## Abstract

This paper makes suggestions for climate policy and defends them based on recent research in economics and the natural sciences. In summary: (i) the optimal carbon tax is rather modest; (ii) the key climate threat is coal; (iii) a carbon tax is to be preferred over a quantity-based system; (iv) the optimal tax on carbon does not appreciably harm growth; (v) subsidies to green technology are beneficial for the climate only to the extent that they make green technology outcompete coal; and (vi) a carbon tax is politically feasible.

## 1. INTRODUCTION

In light of our recent research, in this paper we present our views on how climate policy ought to be conducted. We summarize these views with six points:

1. The optimal carbon tax is rather modest. We judge an appropriate–from a global perspective–tax on carbon to be on the order of 25 cents per litre of gasoline.

2. It’s (almost) all about coal. The (estimated) reserves of oil and natural gas are small relative to those of coal and would only increase global temperatures rather modestly. The using up of a large part of our coal reserves, in contrast, presents a major threat to our climate.

3. A carbon tax is to be preferred over a quantity-based system.

4. The optimal tax on carbon does not appreciably harm growth.

5. Subsidies to green technology are beneficial for the climate only to the extent that they outcompete coal. We also argue that they may not even be necessary if an optimal carbon tax is used.

6. At this moment in time, we judge a carbon tax to be politically feasible. One often hears that carbon taxes are politically infeasible; we argue that they are likely not.

Our paper is designed to explain and support these points. In fact, the bulk of the paper builds a background by reviewing the recent research using integrated assessment models, i.e., models which jointly describe the natural science aspects of climate change with the economic ones. Although our discussion here contains some qualitative arguments, we place significant emphasis on quantitative conclusions from the literature. For this reason, we will briefly summarize the integrated assessment models used and how their parameters are calibrated.

An economic model of climate change driven by the emission of carbon dioxide needs to describe three phenomena and their dynamic interactions: ( 1 ) economic activity, ( 2 ) carbon circulation and ( 3 ) the climate. The economy is needed to model emissions and economic effects of climate change. The carbon circulation is needed to model how emissions translate into carbon dioxide concentrations at different points in time. Finally, one needs to understand how the climate is affected by the carbon dioxide concentration. Of course, all these systems are immensely complicated. In order to combine the mechanisms from natural science into an integrated model useful for conducting economic policy analysis, the different model blocks need to be expressed in a very simple form. The key complication is that, in a model with forward-looking economic agents, the outcome at any point in time depends on expectations about the subsequent future. Loosely speaking: the present depends on the future, a reverse causality that never arises in a natural science model, however dynamically complicated.

In Section 2, we describe a very simple, yet quantitatively reasonable, framework for capturing the key natural science mechanisms that we later embed in our full integrated economy-climate model. This framework, in particular, is simple enough that it can be used in the forward-looking economy-climate model that we use for our policy analysis. On another level, and perhaps more importantly, the description of the climate and carbon cycle determination serves to highlight the inescapable scientific mechanisms resulting in global warming. These mechanisms are not controversial per se but some quantitative aspects are uncertain; this will be highlighted in our presentation.

Section 3 goes over a key element behind the quantitative policy analysis, namely the measurement and modelling of damages from climate change. In Section 4, we then briefly discuss two simple integrated assessment models, one dynamic and one static. The static model captures most of the essence and builds very straightforwardly on the elements of a typical intermediate course in microeconomics; the dynamic model only adds marginally to this setting but allows a formal discussion of discounting, which has been discussed at length in this literature. Section 5 then goes over our policy messages and defends them based on the analysis in the earlier chapter and some additional, less formal arguments. Section 6 concludes.

## 2. THE NATURAL SCIENCE ELEMENTS

We begin by discussing the object of interest–the climate–and then the determinants of the main climate driver, namely, the atmospheric carbon concentration. The damages from global warming also contain natural science elements, but these are discussed in the next section.

### 2.1. The climate

A natural definition of the global climate is the distribution of weather events, i.e., realizations of e.g., temperature, precipitation, wind speed and ice coverage, over time and space. Clearly, a complete description of the global climate is infeasible. However, it turns out that there is a key state variable describing the climate: the global mean surface temperature. In simulations of advanced climate models, one finds that predictions for other climate parameters, like precipitation and regional temperatures, can be well approximated by simple functions of the global mean surface temperature. We will therefore briefly describe a simple model of how the emission of carbon dioxide affects the global mean temperature.

#### 2.1.1. The energy balance and temperature

The earth is heated by incoming short-wave radiation from the sun, and cooled by outgoing long-wave (infrared) radiation. An energy balance model describes how the global mean temperature changes over time as a result of these energy fluxes. The incoming short-wave radiation is 340 Wm −2 when averaged over the surface of the earth, and approximately one-third of this is directly reflected back to space. In equilibrium, the resulting net short-wave radiation must be balanced by the outgoing long-wave radiation.

We now consider what happens after a change in the energy budget. We take as a starting point a pre-industrial equilibrium state in which the incoming and outgoing energy fluxes were equal, and the global mean temperature therefore constant. We analyse a positive perturbation of the energy budget by the amount F (measured in Wm −2 , and called forcing ). Because of the perturbation, the incoming energy flux is larger than the outgoing flux, which leads to increasing temperature. This is described by the equation

(1)
$dTdt=σ(F−κT),$
where T ( t ) denotes the increase of the global mean temperature (measured in °C, i.e., centigrades) compared to the pre-industrial steady state. The forcing F ( t ) is determined by the CO 2 concentration through the greenhouse effect, which we will describe later. The term κT describes the fact that a higher temperature leads to a larger outgoing energy flux. 2 The parameter σ determines how quickly the temperature changes due to a given imbalance in the fluxes. It is inversely proportional to the heat capacity of the climate system, which is dominated by the ocean. If F is constant, the solution to Equation (1) with the initial condition T (0) = 0 (the pre-industrial state) is
$T(t)=Fκ(1−e−σκt).$

Asymptotically, as t , the climate approaches a new steady state:

(2)
$T∞=Fκ.$

Much climate research is devoted to determining the key parameters κ and σ . If the circulation and composition of the atmosphere would not change as the temperature changed, κ could be obtained from relatively simple radiation calculations, similarly as when calculating how blackbody radiation depends on the temperature. This gives $κ=3.2 Wm−2/°C$ , which would imply that a perturbation of the energy balance by 1 Wm −2 increases the equilibrium temperature T by 0.3 °C. Sometimes this simple mechanism is referred to as the ‘Planck feedback’. Due to various other feedbacks to be discussed below, κ is likely to be smaller than this value, i.e., the outgoing energy flux increases less with increasing temperature than what is implied by the Planck feedback. A given forcing then results in a larger temperature increase.

#### 2.1.2. Carbon dioxide and the greenhouse effect

Now consider the reason why a higher CO 2 concentration changes the energy balance, i.e., implies a positive forcing. The atmospheric gases are transparent to the solar short-wave radiation, whose maximum intensity is in the visible wavelength range. The most abundant gases, which consist of molecules with one or two atoms (such as nitrogen and oxygen), are also transparent to the outgoing long-wave radiation. However, gases consisting of molecules with three or more atoms, such as carbon dioxide, water vapour and methane, strongly absorb long-wave infrared radiation. Since the outflow of energy has a larger content of infra-red radiation than does the inflow, an increase in the concentration of greenhouse gases has a positive effect on the energy balance: a positive forcing F . Gases with this property are called greenhouse gases. The mechanisms behind this are well understood and easy to verify experimentally. Even a small concentration of such gases has a large effect on the energy balance of the earth.

The effect of the CO 2 concentration on the energy balance is well approximated by the function

(3)
$F=ηln⁡ 2ln⁡(SS¯),$
where S and $S¯$ represent the actual and preindustrial atmospheric CO 2 concentrations, respectively. The present concentration is 400 ppm, and the preindustrial value is 280 ppm. The exact value of the parameter η is not known, but a value of 3.7 Wm −2 may be used. 3 This means that a doubling of the CO 2 concentration leads to the forcing F = 3.7 Wm −2 . Since the perturbation is related to the relative change in CO 2 concentration, the formula is valid regardless of the units used for the CO 2 concentration. We will use the unit GtC, billions of tons of carbon in the atmosphere as a whole. The present value of S is then approximately 840GtC, and the value of $S¯$ is 600GtC.

