3.1 Relevance of Climate-related Policies

Empirical evidence highlights the increasing relevance of climate-related policies among both fiscal and monetary authorities. Governments have progressively integrated climate change mitigation into their agendas, employing a range of instruments such as Pigouvian taxes (Baiardi and Menegatti, Reference Baiardi and Menegatti2011) and public investments in abatement technologies (Porter and van der Linde, Reference Porter and van der Linde1995; Brock and Taylor, Reference Brock and Taylor2010). As shown in Figure 1a, the number of environmental policies implemented across OECD countries from 1990 to 2022 exhibits a clear upward trend. Existing literature emphasizes that these instruments typically take the form of direct regulation (e.g., setting minimum standards or prohibiting inputs) and the taxation of polluters. In parallel, financial markets and monetary authorities have increasingly acknowledged climate-related risks. Survey evidence shows that while regulatory risk is perceived as the dominant concern in the near term, physical climate risks are expected to become more important in the long run, with asset prices widely believed to underestimate these risks (Stroebel and Wurgler, Reference Stroebel and Wurgler2021). Central banks, in particular, are devoting greater attention to the potential threats climate change poses to financial stability. This growing emphasis is also reflected in their public communication, as shown by Campiglio et al. (Reference Campiglio, Deyris, Romelli and Scalisi2025), who document a significant rise in climate-related discourse across central banks over recent decades. Figure 1b illustrates this shift: while Federal Reserve officials have integrated climate-related topics only moderately, nearly all recent ECB speeches include explicit references to climate change, indicating a stronger commitment to embedding environmental considerations into European monetary discourse.

Figure 1.

Fiscal and monetary policy makers and climate change.

Notes: Panel (a) displays the annual aggregate number of climate mitigation policies adopted across various sectors or policy areas within OECD countries from 1990 to 2022. The policies are categorized into market- and non-market-based instruments, covering the electricity, building, transport, and industry sectors, as well as cross-sectoral and international policies (Source: OECD Climate Action Dashboard). Panel (b) illustrates the proportion of speeches by officials from the Federal Reserve (light gray) and ECB (dark gray) that addressed green topics, spanning from 1999 to May 2023 (Source: Speech database of Fed and ECB).

3.2 Sectoral Heterogeneity and the Polluting Brown Sector

Our model structure embeds the environmental externality into an OLG framework with endogenous credit cycles, in which sectoral heterogeneity is a core element. In our framework, the labor-intensive sector is non-polluting, whereas only the capital-intensive sector generates emissions. This distinction between a green and a brown sector is motivated by three considerations: capital intensity, financing frictions, and the environmental footprint of final consumption.

First, empirical evidence suggests a positive association between capital intensity and environmental pollution. Studies such as Antweiler et al. (Reference Antweiler, Copeland and Taylor2001), Cole and Elliott (Reference Cole and Elliott2003), Cole et al. (Reference Cole, Elliott and Shimamoto2005), Dietzenbacher and Yan (Reference Dietzenbacher and Yan2024) find that sectors with higher physical capital requirements, such as heavy industry and fossil fuel extraction, tend to generate disproportionately high emissions. In line with this evidence, we model the brown sector as highly capital-intensive, relying exclusively on capital in its production technology, while the green sector combines capital and labor. Cole and Elliott (Reference Cole and Elliott2005) also highlight the importance of high capital expenditures in relation to pollution. In a broader study that includes additional explanatory variables, Agnolucci and Arvanitopoulos (Reference Agnolucci and Arvanitopoulos2019) confirm the role of high capital expenditures among other contributing factors, although they cannot corroborate the role of physical capital input.

Second, in our model one sector requires external finance through collateralized credit. Above, we argued that monetary authorities and financial markets increasingly account for climate-related risks and polluting (brown) sectors typically face greater difficulties in accessing finance. Consistent with this stylized fact, we assume that projects relying on collateral-backed credit are those in the brown, polluting sector, whereas green projects are not subject to such financing frictions. Recent empirical evidence supports this assumption, showing an increasing divergence in financing conditions: sustainable investing has significantly lowered the cost of capital for green projects while raising relative financing costs for brown firms (Gormsen et al., Reference Gormsen, Huber and Oh2024; Chen and Takahashi, Reference Chen and Takahashi2024).

In our model we interpret these financing frictions as potential instruments for green financial regulation. One prominent example is the adjustment of collateral frameworks by central banks, such as the ECB’s planned 2024 reform that explicitly restricts collateral eligibility for assets issued by firms with high carbon footprints (Beckmann et al., Reference Beckmann, Gern, Jannsen and Sonnenberg2023).Footnote ³ Another example is Giovanardi et al. (Reference Giovanardi, Kaldorf, Radke and Wicknig2023), who analyze the use of “green haircuts” as financial policy tools (and find them beneficial, albeit potentially less effective than direct carbon taxation). Unlike their setup, which treats pollution at an aggregate level, our model explicitly differentiates between sectors. For financing projects in the brown sector, investors must pledge their expected returns as collateral, with financial constraints applying specifically to this sector. This is similar to Raberto et al. (Reference Raberto, Ozel, Ponta, Teglio and Cincotti2019), who examine how differentiated capital requirements can redirect investments toward greener sectors, aligning closely with our focus on sector-specific financial regulation.

Third and finally, the two sectors differ fundamentally in how their outputs are used. Output from the green sector serves as an investment good, funding both green and brown projects, as well as a consumption good for older agents who previously invested in green projects. In contrast, output from the brown sector is exclusively used for final consumption by older agents who have either invested directly in brown projects or provided credit to them. This asymmetry is particularly relevant from an environmental standpoint, as empirical studies consistently demonstrate that final consumption activities generate a significantly larger ecological footprint than investment activities, especially when consumption involves energy-intensive goods and services (e.g., Wiedmann et al., Reference Wiedmann, Minx, Barrett and Wackernagel2006; Ivanova et al., Reference Ivanova, Stadler, Steen-Olsen, Wood, Vita, Tukker and Hertwich2016). Thus, our model structure reflects the stylized fact that the brown sector produces exclusively for consumption and thus contributes disproportionately to environmental degradation.

Taken together, these arguments justify modeling the capital-intensive and consumption-oriented sector that is subject to financial frictions as the polluting sector. We acknowledge that alternative approaches exist for modeling sectoral heterogeneity and pollution profiles, but our framework deliberately focuses on the interaction of financial market frictions with fiscal and financial regulation. This focus allows us to analyze how credit dynamics and regulatory instruments jointly shape both macroeconomic stability and environmental outcomes.

