Climate change and damages

Before we describe the macroeconomic environment, we present our assumptions on the climate system, on pollution, and how it feeds back in the economy by destroying stocks of available capital. Polluting non-renewable resources are used as inputs in production. Let S _t denote the stock of non-renewable resources available in time t and R _t the resource extraction. Extracting and burning fossil fuels in time t depletes the resource stock and simultaneously adds to the existing stock of pollution P _t according to:

(1)$$\begin{equation} \dot{S}_{t}=-R_{t},\quad{\rm given}\quad S_{0}>0, \end{equation}$$

and

(2)$$\begin{equation} \dot{P}_{t}=\phi R_{t},\quad{\rm given}\quad P_{0}>0, \end{equation}$$

with ϕ > 0 the pollution intensity of the non-renewable resource. Resource extraction is decreasing over time leading to a decreasing time profile for the resource stock as in figure 1. Combining the above two equations leads to a linear relationship between the stock of pollution and the stock of non-renewable resources; the P(S) line in figure 2:Footnote ¹⁶

(3)$$\begin{equation} P_{t}=P_{0}+\phi(S_{0}-S_{t}). \end{equation}$$

As the stock of non-renewable resources gets depleted, pollution increases. When the whole stock is depleted, pollution gets its maximum value P _max = P ₀ + ϕ S ₀. We use pollution stock as a measure of climate change. The linear relationship of equation (2), between the change in the variable responsible for climate change and greenhouse gas emissions, is well founded in the literature.Footnote ¹⁷ Natural disasters caused by man-made pollution are increasingly harming economic activity by destroying available stocks of capital. In our model, climate change damages are measured as a percentage of the available stock of capital in each period. We will assume that pollution feeds back in the economy through a sigmoid damage function D(P), according to:

(4)$$\begin{equation} D(P_{t})=\delta_{0}+\delta_{1}\left( 1-\frac{1}{1+\delta_{2}(P_{t} -P_{0})^{\eta}}\right), \end{equation}$$

with δ₀ ∈ [0, 1], the natural depreciation of the capital stock, δ₁ ∈ [0, 1 − δ₀], δ₂ > 0, scaling parameters, and η ≥ 1 a convexity parameter. A similar functional form is used in the latest DICE-2016R model but there damages reduce aggregate productivity and not capital stock; see Nordhaus (Reference Nordhaus2017). With this damage function we make sure that damages as a percentage of the stock of capital are bounded between 0 and 1, while at the same time we can calibrate it such that the mapping between pollution and damages is convex for the relevant range of available polluting resources, as typically advocated in the literature (e.g., Golosov et al., Reference Golosov, Hassler, Krusell and Tsyvinski2014).Footnote ¹⁸

Macroeconomic environment

There are two financial assets owned by households: a stock of polluting non-renewable resources S, and capital K. There are also two economic sectors: the manufacturing sector that produces goods readily available for consumption, and the corporate sector that provides goods and services for investments that augment the stock of capital.

In order to produce the consumption good Y, the manufacturing sector combines a part of physical capital K _Y with non-renewable resources R in a Cobb-Douglas fashion:

(5)$$\begin{equation} Y_{t}=AK_{Yt}^{\alpha}R_{t}^{1-\alpha}. \end{equation}$$

Parameters $\alpha \in \lbrack 0,1]$ and A>0 represent the capital expenditure share and the productivity of the manufacturing sector, respectively. According to the Cobb-Douglas specification of (5), we assume that polluting non-renewable resources are essential inputs in the production of the consumer good. In fact, empirical evidence is inconclusive about whether the elasticity of substitution between capital and polluting energy resources is above unity or not. Van der Werf (Reference van der Werf2008) uses industry-level data from 12 OECD countries and consistently finds elasticities of substitution between energy and other inputs which are smaller than unity. Hassler et al. (Reference Hassler, Krusell and Olovsson2015) suggest very low elasticities of substitution between capital and energy in the short run, but high elasticities in the long run, while Papageorgiou et al. (Reference Papageorgiou, Saam and Schulte2017) find elasticities above unity.

Before we draw, however, any pessimistic conclusions about future economic development, note that in our framework there are two ways of substituting towards clean resources and cleaning up the pollution from the dirty resources at finite cost. First, capital K is clean and is assumed to include renewable energy capital such as solar panels, dams and windmills; the availability of renewable energies clearly requires long-run capital investments. Second, we explicitly analyze the policy of cleaning up by looking at Carbon Capture and Sequestration (CCS) as already presented in the graphical part of the paper. With CCS we have directly the possibility to clean up the pollution stock at finite costs.

