b=\frac {g -(\tau _L (1-\alpha )+\tau _K \alpha ) }{\frac {1-m-R(\pi )}{a(\pi )}- \frac {\psi g_2}{\pi } +\frac {\mu }{a(\pi )\pi } } \end{equation}

Using (13), it is equivalent to:

(30) \begin{equation} b =\frac {a(\pi )\pi [g -(\tau _L (1-\alpha )+\tau _K \alpha )] }{\pi (\delta -m )-(1-\tau _K)\alpha a(\pi )\pi - \psi g_2 a(\pi ) +\mu }= \frac {g -\tau _L (1-\alpha )-\tau _K \alpha }{X(\pi )-\frac {1-\delta }{a(\pi )}-(1-\tau _K)\alpha }\equiv B_1(\pi ) \end{equation}

with $X(\pi )\equiv \frac {1-m}{a(\pi )}- \frac {\psi g_2}{\pi } +\frac {\mu }{a(\pi )\pi }$ .

Moreover, (27) can be rewritten as:

(31) \begin{eqnarray} b&=&\frac {a(\pi )}{R(\pi )} \left [ \Sigma (1-\tau _L)(1-\alpha ) +\tau _L (1-\alpha )+\tau _K \alpha -g - X(\pi )\right ] \notag \\ &\equiv & B_2(\pi ) \end{eqnarray}

In the following, we assume:

The first part of the assumption implies that the rate of pollution absorption is lower than the depreciation rate of capital. This is consistent in our context, as pollution stock can refer to greenhouse gases whose some will remain in the atmosphere for thousands of years. The second part of the assumption implies that total factor productivity is elastic to pollution over capital ratio, illustrating an important vulnerability to climate change (see IPCC 2022). When pollution per unit of capital increases total factor productivity falls more than proportionally. It implies that $a(\pi )\pi$ is decreasing in $\pi$ . As a result, $X(\pi )$ is an increasing function of $\pi$ .

Example (continued): note that with the example defined in equation (19), $\epsilon _a(\pi )\lt -1$ implies that $\pi \gt g_1A_1/2$ .

This example illustrates the fact that $\epsilon _a(\pi )\lt -1$ could introduce a lower bound $\underline {\pi }\gt 0$ defined by $\epsilon _a(\underline {\pi })=-1$ such that $\epsilon _a(\pi )\lt -1$ for all $\pi \gt \underline {\pi }$ .

Under Assumptions2 and 3, the numerator of (30) is positive and the denominator is increasing in $\pi$ . We thus have $B_1^{\prime }(\pi )\lt 0$ . In addition, to ensure a positive debt along the balanced growth path, we restrict our attention to cases where:

(32) \begin{equation} X(\pi )-\frac {1-\delta }{a(\pi )}\gt (1-\tau _K)\alpha \end{equation}

New debt emissions should be higher than the cost of existing debt. Indeed, using (27) and (28), $X(\pi )=\gamma /a(\pi )$ . It implies that inequality (32) is equivalent to $\gamma \gt R(\pi )$ . Using (30), we have $b[\gamma -R(\pi )]=a(\pi )[g -\tau _L (1-\alpha )-\tau _K \alpha ]$ . Since we assume that the government budget is characterized by a primary deficit, a BGP should be characterized by a growth factor larger than the interest factor.

Since the left-hand side of inequality (32) is increasing in $\pi$ , there exists $\pi _1\gt 0$ such that $X(\pi _1)=\frac {1-\delta }{a(\pi _1)}+(1-\tau _K)\alpha$ if there is a value $\widetilde {\pi }\gt 0$ such that $X(\widetilde {\pi })\lt \frac {1-\delta }{a(\widetilde {\pi })}+(1-\tau _K)\alpha$ . Then, inequality (32) is satisfied for all $\pi \gt \pi _1$ , with $\displaystyle \lim _{\pi \to \pi _1^+}B_1(\pi )=+\infty$ . Note that the existence of $\pi _1\gt 0$ can be compatible with Assumption2.

Our example illustrates that this lemma is satisfied for a non-empty set of parameters.