4.1 Production and investment choices

The model adopts an overlapping generations framework (Diamond, Reference Diamond1965) with two period lives where a unit measure of agents arrives at each point of time $t=0,1,2,\ldots$ .Footnote ⁴ The model features two sectors, a green and a brown sector, and an environment that is subject to pollution. In each period a final good, the numeraire, is produced in the green sector which can be used for investments or for consumption. The green goods sector uses the following Cobb-Douglas technology

\begin{equation*} Y_{t}=Z E_{t}^{\gamma }K_{t}^{\alpha }L_{t}^{1-\alpha }, \end{equation*}

where $Y_{t}$ represents the final good produced at time $t$ , $Z$ denotes total factor productivity, $E_{t}$ is the environmental resource, and $K_{t}$ and $L_{t}$ represent physical capital and labor, respectively. Here, $\gamma \lt 1$ is the elasticity of output with respect to the environmental resource, while $\alpha \lt 1$ represents share of capital in production (i.e., the elasticity of output with respect to production). The environment is a freely usable aggregate resource that positively influences the productivity of the other input factors. It is not privately owned, and agents do not anticipate its aggregate development in their decision-making, nor do they consider the impact of their actions on this resource. Implementing the normalization $Z(1-\alpha )=1$ , competitive factor markets yield the following factor remunerations for capital and labor, respectively, depending on $k_t=\frac {K_{t}}{L_{t}}$ ,

(1) \begin{equation} \rho _{t} = f^{\prime }(k_{t}) = \frac {\alpha }{1-\alpha }E_{t}^{\gamma } K_{t}^{\alpha -1} L_{t}^{1-\alpha } = \frac {\alpha }{1-\alpha } E_{t}^{\gamma } k_{t}^{\alpha -1} \end{equation}

and

(2) \begin{equation} w_{t} = f(k_{t})-k_{t}f^{\prime }(k_{t})= E_{t}^{\gamma } K_{t}^{\alpha } L_{t}^{1-\alpha } = E_{t}^{\gamma } k_{t}^{\alpha }. \end{equation}

Note that $\rho _{t}$ is decreasing and $w_{t}$ is increasing in $k_{t}$ , and the environmental resource $E_{t}$ is not measured in units of labor. The environmental resource enhances the returns to the other two inputs.

Each period, a new measure of agents arrives (is born) and stays active for two periods. In the first period, young agents provide one unit of labor and earn the net worth $w_{t}$ . They save everything and face an investment decision at the end of period one to maximize their consumption at the end of the second period (when they are old). To allocate their net worth, they can choose from three options: investing in one of two projects (green or brown) or supplying credit through competitive, albeit imperfect, capital markets.

Investing in the green sector that uses the final good as capital input for the next period as $k_{t+1}$ , is expected to yield a return equal to the marginal product of capital, $\rho _{t+1},$ under the assumption of perfect foresight. Green projects fuel production processes and have a positive intergenerational spillover: They generate labor demand and therefore income for the next generation. In the present specification and to keep the model tractable, we assume that green projects do not emit pollution and are therefore not subject to the environmental tax.

Our economy also includes a second sector with brown projects as an additional option for allocating net worth. These use only capital inputs and are independently profitable. Compared to green projects, the brown projects need credit to be financed and are thus only available if credit is available. We deliberately assume that brown projects need an indivisible amount of $m$ units of the final good. In the production process, they will be converted into $mB$ units of the final good in $t+1$ with $B\gt 1$ being an exogenously given profitability. Brown projects generate a negative intergenerational externality: they emit pollutants that adversely affect the production process in the subsequent period. Consequently, they are subject to an environmental tax $\theta$ , so that the final return is $mB(1-\theta )$ . Corresponding to the high capital intensity of such projects, they require external funds from the capital market, such that $m\gt w_{t}$ holds. This introduces a third option for allocating net worth. Our model features a financial market where agents may lend their net worth to other agents of the same generation (brown entrepreneurs), with an expected return in equilibrium equal to the marginal product of capital $r_{t+1}=\rho _{t+1}$ .

Confronted with the choice of investment opportunities, agents transitioning from the young to the old generation, assess their optimal choice by comparing the expected returns of the green to the brown sector. Hence, they are only willing to invest in the latter if the respective return is greater or equal to the green sector or the simple lending activity via the competitive capital markets. Thus, the agents’ choice is determined by a profitability assessment resulting in the profitability constraint (PC)

(3) \begin{equation} B(1-\theta ) \geq r^{e}_{t+1}=\rho ^{e} _{t+1}= \frac {\partial f(k_{t+1},E_{t})}{\partial k_{t+1}} = \frac {\alpha }{1-\alpha }E_{t}^{\gamma } k^{\alpha -1}_{t+1}. \end{equation}

Note that the expected return to green projects $\rho ^{e} _{t+1}$ depends on $E_{t}$ , since individual agents’ decisions do not take into account any changes in the aggregate environmental resource. It depends on $k_{t+1}$ since agents have perfect foresight with respect to their own individual capital stock. In the financial markets agents expect to earn by lending as much as by investing in green projects, thus $ r^{e}_{t+1}=\rho ^{e} _{t+1}$ .

4.2 Regulation and the environment

As brown projects also produce pollution and thus damage the environmental resource, they become subject to (exogenous) regulation. Our model tracks two distinct dimensions of regulation, which we deem financial and fiscal regulation. The fiscal regulator may impose a Pigouvian-style tax $0\lt \theta \lt 1$ on the brown projects’ return, $B$ , such that the net return agents receive is equal to $(1-\theta )B$ . This tax will then be used to invest either in an abatement technology or in direct improvements of the environment, which we will discuss below. Both are considered to exert a quality improving effect on the environment.Footnote ⁵ Note that we implicitly assume that the fiscal regulator runs a balanced budget.