In turn, the corporate sector, responsible for providing the investment good I, has a linear technology in physical capital K _I:

(6)$$\begin{equation} I_{t}=BK_{It}, \end{equation}$$

with B>0 a productivity parameter. The pollution stock is responsible for climate change and damages to the existing stock of capital through function D(P) defined in (4).Footnote ¹⁹ Capital accumulation reads:

(7)$$\begin{equation} \dot{K}=I_{t}-D(P_{t})K_{t}\quad{\rm given}\quad K_{0}>0. \end{equation}$$

Finally, households receive rents from physical capital and non-renewable resources and balance their income with expenditure on the consumption good C, and on the investment good H, the latter being the equivalent of savings in the present economy. In equilibrium, households demand the total consumption and investment goods, i.e., C = Y and H = I respectively, while capital is exactly shared between the manufacturing and the investment sector, i.e., K _Y + K _I = K; GDP in this economy reads p _YY + p _II, with p _Y, p _I the prices of the consumption and the investment good, respectively. Let us now proceed by describing the market economy and the equilibrium.

Firms

Let the consumption good Y be the numeraire, i.e., p _Y = 1. Firms in both sectors maximize instantaneous profits according to:

(8)$$\begin{equation} \max_{K_{Y},R}\{Y_{t}-p_{Kt}K_{Yt}-p_{Rt}R_{t}\}\quad{\rm and}\quad \max_{K_{I}}\{p_{It}I_{t}-p_{Kt}K_{It}\}, \end{equation}$$

with p _K the rental price of capital, p _R the price of non-renewable resources, and p _I the price of the investment good, i.e., the price of investment in new forms of capital. Let us define with ϵ ≡ K _Y/K the share of total available capital employed by the manufacturing sector. With (5) and (6), this maximization gives the demand curves for non-renewable resources and capital in the two sectors:

(9)$$\begin{equation} (1-\alpha) \frac{Y_{t}}{R_{t}}=p_{Rt}, \quad\alpha\frac{Y_{t}}{\epsilon_{t} K_{t}} = p_{Kt}, \quad p_{It} B = p_{Kt}. \end{equation}$$

The last two equations imply a no-arbitrage condition for the use of capital between sectors: employing the marginal unit of capital in the two sectors should yield the same return.

Households

Households allocate their rental income from physical capital and non-renewable resources between expenditure on consumption C and on additional capital formation H. Let T represent generic non-distorting lump-sum transfers. Then, the income balance reads:

(10)$$\begin{equation} p_{Kt} K_{t} + p_{Rt} R_{t} + T_{t} =C_{t}+ p_{It} H_{t}. \end{equation}$$

Expenditure on capital formation adds to the existing stock of capital according to:

(11)$$\begin{equation} \dot{K}= H_{t} - D(P_{t}) K_{t}, \end{equation}$$

although agents do not internalize damages to capital accumulation through higher levels of pollution. Total wealth reads W = p _I K + p _R S. Time-differentiation of total wealth using (1), (10), (11), and the fact that p _K = p _I B from (9), leads to the household's dynamic budget constraint, according to:

(12)$$\begin{equation} \dot{W}_{t}= \theta_{t} W_{t} \hat{p}_{Rt} +(1-\theta_{t})W_{t} ( \hat{p}_{It}+B- D(P_{t}) ) -C_{t}+T_{t}, \end{equation}$$

with θ ≡ p _R S/W, the share of the individual's resource wealth in the total assets; hats denote growth rates, i.e., $\hat {p}=\dot {p}/p$. Finally, the representative household chooses the time path of consumption C and asset allocation θ in order to maximize lifetime utility:

(13)$$\begin{equation} \int_{0}^{\infty}U(C_{t})e^{-\rho t}dt, \end{equation}$$

subject to the budget constraint (12); ρ > 0 is the intergenerational discount rate. We will assume throughout that households have constant relative risk aversion (CRRA) preferences according to U(C) = C ^1−σ/(1 − σ), with σ > 0 the inverse of the elasticity of intertemporal substitution. Following the empirical literature, we will focus on the case of σ > 1. With r being the aggregate rate of interest, the household optimization yields:Footnote ²⁰