Example (continued): using the example defined in equation (19), we illustrate the existence of the bound $\pi _1$ . Inequality (32) writes $F(\pi )\gt (1-\tau _K)\alpha$ , with:

(33) \begin{equation} F(\pi )\equiv \frac {\pi [g_1A_1(\delta -m)+\psi g_2]+g_1A_1(\mu -\psi g_2)}{\pi (g_1A_1-\pi )} \end{equation}

Since $F(\pi )$ is an increasing function and $F(g_1A_1)=+\infty$ , there exists $\pi _1\in (g_1A_1/2, g_1 A_1)$ if $F(g_1A_1/2)\lt (1-\tau _K)\alpha$ . This happens if $\delta -m \lt A_1(1-\tau _K)\alpha /2$ and

(34) \begin{eqnarray} g_1\gt \frac {4 \mu -2\psi g_2 A_1}{A_1 [(1-\tau _K)\alpha A_1-2 (\delta -m)]}\equiv \underline {g_{1}} \left ( g_{2}\right ) \end{eqnarray}

Using (31), a steady state with positive debt per unit of capital should satisfy:

(35) \begin{equation} \Sigma (1-\tau _L)(1-\alpha ) \gt g-\tau _L (1-\alpha )-\tau _K \alpha +X(\pi ) \end{equation}

Since the left-hand side of this inequality is constant and the right-hand side is increasing in $\pi$ , there exists $\pi _2\gt 0$ such that inequality (35) is satisfied for all $\pi \lt \pi _2$ , with $B_2(\pi _2)=0$ . In this case, we also deduce that under Assumption3, we have $B_2^{\prime }(\pi )\lt 0$ . For this, we need to have a value $\widehat {\pi }\gt 0$ such that $\Sigma (1-\tau _L)(1-\alpha ) \lt g-\tau _L (1-\alpha )-\tau _K \alpha +X(\widehat { \pi })$ .

Lemma 2. Under Assumptions 1 – 3 , assume that there exists a value $\widehat {\pi }\gt 0$ such that $\Sigma (1-\tau _L)(1-\alpha ) \lt g-\tau _L (1-\alpha )-\tau _K \alpha +X(\widehat {\pi })$ . The interval $(\pi _1, \pi _2)$ is non-empty if the following inequality is satisfied:

(36) \begin{equation} \Sigma (1-\tau _L)(1-\alpha ) \gt g-\tau _L (1-\alpha )-\tau _K \alpha +\frac { 1-\delta }{a(\pi _1)}+(1-\tau _K)\alpha \end{equation}

where $\pi _2$ is defined by $\Sigma (1-\tau _L)(1-\alpha ) =g-\tau _L (1-\alpha )-\tau _K \alpha +X(\pi _2)$ .

Example (continued) : in our example, let $\widehat {\pi }=g_1A_1$ be such that $a (\widehat {\pi })=0$ . In this case, we have $X(\widehat {\pi })=+\infty$ , which ensures the first inequality in the lemma.

Note that in our example, we have $\pi _1$ higher but arbitrarily close to $g_1A_1/2$ if $g_1$ tends to $\frac {4 \mu -2\psi g_2 A_1}{A_1 [(1-\tau _K)\alpha A_1-2 (\delta -m)]}$ . Therefore, inequality (36) is satisfied if $\Sigma (1-\tau _L)(1-\alpha ) \gt g-\tau _L (1-\alpha )-\tau _K \alpha +\frac { 2(1-\delta )}{A_1}+(1-\tau _K)\alpha$ . This last inequality is satisfied if $\tau _K$ and $A_1$ are high enough and the primary deficit is not too important. It proves the existence of $\pi _2$ and of a non-empty interval $(\pi _1, \pi _2)$ .

Since $B_1(\pi _1)=+\infty \gt B_2(\pi _1)$ and $B_1(\pi _2)\gt B_2(\pi _2)=0$ , the economy may be characterized by an even number (two) of steady states.

