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Environmental and Resource Economics

, Volume 70, Issue 4, pp 781–806 | Cite as

Optimum Growth and Carbon Policies with Lags in the Climate System

  • Lucas Bretschger
  • Christos Karydas
Article

Abstract

We study the optimal carbon tax in an economy in which climate change, stemming from polluting non-renewable resource, affects the economy’s growth potential. Our main contribution is to introduce and explore the natural time lag of the climate system between emissions and damages to capital accumulation in an endogenous growth setting. This allows us to investigate how optimal climate policy, and its interplay with climate dynamics, affect long-run growth and the transition of the economy towards it. Without pollution decay, a higher speed of emissions diffusion steepens the growth profile of the economy. With pollution decay, this leads to lower short-run but higher long-run economic growth during transition. Poor understanding of the emissions diffusion process leads to suboptimal carbon taxes, resource extraction and growth.

Keywords

Climate policy Non-renewable resource dynamics Pollution diffusion lag Optimum growth 

JEL Classification

Q54 O11 Q52 Q32 

1 Introduction

Climate change has certain characteristics that impede the implementation of optimal environmental policies: it has a global dimension, necessitating difficult international negotiations and agreements; it requires mitigation policies that create economic costs and benefits which are substantial and unevenly distributed across different countries, and finally; it asks for a policy design that necessitates consideration of a very long time horizon. This poses a major challenge for a usually myopic political decision making process: past environmental policies were mostly implemented after major environmental damages had been publicly observed, creating political necessity to act.1

The effects of climate change will only be fully visible after several decades because greenhouse gas emissions cause economic damages with a major time lag. The Stern Review states “climate models project that the world is committed to a further warming [...] over several decades due to past emissions.”, (Stern 2007, p. 15). Looking into the future and the potentially large damages from climate change, one would expect a time lag of about 50 to 150 years, depending on the scenario followed, (Stern 2007, p. 178). A certain degree of uncertainty remains in any case, an example of which is prominently given in the new IPCC fifth assessment report: “ ...due to natural variability, trends based on short records are very sensitive to the beginning and end dates and do not in general reflect long-term climate trends. As one example, the rate of warming over the past 15 years [...] is smaller than the rate calculated since 1951” (IPCC 2013). The existence and form of this delay in the natural system has major implications for optimum growth and carbon policies, which we study in this paper.

The model is motivated by the evidence that natural disasters have a substantial impact on the economy, destroying part of its physical capital stock (Stern 2013; Bretschger and Valente 2011). At the same time, economic growth exacerbates the impact of natural disasters as the economy accumulates capital, so that each new event has a higher damaging potential. Since 1900, reported economic damages related to weather phenomena and climate change such as floods, droughts, storms, extreme temperatures, and wildfires account for about \(75\%\) of all the natural disasters recorded (EM-DAT The International Disasters Database 2015). Moreover reported damages have increased greatly since the late 1980s.

This paper develops a theoretical model of a growing economy that is harmed by climate change. The model framework used in the paper is based on the endogenous growth approach of Rebelo (1991), enhanced by a polluting non-renewable resource as an essential input to production. We incorporate relevant features such as carbon emissions from non-renewable resources, the slow adjustment of the stock of pollution to emissions, and climate change that affects capital depreciation. Using this endogenous growth setup we characterize the optimal carbon tax when climate change affects the economy’s growth potential. We also study how climate dynamics interact with resource extraction and growth in the case of optimal and suboptimal policies. Our main contribution in the theoretical literature is twofold.

First, with our specification of damages in capital accumulation—linear to the level of pollution—and logarithmic utility, the optimal tax is proportional to current consumption, in line with the literature; for instance Gerlagh and Liski (2012), Golosov et al. (2014), Grimaud and Rouge (2014), van den Bijgaart et al. (2016). In the case of a more general CRRA utility, it asymptotically approaches this behavior. Climate change policy postpones resource extraction and consumption, and induces economic growth to start from a higher level, converging asymptotically to a lower positive constant, the latter being unaffected by policy. If all carbon in the atmosphere is removed through carbon decay, there is no climate problem in the long run; if carbon decay is absent, the long-run growth rate is affected by cumulative extraction.

Second, we introduce in continuous time a well-specified time lag between emissions from polluting non-renewable resources and the damages they cause. With our specification, a unit of emissions follows a diffusion process in which it only gradually increases the stock of harmful pollution; taken together with carbon decay this allows for a hump-shaped impulse response function. This process proves to be crucial for the transition of the economy towards its steady state: without pollution decay, a higher speed of emissions diffusion steepens the growth profile of the economy; with pollution decay this leads to lower short-run but higher long-run economic growth during transition. It follows that poor understanding of the emissions diffusion process can lead to suboptimal carbon taxes, resource extraction and growth. We use this result to argue that if emission taxes are not set by the social planner but by a regular political process, there is a risk of setting tax rates at too low a level.

To the best of our knowledge, this is the first contribution which combines endogenous growth with polluting non-renewable resources to derive the impact of a time lag in pollution dissemination in terms of closed-form solutions. Several contributions have studied the dynamic response of the economy to pollution. Withagen (1994) shows that the introduction of pollution from non-renewable resources in the utility function delays optimum resource extraction. Hoel and Kverndokk (1996) abstract from the finiteness of non-renewable resources by focusing on the economic recoverability of the resource stock. They also note that in the presence of greenhouse effects it will be optimal to slow down extraction and spread it over a longer period. Tahvonen (1997) additionally allows for a non-polluting backstop technology and defines different switching regimes between non-renewable resources and the backstop, which depend on initial pollution and the price of non-renewable resources and the backstop. These models, in partial equilibrium, abstract from capital accumulation, which is crucial for growth, and capital destruction due to climate change, which represents climate damages in a more realistic way.

Sinclair (1994) argues that ”If global warming is taken to be a serious phenomenon, [...] interest rates need to be co-endogenized with other relevant variables”, and studies the impact of environmental pollution in general equilibrium. The impact of pollution on growth has also been studied by Bovenberg and Smulders (1995) and Michel and Rotillon (1995).2 In a Ramsey growth model, van der Ploeg and Withagen (2010) analyze optimal climate policy based on the social cost of carbon and the existence of renewable resources. Ikefuji and Horii (2012) develop a model with capital destruction due to climate change and conclude that growth is sustainable only if the tax rate on the polluting input increases over time. Contrary to our model they abstract from resource finiteness and the inherent time lag in climate change. Using an endogenous growth model, Bretschger and Valente (2011) show that less developed countries are likely to be hurt more than developing ones, with greenhouse gas emissions inducing negative growth deficits and possible unsustainability traps. Grimaud and Rouge (2014) analyze how the availability of an abatement technology affects optimal climate policies using an endogenous growth model based on the expansion-in-varieties framework and show that when such a technology is available, the optimal carbon tax that postpone resource extraction is uniquely determined. Another related paper is Golosov et al. (2014), which introduces non-renewable resources as in our model but abstracts from any capital stock.3 Including the stock of capital is crucial for our approach to capture both endogenous growth and climate damage.

Time lags in the climate system are usually implemented in integrated climate assessment models. Prominent examples are Nordhaus (1992, 2011) that calibrate a Ramsey growth model to show a significant Pareto-improvement due to climate mitigation investment. Most theoretical models on climate change have sidestepped time lags in the climate system. Important exceptions are the contributions of Gerlagh and Liski (2012) and van den Bijgaart et al. (2016). In the former the authors rely on the assumption of full capital depreciation in each period and using quasi-hyperbolic preferences find that the equilibrium carbon price exceeds the imputed externality cost by multiple degrees of magnitude. The latter derives the social cost of carbon in closed-form for a general neoclassical economy whose development is approximated by a balanced growth path.

The remainder of the paper is organized as follows. Section 2 presents the climate dynamics and the technologies of our economy. In Sect. 3 we characterize the social cost of carbon, i.e. the first best (Pigouvian) per-unit tax that restores the socially optimal allocation. In Sect. 4 we solve for the decentralized equilibrium. Section 5 analyzes the effect of climate dynamics and different taxation policies on economic growth. In Sect. 6 we provide simulations in the case of a general CRRA utility and explain our results. Section 7 concludes.