(14)$$\begin{align} \hat{C}_{t} & =\frac{1}{\sigma}(r_{t}-\rho), \end{align}$$

(15)$$\begin{align} \hat{p}_{Rt} & =r_{t}=\hat{p}_{It} +B - D(P_{t}). \end{align}$$

Equation (14) is the usual Keynes-Ramsey condition for consumption growth. Equation (15) is a no-arbitrage condition between assets: accounting for depreciation, each asset should yield the same marginal return in equilibrium. In this closed economy, this return is the risk-free rate of interest r. Note that the first equation of (15) is the Hotelling rule for the price evolution of the non-renewable resource: the appreciation in the resource's marginal profitability – the resource price when no extraction costs are considered – should yield indifference between investing the rents of immediate extraction at a risk-free return r, or extraction next period at a price grown by the same rate. Finally, the above optimization must be augmented by the appropriate transversality condition, which reads:

(16)$$\begin{equation} \lim_{t\rightarrow\infty}\lambda_{t}W_{t}e^{-\rho t}=0, \end{equation}$$

with variable λ = C ^−σ, the shadow price of total wealth.Footnote ²¹

Equilibrium

In equilibrium, total demand for consumption and investment goods should equal their total supply, i.e., C = Y and H = I. Given positive K ₀, S ₀, non-negative P ₀, the dynamics of resource depletion, of pollution, and of capital accumulation (i.e., (1), (2) and (7)), along with the first order conditions for firms and households (i.e., equations (9), (14), (15)) and the transversality condition (16), completely characterize the dynamic behavior of the decentralized economy.

Solving the basic model

Let u ≡ R/S be the resource depletion rate. Log-differentiating equations in (9) using $\hat {R}=\hat {u}-u$ from (1) and the definition of u, and (7) with I = B(1 − ϵ)K, leads to:

(17)$$\begin{align} \hat{u}_{t} & =u_{t}-(\sigma-1)\hat{C}_{t} - \rho, \end{align}$$

(18)$$\begin{align} \hat{\epsilon}_{t} & =B \epsilon_{t} - (\sigma-1)\hat{C}_{t} -\rho.\end{align}$$

Finally, log-differentiating the production function (5) for C = Y in equilibrium, using (7) with I = B(1 − ϵ)K as before, and $\hat {\epsilon }$ and $\hat {u}$ from above, gives the time evolution of the consumption growth rate according to:

(19)$$\begin{equation} \hat{C}_{t}=\underbrace{\frac{\alpha B}{\sigma}}_{\Omega-{\rm productivity} }-\underbrace{\frac{\rho}{\sigma}}_{\Delta-{\rm discounting}}-\underbrace {\frac{\alpha D(P_{t})}{\sigma}}_{\Lambda-{\rm depreciation}}.\end{equation}$$

Expression (19) allows us to study the different effects of productivity, depreciation and discounting on the growth rate on consumption, along the time horizon. The effects of Ω, Λ, and Δ are used throughout the-text to determine the growth rate of consumption through transition and in the steady state, as in figure 5. For any given damage function D(P), the dynamic system of (17) and (18), with $\hat {C}$ from (19), along with the resource and climate dynamics (1), (3), and the transversality condition (16), are sufficient to completely characterize the decentralized economy. As shown in Bretschger and Karydas (Reference Bretschger and Karydas2018), the dynamic system features a saddle-path stability, while it reaches a BGP when polluting resources get depleted, both asymptotically. The steady state values in our economy read:

(20)$$\begin{align} & S_{\infty} = 0, \end{align}$$

(21)$$\begin{align} & P_{\infty} = P_{max}=P_{0}+\phi S_{0}, \end{align}$$

(22)$$\begin{align} & \hat{C}_{\infty} =g_{C}=\frac{1}{\sigma}(\alpha B-\alpha D(P_{\infty })-\rho), \end{align}$$

(23)$$\begin{align} & u_{\infty} = (\sigma-1)g_{C}+\rho, \end{align}$$

(24)$$\begin{align} & \epsilon_{\infty} = (1/B)((\sigma-1)g_{C}+\rho). \end{align}$$

Our model of endogenous growth with nonlinear damages and CRRA utility does not allow for an analytical solution of the transition towards the steady state; we, therefore, rely on numerical simulations. We will solve the model by numerical differentiation using the Runge-Kutta method. Figure A1 of the online appendix shows graphically the outcome of the simulations for our baseline model.Footnote ²²

Carbon taxation

Carbon taxes are the most widely studied policy instrument. This policy in the present macroeconomic context has been extensively analyzed, both for the social optimum and the decentralized equilibrium, in Bretschger and Karydas (Reference Bretschger and Karydas2018), who focus on the lags in the climate system between the flow of emissions and damaging pollution. The following results can be retrieved as the limiting case of no-lags in the decentralized equilibrium of the aforementioned contribution.