The other dimension concerns an exogenous financial regulator who may intervene in the credit market. This intervention restricts agents in their borrowing and lending decisions. The regulatory authority assesses the state of pollution and sets the pledgeability ratio $\mu$ . Put differently, the authority enforces an environmental haircut, $(1-\mu )$ , for financing brown projects. Agents may pledge only a fraction $\mu$ of the expected net project revenue from brown projects, $mB(1-\theta )$ , as collateral. Thus, the regulatory parameter $0\lt \mu \lt 1$ makes financing harder, tightening the collateral requirement. Agents who would like to start a brown project have to borrow $m-w_{t} = m-E_{t}^{\gamma } k_{t}^{\alpha }$ . Thus, they can initiate brown projects if and only if the regulatory constraint (RC) is satisfied:

(4) \begin{equation} \mu mB(1-\theta )\geq r^{e}_{t+1}(m-E_{t}^{\gamma } k_{t}^{\alpha })\qquad \text{or} \qquad \frac {\mu mB(1-\theta )}{m-E_{t}^{\gamma } k_{t}^{\alpha }}\geq r^{e}_{t+1}. \end{equation}

That is, if the expected return (i.e., the collateral) that brown entrepreneurs offer to the lender (left side) is at least as high as the expected return in financial markets (right side). Financial regulations – specifically, the environmental haircut $(1-\mu )$ – thus tighten financing conditions for brown projects and channel more funding into green projects.Footnote ⁶

In general, the RC sets a tighter limit compared to the PC in equation (3), if

(5) \begin{equation} \frac {\mu mB(1-\theta )}{m-E_{t}^{\gamma } k_{\mu }^{\alpha }} \leq B(1-\theta ) \end{equation}

holds. The regulatory and PC are equal if

(6) \begin{equation} k_{\mu } = \left ( (1-\mu )mE_{t}^{-\gamma }\right )^{1/\alpha }. \end{equation}

What a lender can expect from providing finance to a brown project, i.e., $R(k_{t},E_{t})$ , depends on which of the two constraints is tighter:

(7) \begin{equation} R(k_{t},E_{t})\equiv B(1-\theta ) \min \Bigl \{\frac {\mu }{1-E_{t}^{\gamma } k_{t}^{\alpha } /m},1\Bigr \}= \left \{ \begin{array}{ll} \frac {\mu B(1-\theta )}{1-\frac {E_{t}^{\gamma } k_{t}^{\alpha }}{m}} & \quad \text{if } k_{t}\leq k_{\mu } \quad { \text{(if RC is tighter)}} \\ B(1-\theta ) & \quad \text{if } k_{t}\geq k_{\mu } \quad {\text{(if PC is tighter)}. }\end{array} \right . \end{equation}

For modeling the environmental resource, we resort to a model in style of Smulders and Gradus (Reference Smulders and Gradus1996) and Caravaggio and Sodini (Reference Caravaggio and Sodini2018), where the environment is an input for production and economic activity damages this resource. For our specification, the environmental resource $E$ , which is an input for green projects, evolves according to the following dynamic equation

(8) \begin{equation} E_{t+1} =\dfrac {\bar {E}+[\varphi tax_{t}]^{\varepsilon _{I}}}{1+\frac {1}{1+((1-\varphi ) tax)^{-\varepsilon _{A}}}\Omega X_{t}}. \end{equation}

The natural level of the resource is given by $\bar {E}\gt 0$ and $tax _{t} = \theta m B X_t$ denotes the total tax accrued by the number of brown projects $X_t = \frac {k_{t}^{\alpha }E_{t}^{\gamma } - k_{t+1}}{m}$ . Parameter $\varphi$ is the fraction of taxes used for environmental improvement and thus $(1-\varphi )$ is used for abatement, with the respective semi-elasticities $\varepsilon _{I}\gt 0$ and $\varepsilon _{A}\lt 0$ . Finally, $\Omega X_t$ is the total pollution produced by brown projects that weigh on the environmental quality. Note that we do not model any accumulation or decumulation of environmental stocks over time. This assumption is consistent with our focus on short-run business-cycle dynamics and parallels the treatment of physical capital in the OLG framework, which fully depreciates each period. The natural level of the resource, $\bar {E}$ , defines the level of the resource if there are no brown projects at work generating emissions or taxes. Once brown project activity starts, this level decreases by total pollution $\Omega X_t$ . While damaging the resource, taxes accrue in the presence of a $\theta \gt 0$ , which then can be used twofold. First, introducing measures to improve the resource directly, e.g., by reforestation. The respective semi-elasticity $\varepsilon _I$ steers the effectivity of the improvement measures. As a second measure, abatement has the potential to directly diminish the emitted pollutants, as for instance filters. Again, the attached semi-elasticity, $\varepsilon _{A}$ , steers the ”productivity” of this abatement technology. This specification closely follows the argumentation in Brock and Taylor (Reference Brock and Taylor2010). To summarize, our specification features three avenues for regulation, with different general equilibrium and environmental effects. The environmental haircut $(1-\mu )$ and the Pigouvian tax $\theta$ represent financial and fiscal regulation, respectively, and are thus our two main policy parameters. The parameter $\varphi$ distributes the accrued taxes to either improve the environment directly or install abatement devices.

4.3 Short-run equilibrium and its dynamics

We are now in a position to characterize the short-run equilibrium and derive its implied dynamics. In the short-run equilibrium $\rho ^{e}_{t+1}=r^{e}_{t+1}$ must hold with equality, as otherwise agents would never be willing to start brown projects due to the lower profitability compared to green projects. Hence, they will simply not be funded as the rate of return of lending is too low. This implies that

(9) \begin{equation} \rho _{t+1}^{e}=r^{e}_{t+1}=\frac {\partial f(k_{t+1},E_{t})}{ \partial k_{t+1}}=\frac {\alpha }{1-\alpha }k_{t+1}^{\alpha -1}E_{t}^{\gamma }\geq R(k_{t},E_{t}). \end{equation}

Now, if $\rho ^{e}_{t+1}=r^{e}_{t+1}\lt R(k_{t},E_{t})$ would hold with strict inequality, it is always desirable to start brown projects (higher returns). However, due to the return of lending being too low, agents will not be able to obtain credit as nobody would provide the required credit. Hence, this cannot be an equilibrium.