A primary deficit ( $g\gt \tau _L (1-\alpha )+\tau _K \alpha$ ) ensures the possibility of having a positive stationary debt per unit of capital in our context where growth is higher than the interest factor. At the same time, if the environmental expenditure is too high ( $g\gt \overline {g}$ ), the primary deficit is too significant to observe stationarity in debt per unit of capital. The share of GDP devoted to environmental issues has to take an intermediate value to observe stationary solutions. In addition, condition (36) has to be satisfied.

Example (continued): using the example defined in equations (2) and (19), we illustrate what the conditions for existence of BGPs imply in terms of policy.

Using Lemmas1 and 2, we have conditions ensuring that the interval $ (\pi _{1},\pi _{2})$ , necessary for the existence of a steady-state $\pi _{i}$ , is non-empty. Adaptation expenditures must satisfy the constraint (34), $g_{1}\gt \underline {g_{1}}\left ( g_{2}\right )$ , where adaptation $g_1$ must not deviate significantly from the minimum value $\underline {g_{1}}(g_2)$ .

Thus, for any tax rates $\tau _{L}$ et $\tau _{K}$ , existence of a primary deficit and (34) determine the precise level of environmental policy expenditures $g_{1}$ and $g_{2}$ . Adaptation $g_{1}$ has to be close to $\underline {g_{1}}\left ( g_{2}\right )$ and mitigation $g_{2}$ adjusts so as to have a weak primary deficit. Using Assumption3, mitigation is also constrained by the condition imposing a maximum threshold $\mu /A_{1}\gt \psi g_{2}$ . Considering that $\mu$ and $A_1$ are close or equal, this last inequality is never binding.

We also need to ensure that savings are high enough to finance the current deficit, investment, and the cost of debt, i.e. inequality (36) holds:

(37) \begin{equation} \frac {\beta }{1+\beta }\left ( 1-\tau _{L}\right ) \left ( 1-\alpha \right ) \gt g_{1}+g_{2}-(\tau _{L}(1-\alpha )+\tau _{K}\alpha )+2\frac {\left ( 1-\delta \right ) }{A_{1}}+\alpha \left ( 1-\tau _{K}\right ) \end{equation}

To understand how considering these different conditions create constraints on the policy instruments, let us define $\kappa \gt 0$ small enough as being the primary deficit: $\kappa =g_{1}+g_{2}-\left [ \tau _{L}\left ( 1-\alpha \right ) +\tau _{K}\alpha \right ]$ . Substituting this expression in inequality (37), we obtain:

\begin{equation*} \tau _{L}\lt \frac {\alpha }{1-\alpha }\frac {1+\beta }{\beta }\tau _{K}+1-\frac {1}{1-\alpha }\frac {1+\beta }{\beta }\left ( \kappa +\alpha +2\frac {1-\delta }{A_{1}}\right ) \end{equation*}

We show that these conditions impose interdependencies between the tax rates $\tau _L$ and $\tau _K$ , and the size of the primary deficit $\kappa$ . If the tax rate on labor income is positive, it requires a sufficiently high tax rate on capital income. Moreover, the higher the primary deficit is, the higher the capital tax rate. This positive relationship allows to keep the debt burden low enough such as the debt over capital ratio can be stationnary.

The BGP characterized by the lowest pollution to capital ratio ( $\pi _{I}$ ) has the highest level of debt over capital ( $b_{I}$ ), while the one with the highest pollution-to-capital ratio ( $\pi _{II}$ ) is also defined by the lowest debt per unit of capital ( $b_{II}$ ). We can note that the decreasing relationship we observe between $\pi$ and $b$ is ensured by the sufficient TFP vulnerability ( $\epsilon _a(\pi )\lt -1$ ). This property is specific to our analytical framework and is explained in detail in the following section.

4.2 Balanced growth and the role of TFP sensitivity to pollution

We want to clearly understand the links between pollution to capital, debt to capital, and growth that come from the comparison of the two BGPs.