Let τ represent given per-unit taxes on emissions ϕ R, with ϕ > 0 the emissions intensity of the non-renewable resource. The first-order conditions for firms in the manufacturing sector read:

(25)$$\begin{equation} (1-\alpha) \frac{Y_{t}}{R_{t}}=p_{Rt}+\phi\tau_{t}, \quad\alpha\frac{Y_{t} }{\epsilon_{t} K_{t}} = p_{Kt}, \quad p_{It} B = p_{Kt}. \end{equation}$$

What changes is only the optimality condition for the employment of the non-renewable resource: its marginal cost is augmented by the effective tax rate ϕτ. Let now ψ ≡ p _R/(p _R + ϕτ) be the share of producer price p _R in the consumer price of the non-renewable resource, p _R + ϕτ. Equation (18) continues to hold while the equivalent of (17) becomes:

(26)$$\begin{equation} \hat{u}_{t}=u_{t}-(\sigma-1)\psi_{t} \hat{C}_{t}-\rho\psi_{t}- (1-\psi_{t}) (\hat{\tau}_{t}-\hat{C}_{t}). \end{equation}$$

The dynamics of the tax are obviously of importance for the results. As is usually the outcome of such models with polluting non-renewable resources, the optimal tax is proportional to consumption when σ = 1 (e.g., Golosov et al., Reference Golosov, Hassler, Krusell and Tsyvinski2014; Grimaud and Rouge, Reference Grimaud and Rouge2014), or it asymptotically becomes so in the long run for σ ≠ 1 (e.g., Golosov et al., Reference Golosov, Hassler, Krusell and Tsyvinski2014; Bretschger and Karydas, Reference Bretschger and Karydas2018). Moreover, it is well established in the literature of the economics of non-renewable resources that any per unit tax that grows at a rate lower than the rate of interest delays resource extraction (e.g., Dasgupta and Heal, Reference Dasgupta and Heal1979).Footnote ²³ In light of the above, we will only study taxes that grow with consumption, i.e., $\hat {\tau }=\hat {C}$. With this conjecture, by log-differentiating the first equation of (25) we get:

(27)$$\begin{equation} \hat{\psi}_{t}=(1-\psi_{t})( (\sigma-1) \hat{C_{t}}+\rho). \end{equation}$$

From (14) and (15), p _R grows at rate r, higher than C, and therefore τ, implying that ψ goes to unity as time goes to infinity. Following the same procedure as before with $\hat {\tau }=\hat {C}$, consumption growth reads:

(28)$$\begin{align} \hat{C}_{t} & =\underbrace{\frac{\alpha B}{1+(\sigma-1)(\alpha+(1-\alpha)\psi _{t})}}_{\Omega-{\rm productivity}}-\underbrace{\frac{(\alpha+(1-\alpha )\psi_{t})\rho}{1+(\sigma-1)(\alpha+(1-\alpha)\psi_{t})}}_{\Delta -{\rm discounting}}\notag \\ & \quad -\underbrace{\frac{\alpha D(P_{t})}{1+(\sigma -1)(\alpha+(1-\alpha)\psi_{t})}}_{\Lambda-{\rm depreciation}}, \end{align}$$

which asymptotically converges to (19) in the steady state for ψ = 1. For a given damage function D(P), and a carbon tax that grows with consumption, the dynamic system of (18), (26) and (27), with $\hat {C}$ from (28), along with the resource and climate dynamics (1), (3) and the transversality condition (16), are sufficient to completely characterize the evolution of decentralized economy.