Combining Equation (2) with Equation (3) , we find a relation between the CO 2 concentration and the steady state temperature:

$T∞=η/κln⁡ 2ln⁡(SS¯).$

The ratio η/κ is the heating that would arise in steady state after a doubling of the CO 2 concentration. Using the Planck feedback gives $η/κ≈1.2°C$ . This is a modest sensitivity, and very likely a too low estimate of the overall sensitivity of the global climate. The reason is that there are many other feedbacks. For example, a higher temperature will increase the atmospheric water vapour concentration, which adds to the forcing from CO 2 . A higher temperature will also change the size of the global ice cover and cloud formation, both having an effect on the energy budget. Formally, we can include the feedbacks in the energy budget by adding a term xT , giving

$dTdt=σ(F+xT−κT),$
where we now think of κ as solely determined by the Planck feedback. The steady state temperature is now given by
(4)
$T∞=ηκ−x1ln ⁡2ln⁡(SS¯)$

The coefficient $λ≡η/(κ−x)$ is called the equilibrium climate sensitivity and captures the response in the global mean temperature to a doubling of the CO 2 concentration. 4 Theoretically, we cannot rule out either x < 0 or $x≥κ.$ In the latter case, dynamics would be unstable and λ not well defined. This does not seem to be consistent with historical evidence. Also x < 0 is hard to reconcile with the observation that relatively small historical changes in the forcing appear to have had substantial effects on the climate. However, within these bounds a large degree of uncertainty remains. According to the latest judgements of the evidence, IPCC sets a likely range for λ to $3°C±1.5°C$ .

### 2.2. The carbon cycle

The global carbon circulation system is, of course, very complicated. However, we argue that a simple summary of how carbon depreciates is sufficient to give important insights into how the carbon circulation affects the economics of climate change. We base our summary description on IPCC ( 2007 a ) and Archer (2005) who claim that:

• one share (about 50%) of the emitted CO 2 leaves the atmosphere quite quickly (within a few years to a few decades),

• another share (around 20–25%) stays very long (thousands of years) until CO 2 acidification has been buffered, whereas

• the remainder decays with a half-life of a few centuries.

We model this by specifying a carbon depreciation function $d(s)$ such that 1 – d ( s ) describes the share of the emitted carbon that remains in the atmosphere after s units of time. Targeting the above simple summary, we set

(5)
$1−d(s)=ϕL+(1−ϕL)ϕ0(1−ϕ)s/10,$
and the parameters ${ϕL,ϕ0,ϕ}={0.2,0.38,0.023}$ for s measured in years. Figure 1 depicts the function $1−d(s).$

Figure 1.

Share of an emitted unit of carbon that remains in atmosphere against time since it was emitted

Figure 1.

Share of an emitted unit of carbon that remains in atmosphere against time since it was emitted

It is important to note that the linear depreciation structure is a simplification. Specifically, the rate of depreciation as well as the share that stays thousands of years depends on the size of emissions. To give an illustration of how sensitive the depreciation is to the emission scenario, IPCC (2013) shows that while around 20% of an emitted pulse of 100–1,000GtC remains in the atmosphere after 2000 years (in line with the summary above), almost 40% remains in the atmosphere after 2000 years of a pulse of 5000 GtC, i.e., a much more dramatic scenario. 5 Even after 10,000 years, the share in the atmosphere is above 20%. The half-life of the third share in the summary above is about twice as large in this scenario. 6 Later, we will use the simple linear model of carbon depreciation, but it is important to note that the parameters may need to be adjusted if extreme emission scenarios are considered.

## 3. DAMAGES FROM CLIMATE CHANGE

Climate change is believed to significantly affect human welfare, and to do so for a long time to come. However, the assessment and, more generally, quantification of this belief, is a huge challenge. In fact, we believe that the measurement of ‘damages’ from climate change is the single weakest element of the climate-economy package that scientists have to offer as a background for policymakers. Unfortunately, the missing knowledge includes both qualitative and quantitative parts. The qualitative parts involve the forms the damages take, and these forms are important for understanding how easy it might be to adapt to climate change. Depending on whether the bulk of the damages are due to a rising sea level or a ‘mere’ temperature increase, very different responses are appropriate. The quantitative issue is, moreover, central: how much emissions should be reduced (or how high taxes should be) will naturally depend on the size of the damages. There may be strong non-linearities in damages, so that small temperature increases are not so costly whereas increases above some threshold are close to disastrous; the existence of such thresholds are obviously crucial for policymakers. Important irreversibilities, for example due to loss of eco-diversity, are also possible. Damages, moreover, involve heterogeneous impacts across the world (and more generally across groups), making distributional concerns important. But as a summary statement, it is fair to say that we know very little–in fact, we have only recently begun to accumulate knowledge–in all these areas. This needs to be kept in mind throughout the policy discussion.

Damage measurements can perhaps usefully be put into two categories: top-down and bottom-up. The former looks at data on observable aggregates (such as output or mortality) and tries to relate them to climate, or merely temperature. An advantage of this approach is that one looks at aggregates, thus obtaining relatively broad-measured impacts. A disadvantage is that the method does not examine the mechanisms through which climate affects the aggregate: it does not indicate the specifics of the channel nor whether the link between the climate and the aggregate involves adaptation (and, to the extent there is adaptation, how costly adaptation was). The bottom-up approach looks at specific damages (say, output by narrowly defined sector or population segment), allowing a more careful study of mechanisms and adaptation. On the other hand, climate policy should be based on taking all damages into account, and thus coverage becomes a major issue. So far, there are relatively few studies of different disaggregated impacts, and they tend to be region-specific: the ‘world map of damages’ is so far to a large extent full of unchartered territory. We will briefly go over the main results coming out of (a subset of) this literature, each one in turn. A key output of this discussion is a (qualitative and quantitative) formulation of a ‘damage function’ that will be used in the integrated assessment model later.

### 3.1. Top-down studies

Researchers have used both cross-sectional and panel data. Cross-country regressions of outcome variables such as GDP on country-specific temperature suggest a clear negative relation, at least for sufficiently high initial temperatures. Of course, omitted-variable bias can be important in such regressions (as for example institutional quality appears to be rather strongly correlated with temperature). Mendelsohn et al. (1994) argue, for the case of agriculture, that a regression using regions within a country, with a fixed effect per country, allows a sensible control for institutions–under the assumption that the institutions within a country are very similar–and thus climate variability across regions within a country allows identification of a negative effect of temperature on output.

An influential study of temperature variations over time, in a broad cross-section of countries, is Dell et al. (2012) . Their focus is more on short-term variations in temperature, and climate change is therefore arguably not captured well by these regressions. They find rather small effects of temperature increases on output, but they do find effects on the growth rate of output. This finding is potentially important, since growth-rate effects imply much more potent effects on human welfare. An additional finding in this study is that of heterogeneous losses from temperature change: the growth-rate losses are only observed in countries that are poor. A later study using shorter, though more disaggregated, data, is that in Krusell and Smith (2014) who find growth rate effects of the same sign, but these are statistically insignificant; moreover, they find significant negative level effects on output of higher temperature and no heterogeneity in their estimates across poor and rich regions. Thus, these studies taken together clearly show negative effects of temperature increases of a magnitude similar to that found in Mendelsohn et al. (1994) , but there is still significant uncertainty about the specific effects. A longer panel, thus potentially identifying climate, as opposed to temperature, change is that in Bluedorn et al. (2010) . This paper finds rather weak, and statistically uncertain, effects of climate on current income, but non-monotonic effects historically.

### 3.2. Bottom-up studies

Nordhaus’s main calibration of his aggregate, as well as disaggregated, damage functions (we will discuss these functions below) is based on adding up detailed microeconomic estimates of the effects of temperature change. These damages take a variety of forms (e.g., effects on agriculture, sea-level rise, health, and non-market amenity impacts) and amount to a total of 0.48% of output for a 2.5-degree warming. Ciscar et al. (2011 , 2014 ) report detailed estimates for the European Union, covering a number of sectors in great detail. At a business-as-usual scenario leading to a 3.5-degree warming globally, damages in the EU due to climate change is estimated to 1.8% of GDP in the year 2080. This study is very ambitious in its coverage, but it is of course not possible to exclude the possibility that important impacts are missed.

Nordhaus’s damage function, however, relies not only on the bottom-up estimate but also on survey evidence. Here, researchers were asked to estimate probabilities of various pre-specified events. This survey resulted in a probability of 6.8% that the damages from heating of 6 centigrades are catastrophically large, defined as a loss of 30% of GDP. Nordhaus, moreover, calculates the willingness to pay for such a risk using a coefficient of relative risk aversion of 4 and adjusts the estimate up accordingly for use in an integrated assessment framework where uncertainty is not taken into account. Nordhaus thus adds the bottom-up information to that in the survey when selecting the parameters in his damage function. We will describe this damage function in the following section.