2 The Basic Model

2.1 Climate System

Producers of consumption goods use polluting non-renewable resources, \(R_{t}\), which generate a flow of emissions \(\phi R_{t}\); \(\phi \ge 0\) denotes the carbon content of the resource and t the time index. Emissions add to the stock of harmful pollution \(P_{t}\), which depreciates at rate \(\theta \ge 0\). In our model the pollution accumulation process differs from the usual assumption of instantaneous emissions diffusion. We realistically assume that emissions slowly diffuse into the stock of harmful pollution, reflecting the inherent time lag of the climate system.

Take first the usual assumption of instantaneous emissions diffusion and let \(Z_t=\phi R_t\) be the flow of emissions that effectively adds to the stock of pollution according to \(\dot{P}_t=Z_t+\theta \left( \bar{P}-P_t\right) \); \(P_0\) given. A dot denotes the time derivative. Thus, at each date t, the stock of carbon increases by the flow of emissions, \(Z_t\), and decreases by the natural removal \(\theta \left( \bar{P}-P_t\right) \); with \(\bar{P} \in (0,P_0]\) we proxy the long-run level of carbon concentration when \(\theta \ne 0\); we set it to \(P_0\) without loss of generality.4

Let us now include a distributed time lag formulation for the flow of emissions, i.e. \(Z_{t}\equiv \int _{-\infty }^{t}\kappa e^{-\kappa (t-s)}\phi R_{s}ds\). Variable \(Z_t\) represents now the history of man-made emissions that effectively adds to the stock of pollution with a lag. Parameter \(\kappa \ge 0\) is the speed of this diffusion process; limiting cases are instantaneous diffusion (\(\kappa \rightarrow \infty \)), i.e. \(Z_t=\phi R_t\), and no diffusion (\(\kappa \rightarrow 0\)), i.e. \(Z_t=0\) at all times. We show in “Appendix 1” that, given \(P_0 \ge 0\) and \(Z_0=0\), the dynamic evolution of the climate system follows
$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{P}_{t}=Z_{t}+\theta \left( \bar{P}-P_{t}\right) , \\ \dot{Z}_{t}=\kappa \left( \phi R_{t}-Z_{t}\right) . \end{array}\right. } \end{aligned}$$
(1)
From the solution of (1), the marginal increase in the stock of carbon in period \(\nu \) from a marginal unit of emissions in period \(t \le \nu \), i.e. its impulse response, reads:
$$\begin{aligned} \frac{dP_{\nu }}{{d\left( \phi R_{t}\right) }}\equiv f_{\nu t}=\kappa \frac{e^{-\theta (\nu -t)}-e^{-\kappa (\nu -t)}}{\kappa -\theta }>0,\qquad \text { for all }\nu \ge t. \end{aligned}$$
(2)
The impulse response function (2) is hump-shaped with a peak at \(\nu -t=\ln (\kappa /\theta )/(\kappa -\theta )\); see Fig. 1. The maximum emissions-damage response reads \((\kappa /\theta )^{\frac{1}{1-\kappa /\theta }}\) and is therefore a monotonically increasing concave function in \(\kappa /\theta \), which converges to unity as \(\kappa /\theta \) grows to infinity. For a constant speed of emissions diffusion \(\kappa \), a decrease in the decay rate \(\theta \) increases the maximum emissions-damage response and shifts it towards the future; see Fig. 1a. Conversely, for constant decay rate \(\theta \) , an increase in \(\kappa \) increases the emissions-damage peak, shifts it towards the present but puts a relatively larger damaging impact on the short run in comparison to the long run; see Fig. 1b.
Fig. 1

Marginal increase in the stock of carbon following a marginal increase in emissions for different values of \(\kappa \) and \(\theta \). Left for constant \(\kappa \), right for constant \(\theta \). a \(\kappa =0.02\), \(\theta _{\text {solid}}=0.01\), \(\theta _{\text {dashed}}=0.007\). b \(\theta =0.01\), \(\kappa _{\text {solid}}=0.02\), \(\kappa _{\text {dashed}}=0.03\)

Gerlagh and Liski (2012) arrive at the discrete-time equivalent expression of (2). Using their values for the parameters of the climate system, \(\kappa =0.02\) and \(\theta =0.01\), we confirm their result of a peak emissions-damage response of about 70 years. It is important to note that the speed of emissions diffusion has a dual effect on the marginal damages from the extraction and use of the polluting resource: a level effect on the magnitude of marginal damages and a delay/discounting effect. This can be seen as follows.

For very small time intervals \(\nu -t\), Eq. (2) can be approximated by \(f_{\nu t}\approx \kappa (\nu -t)\); a marginal unit of resources extracted and burned increases harmful pollution within this small time interval by \(\kappa \). It follows that in the very short run our specification has a relatively larger damaging impact of a marginal increase in emissions the larger \(\kappa \) is. For a given decay rate \(\theta \), this will lead to a higher peak of the pollution response, closer to the current date. For longer time periods, the marginal increase of harmful emissions will result in a marginal increase in the stock of pollution determined by the adjustment term \(\frac{e^{-\theta (\nu -t)}-e^{-\kappa (\nu -t)}}{\kappa -\theta }\), which accounts for pollution decay and the slow diffusion of emissions into the stock of harmful pollution. If pollution decay is disregarded, i.e. \(\theta =0\), the damage response accounts only for the probability that a marginal unit of emissions emitted in period t has reached the stock of pollution in period v, \(f_{vt}=1-e^{-\kappa (v-t)}\). These effects of \(\kappa \) and \(\theta \) have a big impact on the transition of the economy towards its steady state, which we study here.5

2.2 Aggregate Economy

Markets are fully competitive. Production in each period t is based on constant returns to scale technologies and on two inputs: capital \(K_t\), and polluting non-renewable resources \(R_t\). The stock of capital is a generic reproducible factor in this economy that includes both physical and human capital; we will call it “capital” for convenience. As proposed by Stern (2013) and Bretschger and Valente (2011) physical capital is exposed to climate disasters. Natural events like floods, droughts, wildfires, and extreme temperatures caused by anthropogenic climate change can destroy buildings, equipment, crops, roads, and public infrastructure; this puts a natural drag on economic growth, since part of the economic resources have to be allocated to fixing these damages. Conversely, there are durable forms of capital like human skills that cannot be depleted. We will therefore allow for only a part \(\eta \) of the capital stock to be affected by harmful pollution.6

Following Rebelo (1991) there are two production sectors in this economy: the consumer goods sector and the investment sector. The consumer good is the numeraire, and is produced with both inputs; the investment good sector is assumed to be capital intensive and uses only capital. The economy features the following aggregate production functions for the consumption good \(Y_t\), and the investment good \(I_t\),
$$\begin{aligned} Y_{t}= & {} A (\epsilon _t K_{t})^{\alpha }R_{t}^{1-\alpha }, \end{aligned}$$
(3)
$$\begin{aligned} I_{t}= & {} B(1-\epsilon _t)K_{t}, \end{aligned}$$
(4)
where \(\epsilon _t \equiv K_{Yt}/K_{t}\in [0,1]\) is the aggregate fraction of capital devoted to the consumption good, and \(R_{t}\) the total demand for the non-renewable resource. Investment leads to capital accumulation according to
$$\begin{aligned} \dot{K}_{t}=I_t-D(P_t)\eta K_t, \end{aligned}$$
(5)
with \(K_{0}>0\), and \(\eta \) the share of non-durable capital, which we assume to be constant. The part of capital that is exposed to wear decays according to the damage function \(D(P_t)\) due to natural depreciation and higher pollution levels. Costless resource extraction \(R_{t}\) depletes the existing stock of the non-renewable energy resource \(S_{t}\) (with \(S_{0}>0\)), according to the standard law of motion and the stock constraint
$$\begin{aligned} \dot{S}_{t}=-R_{t},\qquad \int _{0}^{\infty }R_{t}dt\le S_{0}. \end{aligned}$$
(6)
Finally, the economy admits a representative household with preferences \(U(C_t)\) that owns all the financial wealth, i.e. capital and energy resources. In the general CRRA form we have \(U(C_t)=\frac{C_t^{1-\sigma }}{1-\sigma }\) while with \(\sigma =1\) we get the logarithmic form, i.e. \(U(C_t)=\log (C_t)\); parameter \(\sigma \) is the inverse of the elasticity of intertemportal substitution.