First, we turn our attention to the growth factor $\gamma$ . From (28), we see that it is a declining function of the debt per capital ratio $b$ , through a priori a usual crowding-out effect of debt on investment. Moreover, as pollution generates negative external effects on production and therefore also on savings, the growth factor also depends negatively on the pollution per capital ratio $\pi$ . In our context of TFP vulnerability ( $\epsilon _a(\pi )\lt -1$ ), a BGP with a higher level of $\pi$ is characterized by a lower debt per capital ratio $b$ . Therefore, at the BGPs, the relationship between $\gamma$ and $\pi$ (or $b$ ) seems ambiguous.

To avoid this ambiguity, we exploit the fact that the growth of capital is equal to the growth of the pollution stock. Then, using (27) and (28), the growth factor can be expressed as a function of $\pi$ :

(38) \begin{equation} \gamma = 1-m+\frac {\mu -\psi g_2 a(\pi )}{\pi } \end{equation}

Hence, we note that the growth factor is higher than 1, i.e. growth is positive, as soon as $m$ is low enough. The growth factor increases with the pollution flow but decreases with the current pollution stock. Indeed, since mitigation decreases with pollution through its effect on TFP, $\pi$ has two opposite effects on growth, a positive one through pollution flows and a negative one through the pollution stock.

Therefore, growth is a decreasing function of pollution over capital ratio (and hence an increasing function of $b$ ) if and only if the elasticity of TFP with respect to $\pi$ satisfies:

(39) \begin{equation} -\epsilon _a(\pi )\lt \frac {\mu -\psi g_2 a(\pi )}{\psi g_2 a(\pi )} \end{equation}

which may be in accordance with Assumptions2 and 3. We thus deduce that:

In case 1, the growth rate decreases with $\pi$ , so it is lower in the state with low debt and high pollution ( $\pi _{II}$ , $b_{II}$ ). This configuration is characterized by a not excessive TFP vulnerability to pollution. When the production is not too sensitive to pollution through the TFP, a higher level of $\pi$ means a lower pollution flow over pollution stock, which implies lower growth.

In case 2, the high TFP vulnerability explains that growth is higher in the state ( $\pi _{II}$ , $b_{II}$ ). Indeed, a higher level of pollution over capital implies a strong increase in the pollution flow because of the decrease in public mitigation. Then, the pollution flow over the pollution stock increases, which implies higher growth.

This result is important as it reveals that as long as we consider a not excessive TFP vulnerability to pollution (case 1 of Corollary1), all other things being equal, a BGP characterized by a lower pollution level per unit of capital is associated with higher capital growth. Recall that this BGP also has a higher level of debt over capital. A direct implication of Corollary1 is that when TFP vulnerability is not excessive, a BGP with higher growth means a BGP with higher debt over capital. In contrast, when TFP vulnerability is very high, a BGP with lower growth means a BGP with higher debt over capital. Therefore, there is a crowding-in effect of debt on growth in the first case and a crowding-out effect in the second one.

Using (26) and (28), the intertemporal budget constraint evaluated at a BGP can be written:

(40) \begin{equation} b [\gamma -R(\pi )]=a(\pi )[g-\tau _L (1-\alpha )-\tau _K \alpha ] \end{equation}

When pollution over capital is low, the TFP and, therefore, the interest factor are high. This means that both the primary deficit and debt services are high. Debt over capital is high, even if growth is higher than at the steady state with a higher ratio of pollution over capital (case 1 of Corollary1). This explains that higher debt can be compatible with higher growth. This is an interesting result regarding the macroeconomic literature that mainly finds that public debt usually has a crowding-out effect on growth (see the seminal contribution by Diamond, Reference Diamond1965), except in the presence of some financial imperfections (see Woodford, Reference Woodford1990). Using (40), if the TFP is constant and, therefore, the interest factor too, we immediately deduce that public debt over capital and growth are inversely related. Public debt always has a crowding-out effect on growth.