### 3.3. Damage functions

In this paper, we will primarily discuss climate policy from a global perspective and not so much address issues of distribution. This is not because we do not believe they are important; on the contrary, we do, both from the perspective of constructing an aggregate of the social cost of carbon and from a political-economy perspective. 7 However, discussing heterogeneity carefully would necessitate an extension of the analysis which is hard to fit into the present paper. Hence, we now focus on how global damage functions used in the integrated assessment literature are usually modelled and calibrated. 8

#### 3.3.1. Nordhaus’s damage function

Though damages appear in many places in the economy, Nordhaus early on adopted what has become the industry standard, namely a formulation where all damages appear in a factor that multiplies the aggregate production function of the economy. That is, a damage is then expressed as lower total-factor productivity, TFP. To cut to the specifics, the multiplicative damage factor, D , that Nordhaus uses in his most recent work is

(6)
$D(T)=1−11+0.00267T2≈0.0267T2.$

This expression is increasing in T , global temperature, so that output is $(1−D(T))Y$ net of damages (where Y defines output under no damages), no matter how (in what sectors, with what combination of production factors, etc.) output is produced. Also note that D ( T ) is convex in the relevant region (more precisely, below T = 11.2), so that the marginal damage factor is higher for higher temperatures. 9 It is broadly believed that damages increase at a higher rate as the global temperature rises and Nordhaus’s formulation is thus consistent with these beliefs. It should be added, however, that the size (and even presence) of the convexity has not been firmly established yet empirically.

It should be noted here that in terms of modelling, several other forms of damages (such as direct utility losses from higher temperature or higher depreciation of the capital stock) have the same analytical implications as the formulation with TFP damages. For a discussion of this equivalence, see Gars (2012) .

#### 3.3.2. A damage function expressed in terms of carbon concentration

It turns out, for the construction of a complete integrated assessment model, that a very valuable simplification can be achieved as follows: one can describe damages directly as a function of the level of atmospheric carbon concentration, rather than as a two-step function describing first how carbon concentration maps into temperature and then applying the damage functions above. The reason why this is a simplification is that the direct carbon-damage formulation can be calibrated with a functional form that is very analytically convenient. We noted above that there is a convexity in the temperature–damage relationship but a concavity in the carbon–temperature relationship ( Equation (3) ) above and these two together imply that, over the range of carbon concentration values that are empirically relevant, a linear-in-log relationship is a good approximation:

(7)
$D(T(S))≈1−e−γ(S−S¯),$
where γ is a constant. We refer to γ as an elasticity parameter because
$11−D ∂D(T(S))∂S=γ.$

That is, a marginal unit of carbon in the atmosphere has a constant proportional impact on output net of damages given by γ . Two different approximations have been made in Equation (7) . The first one is that T is treated as a function of S ; in effect, T is replaced by T . This means that the thermal inertia of the ocean is neglected, so that the temperature adjusts instantaneously to the CO 2 concentration. This approximation, which means that we model the effects of emissions on global warming as too large (over the transition period) appears acceptable since the focus is on the long run. More specifically, a key purpose of the model is to analyse the social cost of carbon. Our assumption implies that damages from an emitted unit of carbon accruing near in time after the emission are somewhat exaggerated. Unless the subjective discount rate is high, this is not of large quantitative importance, however. 10 The second approximation is that the functional form of $D(T∞(S))$ is simplified; the ‘error’ here is thus relative to Nordhaus’s formulation but it is of second order over a large range of emission scenarios. 11

It will, quite naturally, turn out that the calibration of γ is key for determining the social cost of carbon. What is so useful about this functional form is that the marginal damage elasticity does not depend on the current level of carbon concentration, output, or any other variable and the marginal damage elasticity is the key element in calculating the optimal carbon tax, as we shall see below.

#### 3.3.3. Remarks

Before proceeding, let us make a few remarks of caution against the backdrop of our initial point: that damage measurements is the area we know the least about.

An overall worry in damage measurement is that the historical range of climate variation–in any given region–for which there is useful economic data (on, say, output or mortality) is very limited compared to the increases in global temperatures that will likely result if a significant fraction of the remaining fossil fuels is used up. Over the last one hundred years, for example, we have seen the global temperature climb by about one centigrade but many projections forward involve increases of, say, five degrees. The only approach to damage measurements so far that allows any form of insight into what would occur at five or more degrees of warming is the cross-sectional top-down method referred to initially, since here one compares the outcome variables (like output) between regions in the world today with very different average temperatures. In fact, it is quite well known that there is a strong negative correlation between average temperature and GDP per capita, as depicted in Figure 2 . 12

Figure 2.

Average temperature 1950–1959 (blue) and 1996–2005 (red) versus GDP (PPP). Reprinted from Dell et al. (2012)

Figure 2.

Average temperature 1950–1959 (blue) and 1996–2005 (red) versus GDP (PPP). Reprinted from Dell et al. (2012)

From a natural science perspective, it may also be that five degrees of warming would involve potentially irreversible non-linearities that imply that the mapping from carbon concentration to climate variables such as temperature (e.g., through feedback effects) becomes more convex. In this case, our approximation to the damage function above will become worse. 13

There are potential non-linearities in damages. For example, humans appear to appreciate biodiversity intrinsically and biodiversity may involve tipping points. It may also be that ability to adapt to climate change is powerful within, but not beyond, a certain range of temperatures. The human body, for example, can handle (survive and be productive in) temperatures that are much higher than those in Europe today, but clearly there is a limit on temperature above which humans cannot survive or work productively. 14 If more were known about these processes and it would be possible to map them into observables, one would adjust the aggregate damage function appropriately.

One reason why some argue that damages are bound to be limited is adaptation in the form of migration: humans can, if temperatures rise enough, always migrate to colder areas. However, migration is associated with costs. Obviously, moving New York City away from the coast (which may be needed if the sea level rises enough) would be very costly, but there are also other costs of migration, especially in poorer countries. Harari and La Ferrara (2012) document how migration caused by poor agricultural outcomes can cause armed conflict, perhaps by ethnic violence. If people need to move across borders, one can easily imagine political and military conflicts. It is very difficult to assess these costs. One approach (followed, for example, by Desmet and Rossi-Hansberg (2015) , who study migration in a theoretical model of climate and the economy) is to imagine that costs are U-shaped, i.e., that there is an ideal temperature for every location. This is an interesting way forward and could be combined with costs of ‘crossing borders’. 15

## 4. THE INTEGRATED ASSESSMENT MODEL

The purpose of this section is not to go through the details of the typical integrated assessment models in the literature but rather to give one example of a model which can be understood based on basic microeconomics and which serves a useful framework for policy analysis. Any claims regarding how the model behaves here are substantiated in other papers, to which the reader is referred.

### 4.1. The dynamic economic model

Any economic model that is used for quantitative policy analysis of climate change should, in our view, have some basic properties. It should involve dynamics and long-run analysis. It should be similar, or a good approximation, to our standard frameworks from growth analysis–in this case the Solow model or, rather, versions of that model with (at least some degree of) optimizing saving. 16 It should allow for uncertainty. It should be based on microeconomic principles, so that standard welfare analysis can be conducted. These requirements are straightforwardly satisfied in a dynamic neoclassical model of the kind used in modern growth and business-cycle theory. While our model is not designed for business cycles, it could be altered to accommodate many views on business cycles were one to adapt it to short-run analysis–but that is not the purpose here.

There is a representative consumer in the model (now a stand-in for the average world citizen) with a utility function of a single good that is consumed at different points in time. The utility function involves discounting, a key element in evaluating policy, as well as a need for smoothing consumption over time. We abstract from population growth here for simplicity. 17 The consumption good is produced with an aggregate (world) production function of capital, labour, and energy, and it allows for technical change. We will assume that it has unitary elasticity across inputs. This does not appear restrictive in the case of capital and labor but may be restrictive when it comes to energy; in the short run, it seems much harder to substitute. However, over the longer run, technology choice is endogenous and the assumption of unitary elasticity is less inappropriate. 18 Capital is accumulated in a standard Solowian manner, taking consumption and investment to be perfect substitutes (a questionable assumption for short-run analysis but a reasonable one for long-run applications such as this one).

In this exposition we assume, for simplicity, that the energy sector is a pure coal sector and that coal is produced using labour only–the same kind of labour as is used to produce consumption and investment goods. The sole reliance on labour in the coal industry makes for closed-form solutions but is not realistic; however, it is not a serious flaw in the quantitative analysis since the quantitative effects on the main variables of interest are limited given the small share of coal production in GDP. Moreover, we assume that the damages to TFP from climate change appear only in the consumption/investment sector. This simplifies the algebra and is not of quantitative significance, since the energy sector is a rather small part of GDP.