The short-run equilibrium value of $k_t$ determines the level of investment in green projects in the next period, $k_{t+1}$ , and thereby defines the dynamic map. We distinguish three regimes. In the first regime, when capital $k_t$ is low, green projects are more profitable than their brown counterparts; that is, $\rho ^{e}_{t+1} \gt R(k_t, E_t)$ in equation (9). Agents would never run brown projects, and capital will only flow toward green projects. This is due to the PC and the implied law of motion is

(10) \begin{equation} k_{t+1}=\Psi _{L}=E_{t}^{\gamma }k_{t}^{\alpha }. \end{equation}

This mechanism reflects a Solow-type accumulation process. Starting from a low level of investment in green projects, it initially increases, but rising capital reduce its profitability. At some point, green and brown projects become equally profitable, after which agents strictly prefer brown projects due to their higher returns. Now the regulatory constraint becomes binding, as the financial regulator steers the environmental haircut, making financing harder. Thus, with a net worth still too low to provide sufficient collateral, i.e., as borrowers are not able to pledge the full expected return to the lender, agents can still only start green projects and equation (10) continues to apply.

With increasing investment in green projects, the agents net worth increases. If it is sufficiently high (or the financial regulation loose enough), the RC eases. There exists a threshold, $k_{c}$ , where the regulatory constraint (see equations (4) and (9)) is satisfied with equality (and all agents still invest only in green projects). Observing that $R(k_{t},E_{t})=\frac {\mu mB(1-\theta )}{m-E_{t}^{\gamma }k_{t}^{\alpha }}$ and inserting equation (10) into equation (9) allows to determine the respective threshold $k_c$ . That is implicitly given as $k_{c}=\left [\frac {\alpha }{1-\alpha } \frac {1}{\mu B(1-\theta )} \left (1-\frac {k_{c}^{\alpha }E_{t}^{\gamma }}{m}\right ) E_{t}^{\alpha \gamma }\right ]^{\frac {1}{\alpha (1-\alpha )}}$ . In the second regime, the short-run value of investment in green projects is beyond this threshold, i.e., $k_{t}\gt k_{c}$ , brown projects can and will be financed. The credit flow is determined such that the maximum pledgeable rate of return, $R(k_{t},E_{t})$ , equals the expected return on green projects, ensuring that equation (9) continues to hold with equality for $k_{t} \gt k_{c}$ . For $k_{t}\lt k_{\mu }$ , $R(k_{t},E_{t})=\frac {\mu mB(1-\theta )}{m-E_{t}^{\gamma }k_{t}^{\alpha }}$ and the resulting dynamic equation is

(11) \begin{equation} k_{t+1}=\Psi _{M}=\left [\frac {\alpha }{1-\alpha } \frac {1}{\mu B(1-\theta )} \left (1-\frac {k_{t}^{\alpha }E_{t}^{\gamma }}{m}\right )\right ]^{\frac {1}{1-\alpha }} E_{t}^{\frac {\gamma }{1-\alpha }}. \end{equation}

In the third regime, either net worth is sufficiently high or regulation sufficiently loose to overcome the RC. Once capital exceeds the threshold $k_{\mu } = \left ( (1-\mu )mE_t \right )^{1/\alpha }$ , all additional credit flows into brown projects. Hence, investment in green projects is determined by $\rho ^{e}_{t+1}=\frac {\partial f(k_{t+1},\,E_{t})}{ \partial k_{t+1}}=B(1-\theta )$ such that the dynamics follow $\Psi _{R}$ ,

(12) \begin{equation} k_{t+1}=\Psi _{R}=\left [\frac {\alpha }{1-\alpha } \frac {1}{B(1-\theta )}\right ]^{\frac {1}{1-\alpha }} E_{t}^{\frac {\gamma }{1-\alpha }}. \end{equation}

We are now able to specify the full dynamic equation, by putting above derivations together. The three regimes, indicated by $\Psi _{i},\,i\in (L,M,R)$ , are separated by the thresholds $k_{c}$ and $k_{\mu }$ .

(13) \begin{equation} \begin{array}{ll} k_{t+1}= & \left \{ \begin{array}{l@{\quad}ll} \Psi _{L}=k_{t}^{\alpha }E_{t}^{\gamma } & \text{if } & k_{t}\leq k_{c} \\ \Psi _{M}=\left [ \frac {A}{\mu }\left ( 1-\frac {k_{t}^{\alpha }E_{t}^{\gamma }}{m}\right ) \right ] ^{\frac {1}{1-\alpha }}E_{t}^{\frac {\gamma }{1-\alpha }} & \text{if } & k_{c}\leq k_{t}\leq k_{\mu } \\ \Psi _{R}=\left ( \frac {A}{\mu }\right ) ^{\frac {1}{1-\alpha }}E_{t}^{\frac {\gamma }{1-\alpha }} & \text{if } & k_{t}\geq k_{\mu }\end{array}\right . \\ & \\ E_{t+1}= & \ \ \ \Phi =\dfrac {\bar {E}+[\varphi \theta m BX_{t}]^{\varepsilon _{I}}}{1+\frac {1}{1+((1-\varphi )\theta m BX_{t})^{-\varepsilon _{A}}}\Omega X_{t}} \ \ \ \ \ \forall k_{t}\end{array} \end{equation}

where

(14) \begin{equation} X_{t}=\frac {k_{t}^{\alpha }E_{t}^{\gamma }-k_{t+1}}{m} \end{equation}

denotes the number of brown projects and

(15) \begin{equation} A=\frac {\alpha }{1-\alpha }\frac {1}{B(1-\theta )} \end{equation}

collects a parameter cluster for future reference.

The map is thus piecewise-smooth with the two variables $k_{t}$ and $E_{t}$ . The first and second regime are separated by

(16) \begin{equation} k_{c}=\left [ \frac {\alpha }{1-\alpha }\frac {1}{\mu B(1-\theta )}\left ( 1-\frac {k_{c}^{\alpha }E_{t}^{\gamma }}{m}\right ) E_{t}^{\alpha \gamma }\right ] ^{\frac {1}{\alpha (1-\alpha )}} \end{equation}

The threshold between the second and third regime is

(17) \begin{equation} k_{\mu }=[m(1-\mu )E_{t}^{-\gamma }]^{1/\alpha }. \end{equation}

Note that both thresholds depend on $E_t$ and are thus time-dependent. We restrict the twelve parameters as follows: $\varepsilon _{A}\lt 0$ ; $0\lt \alpha ,\gamma ,\mu ,\varphi ,\theta \lt 1$ ; $\Omega ,\bar {E},\varepsilon _{I}\gt 0$ and $m,B\gt 1$ .

4.4 Global dynamics: analytical results

This and the subsequent subsections examine the interaction between parameters and how these interactions influence the system’s dynamics and stability from both analytical and numerical perspectives. The economic interpretation and discussion are reserved for Section 5.