Assumption 1

The utility function is logarithmic, i.e. \(U(C_t)=\mathrm{log}(C_t)\).

The assumption of log-utility (\(\sigma =1\)) is the most widely used case in the literature of endogenous growth with polluting non-renewable resources, as it allows for closed-form solutions and a full characterization of the macroeconomic model features. We will also use it in the basic approach so that we can directly compare our results with the relevant literature. As an extension, we treat and discuss the case of \(\sigma \ne 1\) in Sect. 6. In particular, we will derive how the interplay between the time lag in emissions diffusion and the substitution and income effect that arise in the non-log-utility case affect the dynamics and the steady state of the economy. We will show that when \(\sigma =1\) and capital damages are linear to the stock of pollution, the optimal emissions tax rate grows with consumption while this condition is asymptotically reached with \(\sigma \ne 1\).

Assumption 2

Capital damages are linear to the level of pollution, i.e. \(D(P_t)=\delta +\chi P_t\).

Parameter \(\delta \) is the natural depreciation of the capital stock and \(\chi \) the damage sensitivity to pollution; see Ikefuji and Horii (2012) for a similar specification.

2.3 Discussion About the Model

In equilibrium, aggregate demand for the consumption good must equal its total supply, i.e. \(C_{t}=Y_{t}\). The resource stock is finite and extraction and use of the non-renewable resource has to stop in finite or infinite time. This puts an upper bound on pollution and capital damages under all assumptions regarding the decay of the pollution stock. Furthermore, due to the specification of the production function (3), and the fact that damages due to pollution accumulation are bounded, the resource stock is essential in the sense that an additional unit of resources used in the production of the consumption good is always welfare enhancing. Accordingly, resource extraction will be positive in each time period and the resource stock will only be asymptotically depleted so that (6) holds with equality; see for example Daubanes and Grimaud (2010) for a similar argumentation. Following the same logic, the share of capital allocated to the consumption good sector has to obey \(\epsilon _{t} \in (0,1)\) for all \(t\ge 0\); formal proofs are given in “Appendix 3”. In the face of pollution, the economy at hand is always in transition. The only possible steady state is the one where resources are asymptotically depleted and pollution asymptotically reaches its steady state value, \(P_{\infty }=P_{0}+\phi S_{0}\), if \(\theta =0\), or \(P_{\infty }=\bar{P}=P_0\), if \(\theta >0\). When that happens the growth rate of resource extraction, \(g_{Rt}\equiv \dot{R}_{t}/R_{t}\), and the share \(\epsilon _{t}\) must have also reached their steady state values.7 It follows from our specifications for the consumption good and capital accumulation that in the steady state the economy asymptotically reaches a balanced growth path which can be defined as follows:

Definition 1

An equilibrium path is an asymptotic balanced growth path, if capital allocation and the growth rate of resource extraction are asymptotically constant, i.e. \(\lim _{t\rightarrow \infty }\epsilon _t=\epsilon _{\infty }\), and \(\lim _{t\rightarrow \infty }\dot{R}_{t}/R_{t}=g_{R\infty }\); then \(\lim _{t\rightarrow \infty }\dot{K}_t/K_t=g_{K\infty }\) and \(\lim _{t\rightarrow \infty }\dot{C}_t/C_t=g_{C\infty }\), asymptotically constant.

Below we solve the planning problem and characterize the social cost of carbon. In Sect. 4 we show that this is the first-best carbon tax that optimally corrects for the externality.

3 Social Optimum

The social planner chooses the fraction of capital allocated to the consumption good, \(\epsilon _t\), and the resource extraction, \(R_t\), in order to maximize \(\int _0^\infty U(C_t) e^{-\rho t}dt\) with \(C_t=Y_t\), subject to Eqs. (1), (3)–(6). Let \(\lambda _{Ct}, \lambda _{St}, \lambda _{Zt}\) be respectively the shadow prices for the consumption good \(C_t\), the stock of the non-renewable resource \(S_t\), and the history of lagged emissions \(Z_t\). The first-order condition for resource extraction follows:
$$\begin{aligned} (1-\alpha )\frac{C_t}{R_t}=\frac{\lambda _{St}}{\lambda _{Ct}}-\phi \kappa \frac{\lambda _{Zt}}{\lambda _{Ct}}. \end{aligned}$$
(7)
According to Eq. (7), in each point of time, the marginal benefit from extracting and using the resource (left-hand-side) equals the marginal cost of resource use (right-hand-side), in terms of the consumption good. The cost consists of the scarcity cost of the exhaustible resource, \(\lambda _{St}/\lambda _{Ct}\), i.e. its producer price in a competitive market, and of the social cost of carbon (SCC), i.e. the marginal externality damage of an additional unit of emissions, \(X_t\equiv -\phi \kappa \lambda _{Zt}/\lambda _{Ct}\). \(X_t\) captures the externality from carbon emissions and as we show in Sect. 4 is equal to the optimal Pigouvian tax. We prove in “Appendix 2” that it can be written as
$$\begin{aligned} X_t=C_t \frac{\alpha \eta \phi }{\rho } \kappa \int _t^{\infty }\left[ \int _s^{\infty } D'(P_v)\left( \frac{\bar{\epsilon }}{\epsilon _v}\right) \left( \frac{C_t}{C_v}\right) ^{\sigma -1} e^{-(\rho +\theta )(v-s)}dv \right] e^{-(\rho +\kappa )(s-t)}ds, \end{aligned}$$
(8)
with \(\bar{\epsilon }=\rho /B\). The intuitive explanation of (8) is the following. The remaining portion in year \(\nu \ge s\), after decay, of a marginal unit of emissions from year t, that has reached the stock of pollution in year \(s\ge t\), has a negative impact in all years \(\nu \ge t\). The first term inside the square brackets is the marginal damage of pollution on capital accumulation, \(D'(P_v)\). The second term comes from the utility denominated shadow price of capital and is responsible for allocating capital between the consumption and the investment sector, while the third term reflects preferences of agents regarding intertemporal consumption smoothing. The exponential terms reflect the delay/decay structure of the climate system: \(e^{-\theta (\nu -s)}\) is the share of emissions remaining in year \(\nu \) from emissions that reached the stock of pollution in year s, while \(\kappa e^{-\kappa (s-t)}\) accounts for the slow adjustment of the stock of pollution from the marginal unit emitted in year t.

The cost of the externality is greater, when the following are larger: the emissions intensity parameter \(\phi \), the part of non-durable capital \(\eta \), and the share of capital in the production of the consumption good \(\alpha \). It is also greater when the following are smaller: the discount rate \(\rho \), the pollution decay \(\theta \), and, ceteris paribus, the path of capital allocated to consumption \(\{\epsilon _v\}_{t}^\infty \), since higher investment translates to a larger stock of capital in subsequent periods, creating larger damaging potential in the future.8 Finally, we point out the effect of the slow emissions diffusion as a suppressing factor on the magnitude of marginal damages by the multiplicative term in the beginning of (8). As discussed in Sect. 2.1, the speed of emissions diffusion has a dual effect on the marginal damages from the extraction and use of the polluting resource: (i) a level effect on the magnitude of marginal damages and (ii) a delay effect.

At this point, a direct comparison of our results to the literature seems appropriate. A similar expression to (8) has been found in van den Bijgaart et al. (2016). There the authors consider a general neoclassical economy, with climate dynamics similar to ours, where climate change destroys part of the final output. We show instead that similar results can be obtained in an endogenous growth framework, and in the case where pollution harms capital accumulation. Moreover, in several models of growth with polluting non-renewable resources the marginal externality damage is a linear function of the consumption good all along the optimal path, irrespective of whether lags in emissions dissemination are considered or not; see for example Gerlagh and Liski (2012) and Golosov et al. (2014).