Thus, using standard notation, the utility function of the representative world consumer is

$E0∑t=0∞(11+ρ)tu(ct),$
where t can be thought of as year t (and t = 0 is normalized as ‘now’), u is a strictly concave and increasing function of consumption, c , capturing a need to consumption-smooth as well as risk-insure, and ρ is the subjective discount rate. Using the damage function ( Equation (7) ), the resource constraint for the consumption/investment good reads
$ct+kt+1=e−γt(St−S¯)Atktαn1t1−α−νEtν+(1−δ)kt,$
with k denoting capital, A an exogenous TFP component that is possibly growing over time, n1 labour used in this sector, and E the energy (coal) input (and α and ν are exogenous share parameters); and that for coal reads
$Et=χtn2t,$
with n2 denoting labour used in this sector and χ an exogenous productivity factor that like A may grow over time. Market clearing for labour occurs when $n1t+n2t=1$ (we normalize labour to 1). Carbon in the atmosphere evolves according to a linear depreciation schedule, so that
$St−S¯=∑s=−TtEs(1−dt−s),$
where $1−dt−s$ is given by ( Equation (5) ). In this model, Greek letters are exogenous parameters of which γ t may be random; in addition, A t may be random as well.

It is straightforward to define what the socially optimal allocation is: a planner chooses sequences of consumption and energy subject to the above restrictions to maximize the stated objective function. One can similarly define a dynamic (stochastic) competitive equilibrium where all firms (including coal producers) make zero profits and consumers maximize their utility subject to budget constraints allowing saving. Importantly, in the market equilibrium, no agent takes the externality–how emissions E affect S and hence productivity–into account; the social planner, in contrast, does take this into account.

For this model, and much more general versions of it, it is straightforward (see Golosov et al., 2014 ) to derive the marginal social cost of emitting carbon at the optimal allocation, the OSCC (the Optimal Social Cost of Carbon). This formula says that the OSCC is the appropriately discounted value of current and future externality damages caused by a current emission of a unit of carbon. Appropriately discounted involves both the discount rate ρ and any other element due to non-logarithmic curvature and consumption growth; in the case of logarithmic utility discounting involves only ρ , regardless of the rate of consumption growth. Computing the current and future externality damages involves two factors. First, one has to figure out, for any future date s periods after the emission, how much of the initial emitted unit is still in the atmosphere. The answer is given by the depreciation parameter for carbon s periods out. The second factor, to be multiplied with the first, is simply the marginal externality damage on production of carbon present at that future date s . Finally, sum these damages across all future dates (and states, in case there is uncertainty).

Golosov et al. (2014) show that, under assumptions that are viewed as quantitative reasonable in the macroeconomic literature on growth, the formula simplifies radically. In particular, it turns out that we can express the OSCC for emissions at time t –or, equivalently, the implied optimal period- t tax à la Pigou, τ t –as

(8)
$τt=γδˆyt,$
where y t is output of consumption and investment goods at t , γ is the (expected) damage elasticity parameter introduced in ( Equation (7) ), and $δˆ$ is the appropriate combination of preference discounting and carbon depreciation. 19 The formula reveals that the optimal tax in dollars per ton is proportional to global output. This may seem counterintuitive, but is explained by the fact that the damage (the externality) is proportional to global output. Thus, the externality brings the finite size of the earth into the model; without the externality the optimal prices and rent would be unaffected by a doubling of c , k and n . The constant of proportionality $γδˆ$ in Equation (8) is given by structural parameters, independent of variables such as production inputs, the atmospheric carbon concentration (now and later), technology (now and later), and so on.

Calibrating the model requires assigning values to all its parameters (preferences, technology, etc.). However, only a (rather small) subset of these parameters are needed to find the value for the OSCC. We shall look at a calibration below in order to gauge what an optimal tax ought to be.

What is the optimal level of carbon emissions? It turns out that even in the simple model, the answer depends on all details of the model; even in its most stripped-down form, this is evident. For example, if labour productivity in the coal sector is very high, optimal coal use is higher, simply because its private cost is lower. Hence, in order to determine the optimal quantity, the planner needs to know the cost structure in the coal industry. The static model below will illustrate.

### 4.2. A simple static model

Now consider a conceptually much simpler model without a time dimension: utility is given by

$u(c),$
consumption from a static resource constraint by
$c=e−γ(S−S¯)Akαn11−α−νEν,$
where k is a fixed, exogenous factor, and energy (in the form of coal, again) from
$E=χn2,$
with labour market resources satisfying $n1+n2=1$ . Finally, carbon in the atmosphere is given by
$S−S¯=ϕE,$
where $ϕ$ represents the fraction of emissions ending up in the atmosphere, thus allowing depreciation within the static model. Clearly, this is the most straightforward static version of the dynamic model above. It should be pointed out that the simple model cannot formally be thought of as a steady state to, or a long-run outcome of, the dynamic model, but nevertheless the dynamic and static models are very similar and give rise to policy implications that have the same form. Of course, the parameters need to be reinterpreted; one can perhaps think of the static model as one of the outcome over a one hundred-year period (with no discounting during this period and infinite discount on any future after that, with $ϕE$ reflecting the average carbon addition to the atmosphere during the century if E is emitted every year for a hundred years, and so on).

Solving for a competitive equilibrium with a unit tax τ on carbon (i.e., for every unit of E purchased, the firm has to pay τ units of consumption) really just involves setting the after-tax marginal product of labour to be the same across the two sectors. This condition gives, after a minor amount of algebra,

$(1−α−ν)Ae−γϕEkα(1−Eχ)−α−νEν(νAe−γϕEkα(1−Eχ)1−α−νEν−1−τ)χ=1.$

This is one equation in one unknown: coal use E . The equation has an easy solution when there are no taxes ( τ = 0); otherwise, it is a non-linear equation in E that has to be solved numerically. In the case without taxes, the equation becomes

$(1−α−ν)E=ν(1−Eχ)χ ⇒ E=χν1−α.$

This equation is fairly simple: neither the TFP parameter A nor capital, k , or the damage and carbon depreciation parameters, γ and $ϕ$ , end up mattering for the determination of coal use/the energy provision. What is important is productivity in the coal sector ( χ) and the relative cost share of energy in production ( $ν/(1−α)$ ). In contrast, what is socially optimal is given by

$γϕ+1−α−νχ−E=νE,$
now naturally also involving both γ and $ϕ$ . 20 These two equations are not the same but we notice, going back to the equation determining market energy use for an arbitrary energy tax τ , that if the tax is set to satisfy
$τ=γϕAe−γϕEkα(1−Eχ)1−α−νEν=γϕc=γϕy$
then the market allocation would be socially optimal. Thus, $γϕy$ is the Pigou tax in this case. This is intuitive: this amount is precisely the OSCC as given by the damage externality caused by coal use.

We will return to implementation below but it is important to note here that the OSCC is proportional to output through only the product of γ and $ϕ$ , i.e., the damage elasticity and carbon depreciation. We saw above that another parameter matters as well in the dynamic analysis–discounting, as given by ρ there, but because the present analysis is static this parameter is not present–but we must note here that the nature of the solution is extremely similar across the static and the dynamic setting. Thus, both analyses point to the fact that a carbon tax requires relatively little information to implement. In contrast, a quantity regulation, hence going straight at what E ought to be, is more demanding–it requires knowledge also of details about coal production and how it impacts on GDP. Moreover, with slight (and reasonable) extensions of this setting, one realizes that the formula for the optimal tax is barely affected by population growth and other technology parameters (especially concerning green energy) that instead are crucial quantitatively when regulating quantities.

## 5. POLICY ANALYSIS

We now provide a sequence of remarks on climate policy. The sequencing is put together in an order that, roughly speaking, has decreasing ties to the formal analysis above. For example, our first point is a quantitative evaluation of what the optimal carbon tax ought to be, whereas the final points have to do with political challenges in implementing different kinds of policies, a subject on which the above analysis is silent since it does not directly touch on politics (some insights from the formal analysis do, however, have implications for a political economy analysis).