Depending upon the regime, the map has a unique fixed point with distinct characteristics. The following propositions outline the key properties of the fixed points in the left and in the right regime. Due to the analytical complexity, we do not provide explicit results for the fixed point in the middle regime. However, the properties of the fixed points in the left and right regimes have implications for the fixed point in the middle regime.

Proposition 1. In the left regime, the fixed points of $k$ and $E$ is given as

\begin{equation*} \begin{array}{lll} k_{L}^{\ast }=\Psi _{L}\left ( k_{L}^{\ast },E_{L}^{\ast }\right ) & & k_{L}^{\ast }=\left ( E_{L}^{\ast }\right ) ^{\frac {\gamma }{1-\alpha }} \\ & & X_{L}^{\ast }=0\\ E_{L}^{\ast }=\Phi \left ( X_{L}^{\ast }\right ) & & E_{L}^{\ast }=\bar {E} \end{array} \end{equation*}

Property 1: The fixed point $L$ is (locally) stable for $\alpha \lt 1$ , which holds via assumption.

Property 2: The fixed point $L$ collides with $k_{c}$ , given in equation ( 16 ) at

\begin{equation*} \mu ^{BC-Lkc}=A\left ( 1-\frac {\bar {E}^{\frac {\gamma }{1-\alpha }}}{m}\right ). \end{equation*}

Property 3: The fixed point $L$ potentially collides with $k_{\mu }$ as defined in ( 17 ) at

\begin{equation*} \mu ^{BC-Lk\mu }=1-\frac {\bar {E}^{\frac {\gamma }{1-\alpha }}}{m}. \end{equation*}

Note that for $A=\frac {\alpha }{1-\alpha }\frac {1}{\left ( 1-\theta \right ) B}\lt 1$ , $\mu ^{BC-Lkc}\lt \mu ^{BC-Lk\mu }$ . In that range, $\mu ^{BC-Lkc}$ is the relevant threshold. $A=\frac {\alpha }{1-\alpha }\frac {1}{\left ( 1-\theta \right ) B}=1$ defines a critical value

\begin{equation*} \theta ^{crit}=1-\frac {\alpha }{\left ( 1-\alpha \right ) B}. \end{equation*}

Proposition 2. In the right regime, the fixed points of $k$ and $E$ are implicitly given as

\begin{equation*} \begin{aligned} k_{R}^{\ast } &= \Psi _{R}\left ( k_{R}^{\ast }, E_{R}^{\ast } \right ) = A^{\frac {1}{1 - \alpha }} \left ( E_{R}^{\ast } \right )^{\frac {\gamma }{1 - \alpha }} \\ X_{R}^{\ast } &= \frac {1}{m} A^{\frac {\alpha }{1 - \alpha }} (1 - A) \left ( E_{R}^{\ast } \right )^{\frac {\gamma }{1 - \alpha }} \\ E_{R}^{\ast } &= \Phi \left ( X_{R}^{\ast } \right ), \end{aligned} \end{equation*}

where $ A = \frac {\alpha }{1 - \alpha } \frac {1}{(1 - \theta ) B}$ .

Property 1: $ X_{R}^{\ast } \gt 0$ for $ \theta \lt \theta ^{\text{crit}}$ . For $ \theta = \theta ^{\text{crit}}$ , we have $ X_{R}^{\ast } = 0$ , and the $ R$ fixed point collides with the $ L$ fixed point. For $ \theta \gt \theta ^{\text{crit}}$ , the $ R$ fixed point no longer exists.

Property 2: The $ R$ fixed point collides with $ k_{\mu }$ at $ \mu ^{BC-Rk\mu }$ , which is implicitly defined by

\begin{equation*} \left ( m\left ( 1-\mu ^{BC-Rk\mu }\right ) \right ) ^{\frac {1-\alpha }{\gamma }}A^{\frac {-\alpha }{\gamma }}=\frac {\bar {E}+\left ( \varphi \theta mB\left ( 1-\mu ^{BC-Rk\mu }\right ) \left ( 1-A\right ) \right ) ^{\epsilon _{I}}}{1+\frac {\Omega \left ( 1-\mu ^{BC-Rk\mu }\right ) \left ( 1-A\right ) }{1+\left [ \left ( 1-\varphi \right ) \theta mB\left ( 1-\mu ^{BC-Rk\mu }\right ) \left ( 1-A\right ) \right ] ^{-\epsilon _{A}}}}. \end{equation*}

Property 3: The $ R$ fixed point collides with $ k_{c}$ at $ \mu ^{BC-Rkc}$ , which is implicitly defined by

\begin{equation*} \left ( m\left ( 1-\mu ^{BC-Rkc}A^{\alpha -1}\right ) \right ) ^{\frac {1-\alpha }{\gamma }}A^{\frac {-\alpha }{\gamma }}=\frac {\bar {E}+\left ( \varphi \theta mB\left ( 1-\mu ^{BC-Rkc}A^{\alpha -1}\right ) \left ( 1-A\right ) \right ) ^{\epsilon _{I}}}{1+\frac {\Omega \left ( 1-\mu ^{BC-Rkc}A^{\alpha -1}\right ) \left ( 1-A\right ) }{1+\left [ \left ( 1-\varphi \right ) \theta mB\left ( 1-\mu ^{BC-Rkc}A^{\alpha -1}\right ) \left ( 1-A\right ) \right ] ^{-\epsilon _{A}}}}. \end{equation*}

Analogous to Proposition 1, note that $\mu ^{BC-Rk\mu } = \mu ^{BC-Rkc} A^{\alpha - 1}$ . For $A = \frac {\alpha }{(1 - \alpha )(1 - \theta ) B} \lt 1$ , it follows that $\mu ^{BC-Rkc} \lt \mu ^{BC-Rk\mu }$ . Hence, in this range, $\mu ^{BC-Rk\mu }$ constitutes the relevant threshold. Moreover, for $\theta = \theta ^{\text{crit}} = 1 - \frac {\alpha }{(1 - \alpha )B}$ (i.e., $A = 1$ ), all thresholds coincide: $\mu ^{BC-Rkc} = \mu ^{BC-Rk\mu } = \mu ^{BC-Lkc} = \mu ^{BC-Lk\mu } = 1 - \frac {\bar {E}^{\frac {\gamma }{1 - \alpha }}}{m}=\mu ^{crit}$ .