Now, we want to understand precisely why a high TFP vulnerability to pollution is the source of a negative link between $\pi$ and $b$ when we compare both BGPs. We start by examining the extreme case in which TFP is not sensitive to pollution damage (i.e. $\epsilon _a(\pi )=0$ ). It implies that $a(\pi )=a$ and $R(\pi )=R$ are constant. Equations (30) and (31) thus become:

(41) \begin{align} B_1(\pi ) &=\frac {a[g -(\tau _L (1-\alpha )+\tau _K \alpha )] }{\delta -m -(1-\tau _K)\alpha a +\frac {\mu - \psi g_2 a}{\pi } } \end{align}

(42) \begin{align} B_2(\pi )&=\frac {a}{R} \left [ \Sigma (1-\tau _L)(1-\alpha ) +\tau _L (1-\alpha )+\tau _K \alpha -g - \frac {1-m}{a}- \frac {1}{\pi }\left (\frac {\mu }{a}-\psi g_2\right )\right ] \\[8pt]\nonumber\end{align}

with $B_1^{\prime }(\pi )\gt 0$ and $B_2^{\prime }(\pi )\gt 0$ . In that case, if there still exist several steady states, the one with the highest level of pollution over capital will be also characterized by the highest level of debt over capital. Comparing equations (41) and (42) with equations (30) and (31) provides insights into the differences that arise.

Equation (41) represents debt over capital as the ratio of the primary deficit over the new debt emission, which increases with growth, minus the cost of debt services measured by the interest factor. As previously mentioned, the growth of capital is equal to the growth of pollution at a BGP, which explains that it is decreasing in the pollution stock over capital and hence that there is a positive relationship between $\pi$ and $b$ . When productivity is negatively affected by pollution over capital, several adding effects may imply a reversal of this link. These effects can be perceived by using equation (30). Production being affected negatively by $\pi$ , a higher $\pi$ implies a lower primary deficit $a(\pi )[g-\tau _L(1-\alpha )-\tau _K \alpha ]$ . In addition, the cost of debt reimbursement goes down with productivity loss. These adding effects, due to endogenous productivity, are important when vulnerability to pollution is high enough. In that case, we have a negative relationship between debt over capital and pollution over capital. Equation (42) comes from the equilibrium on the asset market taking into account that capital growth is equal to pollution growth and considering the government budget constraint. Debt per unit of capital is equal to the difference, discounted by the interest factor, between savings and the sum of the primary deficit and the increase in capital (which is here equal to pollution growth). When productivity is constant, the only effect of $\pi$ on $b$ is positive and is due to its negative effect on pollution growth, as previously mentioned. When the TFP decreases with pollution over capital, several adding effects may overturn the link between debt over capital and pollution over capital. These effects can be perceived by using equation (31). First, a lower TFP decreases income discounted by the interest factor. Second, growth per unit of TFP increases with $\pi$ , which has a crowding-out effet on debt.

To summarize, TFP vulnerability to pollution implies that higher pollution to capital reduces product so that the debt to capital ratio reduces too because less deficit has to be financed and less saving is available to buy public debt. Turning to the analysis of the dynamics is now essential to determine toward which equilibria the economy will converge.

5.1 Stability and sustainability

The dynamics are driven by equations (24) and (25). They can alternatively be driven by a combination of (24) and (25), and equation (25).

Let $\Phi _t\equiv \frac {b_t}{\pi _t}=\frac {B_t}{P_t}$ . Using (24) and (25), we obtain:

(43) \begin{equation} \Phi _{t+1} = \frac {\frac {R(\pi _t)}{a(\pi _t)} \Phi _t+\frac {g -\tau _L (1-\alpha )-\tau _K \alpha }{\pi _t}}{X(\pi _t)} \end{equation}

Then, $\Phi _{t+1}\geqslant \Phi _t$ is equivalent to:

(44) \begin{equation} \Phi _t \leqslant \frac {g -\tau _L (1-\alpha )-\tau _K \alpha }{\pi _t [X(\pi _t)- \frac {1-\delta }{a(\pi _t)}-(1-\tau _K)\alpha ]} \equiv B_3(\pi _t) \end{equation}

with $B_3(\pi _t)=B_1(\pi _t)/\pi _t$ , $B_3(\pi _1)=\infty$ , $B_3(\pi _2)\gt 0$ and $B_3^{\prime }(\pi _t)\lt 0$ .