The linearity of the marginal externality damage in the consumption good stems from three factors: first, from the log-utility assumption; second, from the damage specification; third, from a constant savings rate at all times. While in the case of the general neoclassical economy one needs to impose the last condition (by assuming full capital depreciation in each period), as in the aforementioned contributions, in the case of endogenous growth, as in Grimaud and Rouge (2014) or in the present paper, this condition is immediately satisfied with logarithmic utility: take (8) with Assumption 1, i.e. \(\sigma =1\). We show in “Appendix 2” that in this case \(\epsilon _t=\bar{\epsilon }\) at all times, i.e. there will be a constant fraction of capital allocated to investment; the equivalent of the constant savings rate in the neoclassical economy. Equation (8) now reads
$$\begin{aligned} X_t=C_t\frac{\alpha \eta \phi }{\rho }\kappa \int _t^\infty \left( \int _s^\infty D'(P_v)e^{-(\rho +\theta )(v-s)}dv\right) e^{-(\rho +\kappa )(s-t)}ds. \end{aligned}$$
The linearity of \(X_t\) in \(C_t\) is granted if \(D(P_t)\) is also linear in pollution. Applying Assumption 2 readily leads to the following proposition:

Proposition 1

Given Assumptions 1 and 2, the marginal externality damage of emissions is proportional to the consumption good and given by
$$\begin{aligned} X_t=\tilde{X} C_t \qquad \text {with} \qquad \tilde{X}=\kappa \frac{\alpha \eta \phi \chi }{\rho (\rho +\theta )(\rho +\kappa )}; \end{aligned}$$
(9)
\(\tilde{X}\) is an increasing and concave function of \(\kappa \), independent of time.

Proof

See last paragraph above. \(\square \)

In the log-utility case, with linear and separable damages due to climate change in the utility function, as in Grimaud and Rouge (2014), or multiplicative exponential damages in the production function, as in Gerlagh and Liski (2012) and Golosov et al. (2014), or even linear damages in capital accumulation, as in the present approach, the social cost of the externality, \(X_t\), is a linear function of the consumer good. In addition, we find that the fraction \(\tilde{X}\) is increasing in the speed of adjustment between emissions and pollution, \(\kappa \), in a concave way reaching its upper limit, \(\frac{\alpha \eta \phi \chi }{\rho (\rho +\theta )}\), as \(\kappa \rightarrow \infty \).

Below we proceed by characterizing the decentralized equilibrium. We show that \(X_t\) is the Pigouvian tax needed to optimally correct for the externality, and study how the economy responds to more general taxation policies.

4 Decentralized Equilibrium

4.1 Firms

Each sector in the economy is populated by a unit mass of competitive firms \(j \in [0,1]\). Specifically, the production of consumption good \(Y_{jt}\) uses capital \(K_{Yjt}\), and resources \(R_{jt}\), according to \(Y_{jt}=AK_{Yjt}^{\alpha }R_{jt}^{1-\alpha }\). The production of the investment good \(I_{jt}\) reads \(I_{jt}=B K_{Ijt}\) . AB are productivity parameters. A producer of consumer good \(Y_{jt}\) solves
$$\begin{aligned} \max _{K_{Yjt},R_{jt}}\left\{ A K_{Yjt}^{\alpha }R_{jt}^{1-\alpha }-p_{Kt}K_{Yjt}-\left( p_{Rt}+\tau _t\right) R_{jt}\right\} , \end{aligned}$$
while one in the investment good sector solves
$$\begin{aligned} \max _{K_{Ijt}}\left\{ p_{It}B K_{Ijt}-p_{Kt}K_{Ijt}\right\} , \end{aligned}$$
with \(p_{Kt}\) the rental price of capital, \(p_{It}\) the price of investment, \(p_{Rt}\) the producer price of the non-renewable resource and \(\tau _t\) a per-unit tax on resource extraction. Because production has constant returns to scale, firms face identical factor input ratios. Hence, the economy admits a representative firm active in both sectors with \(Y_{t}\equiv \int _{0}^{1}Y_{jt}dj\) for total production, \(K_{t}\equiv \int _{0}^{1}\left( K_{Yjt}+K_{Ijt}\right) dj\) for the total stock of capital demanded, and \(\epsilon _t=K_{Yt}/K_t\), the aggregate fraction of capital allocated to the consumption good. The first order conditions of these maximizations give the demand functions for non-renewable resources and capital in the consumption good sector, and a no-arbitrage condition which equates returns from the two usages of capital in this economy, i.e. in the consumption good sector and in the investment sector, namely,
$$\begin{aligned} p_{Rt}+ \tau _{t}=(1-\alpha )\frac{Y_{t}}{R_{t}},\qquad p_{Kt}=\alpha \frac{Y_{t}}{\epsilon _{t}K_{t}}, \qquad p_{Kt}=p_{It} B. \end{aligned}$$
(10)

4.2 Households

There is a continuum of infinitely lived households \(i \in [0,1]\) that have the option to allocate their income to consumption, through the consumption good sector, or to additional capital formation, through the investment sector. The representative household i owns a share of the stock of energy resources \(S_{it}\), and capital, \(K_{it}\). In each time period a share of resources \(R_{it}\) is extracted and sold to firms at a price \(p_{Rt}\). Furthermore, \(K_{it}\) is rented to firms at prices \(p_{Kt}\). With \(T_t\) denoting lump-sum transfers, individual income amounts to \(p_{Kt}K_{it}+p_{Rt}R_{it}+T_{t}\) while expenditures equal \(C_{it}+p_{It}H_{it}\), with \(C_{it}\) denoting the flow of consumption and \(H_{it}\) reflecting the purchase of additional capital through the investment sector at price \(p_{It}\). Capital and resource stocks evolve according to
$$\begin{aligned} \dot{K}_{it}=H_{it}^{K}-D(P_t)\eta K_{it}, \qquad \dot{S}_{it}=-R_{it}, \end{aligned}$$
(11)
while income equals expenditure, so that the income balance reads
$$\begin{aligned} p_{Kt}K_{it}+p_{Rt}R_{it}+T_{t}=C_{it}+p_{It}H_{it}. \end{aligned}$$
(12)
Differentiating the household’s assets, \(a_{it}=p_{It}K_{it}+p_{Rt}S_{it}\) with respect to time, using (11), (12), and the fact that \(p_{Kt}=p_{It}B\) from (10), yields the household’s dynamic budget constraint
$$\begin{aligned} \frac{\dot{a}_{it}}{a_{it}}=\beta _{it}^{S}\frac{\dot{p}_{Rt}}{{p}_{Rt}}+ \left( 1-\beta _{it}^{S}\right) \left[ \frac{\dot{p}_{Kt}}{{p}_{Kt}}+B-\eta D(P_{t}) \right] -\frac{C_{it}}{a_{it}}+\frac{T_{t}}{a_{it}}, \end{aligned}$$
(13)
with \(\beta _{it}^{S}\equiv p_{Rt}S_{it}/a_{it}\), the share of the individual’s resource wealth in her total assets. The household’s objective is to choose the time path of consumption and share \(\beta _{it}^{S}\) which maximize its lifetime utility
$$\begin{aligned} \int _{0}^{\infty }U(C_{it})e^{-\rho t}dt, \end{aligned}$$
subject to the budget constraint (13). In the general CRRA form we have \(U(C_{it})=\frac{C_{it}^{1-\sigma }}{1-\sigma }\) while with \(\sigma =1\) we get the logarithmic form, i.e. \(U(C_{it})=\log (C_{it})\). From combining the first order conditions of household optimization we find
$$\begin{aligned} \sigma \frac{\dot{C}_{it}}{C_{it}}= & {} r_{t}-\rho , \end{aligned}$$
(14)
$$\begin{aligned} \frac{\dot{p}_{Rt}}{{p}_{Rt}}= & {} r_{t}, \end{aligned}$$
(15)
$$\begin{aligned} \frac{\dot{p}_{Kt}}{{p}_{Kt}}+B- D(P_{t})\eta= & {} r_{t}. \end{aligned}$$
(16)
These are the Keynes–Ramsey rule for consumption growth, the Hotelling rule for resource price development, and the return on investing in capital formation, with \(r_{t}\) being the economy-wide interest rate. By equating (15) with (16) we see that both assets, i.e. non-renewable resources and capital, should yield equal returns. The optimization is complemented by the appropriate transversality condition, reading
$$\begin{aligned} \lim _{t\rightarrow \infty }a_{it}C_{it}^{-\sigma }e^{-\rho t}=\lim _{t\rightarrow \infty }\left( \frac{p_{Kt}}{B}K_{it}+p_{Rt}S_{it}\right) C_{it}^{-\sigma }e^{-\rho t}=0. \end{aligned}$$
(17)
Finally we need to impose the restriction that \(\chi \) satisfies \(\alpha \left( B-\eta \left( \delta +\chi P_\infty \right) \right) >\rho \) so that households have enough incentives to invest in capital formation.