### 5.1. The optimal tax on carbon

As shown above and by Golosov et al. (2014) , under assumptions that are viewed as quantitatively reasonable in the macroeconomic literature on growth, a simple formula for the OSCC can be derived. The formula deserves to be repeated here since we will now give it quantitative content:

$τt=γδˆyt,$
where y t is global output of consumption and investment goods at t , γ is the (expected) damage elasticity parameter in ( Equation (7) ), and $δˆ$ is the carbon duration using the subjective discount rate. Specifically,
$δˆ=∑s=0∞(11+ρ)s(1−d(s)).$

We then obtain

$δˆ=ϕL(1+ρ)ρ+(1−ϕL)ϕ0ϕ+ρ,$
using the formulation of carbon depreciation in ( Equation (5) ). 21Golosov et al. (2014) calibrated γ to $2.4×10−5$ based on Nordhaus (2007) . As discussed above, there is substantial uncertainty about the damage function. IPCC ( 2007 b ), reports expected damages at 4 degrees heating to be 1–5% of GDP as shown in Figure 3 .

Figure 3.

Global damage estimates. Dots are from Tol (2009) . The solid line is the estimate from the DICE-2013R model. The arrow is from the IPCC (2007b ) page 17. Reprinted from Nordhaus and Sztorc (2013)

Figure 3.

Global damage estimates. Dots are from Tol (2009) . The solid line is the estimate from the DICE-2013R model. The arrow is from the IPCC (2007b ) page 17. Reprinted from Nordhaus and Sztorc (2013)

Let us consider a calibration of γ based on the upper end of this range. Using the Arrhenius Equation (4) with a climate sensitivity of 3 °C, we find that to obtain an increase in the global mean temperature by 4 °C, the atmospheric CO 2 concentration is $600e43ln⁡ 2=1512$ GtC. Finally, we use the damage function $D(1512)=1−eγ(1512−600)=0.05$ to solve for γ , giving $γ=5.624×10−5.$ It may be noted that the calibration of γ in Golosov et al. (2014) corresponds to substantially lower damages, but still within the range reported by IPCC ( 2007 b ), namely 2.2% at 4 °C. Setting global GDP to 650 trillion euro per decade and using the carbon depreciation parameters described above, the formula for the optimal tax per GtC is given by the following expression. 22

$5.624×10−5·650×1012(0.2(1+ρ)ρ+(1−0.2)0.380.023+ρ).$

In Figure 4 , we plot the optimal tax per ton of carbon against the subjective yearly discount rate. We also plot the tax for the more optimistic calibration of γ from Golosov et al. (2014) .

Figure 4.

Social cost of carbon (optimal tax) in 2015 Euros per ton of carbon as a function of the subjective discount rate for two calibrations of the damage elasticity

Figure 4.

Social cost of carbon (optimal tax) in 2015 Euros per ton of carbon as a function of the subjective discount rate for two calibrations of the damage elasticity

To obtain some perspective on the tax, it may be helpful to note that a litre of gasoline contains about 0.64 kg of carbon. A tax of, say 400 Euro/ton carbon, which is about the same level as the current tax in Sweden, therefore corresponds to 0.25 euro per litre of gasoline.

#### 5.1.1. Risk

An important finding in Golosov et al. (2014) is that, despite the presence of uncertainty and risk aversion, what matters for the optimal tax is only the expected value of the damage elasticity γ and not the degree of uncertainty about it. 23 This is important since there is substantial uncertainty around all the underlying mechanisms determining the relation between CO 2 concentration and damages. It is, however, at least as important to realize the caveats to this result. To derive an approximately constant damage elasticity, we used the logarithmic relationship between CO 2 concentration and temperature in ( Equation (4) ) and the moderately convex relationship between temperature and damages in ( Equation (6) ). Deviations from these smooth relationships could make risk an important factor. Examples of such deviations would be thresholds in the climate system or carbon circulation. It could be the case that at a particular temperature some positive feedbacks in the energy budget suddenly become much more potent. Similarly, the carbon circulation system could abruptly change when some level of concentration is passed. 24 Similarly, it may be the case that the quadratic damage function severely underestimates the convexity of damages as a function of the global mean temperature. In any of these cases, uncertainty starts to matter. A study arguing that it is the tails, and not the mean, that matter is Weitzman (2009) . In Weitzman (2012) , a damage function that becomes extremely convex at high levels of global warming due to a term with a power of 6.754 in temperature is used. This function is chosen to capture an assumption that at 12 °C heating, the loss of GDP is 99%. Weitzman argues that it is hard to rule out a climate sensitivity of 12. In an example, he sets the probability of this event to around 1%. In this case, a mere doubling of the CO 2 concentration may have such dramatic consequences that they must be avoided, even when the costs are high. Also quite unlikely events may warrant forceful policy intervention if these events are sufficiently bad. A high carbon tax, or a very tight emission quota, can be an insurance against a catastrophe. Of course, in this case most likely, paying the insurance premium will ex post turn out to be of no value, but nevertheless it is worth it ex ante .

While we agree with the logic of Weitzman’s argument we also insist that policy be based on quantitative evaluations: it is not sufficient to refer to abstract arguments. There are many potential catastrophes of other forms that cannot be ruled out–a killer flue, super tsunamis, comets colliding with earth, giant volcano eruptions, etc.–and these could potentially be as far-reaching and damaging to mankind as CO 2 emissions. Although it is an extremely difficult task, our view is that to set policy and devote resources, policymakers must, with the help of scientists, try to quantify and weigh different such risks against each other (and against other areas of spending) in order to then be able to devise a desirable, cost-effective policy mix. So until measurements are available that go beyond the mere idea about how tail uncertainty can be disastrous (if it is large enough and risk aversion is high enough), we prefer to base our analysis on available scientifically based estimates, of course allowing robustness around them. The notion ‘prudence’ is often mentioned in this context and, clearly, prudence is called for, but the question is how much–a question that currently has no good answer. Another aspect of this issue is that, precisely because many suspect that non-linearities and irreversible mechanisms exist and are central, much more research on it is needed.

Above, we have argued that the optimal tax is fairly modest and that it is independent of the emission scenario. Our trust in this depends on how difficult it will turn out to be to reduce emissions. If it turns out to be much more difficult than we expect, the optimal tax calculated with our formula may lead to more emissions and thus more global warming. At some level, our trust in the formula will then fade. The same thing applies if it turns out that the climate sensitivity is much higher than expected. 25 The implication of this is that international negotiations on climate change should first establish a global carbon tax at a reasonable Pigouvian level. Most likely, such a carbon tax would be effective in the sense of curbing climate change. However, if we obtain indications that this is not the case, and emission forecasts risk taking us into carbon concentrations and temperatures about whose consequences we have very little knowledge, more forceful measures should be considered. Without such indications, however, if the immediate focus is on very strong measures we fear that nothing will be achieved in the negotiations.

### 5.2. It’s (almost) all about coal

A conventional oil or gas reserve is an asset with a positive value. As we all know, finding oil reserves has made countries rich. This reflects the fact that the average extraction cost for conventional oil and gas for a long time has been much lower than the price–the price has a rent component. As long as a tax does not eliminate this rent, i.e., does not drive the price net of taxes below the extraction cost, extraction remains profitable and is then likely to continue. Even quite high global carbon taxes are unlikely to eliminate the rent for a large share of existing conventional oil and gas reserves. These will therefore be exploited regardless of whether global carbon taxes are introduced or not.

We should note that if the carbon tax is set to reflect the damages caused by emissions, exploiting these reserves is socially efficient if it is privately profitable. That is, in such a case the social value of using the reserves is higher than keeping them unexploited also when the externalities are included in the calculations. The current estimate of existing conventional oil and gas reserves indicates that indeed these reserves are not large enough to pose a substantial threat to the global climate. Current estimates of oil and gas resources indicate a stock of 300GtC. 26

Currently, the amount of carbon in the atmosphere is about 840GtC. Assuming, fairly conservatively, that half of carbon emissions stay in the atmosphere for an economically long horizon, emissions of 300GtC would lead to an increase in the carbon concentration of 18%. Using Equation (4) with a climate sensitivity of 3 °C, this leads to an increase of the global mean temperature of $3ln⁡1.18ln⁡ 2≈0.7°C$ . This is certainly not trivial, but neither does it appear to be a major threat.

For coal, the situation is very different. First, no countries become rich by finding coal reserves. This is due to the fact that extraction costs are close to market prices–rents are negligible. This in turn means that a relatively small reduction in the price before taxes makes a lot of coal extraction unprofitable. In contrast to oil, a carbon tax therefore has the potential to have a large effect on extraction.