In Propositions 1 and 2, we derived analytic expressions for the border collision lines that partition the parameter plane. The respective fixed point exists for parameter combinations within one partition, while for parameter combinations in the other partition, the corresponding map exits its defining range (see equation (13)), and the fixed point no longer exists. Although we do not have explicit analytic results for the middle regime, these boundaries have implications for it: parameter combinations for which neither the left nor the right regime applies correspond to dynamics governed by the middle regime.

To prepare the ground for the numerical analysis, the following corollary summarizes the properties of the relevant critical lines and critical points.

4.5 Global dynamics: numerical analysis

We proceed with a numerical analysis to illustrate the analytically derived bifurcation lines and to gain further insight into the dynamics. The following parameter are set throughout the manuscript:

(18) \begin{equation} B = 2, m=1.05, \alpha = 0.33, \gamma = 0.4, \Omega = 1, \bar {E} = 0.8, \varepsilon _{I} = 0.5, \varepsilon _{A} = -0.5. \end{equation}

We follow the parameter specification of Matsuyama et al. (Reference Matsuyama, Sushko and Gardini2016) with respect to the exogenously given profitability of brown projects, $B$ , and the minimum investment size, $m$ . These parameters are kept constant throughout our analysis. However, as shown in Corollary 1, variations in their values would affect the results in predictable ways. The capital share is set to $\alpha = 0.33$ – the same value used by Matsuyama et al. (Reference Matsuyama, Sushko and Gardini2016), which is consistent with recent stylized empirical evidence (Karabarbounis, Reference Karabarbounis2024). This parameter also represents the elasticity of output with respect to physical capital. Assuming that the elasticity of output with respect to natural capital (i.e., the environmental resource) lies in a comparable range, we set $\gamma = 0.4$ (following Caravaggio and Sodini, Reference Caravaggio and Sodini2018). Furthermore, we normalize emissions by setting $\Omega = 1$ , implying proportionality between emissions and the number of brown projects. The natural environmental level, $\bar {E}$ , and the semi-elasticities $\varepsilon _{I}$ and $\varepsilon _{A}$ are treated as fixed constants. These parameters act primarily as scaling factors; they are not directly observable but are chosen to ensure that the key dynamic mechanisms of the model are clearly visible. As noted, Corollary 1 also indicates how variations in these parameters would influence the results.

The policy parameters – the pledgeability ratio $\mu$ (thus the environmental haircut $1-\mu$ ), the environmental tax rate $\theta$ , and $\varphi$ , which governs the use of environmental tax revenues – are central to our analysis. We therefore vary them over the entire range. In particular, we consider $0\leq \mu , \theta \leq 0.8$ (the upper bound is set below unity because higher values no longer influence the dynamics). For $\varphi$ , which governs the allocation of environmental tax revenues, we primarily set $\varphi = 0$ , corresponding to full allocation to emissions abatement. In selected scenarios, we also consider $\varphi = 0.5$ , representing an equal split between abatement and environmental quality improvements. Finally, we explore the full range of allocation schemes, $0 \lt \varphi \lt 1$ , to highlight the non-trivial impact of tax allocation on the model’s outcomes.Footnote ⁷

By varying only the environmental policy parameters while keeping all other parameters fixed, we establish a consistent framework that isolates the effects of policy changes and allows us to assess their impact on the system’s dynamics.Footnote ⁸ This approach facilitates a clearer understanding of the interactions of financial and fiscal environmental regulation in a nonlinear dynamic general equilibrium model, where the financial market is subject to endogenous instabilities. Overall, our parameter exploration follows the approach advocated by Cai and Lontzek (Reference Cai and Lontzek2019), who caution against relying on a single calibration in climate-economic models due to inherent uncertainties. They emphasize that while optimal carbon taxation should ideally reflect the social cost of carbon, substantial variation in economic and climatic conditions leads to a wide range of plausible outcomes.

We proceed with a numerical analysis of the interaction between the two policy parameters, $\mu$ and $\theta$ , within the range $ 0 \leq \theta , \mu \leq 0.8$ , under the assumption that all environmental tax revenues are allocated solely to emissions abatement (i.e., $\varphi = 0$ ). Figure 2 presents the results: panel (a) shows a two-dimensional bifurcation diagram, while panel (b) displays the bifurcation lines corresponding to Propositions 1 and 2. In the bifurcation diagram, parameter combinations leading to convergence toward a fixed point are marked in red. In contrast, convergence to periodic cycles – specifically period-2, period-3, and period-4 cycles – is indicated in blue, yellow, and green, respectively.

Figure 2.

The left panel (a) presents a two-dimensional bifurcation diagram in the $[\mu , \theta ]$ parameter plane for $\varphi = 0$ , illustrating the system’s stability regions and dynamic behavior. The color coding indicates convergence to different attractors: fixed point (red), period-2 cycle (blue), period-3 cycle (yellow), and period-4 cycle (green), with higher-order cycles denoted by white areas. The labels $\Psi _i$ ( $i \in {L, M, R}$ ) denote the regime of equation (13) in which the corresponding fixed point is located. The right panel (b) displays the bifurcation lines derived in Propositions 1 and 2, showing the boundaries between distinct dynamic regimes.

The bifurcation lines delineate three distinct regions, $L$ , $M$ , and $R$ , each corresponding to one of the three possible regimes where the respective fixed point exists. The middle regime is defined as the parameter region where neither the left nor the right regime applies. We have analytically shown that the left fixed point is stable. Numerical analysis further reveals that the right fixed point is also stable, while the middle fixed point loses stability at lower values of $\mu$ (higher values of $\theta$ ), leading to cyclical dynamics.

We continue with a one-dimensional bifurcation analysis, presented in Figure 3, which displays bifurcation diagrams of the financial regulation parameter $\mu$ on $k_t$ for fixed values of the environmental tax rate $\theta$ . Specifically, the left panel (a) shows the case with $\theta = 0.4$ , and the right panel (b) corresponds to $\theta = 0.7$ , with $\varphi = 0$ in both cases. We use two sets of initial conditions to compute the bifurcation diagrams, shown in blue and red, respectively. The enlargement in panel (b) highlights the coexistence of a period-3 with a chaotic interval, respectively. These diagrams illustrate instances of multistability, where multiple cyclical attractors coexist within the same parameter regime. This coexistence highlights the system’s capacity for complex dynamic behavior, in which different initial conditions or slight parameter variations may lead to convergence toward distinct long-run outcomes, including periodic or potentially chaotic trajectories.