4.3 Equilibrium

In equilibrium total demand for the consumption good must equal its total supply, i.e. \(C_t=\int _0^1 C_{it}di=Y_t\). Given the initial values \(K_{0},S_{0},P_{0}\) and the dynamic evolution of the tax rate, the dynamics of the climate system (1), capital accumulation (5), resource depletion (6), the first order conditions for the representative firm (10), the aggregate version of the Keynes–Ramsey rule (14), the Hotelling rule for the price evolution of the non-renewable resource (15), the return on investment in capital formation (16), and the transversality condition (17), completely characterize the dynamic behavior of the decentralized economy.

4.4 The Pigouvian Tax

In Sect. 3 we characterized the socially optimal solution and derived the expression for the social cost of carbon, \(X_t\). Here we show that this is in fact the Pigouvian tax in the decentralized equilibrium that produces the first-best allocation.

As shown in “Appendix 5”, with \(\sigma =1\), the capital share \(\epsilon _t\) immediately jumps to its optimal steady state value \(\bar{\epsilon }=\rho /B\) also in the decentralized case. By comparing the social planner’s optimality condition (7) with its equivalent from (10), using \(C_t=Y_t\), it is straightforward to see that the resource extraction will follow its optimal path if the producer’s price for the non-renewable resource equals its scarcity rent (\(p_{Rt}=\lambda _{St}/\lambda _{Ct}\)), and if the per-unit carbon tax equals the marginal externality damage of emissions found in (9) (\(\tau _t=X_t\)). This is the optimal tax which we denote by \(\tau _t^o\).

Since \(\tau _t^o \equiv X_t\), when Assumptions 1 and 2 are satisfied, the optimal tax is a constant fraction of the consumption good. The important point about this result is that it provides appropriate incentives to the economy to stretch the path of resource extraction. To be more precise, the per-unit tax that postpones extraction has to grow at a slower rate than the price of the non-renewable resource. Then, the unit price paid for the resource by consumers increases less rapidly than the price received by producers, which grows at the market’s interest rate, giving them the incentive to postpone extraction: with \(\sigma =1 \) the price received by producers \(p_{Rt}\), grows at the rate \(r_{t}\) [from (15)] while \(\tau _t^o\) grows with consumption, i.e. at \(r_{t}-\rho \) [from the aggregate version of (14)].9

Furthermore, it is a known result from the theory of non-renewable resource taxation that any term in the optimal per-unit tax that grows with the interest rate has no effect on the extracting behavior of the economy, suggesting that there is an infinite number of optimal taxes that give the same resource extraction incentives; see Dasgupta and Heal (1979), and Gaudet and Lasserre (2013).10 We show in “Appendix 6” that this is also the case here.

4.5 Response to Taxation

In light of the previous discussion, we will only study taxation policies proportional to consumption according to the following assumption:

Assumption 3

All taxes considered are proportional to consumption: \( \tau _t =\tilde{\tau } C_t\), with \(\tilde{\tau }\) constant.

Proposition 2

Suppose that Assumptions 12, and 3 apply. Then in a decentralized equilibrium,
  1. (i)

    the fraction of consumption \(\tilde{\tau }\) determines the dynamics of resource extraction; a higher value stretches resource extraction to the future; \(\tilde{\tau }=0\) (no tax) results in the fastest equilibrium extraction,

     
  2. (ii)

    economic growth starts from a higher level, the higher \(\tilde{\tau }\) is, converging asymptotically to a positive constant \(g_{C\infty }\), which is lower than initial growth and unaffected by policy.

     

Proof

(i) Following the same procedure as in “Appendix 6”, the time path of resource extraction and its growth rate can be calculated to be only dependent on \(\tilde{\tau }\) as,
$$\begin{aligned} R_{t}(\tilde{\tau })= & {} \frac{1-\alpha }{\tilde{\tau }\left[ 1+e^{\rho t}\left( e^{\frac{S_{0}\rho }{1-\alpha }\tilde{\tau }}-1\right) ^{-1}\right] }>0, \end{aligned}$$
(18)
$$\begin{aligned} g_{Rt}(\tilde{\tau })= & {} \frac{-\rho }{1+e^{-\rho t}\left( e^{\frac{S_{0}\rho }{1-\alpha }\tilde{\tau }}-1\right) }<0. \end{aligned}$$
(19)
With our assumptions \(\tilde{\tau }\) is decisive for the dynamics of resource extraction: with tax, \(g_{Rt}>-\rho \) and \(g_{R\infty }=\lim _{t\rightarrow \infty } g_{Rt}=-\rho \); zero tax entails the fastest resource depletion, \(g_{Rt}=-\rho \) in all time periods. When environmental policy is implemented, resource extraction is stretched to the future: \(dg_{Rt}/d\tilde{\tau }>0\) (i.e. a flatter resource extraction profile)
(ii) By log-differentiating (3), with \(\epsilon _t=\bar{\epsilon }=\rho /B\), using (5), we get the growth rate of consumption in the decentralized equilibrium, \(g_{Ct}\equiv \dot{C}_t/C_t\), as a function of \(\tilde{\tau }\)
$$\begin{aligned} g_{Ct}(\tilde{\tau })=\alpha \left[ B-\rho -\eta D\left( P_t\right) \right] +(1-\alpha )g_{Rt}(\tilde{\tau }). \end{aligned}$$
(20)
With \(\epsilon \) jumping immediately to its optimal steady state and \(P_0\) given, differentiating (20) at \(t=0\) w.r.t. \(\tilde{\tau }\) implies \(dg_{C0}/d\tilde{\tau }=(1-\alpha )dg_{R0}/d\tilde{\tau } >0\), i.e. a higher value for \(\tilde{\tau }\) induces the economy to start from a higher level of economic growth, converging to the positive constant \(g_{C\infty }=\alpha \left[ B-\rho -\eta D\left( P_\infty \right) \right] -(1-\alpha )\rho \). Furthermore, because \(P_\infty = P_0+\phi S_0\), if \(\theta =0\), or \(P_0\) if \(\theta >0\), and \(g_{R0}>-\rho \), the steady state level of economic growth is always lower than initial growth. \(\square \)

Two things are worth noting here. First, resource extraction is independent of climate damages. In general since pollution affects capital accumulation and the interest rate, one would anticipate damages to affect the path of resource extraction. This is not the case in the present setup due to logarithmic preferences: consider for convenience the FOC for \(R_t\) in (10) with a given ad-valorem tax, \(\pi _t\), i.e. \((1-\alpha )C_t/R_t=\pi _t p_{Rt}\). Log-differentiating this expression using the log-differentiated version of the second FOC in (10) along with (5), and (14)–(16) leads to \(\sigma g_{Rt}=-\left( \rho +(1+\alpha (\sigma -1))g_{\pi t}+\alpha (\sigma -1)(B-\eta D(P_t)\right) \); with \(\sigma \ne 1\) resource extraction responds to pollution. With \(\sigma =1\), however, we get \(g_{Rt}=-\rho -g_{\pi t}\).11 In general this result, as well as the fact that \(\epsilon \) jumps immediately to its steady state, is the outcome of the substitution and income effect that arise due to pollution exactly offsetting each other when \(\sigma =1\); we study this in more detail in Sect. 6.