Second, coal reserves are substantially larger than oil and gas reserves. Official global coal reserves are 640GtC. 27 However, it is likely that this is a substantial underestimate. Since coal is priced close to extraction cost, the value of searching for new coal mines is limited. In fact, Rogner (1997) estimates coal reserves to be 3,500GtC with a marginal extraction cost curve that is quite flat. 28

Thus, the conclusion is that a carbon tax is unlikely to have a large effect on the use of conventional oil but that this is not a major problem. On the other hand, a carbon tax is likely to have a large effect on coal use and limiting coal use is therefore of utmost importance. As an example, the Swedish CO 2 tax is SEK 4,110 (430 euros or, in USD, $485) per ton of carbon. 29 The average price of oil in 2015 was approximately$100 per barrel, corresponding to $733 per ton of oil, and the average price of coal in northwestern Europe the same year was$75 per ton. In per cent of the fuel price, the Swedish CO 2 tax was thus 55% for oil and 460% for coal. 30 Clearly, such a tax makes coal use uneconomical while gasoline use has not collapsed in Sweden. Also a more modest tax, of say $100 per ton of carbon, would likely have very large effects on coal use but only modest ones on oil use. The fact that the coal price is fairly close to the extraction cost in combination with the fact that coal supplies are fairly evenly spread over the major regions of the world explains why global trade in coal is limited. Figure 5 shows coal production for the major regions of the world in the left panel. Consumption is shown in the right panel. As we can see, the two panels are fairly similar implying that trade between the regions is limited. This contrasts sharply with oil, where global trade is very important. As is seen in Figure 6 , oil production and consumption do not correspond to each other. These differences between coal and oil also have important policy implications. Given segmented markets, a reduction in coal use in one region of the world is not likely to affect the price and the use in other regions. The market for oil is not segmented in this way, implying that a reduction in demand in one region is likely to affect the world market price negatively and thus increase consumption in other regions. Figure 5. Coal production and consumption. Reprinted from BP (2015) Figure 5. Coal production and consumption. Reprinted from BP (2015) Figure 6. Oil production and consumption. Reprinted from BP (2015) Figure 6. Oil production and consumption. Reprinted from BP (2015) ### 5.3. Taxes versus cap-and-trade Climate change driven by emission of greenhouse gases is an almost perfect example of an externality. The benefits from using fossil fuel are private to the emitter, but since CO 2 quickly mixes in the atmosphere, it has global effects that are independent of who emits and where the emission takes place. Such a text book case of an externality implies that unregulated markets will not lead to a socially efficient level of emissions. In theory, the market failure due to the externalities induced by CO 2 emissions can be solved by quantity restrictions as well as with Pigouvian taxes. 31 To illustrate this point, consider a simple static case when emissions have private benefits and social costs. The private benefits represent the net value to the user of burning fossil fuel in excess of the costs associated with producing the fuel, for example due to extraction and refining. These benefits accrue to the user (consumer surplus) and to the producer (profits) in shares that are determined by market conditions. Here, we are only concerned with the sum of these benefits, not how they are split. In an unregulated and efficient market, profit opportunities are exploited and this will imply that marginal private benefits will be driven down to zero. Emissions also have social costs. These are assumed to be external, i.e., they are neither borne by the emitter nor the fuel producer. Instead, they are borne by a large number of agents spread around the world. The market transaction will therefore be undertaken as if there were no social costs. 32 Let us now depict this graphically. In Figure 7 , the blue solid curve represents the marginal private value of emissions and the red solid curve the social cost. In an unregulated market, emissions will be Q LF since all private gains will be exploited. However, the social optimal emission quantity is Q *, where the marginal private benefits are equal to the marginal social costs. The optimal allocation can be implemented by either a tax per unit of emissions equal to τ * or a quantity restriction (a quota) at Q *. A way of implementing the quantity restriction is to require all emitters to purchase an emission permit that is provided in the quantity Q *. If these permits are traded, their unit price will be τ *. Note that in this case, information about the position of both curves is required to find Q * as well as τ *. Figure 7. Marginal private value of emissions (blue) and marginal social cost (red) Figure 7. Marginal private value of emissions (blue) and marginal social cost (red) Now, recall the discussion in Section 4, where we concluded that a reasonable approximation of the optimal tax is independent of the emission quantity (because a concavity in the emission-to-carbon concentration mapping cancels with a convexity in the carbon concentration-to-damage mapping, thus making the emissions–damage relationship linear). Moreover, this approximation is also remarkably robust. With this result, a graphical representation of marginal costs and benefits of emissions instead can be depicted as in Figure 8 . 33 There, the curve representing the marginal social cost of emissions is horizontal. Now, it is no longer the case that information about the position of both curves is necessary in order to find τ *. In fact, no information about the private value of emissions is needed. Equivalently, changes in the curve representing the marginal private value of emissions, e.g., from the solid to the dashed line in Figure 8 , will change Q * but not τ *. Figure 8. Marginal private value and marginal social cost (constant) Figure 8. Marginal private value and marginal social cost (constant) The finding that the marginal cost curve is flat and its implication that τ * is independent of the private benefits of emissions point to important benefits of using taxes rather than quotas. First, less information is required. This is a value in itself and can also make it easier to come to agreements about the right policy. Individual countries have incentives to misreport their own marginal value of emissions since in a multi-country version of the analysis, the optimal allocation of emission quotas between countries depends on the individual marginal values. Such an incentive does not exist if the policy instrument is a tax. Second, the marginal value of emissions varies over time, in part due to changes in economic activity, which shifts the blue curve in Figure 8 . As the global financial crisis hit the economies of the world, for example, it shifted to the left, and since the stipulated quota Q * did not shift, this led to a collapse of the price of emission rights in Europe. This is illustrated by the new price $τ∗′$ in Figure 8 , far below the marginal social cost. The gap is caused by the failure to predict the appropriate quota, which should have been reduced to $Q∗′$ . With the price collapse regulation itself collapsed. 34 The high volatility of emission rights have been observed in the past also for other pollutants such as e.g., sulphur dioxide. 35 It should be pointed out that for the climate, what matters is not so much whether the price of carbon fluctuates but what its average price is–since the climate is so slow-moving. However, unpredictability of prices is really undesirable for business, i.e., from a pure economics perspective. A case in point is the massive investments in coal made by the Swedish (state-owned) company Vattenfall that turned out to be extremely unsuccessful after the fall in energy prices, and in the value of the free emissions allowances that were obtained with these investments because of grandfathering, so much so as to be a contender for the worst business deal in Swedish history. To the extent that different emission permit markets are not perfectly integrated, the trading system can also lead to large discrepancies in the emission price faced by different agents. This will lead to unnecessarily high costs of achieving a given amount of emission reduction. 36 Third, an efficient transition to a less fossil fuel-dependent economy requires long-term policy commitments. However, predicting the future value of emissions is difficult since, for example, the speed at which alternative energy sources are developed as well as GDP growth rates are hard to predict. Unconditional commitments to a path of emissions are therefore hard to make credibly, especially for economies that change rapidly, and if they were made, they could turn out to be quite suboptimal. Conditional commitments, where emission paths would be made depend explicitly on the factors that drive emission values, seem too complicated to be a way forward. To introduce and commit to keeping a reasonable level for a carbon tax has much fewer of these problems. We also think that a conditioning of the future tax on new information about the global flow cost γ , carbon duration D , and global GDP, is likely to be credibly implementable. ### 5.4. The optimal tax will not be very harmful A common question asked by practitioners and the informed public is whether the proposed carbon tax will harm growth, or welfare, greatly, while of course having benefits in the form of a more agreeable climate. One could attempt to answer this question with an empirical study that convincingly would identify the effects of carbon taxation on growth. One could alternative use (quantitatively restricted) theory to make predictions. We will briefly discuss both. Our overall conclusion is a ‘no’, given our (admittedly tentative) evidence. In terms of empirical evidence, let us first specify the questions we want to answer. One question concerns a single, and perhaps small, country adopting a tax when other countries do not. Another question involves the effects of a world-wide common tax. The first of these questions–which includes the phenomenon of ‘leakage’ of economic activity and energy use from high-tax to low-tax countries–could potentially be answered empirically, but we are not aware of any studies that convincingly establish causality. The second question seems even harder to answer. However, let us make some observations. First, from our own perspective–that of Sweden–let us point out that a carbon tax on a high level has been in place now for 25 years and that there is no indication that Sweden has done particularly poorly over the same period. 37 This, of course, can be due to other counteracting factors, such as a wave of deregulation and lower tax rates on income. Nevertheless, it is hard to imagine that our carbon taxes really have been very detrimental to Swedish growth. 38 Second, there is ample evidence that there is major scope for straightforward energy-saving measures (such as cheap insulation, ‘closing windows and shutting off machines’, etc.) and technological advances directed at saving energy. For examples, the following chart illustrates how the energy efficiency, measured by GDP in US dollars per unit of energy, is very different even across developed economies: for example, it differs by a factor of five between Iceland (low efficiency) and Switzerland (high efficiency) as seen in Figure 9 . These differences, we suspect, likely reflect differences in energy prices across countries (and these energy prices in turn surely reflect supply factors as well as policy). Thus, there appears to be great scope for energy saving. A second empirical argument for energy saving is contained in Hassler et al. (2012) who use post-war US post-war data to back out a time series for energy-saving technical change; we reproduce the graph below in Figure 10 . As can be seen below, this series was essentially flat until the oil price shocks hit and then started growing substantially and persistently. The effects, moreover, are quantitatively large. Figure 9. Energy efficiency. GDP US$(2005 PPP) per unit of energy (kg oil equivalent)

Source : Worldbank, World Development Indicators Online.