Using (25) and (31), $\pi _{t+1}\geqslant \pi _t$ rewrites $b_t\geqslant B_2 (\pi _t)$ . This is equivalent to $\Phi _t\geqslant B_2 (\pi _t)/\pi _t \equiv B_4 (\pi _t)$ , given by:

(45) \begin{eqnarray} B_4(\pi _t) &=&\frac {a(\pi _t)}{R(\pi _t)\pi _t} \left [ \Sigma (\pi _t)(1-\tau _L)(1-\alpha ) +\tau _L (1-\alpha )+\tau _K \alpha -g - X(\pi _t)\right ] \end{eqnarray}

with $B_4^{\prime }(\pi _t)\lt 0$ , $B_4 (\pi _1)\gt 0$ and $B_4 (\pi _2)=0$ .

We can draw a phase diagram using these different ingredients and the results of the previous section. The stationary values of debt over pollution are $\Phi _I= \frac {b_I}{\pi _I}$ and $\Phi _{II}= \frac {b_{II}}{\pi _{II}}$ . Since $B_3(\pi _t)$ and $B_4 (\pi _t)$ are both decreasing and $\pi _I\lt \pi _{II}$ , we deduce that $\Phi _I\gt \Phi _{II}$ . We further note that $B_3^{\prime }(\pi _I)\lt B_4^{\prime }(\pi _I)$ , while $B_3^{\prime }(\pi _{II})\gt B_4^{\prime }(\pi _{II})$ .

A qualitative picture of the dynamics is represented in Figure 2. We conjecture that the steady state $(\pi _I, \Phi _I)$ is stable, whereas the steady state $(\pi _{II}, \Phi _{II})$ is a saddle. Since the two dynamic variables $\pi _t$ and $\Phi _t$ are predetermined, a saddle is generically unstable. We will now confirm this conjecture by the analysis of local dynamics.

Figure 2. Dynamics with sustainability.

This proposition shows that the economy never converges to the steady state with a high level of pollution over capital. As pollution, debt, and capital are all predetermined, the saddle ( $\pi _{II}, b_{II}$ ) is never achieved and therefore delimits a pollution trap, as we will discuss later. Indeed, when initial conditions are characterized by too high levels of pollution and debt with respect to capital, the economy cannot converge to a long-run BGP. Both $b_t$ and $\pi _t$ increase across time. The too high level of pollution over capital implies a too low TFP and GDP growth to be compatible with the convergence to a stable and sustainable BGP. When pollution over capital is not too high, the economy would converge to the stable BGP characterized by the low level of pollution over capital and the highest level of debt over capital or pollution $(\pi _I, b_I)$ , as represented in Figure 2. Such a dynamic path could experience oscillations converging to the steady state.Footnote ¹¹ Depending on the level of pollution over capital, the convergence to this steady state does not require a so low debt relative to capital. The stability of this BGP requires a not too strong TFP vulnerability to pollution and a high tax rate on labor income. If the first condition is not fulfilled, any increase in pollution over capital implies a strong decrease in GDP and, therefore, investment in future capital due to the high TFP vulnerability to pollution. Future pollution over capital increases even more, generating an explosive dynamic path (see for instance equation (25)). Under the last condition, savings are low enough to prevent an unsustainable buildup of debt and capital.

5.2 Endogenous tipping zone (ETZ)

Since the steady state $(\pi _{II}, b_{II}, \Phi _{II})$ is a saddle, it has one stable and one unstable manifold. By inspection of Figure 2, we observe that the stable manifold of this steady state delineates a zone such that when the economy is on the right side of this manifold, pollution will be explosive. We call this zone an Endogenous Tipping Zone (ETZ), because it is endogenously defined by the dynamic behavior of the economy. In the following proposition, we show that:

Figure 3. Endogenous tipping zone (ETZ).

The stable manifold $(SM)$ of the steady state $(\pi _{II}, b_{II}, \Phi _{II})$ is clearly represented in Figure 3. Since the two dynamic variables are predetermined, the economy is generically on the left or the right side of this decreasing curve. If the initial conditions are such that the economy is on the left side of $(SM)$ , the dynamics could be characterized by convergence to the stable steady state with an increase in the long run of debt over pollution and capital and a decrease in pollution per unit of capital.