The steady-state values of all variables (except ψ) are the same as before, i.e., equations (20)–(24). Hence carbon taxes affect the starting point and the transition of control variables (ϵ, u) but not the steady state of the economy. Resource taxation delays extraction and stretches the depletion of the resource stock to the future as can be seen in the left panel of figure A2 of the online appendix. During transition, pollution and damages are therefore always lower than in the baseline case, while consumption growth is always higher. Every variable converges to its long-run equilibrium, which is the same as the baseline case. Finally, due to carbon taxation, the drag of resource extraction on growth is also lower in the beginning, which induces the growth rate of consumption to start from a higher level.Footnote ²⁴

In our setup, the social planner's solution would most closely resemble the above case of resource taxation. Resource extraction would be stretched further in time (i.e., a flatter extraction profile) and so would pollution accumulation. This would also induce a flatter growth profile towards the steady state (i.e., higher growth at all times in comparison to the no-policy case), where resources are being asymptotically depleted and pollution reaches its maximum value.

Decommissioning of the resource stock

A specific feature of models with non-renewable resources (abstracting from increasing extraction costs and backstop technologies) is that the optimal plans of resource owners lead to full exhaustion of the resource stock. As we have shown above, resource taxation simply shifts extraction to the future, without altering the total stock of carbon ultimately emitted into the atmosphere, i.e., P _max remains the same. However, a lower maximal pollution stock would exactly be needed for a climate policy, in line with our long-term targets, i.e., a global warming of 2°C – or even 1.5°C – by the end of the century. It is by now well understood among natural scientists and resource economists that some of the carbon assets must indeed be left in the ground to meet the internationally agreed temperature targets (Meinshausen et al., Reference Meinshausen, Meinshausen, Hare, Raper, Frieler, Knutti, Frame and Allen2009; McGlade and Ekins, Reference McGlade and Ekins2015).

This section examines decommissioning of the existing resource stock S as a policy option. We will construct a simple thought experiment examining the problem from the side of the representative resource owner that faces a given expropriation policy each period with probability 1.Footnote ²⁵ When this policy is effective, it reduces the available stock of non-renewable resources by N ∈ [0, S]. We will further assume that the policy maker chooses the time path of policy N _t which aims at decommissioning in total χ ∈ [0, S ₀] units of polluting resources:

(29)$$\begin{equation} \int_{0}^{\infty} N_{t} dt= \chi. \end{equation}$$

According to the above, the resource stock dynamics now follow:

(30)$$\begin{equation} \dot{S}_{t}= -R_{t}- N_{t}, \end{equation}$$

such that long-run pollution levels reach P _max,decom = P ₀ + ϕ(S ₀ − χ). Following the same procedure as in our baseline case, the appropriate dynamic budget constraint for the representative household reads:

(31)$$\begin{equation} \dot{W}_{t}= \theta_{t} W_{t} (\hat{p}_{Rt}-x_{t}) +(1-\theta_{t})W_{t} (\hat{p}_{It}+B- D(P_{t})) -C_{t}+T_{t}, \end{equation}$$

with θ ≡ p _R S/W, the share of the individual's resource wealth in the total assets, and x ≡ N/S the expropriation rate. The effect of policy x reduces the profitability of the resource stock and alters the portfolio composition between stocks of capital. Accordingly, the no-arbitrage condition between assets, equation (15), now becomes:

(32)$$\begin{equation} \hat{p}_{Rt} - x_{t}=r_{t}=\hat{p}_{It} +B -D(P_{t}). \end{equation}$$

The RHS of the equation that deals with the stock of physical capital remains the same, while the LHS changes by the x term, the policy premium. The basic intuition is unchanged: adjusting for risk and depreciation, every asset should yield the same return. Accordingly, the resource owner should be compensated for the external political expropriation as proxied by parameter x, i.e. $\hat {p}_{R}=r+x$. The first-order conditions for firms, equations (9), and the Keynes-Ramsey rule (14), stay the same. Equation (30) with u ≡ R/S, yields $\hat {R}=\hat {u}-u-x$. Following the same procedure, the differential equations (17) and (18) remain the same, while consumption growth now becomes:

(33)$$\begin{equation} \hat{C}_{t}=\underbrace{\frac{\alpha B}{\sigma}}_{\Omega-{\rm productivity} }-\underbrace{\frac{\rho}{\sigma}}_{\Delta-{\rm discounting}} - \underbrace{\frac{(1-\alpha) x_{t}}{\sigma}}_{\Pi-{\rm policy}} -\underbrace{\frac{\alpha D(P_{t})}{\sigma}}_{\Lambda-{\rm depreciation}}. \end{equation}$$