Second, it sounds counter-intuitive that higher taxation induces the economy to start from a higher point of economic growth. However, according to result (i) of the proposition, it is the constant \(\tilde{\tau }\) that determines the extraction path. Thus, higher taxes that stretch resource extraction to the future impose a lower drag on growth in earlier periods.12

5 Effects of Climate Dynamics on Growth

The level of harmful pollution at each time period, with \(\bar{P}=P_0\), in the general case with pollution decay reads13
$$\begin{aligned} P_{t}=P_{0}+\int _{0}^{t}f_{ts}\phi R_{s}ds, \end{aligned}$$
(21)
with \(f_{ts}\) from (2) and \(R_t\) from (18); see “Appendix 1”.14 Next we discuss the transition process towards the steady state in the decentralized equilibrium. This will depend on the speed of emissions diffusion \(\kappa \), the decay rate \(\theta \), and the policy \(\tau _t\), since these elements govern the dynamics of resource extraction, of the climate system, and in turn affect the growth rate of the economy. We will thoroughly study the case of \(\theta =0\) as only this case allows for a rigorous mathematical analysis. We will then present the results graphically and their intuition based on the presentation of the climate system in Sect. 2.1 and Eq. (21).

5.1 Effects of Climate Dynamics on the Decentralized Equilibrium

The effects of pollution decay in the market solution are given in the following proposition and can be studied graphically in Figs. 2 and 3.

Proposition 3

Suppose that Assumptions 12, and 3 apply. Then in a decentralized equilibrium,
  1. (i)

    without pollution decay, \(\theta =0\), the growth rate of consumption converges monotonically from above towards the steady state, \(g_{C\infty }\); higher \(\kappa \) speeds up the transition process and results in lower economic growth at all times,

     
  2. (ii)

    with positive decay, \(\theta >0\), the growth rate of consumption converges towards the steady state, \(g_{C\infty }\), in a non-monotonic way (i.e. in a U-shaped manner); higher \(\kappa \) leads to a lower minimum growth, which is shifted forward to the present, and in lower short-run but higher long-run economic growth.

     
Fig. 2

Pollution and consumption growth for different \(\tilde{\tau }\) and \(\kappa \), (\(\theta =0\) in both cases)

Fig. 3

Pollution and consumption growth for different \(\tilde{\tau }\) and \(\kappa \), (\(\theta >0\) in both cases)

Proof

(i) When pollution decay is disregarded, \(\theta =0\), pollution starts from \(P_{0}\) and monotonically reaches its higher steady state \(P_\infty =P_{0}+\phi S_{0}\) when resources are asymptotically depleted, i.e. \(\dot{P}_t > 0\) and \(\lim _{t \rightarrow \infty } \dot{P}_t=0\). Moreover from (19), \(\dot{g}_{Rt} < 0\) and \(\lim _{t \rightarrow \infty } \dot{g}_{Rt}=0\). From (20) this leads to \(\dot{g}_{Ct}<0\), with \(\lim _{t \rightarrow \infty } \dot{g}_{Ct}=0\), i.e. growth follows a monotonic path towards its steady state. A higher speed of emissions diffusion under the same tax policy will not affect resource extraction, i.e. \(dR_t/d\kappa =d g_{Rt}/d\kappa =0\); from (18), (19). Moreover with \(\theta =0\), \(df_{ts}/d\kappa >0\) and \(\lim _{t\rightarrow \infty } df_{ts}/d\kappa =0\); from (2). The previous lead to \(d g_{Ct}/d\kappa <0\) and \(\lim _{t\rightarrow \infty } d g_{Ct}/d\kappa =0\); from (20).

(ii) When pollution decay is taken into account, \(\theta >0\), the pollution stock is hump-shaped starting and finishing at \(P_{0}\). From (20), the growth rate of consumption will have an inverse hump shape, i.e. a U-shape. We explained in Sect. 2 that a higher \(\kappa \), will lead, ceteris paribus, to a higher pollution peak which will be also brought closer to the present; moreover, it still holds that \(dR_t/d\kappa =d g_{Rt}/d\kappa =0\); from (18), (19). From the last two points and Eq. (20) it follows that higher \(\kappa \) leads to to a lower minimum growth, shifted forward to the present. \(\square \)

Proposition 3 can be understood intuitively by considering the cases for \(\theta \): if there is no pollution decay, \(\theta =0\), higher speed of emissions diffusion, \(\kappa \), will increase the marginal effect of emissions from all preceding periods on the current pollution level. Taking together the finiteness of the resource, this will speed up the transition process towards the lower steady state resulting in lower economic growth at all times. A lower \(\tilde{\tau }\) would have the same effect on growth: the lower the tax is, the lower the initial level of economic growth and the faster the non-renewable resource extraction in earlier periods; see Proposition 2. Resource depletion is brought forward to date and so does pollution accumulation and its harmful effect on growth. If \(\theta >0\), with higher \(\kappa \), the marginal emissions-damage response will be relatively higher in the short-run but relatively lower in the long-run, and the stock of pollution will follow a lower trajectory towards its steady state in later time periods; see “Appendix 1” and Sect. 2. Since resource extraction will be unaffected when the \(\tilde{\tau }\) fraction stays constant, a higher speed of emissions diffusion, \(\kappa \), will result in economic growth of the decentralized equilibrium being lower in the short run, reaching a minimum level when pollution peaks and converging at a higher rate towards \(g_{C\infty }\). A higher \(\tilde{\tau }\) smooths out such behavior: resource extraction and use is stretched to the future, which, for the same decay structure, will lead to a lower peak of pollution occurring at a later time period. Accordingly, when \(\theta >0\), there are two counter-acting forces on growth from \(\kappa \) and \(\tilde{\tau }\).

5.2 Effects of Climate Dynamics on the Social Optimum

Above we established that there are two counter-acting forces on growth arising from the speed of diffusion, \(\kappa \), and the policy, \(\tilde{\tau }\): when carbon decay is absent, \(\theta =0\), higher \(\kappa \) speeds up the transition process towards the lower steady state; higher \(\tilde{\tau }\) has a mitigating effect: it induces the economy to start from a higher level of economic growth and stretches resource extraction and pollution accumulation to the future which acts positively on growth. When \(\theta > 0\), other things being equal, a higher speed of emissions diffusion induces a relatively higher marginal damaging impact in the short run relative to the long run and leads to a higher pollution peak, closer to the present. Economic growth that has a U-shape, reaches a lower minimum which is also brought forward. A higher \(\tilde{\tau }\) smooths out such a behavior. Since \(\tilde{\tau }^o = \tilde{X}\), from Proposition 1, \(\tilde{\tau }^o\) is increasing in \(\kappa \). Accordingly, the negative “direct” effect of a larger \(\kappa \) through its influence on climate dynamics, is mitigated by a positive “indirect” effect of \(\kappa \) through a higher optimal \(\tilde{\tau }^o\). The previous can be summarized in the following proposition:

Proposition 4

Given Assumptions 1 and 2, in a social optimum solution without pollution decay, \(\theta =0\), a larger \(\kappa \) steepens the growth profile of the economy; with \(\theta >0\), a larger \(\kappa \) creates ambiguous results on the timing and level of minimum economic growth.