Figure 9.

### A politically feasible policy?

Hassler et al. claim that their proposal of a carbon tax is politically feasible. Their claim is supported by three arguments: ( 1 ) Even countries with an avowed distaste for taxes have some areas in which taxes are high, such as US property taxes. ( 2 ) Carbon can be taxed highly, as the presence of high gasoline taxes in Sweden and, for that matter, in most EU countries shows. ( 3 ) The revenue collection of the global carbon tax can always be devolved to the local level in order to harness the inherently greater political appeal of using carbon tax revenues to reduce other taxes or increase transfers at the local level, compared to revenue recycling at higher levels of government. These observations have merit, and they go beyond the argument for policy harmonization in the early debates on global climate policy ( Cooper, 1998 ). Hassler et al. could have added that there is even experimental evidence that supports their third argument: Kallbekken et al. (2011 ) find that more narrow targeting of revenue recycling lowers opposition to an externality-correcting tax. (They also find that not calling it a ‘tax’ helps.) But is the evidence sufficient for building a persuasive case that a globally uniform carbon tax is politically feasible? First, already at the domestic level carbon taxes are typically not uniform, but differ across sources. Sweden is a case in point: The effective carbon tax for stationary industrial sources in Sweden has evolved in very different ways over the last ten years compared to residential or mobile sources due to exemptions and discounts. This has led to considerable domestic carbon tax differentials in Sweden, confirming the rule rather than the exception in OECD countries ( OECD, 2013 ). That rule is that it is difficult to impose a uniform carbon tax even within a single country. Secondly, many political arguments militate against imposing the same uniform carbon tax across countries. At €100 per ton of CO 2 , the first-order redistributive impacts of such a tax on developing countries, even if they were able to collect it, would be tremendous. Take the case of Egypt: Based on World Bank data, the revenues from a $485/tC tax on the estimated 5.8 × 10 7 metric tons of carbon emissions from fossil-fuel burning, cement production and gas flaring in 2008 ( Boden et al., 2011 ) would be roughly the same ($28bn) as the Egyptian government’s total tax receipts in that year (\$25bn). Is it plausible that developing countries will agree to policies with such impacts? In my view, a more likely outcome of an international negotiation process on a carbon tax would be that different countries would be allowed to choose different average carbon tax rates based on differences in per-capita income, historical contributions to atmospheric carbon stocks, etc. International negotiations will therefore not escape the same sticking points we see in negotiations about emission quantities just because they are framed in a tax context. The third, and final, argument against the political feasibility of the proposed tax is that even in a world in which countries, and local communities, nominally accept a uniform global tax rate, the tax-raising entities will still find it in their interest to lower the effective tax rates through variations in tax collection and tax enforcement in order to attract capital. Experiences in federal systems are instructive on this point ( Helland, 1998 ; Fredriksson and Millimet, 2002 ). Taken together, the empirical evidence is therefore not obviously in line with a high political feasibility of a globally uniform carbon tax.

### The best of all possible instruments?

Instrument choice in climate policy involves complex trade-offs between effectiveness, cost, uncertainty, dynamic incentives, flexibility, monitoring and enforcement, equity and acceptability (see e.g., Harrington and Morgenstern, 2004 ). In the context of climate policy, instrument choice is typically narrowed down to a debate of taxes versus cap-and-trade, but this dichotomy does not exhaust the available menu, something every instrument ranking should keep in mind. The narrow debate of taxes versus cap-and-trade has been conducted vigorously, with many observers (see e.g., the various contributions in Hansjürgens (2005 )) coming down in favour of cap-and-trade despite acknowledging the drawbacks of price volatility that Hassler et al. also mention. One candidate explanation for this could be different views on the shape of the marginal damage function. However, Hassler et al’s derivation of an essentially flat marginal damage function only reinforces the existing consensus in the climate policy literature. The better explanation lies therefore with concerns over non-linearities in the climate system once atmospheric carbon stocks exceed yet unknown thresholds. To these concerns cap-and-trade answers by providing the policymaker with control over the quantity of emissions. Hassler et al. acknowledge the presence of these uncertainties, yet come down strongly in favour of a tax-based climate policy on the grounds of balancing ‘prudence’ with what is currently known. As things stand, there is no clear normative framework that favours one or the other position on this. But it is in any case worth pointing out that the disagreement is less about the shape of the damage curve than about the proper way to deal with fundamental uncertainties about the response of the climate system to increases in the atmospheric carbon stocks beyond historic level. In light of these uncertainties, the ability of climate policy to adapt to new knowledge needs to be considered. In terms of flexibility, a comparison of the relative merits of tax versus emissions trading will always hinge on details of policy design: A global tax, changes to which would have to be renegotiated, entails greater regulatory commitment, but provides little flexibility. A centralized authority that controls permit supply for a global carbon market could adjust relatively more quickly as a matter of business. As the literature has been pointing out for some time now ( Harrington and Morgenstern, 2004 ), instrument choice is subtler than ‘tax vs. emissions trading’. The merits of a carbon tax itself depend to a considerable extent on how revenue recycling is carried out ( Goulder and Hafstaed, 2013 ). Finally, it is not clear that tradable permit systems are politically infeasible because the bargaining is over a fixed pie. Existing emissions trading schemes have successfully solved that bargaining challenge.

Hassler et al. make three other points, namely the desirability of preventing the large-scale exploitation of the planet’s coal reserves, the modest growth impacts of their proposed carbon tax and the questionability of R&D subsidies. One can quibble with some of their interpretations of the literature. But as a well crafted and bold statement about the desirable core features of a global climate policy, this paper makes a real contribution to the debate.

## Ingmar Schumacher

### IPAG Business School

In the paper ‘Climate Policy’, the authors John Hassler, Per Krussell and Jonas Nycander forward several important points that they argue should be taken into account by climate policy. They base the arguments that lead to these points on a review of the integrated assessment literature, as well as on the approach and subsequent results derived in Golosov et al. (2014 ). In this discussion I pick out two of their main points, which are ‘[w]e judge an appropriate … tax on carbon to be on the order of ’ 400 euro per ton carbon; as well as the point that climate policy is ‘(almost) all about coal’. With respect to the first point, I discuss more closely the implication of relaxing some of their assumptions, while I hope to show that their second point may potentially be missing some important observations.

The authors rely on several crucial assumptions in order to argue that an appropriate carbon tax is around 400 euro per ton of carbon. In particular, the authors assume a simplified carbon cycle, logarithmic utility, a Cobb–Douglas production function, full depreciation of capital, no population growth, no Total Factor Productivity (TFP) growth, as well as a negative exponential damages function on output. Together with a constant ratio of consumption to output, the authors argue that the optimal carbon tax is proportional to output. I discuss the implication of generalizing some of these assumptions, or putting them in perspective by relating them to other works. 50 In particular, I will look more closely at the linear carbon cycle, whether or not population growth and TFP growth change the results, the role of the discount rate and the intertemporal inequality aversion, and finally asking whether this carbon tax will fulfil the goal they raise (making coal uneconomical).

Most researchers generally agree that simple, clear models help build intuition and understanding. In this sense, the approach and results put forward in Golosov et al. (2014 ) and used in this paper are without doubt very important. The downside is that analytical tractability tends to require a lot of Occam’s razor together with specific functional forms, without which these clear and simple results would often be unachievable. This point is, of course, even more relevant when it comes to such far-reaching topics as economic systems that are studied together with climate feedbacks. For this reason integrated assessment models tend to be big, black boxes. However, these integrated assessment models are so big simply because all ingredients are deemed necessary. Thus, the question is whether certain approximations, such as those taken by the authors, can readily be used to forward a single carbon price.