A given decommissioning policy path reduces the growth rate of consumption by the term Π all along the transition and the steady state. A steeper price path of the resource with an effective policy does not lead to faster extraction as would be the case without the policy, because the total stock of available (polluting) resources is gradually reduced. Climate damages are lower during transition and in the steady state, since P _max,decom = P _max − ϕχ, with P _max = P ₀ + ϕ S ₀. By comparing (19) with (33) as t reaches infinity we see that as long as x _∞ < (D(P _max) − D(P _max,decom))α/(1 − α), long-run economic development is promoted by the policy. The mechanism can be studied in figure A3 in the online appendix.Footnote ²⁶ Given x _∞, steady states read: $P_{\infty }=P_{max,decom}=P_{0}+\phi (S_{0}-\chi )$, $\hat {C}_{\infty }=g_{C}= ({1}/{\sigma })(\alpha B-\alpha D(P_{\infty } )-\rho -(1-\alpha )x_{\infty })$, with S _∞, u _∞, ϵ _∞ as in (20), (23) and (24), respectively. Figure A3 of the online appendix shows graphically the dynamic development of the economy.

Abatement

This section deals with abatement as a policy option. We will formally study the case of carbon capture and sequestration (CCS) of figure 7(c).Footnote ²⁷ We will assume that in order to proportionally reduce effective emissions each period by χ ∈ [0, 1], the economy has to devote a part X of the stock of physical capital, i.e.,

(34)$$\begin{equation} \chi\phi R_{t}=\zeta X_{t}K_{t}, \end{equation}$$

with ζ > 0 a scaling parameter with appropriate units. Pollution stock dynamics now follow

(35)$$\begin{equation} P_{t}=P_{0}+\phi(1-\chi)(S_{0}-S_{t}), \end{equation}$$

while the growth rate of physical capital reads

(36)$$\begin{equation} \hat{K}_{t}=B(1-\epsilon_{t})-D(P_{t})-X_{t}. \end{equation}$$

According to the above, abatement expenditure is an external action to the households, reducing the available stock of physical capital each period. Firms are facing the same demand curves, equations (9). The dynamic budget constraint of households changes to

(37)$$\begin{equation} \dot{W}_{t}= \theta_{t} W_{t} \hat{p}_{Rt} +(1-\theta_{t})W_{t} ( \hat{p}_{It}+B- D(P_{t})-X_{t}) -C_{t}+T_{t}, \end{equation}$$

which leads to the appropriate no-arbitrage condition between assets:

(38)$$\begin{equation} \hat{p}_{Rt}=r_{t}=\hat{p}_{It}+B-D(P_{t})-X_{t}. \end{equation}$$

In comparison to (32), due to abatement expenditure, households now expect higher net return from physical capital, i.e., $\hat {p}_{I}+B=r+D(P)+X$. Equations (17) and (18) still hold, while with the latter, (34) and (36), the dynamics of abatement expenditure rate X reads:

(39)$$\begin{equation} \hat{X}_{t}=X_{t}-(\sigma-1)\hat{C}_{t}-B(1-\epsilon_{t})+D(P_{t})-\rho. \end{equation}$$

Finally, consumption growth becomes:

(40)$$\begin{equation} \hat{C}_{t}=\underbrace{\frac{\alpha B}{\sigma}}_{\Omega-{\rm productivity} }-\underbrace{\frac{\rho}{\sigma}}_{\Delta-{\rm discounting}} - \underbrace{\frac{\alpha X_{t}}{\sigma}}_{\Pi-{\rm policy}}-\underbrace {\frac{\alpha D(P_{t})}{\sigma}}_{\Lambda-{\rm depreciation}}. \end{equation}$$

Given policy χ, initial conditions S ₀, P ₀, steady states lim _t→∞X _t = 0, lim _t→∞ϵ _t = ϵ _∞, lim _t→∞u _t = u _∞, and equations (35), (17), (18), (39) and (40) are sufficient to completely characterize the dynamic evolution of the economy at hand. Just as in the case of decommissioning, economic growth starts from a lower level due to policy, reaching however a much higher steady state due to lower pollution and damages. The steady states are: $X_{\infty }=0,P_{\infty }=P_{max,abate} =P_{0}+\phi (1-\chi )S_{0},$, $\hat {C}_{\infty }=g_{C} = ({1}/{\sigma }) (\alpha B-\alpha D(P_{\infty })-\rho )$, with S _∞, u _∞, ϵ _∞ as before. Figure A4 in the online appendix graphically presents the results.