Proof

Remember that \(\tilde{\tau }^o= \tilde{X}\), given by (9). Differentiate the social optimum version of (20), \( g_{C}^{o}\equiv g_{C}(\tilde{\tau }^{o})\), w.r.t. \(\kappa \) to get
$$\begin{aligned} \frac{dg_{Ct}^{o}}{d\kappa }=-\alpha \eta \chi \phi \int _{0}^{t}\left[ \underbrace{\frac{df_{ts}}{d\kappa }}_{\text {direct}}R_{s}^{o}+f_{ts} \underbrace{\frac{dR_{s}^{o}}{d\tilde{\tau }^{o}}\frac{d\tilde{\tau }^{o}}{d\kappa }}_{\text {indirect}}\right] ds+(1-\alpha )\underbrace{\frac{dg_{R_{t}^{o}}}{d\tilde{\tau }^{o}}\frac{d\tilde{\tau }^{o}}{d\kappa }}_{\text {indirect}}, \end{aligned}$$
(22)
with \(f_{ts}\) from (2), \(R_{s}^{o}\equiv R_{s}(\tilde{\tau }^{o})\), from (18), and \(g_{Rt}^{o}\equiv g_{Rt}(\tilde{\tau }^{o})\), from (19). The two effects, direct and indirect, in the social optimum, tend to offset each other and in general create ambiguous results about the timing and the magnitude of minimum economic growth when \(\theta >0\). In the case of no pollution decay, \(\theta =0\), since pollution peaks only in the steady state, the direct effect of a larger \(\kappa \) is only about current emissions translating faster into pollution destroying capital. Hence, according to Proposition 2, the economy starts from a higher level of economic growth due to a higher optimal tax, and transitions faster to the lower steady state. A larger \(\kappa \) then only steepens the growth profile of the economy. \(\square \)
For our discussion above on the impact of emissions diffusion, \(\kappa \), on the transition of economic growth towards its steady state in the social optimum we provide Fig. 4 as an illustration. Note also that for the same value of \(\kappa \), the economy starts from a higher level of economic growth for \(\theta =0\). This is due to the discounting character of the pollution decay: from Proposition 1, other things being equal, \(\theta =0\) results in a higher optimal tax than in the \(\theta >0\) case because the discounted value of marginal damages is higher. According to Proposition 2, this induces the economy to start from a higher level of economic growth.
Fig. 4

Optimal level of pollution and consumption growth for different values of \(\kappa \) in the \(\theta =0\) and \(\theta >0 \) case

Below, we discuss as an extension the case of CRRA utility and the interplay between the climate dynamics and the risk aversion of the representative household. Since pollution affects the return on investment in capital formation, with CRRA utility the substitution and income effect that arise do not necessarily cancel out. The Pigouvian tax rule does not anymore start off growing with consumption, even though it asymptotes to such behavior. Whether it starts off below or above the long term value depends on the elasticity of intertemporal substitution.

6 Non-logarithmic CRRA Utility

As we established in Sect. 3, the common feature of our model and those in the literature, of the optimal tax rate being a constant fraction of the consumption good all along the optimal path is a consequence of assuming log-utility function. In this section we will study the case of non-logarithmic utility. Since pollution affects the return on investment in capital formation, with a general CRRA utility the substitution and income effect that arise do not necessarily cancel out. We will see that the Pigouvian tax rule does not anymore start off growing with consumption, even though it asymptotes to such behavior. Whether it starts off above or below its steady state value will depend on the elasticity of intertemporal substitution.

In the steady state where resources have been asymptotically depleted, and the share \(\epsilon \) has already reached its steady state value and consumption grows at a constant rate, the optimal tax rate asymptotically becomes a constant fraction of consumption according to \(\lim _{t\rightarrow \infty } \tau _t^o/C_t=\tilde{\tau }^o\), with
$$\begin{aligned} \tilde{\tau }^o=\kappa \frac{\alpha \eta \chi \phi }{\left[ \frac{\rho +\alpha (\sigma -1) \Theta _\infty }{\sigma }\right] \left[ \frac{\rho +\alpha (\sigma -1)\Theta _\infty }{\sigma }+\theta \right] \left[ \frac{\rho +\alpha (\sigma -1)\Theta _\infty }{\sigma }+\kappa \right] }, \end{aligned}$$
(23)
with \(\Theta _\infty =B-\eta (\delta +\chi P_\infty )\), and \(P_\infty =P_{0}+\phi S_{0}\), if \(\theta =0\), or \(P_\infty =P_{0}\), if \(\theta >0\); see “Appendix 7”.

The assumption of non-logarithmic utility does not allow for further analytical solutions; hence, we will confine ourselves to numerical simulations. To this end, we can rewrite the dynamic system of the social planner in variables which are asymptotically constant on a balanced growth path and then linearize the model in the proximity to the steady state. Our calculation procedure is explained in detail in the “Appendix 7”, while here we present only the main results of our simulation.

We consider a simplified version of the basic model without pollution decay for simplicity, i.e. \(\theta =0\). Pollution starts from \(P_{0}\), asymptotically reaching \(P_{0}+\phi S_{0} \) when resources have been depleted. From the no-arbitrage condition (16) we see that climate change affects the interest rate of the economy. This in principle creates a counteracting substitution and income effect. By combining the budget constraint (13), the Hotelling rule (15) and (16), we get the usual household budget constraint as, \(\dot{a}_{t}=r_{t}a_{t}-c_{t}+T_{t}\). Let’s think of an average interest rate between times 0 and t as \(\bar{r}_{t}=(1/t)\int _{0}^{t}r_{s}ds\) . The propensity to consume out of wealth is determined from15
$$\begin{aligned} \int _{0}^{\infty }e^{-(\bar{r}_{t}(\sigma -1 )/\sigma +\rho /\sigma )t}dt. \end{aligned}$$
(24)
A decreasing average interest rate due to pollution accumulation makes future consumption increasingly expensive compared to consumption today, motivating households to shift consumption from future to the present, i.e. an intertemporal substitution effect. This results in a falling capital share \(\epsilon _t\). On the other hand, agents experience a decreasing interest rate income which tends to reduce consumption levels in all periods. In the latter case, capital allocation in the investment sector is decreasing, indicating an increasing share \(\epsilon _t\). Which effect dominates will depend on the intertemporal elasticity of substitution, \(1/\sigma \). From (24), if \(\sigma >1\), the propensity to consume out of wealth is increasing with falling \(\bar{r}_t\), i.e. the substitution effect dominates. If \(\sigma <1\) the propensity to consume out of wealth decreases with falling \(\bar{r}_t\), i.e. the income effect dominates. If \(\sigma =1\) they both cancel out and the shares jump to their steady state values as in Sects. 3 and 4. A slow diffusion of emissions into the stock of harmful pollution tends to mitigate these effects: if the full effects of pollution on capital accumulation appear with a time lag, the reduction in the interest rate is purely delayed.
The same reasoning can be applied to the demand for the non-renewable resource and by extension to the carbon tax rate. Because of the Cobb–Douglas specification of the consumption good, indicating constant expenditure shares, a forward shift of consumption, for \(\sigma >1\), will result in a relatively higher demand for the non-renewable resource in earlier time periods, disregarding its scarcity. The social planner will then have to set a low \(\tilde{\tau }_t^o\) which is increasing as \(\epsilon _t\) falls. Following (8), a higher \(\kappa \) will have a magnifying effect on the net present value of marginal damages so the tax rate will be shifted upwards. In the appendix we solve for the socially optimal allocation when \(\sigma \ne 1\). The model is then linearized and solved computationally. The choice of the values for the parameters and the initial conditions is explained in “Appendix 7”, while Fig. 5 provides the results to illustrate our previous discussion for the standard case of \(\sigma >1\), as commonly used in the literature of endogenous growth; see Ikefuji and Horii (2012). Finally, as explained in “Appendix 1”, the speed of emissions diffusion is the reciprocal of the mean time lag. We choose a low value to reflect a time lag of 50 years, i.e. \(\kappa =0.02\), and a high value to reflect a time lag of 25 years, i.e. \(\kappa =0.04\); see van den Bijgaart et al. (2016).16
Fig. 5

\(t_0 = 2010, \rho =0.015, \sigma =1.5, \alpha =0.9, \theta = 0, \delta =0.05, B=0.106, \chi =1.7 x 10^{6}\) $/GtC, \(\phi =1, \eta =1, \kappa _{\text {low}} =0.02\) (dashed), \(\kappa _{\text {high}} =0.04\) (solid)

7 Conclusion

We use an endogenous growth model to study the effects of climate change caused by the extraction and use of nonrenewable resources. The central feature of the paper is the inclusion of a lag between greenhouse gas emissions and their effect on the stock of harmful pollution, which follows a well-defined time pattern. The time lag between emissions and their impact on the economy, here on capital accumulation, although important, has in general drawn little attention. The standard assumption in the literature of an instantaneous diffusion is the limiting case in our model.