Figure 3.

Bifurcation diagrams of the financial regulation parameter $\mu$ with respect to capital, for $\varphi = 0$ and two different values of the environmental tax rate $\theta$ . Panel (a) displays the case for $\theta = 0.4$ , while panel (b) shows the scenario for $\theta = 0.7$ , including an enlargement of the interval $\mu \in [0.16, 0.17]$ . colors indicate the two sets of initial conditions used for computation.

Finally, Figure 4 is drawn for $ \varphi = 0.5$ . The top left panel (a) shows a 2D bifurcation diagram, while the top right panel (b) depicts the bifurcation lines, highlighting the boundaries of different dynamical regimes. Note the small gray region, which is magnified in the bottom left panel (c). In this region, where $ \mu ^{BC-Rk\mu } \leq \mu \leq \mu ^{BC-Lkc}$ , both fixed points $ L$ and $ R$ coexist, indicating a parameter range with multi-stable behavior. The red/blue arrow refers to the one-dimensional bifurcation diagram in the bottom right panel (d), with $ \theta = 0.72$ fixed, illustrating the coexistence of attractors in this region. Using one set of initial conditions, the bifurcation diagram (shown in blue) exhibits a sharp transition from $k^{*}_{R}$ to $k^{*}_{L}$ as $\mu$ falls below $0.146$ . In contrast, when using a different set of initial conditions, the bifurcation diagram (shown in red) indicates that $k^{*}_{R}$ persists until $\mu$ declines to $0.113$ . For $\mu \lt 0.113$ , the system eventually converges to $k^{*}_{M}$ . Thus, for $ \mu ^{BC-Rk\mu }(\theta = 0.72) = 0.113 \leq \mu \leq 0.146 = \mu ^{BC-Lkc}(\theta = 0.72)$ , which corresponds to the gray region, we observe the coexistence of the fixed points $L$ and $R$ . This coexistence highlights the system’s capacity for complex behaviors, where multiple stable outcomes can arise depending on initial conditions and specific parameter values. Such findings underscore the rich dynamics and sensitivity to parameter changes present in the model, suggesting that policy measures affecting $ \mu$ or $ \theta$ could lead to unintended shifts in stability. This sensitivity implies that small adjustments in policy parameters could move the system across bifurcation thresholds, potentially triggering shifts from one stable state to another or even to cyclical behavior. These phenomena highlight the need for a cautious design of intervention strategies.

Figure 4.

The upper left panel (a) presents a two-dimensional bifurcation diagram in the $[\mu , \theta ]$ parameter plane for $\varphi = 0.5$ , illustrating the system’s stability regions and dynamic behavior. The color coding indicates convergence to different attractors: fixed point (red), period-2 cycle (blue), period-3 cycle (yellow), and period-4 cycle (green), with higher-order cycles denoted by white areas. The labels $\Psi _i$ ( $i \in {L, M, R}$ ) denote the regime of equation (13) in which the corresponding fixed point is located. The upper right panel (b) displays the bifurcation lines derived in Propositions 1 and 2, showing the boundaries between distinct dynamic regimes. The lower left panel (c) provides an enlargement of the region exhibiting coexisting attractors. The red/blue line along $\theta = 0.72$ marks the bifurcation path illustrated in the lower right panel (d), where vertical lines denote the bifurcation values of $\mu$ .

6.1 Income definitions

Our specification allows us to analyze not only the overall income of the economy but also to differentiate income by agent types. Specifically, we distinguish between the labor income of the young generation and three categories of income for the old generation. The latter includes income from green projects, income from brown projects, and income from lending. Since we model the environmental resource as part of total factor productivity, its returns are not tied to a single income group.

We start our analysis at time $t+2$ , where total income comprises both labor and investment income generated by two generations. The young generation, denoted as $L_{t+1}$ , earns labor income $w_{t+1}$ from their work during period $t+1$ , which extends until time $t+2$ . The old generation, represented by $L_{t}$ , requires closer examination. They receive returns on investments made at point $t+1$ . It is important to consider the prevailing regime described in equation (13) when calculating the respective income for both generations.

First, in a situation where only green projects can be started due to tight regulation or low net worth, i.e., in regime $\Psi _{L}$ with $k_{t}\lt k_{c}$ , the income is based on the factor remunerations. Thus the income is determined by

(19) \begin{equation} I_{t+2}|\Psi _{L}=(\underbrace {k_{t+1}^{\alpha }E_{t+1}^{\gamma }}_{\text{wage young}})\cdot L_{t+1}+(\underbrace {\frac {\alpha }{1-\alpha }k_{t+1}^{\alpha -1}E_{t+1}^{\gamma }w_{t}}_{\text{return old (Green)}})\cdot L_{t} \end{equation}

As net worth increases (or regulation becomes looser), brown projects will be initiated, indicating that we are above the threshold $k_{c}$ and thus in regimes $\Psi _{M}$ or $\Psi _{R}$ . Total income now includes the labor remuneration of young agents as well as investment returns from both green and brown projects, in addition to income from lending. To allocate this income to the respective groups within the old generation $L_{t}$ , we must determine the generational shares of green investors, $L_{t}^{G}$ , brown investors, $L_{t}^{B}$ , and lenders, $L_{t}^{L}$ . In equilibrium, aggregate credit supply must equal aggregate credit demand, which can be expressed as $w_{t}L_{t} = k_{t+1}L_{t+1} + mL_{t}^{B}$ . Here, $L_{t}^{B}$ represents the number of agents investing in brown projects and thus coincides with the number of these projects. By definition, we have $L_{t} = L_{t+1} = 1$ , which allows us to determine $L_{t}^{B}$ .

The return on brown projects, after accounting for taxes, is expressed as $mB(1-\theta ) - (m - w_{t})r_{t+1}$ , where lenders receive interest payments from brown investors. Consequently, green investors earn a profit rate on their invested net worth from the previous period, while brown investors accrue returns and must also cover their credit repayments. The total credit payment is given by $(m - w_{t})r_{t+1}L_{t}^{B}$ , with each lender receiving $(m - w_{t})r_{t+1} \frac {L_{t}^{B}}{L_{t}^{L}}$ from the brown investors. The return for green projects has been already defined above.