If the initial conditions are such that we are on the right side of $(SM)$ , pollution per unit of capital will increase a priori indefinitely. Since the unstable manifold is negatively sloped, pollution will also increase with respect to debt. The economy is in the ETZ if the stock of pollution is sufficiently high with respect to capital. Interestingly, at least around the BGP, the ETZ is delimited by a negatively slopped relationship between $\Phi _t$ and $\pi _t$ . This means that the economy will experience explosive paths for lower pollution levels over capital when the debt relative to pollution is higher. When an economy is already burdened with high debt levels, it may struggle to allocate additional resources toward pollution reduction efforts and adaptation. A vicious cycle is triggered, where a high level of debt does not translate into significant spending on pollution control. Instead, it illustrates an inability to adapt and mitigate adequately, thereby increasing damages and making it more difficult to stabilize debt and pollution per unit of capital. In contrast, the lower the debt over pollution, the higher the level of pollution over capital to have an explosive path. This means that lower public debt and higher capital allow to reach a sustained growing economy more easily.

Considering the damage of pollution stock on production, we provide a theoretical mechanism explaining why, for a given pollution-to-capital ratio, the likelihood of observing explosive paths for pollution is higher in countries with higher debt. Such a detrimental situation is pointed out in the literature, but as we can remark in Zenios (Reference Zenios2024), the papers that attempt to formalize it focus mainly on the impact of climate risks on public spending; he highlights that these risks increase the cost of public debt, making public finances even more vulnerable. We obtain a similar conclusion of debt vulnerability considering a purely deterministic context.

The importance of our result is all the greater as public debt-to-GDP ratios have risen for decades and have reached record levels in a significant number of both developed and developing countries(see Figure 1 and WorldBank, 2023). This trend, combined with the heightened vulnerability to environmental issues, emphasizes the need to propose adapted policy tools to avoid a vicious circle of pollution over debt.

5.3 Unsustainability

The system can also be completely unsustainable. This happens if no steady state is stable, i.e. if the equilibrium $(\pi _{I}, b_{I}, \Phi _{I})$ is unstable. In such a situation, pollution and debt over capital will follow an explosive dynamic paths. As we will see, it can occur if the TFP vulnerability to pollution is high and the tax rates are low. The next proposition provides sufficient conditions for such an undesirable configuration:

Figure 4. Dynamics with unsustainability.

This proposition gives sufficient conditions to have all steady states saddle or unstable. It particularly requires sufficiently low tax rates on capital and labor incomes and a strong TFP vulnerability to pollution. Of course, the BGPs are affected by the level of the tax rates, which means that the stationary values of $\pi _i$ and $b_i$ are not similar in Proposition4 and in Proposition2. This explains that we do not conduct an analysis of bifurcations, but in both configurations, two BGPs coexist. When the tax rates are low, the primary deficit is important, as is the cost of debt. Both effects directly deteriorate public finance. The positive impact of low tax rates on aggregate savings is not sufficient to compensate. Low taxation creates conditions promoting unsustainable debt levels and hence hinders the government’s ability to tackle pollution issues. Moreover, when TFP is highly vulnerable to pollution damage, the economy can never converge to a long-run BGP. As we have already seen, a small increase in pollution over capital implies a strong decrease in production and capital, which implies a larger future increase in pollution over capital. Either pollution and debt will go to infinity, characterizing a vicious circle of debt and pollution or the economy will collapse. There may exist a dynamic trajectory that diverges from the BGP with low pollution over capital to converge to the BGP with high pollution over capital. However, since the variables are predetermined, the economy will not generically experience such a dynamic path.

To summarize this section, high levels of tax rates are key ingredients to rule out an explosive accumulation of pollution and debt. However, the economy may be unsustainable for technological reasons, i.e. a high TFP vulnerability to pollution. Finally, high initial levels of debt and pollution with respect to capital promote instability of the dynamic path. Note that a high initial debt over capital is not a priori a source for unsustainability since it may reinforce the possibility of converging to the stable steady state with low pollution over capital. It will depend on the level of pollution over capital.