Confirming results in the literature, with logarithmic utility, and our specification of damages to capital from the stock of pollution, the Pigouvian tax is a constant proportion of the consumption good in each time period. We therefore focus on general policies proportional to consumption and find that with log-utility, resource extraction is only determined by the tax rate. We also derive the crucial impact of climate dynamics on growth and resource extraction in private and social optimum. As regards optimal policy, the optimal per-unit emission tax rate increases in the dissemination speed; higher dissemination speed induces the economy to start at a higher level of economic growth. When pollution decay is not considered economic growth converges monotonically from above to its lower steady state, which is unaffected by policy; when pollution decay is considered, it may exceed the optimal level in the long-run. Finally, we study the effect of a more general CRRA utility function on the optimal carbon tax. We find that for a relevant value of the elasticity of intertemporal substitution above unity, and no pollution decay, the optimal tax grows initially faster than consumption while they asymptotically reach the same growth rate.

Political action is usually triggered only after environmental damages become visible. Therefore, a wrong perception of the speed of diffusion, e.g. a lower value for \(\kappa \), can lead to a suboptimal taxation policy, i.e. a lower tax rate. We draw from our results in the decentralized equilibrium and note that, in the general case of \(\theta >0\), an environmental policy that mistakenly sets a lower than optimal tax will force the economy to start from a point of lower economic growth, reach faster a relatively lower level of minimum growth but then recover at a faster rate towards the steady state; in the case of pollution decay economic growth might exceed the social optimum during transition in the long run. If no pollution decay is considered, \(\theta =0\), an erroneously set environmental policy will result in a lower than optimal economic growth at all times. In this case the economy will start from a low point of economic growth while resource extraction will be brought forward and the harmful results of extracting and using the polluting non-renewable resource will arrive sooner.

We argue that if emission taxes are not set by the social planner but by a regular political process, there is a risk of setting tax rates at too low a level when actors underestimate the true pollution dissemination speed. Underestimation of climate change and pollution dissemination has different reasons. The usually observed myopia of decision makers and short-run targets like elections are one component. Moreover, climate sciences provide results and predictions which naturally include a certain degree of uncertainty because they concern the very long run. Finally, reactions and decisions might rely on cognitive experience. When environmental damages become visible they have the best conditions to trigger political action. Because this is not yet the case for climate change, the concerns of too little political action appear to be warranted.

Footnotes

  1. 1.

    For example the Montreal Protocol on the ozone layer or the ban of asbestos.

  2. 2.

    For a survey of the literature on the relationship between environmental pollution and growth, see Brock and Taylor (2005).

  3. 3.

    In this paper the closed-form solution of the Pigouvian tax depends on the assumption of constant savings rate all along the optimal path. This can be ensured if capital depreciates fully each period which makes it a flow rather than a stock variable.

  4. 4.

    We thereby assume that even if carbon emissions seize, the stock of harmful pollution will not decrease further than its initial level; see for example Grimaud and Rouge (2014) for an equivalent treatment.

  5. 5.

    Our emissions-damage response does not allow for a thick-tailed concentration of the carbon stock, where some part of the stock stays in the atmosphere for thousands of years, as proposed by natural scientists. We could have captured such a behavior with a richer “multi-box” climate module as in Gerlagh and Liski (2012). This added complexity would however not alter the results of the present paper in any fundamental way while it would make the model less tractable.

  6. 6.

    In a previous version, in order to capture this idea, we differentiated between a physical and a knowledge capital stock, the latter being unaffected by pollution. Physical capital was accumulated as in the present version, while we assumed that creation of new knowledge was knowledge intensive and used only itself as an input. However the widely-used Cobb–Douglas specification for (3), implying constant expenditure shares among inputs, makes the use of two differentiated stocks inessential; the results are qualitatively identical as in the current approach, while the model is now more tractable. For an endogenous growth framework with knowledge capital (but no physical capital) and flow pollution directly affecting utility see Grimaud and Rouge (2005).

  7. 7.

    In general we define \(g_{V}\equiv \dot{V}/V\) the growth rate of variable V.

  8. 8.

    We will discuss the effect of \(\sigma \ne 1\) on the SCC, and thus on the Pigouvian tax, in Sect. 6.

  9. 9.

    In fact this implies an equivalence between the per-unit tax that grows with consumption, as in our case, and a decreasing ad-valorem tax, as usually proposed by growth models with polluting resources, e.g. Groth and Schou (2007). To see this, note that the consumer price for the resource is \(p_{R,t}+ \tau _{t}^o=\pi _t p_{Rt}\), with \(\pi _t \equiv 1+\tau _{t}^o/p_{Rt}\), i.e. a decreasing ad-valorem tax rate.

  10. 10.

    Grimaud and Rouge (2014), however, using a model of endogenous growth with polluting non-renewable resources, show that in the presence of Carbon-Capture-and-Storage (CCS) activity the optimal tax rate is linear in consumption, yet unique. In the presence of a CCS activity agents should be indifferent between instruments as long as they have the same results in protecting from climate change, which uniquely pins down the optimal tax rate.

  11. 11.

    Note also that, consistent with the literature, there are infinite ad-valorem taxes with the same dynamics (decreasing at the same rate) but different levels that give the same incentives to postpone extraction; see Dasgupta and Heal (1979), Grimaud and Rouge (2005) and Gaudet and Lasserre (2013).

  12. 12.

    Note, however, that the opposite applies for the initial level of consumption: using (3) and (18) with \(\epsilon =\rho /B\) and \(K_0\) given, it can be shown that \(dC_0/d\tilde{\tau }<0\), implying that higher initial growth goes together with a lower initial consumption level.

  13. 13.

    The complexity of the climate cycle does not allow for an explicit analytical solution. We can, however, approximate the solution using any mathematical software as an infinite sum of terms according to \(P_{t}=P_{0}+\kappa \frac{1-\alpha }{\kappa -\theta }\tilde{\tau }^{-1}\sum _{n=0}^{\infty }\left( \frac{1}{1-e^{\frac{S_{0}\rho }{1-\alpha }\tilde{\tau }}}\right) ^{n}\left( \frac{e^{-\kappa t}-e^{\rho nt}}{\kappa +\rho n}-\frac{e^{-\theta t}-e^{\rho n t}}{\theta +\rho n}\right) \). The interested reader can validate this expression to get the qualitative features of our climate system.

  14. 14.

    Using (21) in the no-tax case we can readily derive that \(P_{t}=P_{0}+\kappa \rho \phi S_{0}\left[ \frac{e^{-\rho t}}{(\theta -\rho )(\kappa -\rho )}\right. \left. -\frac{e^{-\theta t}}{(\kappa -\theta )(\theta -\rho )}+\frac{e^{-\kappa t}}{(\kappa -\theta )(\kappa -\rho )}\right] .\)

  15. 15.

    See, Barro and Sala-i-Martin (2003), Ch. 2.1.

  16. 16.

    In 2010 global consumption was around 49.8 billion US$ (about \(76\%\) of global GDP); World Bank Indicators, 2015. With this value, our calibration implies a carbon tax in 2010 between 50 $/tC (\(\kappa =0.02\)) and 75 $/tC (\(\kappa =0.04\)).

  17. 17.

    For convenience we will drop in this section the “\(^o\)” upper script, having however in mind that all results refer to the social optimum solution.

  18. 18.

Notes

Acknowledgements

This study has benefitted greatly by the comments and suggestions of Julien Daubanes, Alexandra Vinogradova and Andreas Schäfer. Moreover we are grateful for the suggestions of Sjak Smulders and two anonymous referees. We also thank the discussants at the SURED 2014 conference, Ascona–Switzerland, and at the 5th World Congress of Environmental and Resource Economists (2014), Istanbul–Turkey, for their valueable contribution.

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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.CER-ETH Center of Economic Research at ETH ZurichZurichSwitzerland

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