We define the lenders, $L_{t}^{L}$ , as the residual agents who do not engage in either green or brown projects. From this definition, we can derive the following income fractions and schedules:

(20) \begin{align} L_{t}^{G}=\frac {k_{t+1}L_{t+1}}{L_{t}w_{t}} &\quad\text{each with income}\quad R_{t+1}^{G}=r_{t+1}w_{t}\end{align}

(21) \begin{align} L_{t}^{B}=\frac {w_{t}L_{t}-k_{t+1}L_{t+1}}{m} &\quad\text{each with income}\quad R_{t+1}^{B}=mB(1-\theta )-(m-w_{t})r_{t+1}\end{align}

(22) \begin{align} L_{t}^{L}=L_{t}-L_{t}^{G}-L_{t}^{B} &\quad\text{each with income}\quad R_{t+1}^{L}=(m-w_{t})r_{t+1}\frac {L_{t}^{B}}{L_{t}^{L}}.\\[13pt] \nonumber\end{align}

With these derivations, the aggregated income equations for the the middle and right regimes reads

(23) \begin{align} I_{t+2}|\Psi _{M},\Psi _{R}=(\underbrace {k_{t+1}^{\alpha }E_{t+1}^{\gamma }}_{\text{wage young}})\cdot L_{t+1}+\underbrace {R_{t+1}^{G}\cdot L_{t}^{G}+R_{t+1}^{B}\cdot L_{t}^{B}+R_{t+1}^{L}\cdot L_{t}^{L}}_{\text{income old}}, \end{align}

which completes the income definition. Note that the taxes accrued in period $t$ are given by $mB\theta \cdot L_{t}^{B}$ . Importantly, only the old generation pays taxes; thus, in the following analysis, we will focus on their net income.

6.2 Results

Figure 9 illustrates the one-dimensional bifurcation diagrams concerning the parameter $\mu$ . The parameters are set according to equation (18) with $\varphi =0$ . The diagrams are segmented by the values of $\theta$ , showcasing two distinct settings: the upper left and lower left panels correspond to $\theta = 0.1$ , while the upper right and lower right panels represent the scenario where $\theta = 0.4$ . Each panel delineates various income concepts, allowing for a comprehensive analysis of how changes in the parameter $\mu$ influence income dynamics within the model. The bifurcation structure provides insights into the stability of income distributions and highlights potential transitions between different economic states under varying regulatory conditions.

Figure 9.

Bifurcation diagrams of the financial regulation parameter $\mu$ (horizontal axis) on average total income and average environmental quality (upper panels) and the generational incomes as well as the tax revenue (lower panels). Two distinct parameter settings (with $\varphi = 0$ ) are presented. In the left panels (a) $\theta = 0.1$ , while $\theta = 0.4$ in the right panels (b).

Starting with $\theta = 0.4$ , the top right panel of the figure illustrates that tightening financial conditions, achieved by reducing $\mu$ , leads to an increase in average environmental quality, as previously discussed. However, the relationship between environmental quality and average income is non-monotonic. Initially, as financial restrictions take effect and the system transitions into the $M$ regime, income begins to rise. This increase continues until it reaches a peak, after which income declines. Notably, in the $L$ regime, characterized by the absence of brown projects, income falls to a level that is lower than that observed in the $R$ regime. This dynamic highlights a significant trade-off between environmental quality and income generation, suggesting the emergence of a potential policy dilemma. Policymakers face the challenge of promoting environmental improvements without negatively impacting income levels, highlighting the careful balance required in designing effective economic and environmental policies.

The bottom right panel illustrates the impact on income distribution. First, it is important to note that the fluctuations caused by financial market regulations are less pronounced for the income of the young generation, which derives its earnings solely from working in green projects that are not directly influenced by these regulations. Note that stricter regulations dampen the fluctuations. Second, tightening financial market regulations, by reducing $\mu$ , leads to a reversal in income distribution: at high values of $\mu$ , the net income of the old generation surpasses that of the young, while at lower values of $\mu$ , this relationship is reversed. The old generation loses income, while the young generation gains. These findings suggest that while stricter financial regulations may enhance environmental quality, they could inadvertently flip income disparities between generations. Policymakers should consider support measures or transfers for less well-off generations to mitigate these disparities.

We now focus on the scenario where $\theta = 0.1$ for comparison. The panels on the left reveal similar patterns to those seen in the previous analysis; however, a notable difference emerges: in this case, reducing $\mu$ leads to an increase in income within the $L$ regime, as illustrated in the top left panel. In this context, coordinated policy efforts could enhance the overall stability of the system, suggesting that aligning financial and environmental policies may yield beneficial outcomes for income stability and environmental quality.

As the impact of financial regulation is tied to the tax policy, a more in-depth examination of this relationship is warranted. In Table 1, we present an analysis of simulated values for key economic and environmental variables, highlighting the effects of various parameter settings, specifically for $\mu = 0$ , $\mu = 0.8$ , and two scenarios for $\theta$ at $0.1$ and $0.4$ . In the case where $\mu = 0$ , the system operates within the $L$ -regime, where financial decisions remain unaffected by tax policies. However, as $\mu$ is increased, which reflects a loosening of financial restrictions, the system transitions through the $M$ -regime and ultimately reaches the $R$ -regime, where financial constraints become negligible. This shift leads to an increase in the number of brown projects. Notably, in a higher tax environment, coupled with more stringent abatement measures, the deterioration of environmental quality is mitigated compared to the low tax regime. This enhanced environmental quality not only supports higher wage income and overall wealth but also serves as a foundation for investments by the older generation.

Table 1.

Simulated values for key variables across different parameter settings (with $\varphi =0)$ . The values are presented for $\mu = 0$ , indicating a scenario where no brown projects can be started, and for $\mu = 0.8$ , with two different values of $\theta$ : $0.1$ and $0.4$ . Each column corresponds to the impact of these parameters on capital per worker, environmental quality, the number of brown projects, total income, and the distribution of income among different generations

Consequently, this increased wealth enables a greater capital stock per worker, as well as a rise in the number of brown projects, even if the returns are lower due to taxation compared to the low tax regime. As a result, total income experiences an increase. In this high tax scenario, the income for the young generation rises, while the net income for the older generation declines, with green projects playing a more significant role in augmenting the income of the young. Additionally, it is important to note that income derived from lending significantly decreases; this reduction occurs because the heightened capital per worker results in lower profit rates, thereby diminishing the returns from lending activities.