A major criticism of the approach in Golosov et al. (2014 ), and thus also applicable to Hassler et al. (2016 ), has been the carbon cycle. They assume that an increase in emissions immediately increases temperature, while a more realistic climate system would lead to a delayed response between emissions and temperature (after roughly 80 years). The contributions by Gerlagh and Liski (2012 ), van den Bijgaart et al. (2016 ) and Rezai and van der Ploeg (2015 ) show that in this case carbon prices should be roughly half those predicted by Golosov et al. (2014 ) and Hassler et al. (2016 ). Hence, based on their estimate, this would drop the carbon price to around 200 euro per ton of carbon.

Rezai and van der Ploeg (2015 ) depart from Golosov et al. (2014 ) by allowing for non-multiplicative damages, temperature lags, population growth, persistent growth and an intertemporal elasticity of substitution different from 1. In particular, they find that population growth will not have a significant impact on the carbon price. According to the World Population Prospects: The2015Revision , the world population growth will be declining from currently 1.3% to 0.2% by 2100. This may be too little in order to affect carbon prices. However, it would be interesting to know what happens under GDP convergence, i.e. if the poor catch up to the rich and thus the additional population pollutes at a similar level as the rich do right now.

However, Rezai and van der Ploeg (2015 ) find that growth in TFP of 1% reduces the optimal carbon tax by half now, but increases it later. The reason is that as future generations are better off, it makes sense to ease on climate policy now but raise carbon taxes later. 51 The problem is that it is difficult to know what growth rate of TFP to expect in the future. Still, most measures of TFP suggest that the TFP growth rate is very small. For example, based on the May 2015 version of The Conference Board Total Economy Database we find that world average TFP growth has been roughly −0.52% during the past 35 years, without a clear trend. Based on recent new estimates in the Penn World Tables 8.1, we observe a world average TFP growth rate 52 of 0.5% between 1960 and 2014 with a trend that seems to be declining towards zero. We would thus argue that persistent growth, based on some (unknown, residual) factor other than capital or labour, is unlikely. However, these TFP estimates tend to be based on a flexible production function with non-constant shares or elasticities. As a result, it could very well be that technological break-throughs make factors less complementary or even substitutable, which then has important repercussions for the potential of persistent growth. This is where definitely more work and analysis is needed.

It is important to emphasize that the 400 euro per ton of carbon forwarded in Hassler et al. (2016 ) is implicitly based on a discount rate of roughly 0.35% (see their Figure 4 ). The ‘right’ discount rate has been an extensively discussed parameter and there are basically two schools of thought. The prescriptive view (e.g. Stern, 2007 ) argues that future generations’ utilities must not be discounted and advocates an annual discount rate of 0.1%, 53 while proponents of the descriptive approach (e.g. Arrow et al., 1996 ; Nordhaus, 2014 ) suggest to discount at 1.5% which is calculated based on the actual rate of return and better reflects the opportunity costs. Hassler et al. (2016 ) thus choose a discount rate that more closely corresponds to the prescriptive approach. If we fully follow the prescriptive view, we obtain a carbon tax of 1,100 euro per ton of carbon based on their model, while the descriptive discount rate yields roughly 150 euro per ton of carbon. Together with the more realistic carbon cycle we obtain a carbon price of 550 (for a discount rate of 0.1%) or 75 euro (for a discount rate of 1.5%) per ton of carbon.

Another important assumption in Hassler et al. (2016 ) is the logarithmic utility. In particular, van der Ploeg and Withagen (2014 ) show that for a higher degree of intergenerational inequality aversion 54 the current generations will increase consumption and fossil fuel use simply because this reduces the consumption gap (and thus inequality) to the richer future. Hence the optimal carbon tax starts at a lower level but subsequently rises above the one of the logarithmic case. In the simulations of Rezai and van der Ploeg (2015 ) an intergenerational inequality aversion of 2 reduces the optimal carbon tax by something like one third.

Thus, the point to take away up to now is that the carbon tax rule derived in Golosov et al. (2014 ) and advocated in Hassler et al. (2016 ) is remarkably robust, with the exception of the linear carbon cycle that overestimates the carbon price by roughly a factor of two. However, two key parameters 55 which tend to be widely debated in the literature ( Arrow et al., 2013 ) play a crucial role for the actual level of the carbon tax–the discount rate and the intergenerational inequality aversion. It seems to be a common understanding that empirical estimates of the discount rate can be anything from slightly negative to very large and positive ( Frederick et al., 2002 ). Also, elasticities of intergenerational inequality aversion are suggested to be somewhere between one ( Nordhaus, 1993 ) to 10 ( Campbell and Mankiw, 1989 ). 56 Exploiting these large differences can yield any conceivable carbon price. Nevertheless, what one can argue is that if Hassler et al. (2016 ) were to rely on a descriptive argument, i.e. with the discount rate chosen at $ρ=0.35%$ and intertemporal inequality aversion calibrated to market data, then their chosen intertemporal inequality aversion of θ = 1 would be too low. Assuming the Ramsey rule ( $r=ρ+θg$ , where r is the market interest rate, ρ the discount rate, θ the intergenerational inequality aversion and g the economy’s growth rate) holds, and following OMB (2003 ) we set $r=7%$ (where r is ‘an estimate of the average pretax rate of return on private capital in the US economy’). With an average inflation rate of 2.53% between 1990 and 2015, this yields a net interest rate of 4.47%. US real GDP growth during the same period was roughly 3%, and our equation to solve is $4.47%=ρ+3%θ$ . Hence for θ = 1 as used in Hassler et al. (2016 ), the discount rate should be $ρ=1.46%$ , which is roughly equal to what Nordhaus (2014 ) advocates. Thus, the parameter combination used in Hassler et al. (2016 ) does not seem to fit the descriptive approach, 57 and it should be assumed that they propose the carbon tax based on discounting and inequality aversion parameters that are chosen on ethical grounds. This is certainly a valid approach, but it needs to be made explicit. Furthermore, while Stern’s (2007 ) discount rate of 0.1% is based on the view that future utilities must not be discounted (the 0.1% takes the probability of a large-scale disaster into account), it is neither clear what ethical principles underlie the authors’ discount rate nor their choice of the inequality aversion.

There is a final point that should also be emphasized. In a recent contribution, van der Ploeg and Withagen (2014 ) generalize several aspects in Golosov et al. (2014 ). In particular, they allow for stock-dependent extraction costs, more general depreciation of capital, and elasticities of intertemporal substitution different from one (the logarithmic case). Based upon these generalizations, the authors show that in fossil fuel-abundant economies the result in Golosov et al. (2014 ) prevails, meaning that the optimal carbon tax should indeed be proportionate to output. In contrast, in fossil fuel-scarce economies the authors find that the optimal carbon tax should be an increasing proportion of output, and this result becomes more pronounced for lower levels of the elasticity of intertemporal substitution. So it is useful to know whether or not we are in a situation of fossil fuel abundance or scarcity.

Here Hassler et al. (2016 ) argue that the world’s oil and gas reserves amount to roughly 300 GtC, leading to a warming potential of only 0.7 °C. In contrast, official coal reserves are 640 GtC but could be around 3,500 GtC, potentially increasing global temperature by up to an additional 4.87 °C. As, furthermore, extraction costs for coal are first flat and second close to market prices, the authors argue that climate policy is (mostly) about coal. Due to the profit margin they expect a carbon tax to have little impact on oil or gas, but due to the pricing at marginal extraction costs any carbon tax will directly feed into the price of coal. One argument here, by the authors, is that, in per cent of the fuel price, the Swedish CO 2 tax was 55% for oil and 460% for coal, and thus coal may be more easily priced out of the market. This, however, may be a fallacy, since it is always the opportunity costs that matter. Only if the carbon price is so high such that fossil fuels are uncompetitive compared to the greener alternatives will we see a substitution away from these. In fact, in a later section the authors note this point themselves.

In addition, if we do the same exercise for oil and gas as Hassler et al. (2016 ) did for coal and take the unconventional or potential oil and and gas reserves into account, then this may yield additional total (unconventional) oil reserves of 1,000 GtC and unconventional gas reserves of up to 107 GtC, yielding an additional warming potential of 2.16 °C. Oil and gas should still be viewed as a scarce resource and thus there is certainly reason to believe that the result in van der Ploeg and Withagen (2014 ), namely that the optimal carbon tax should be an increasing proportion of output, applies. Clearly we cannot afford to neglect the role of oil and gas for climate policy. This is also important for two reasons. Especially oil is used mostly in transportation, and as such it is not possible to point-source capture the carbon emissions. Furthermore, electric cars are not (yet) sufficiently cheap and do not have the same range in order to be a viable substitute. Instead, especially for coal which is mainly used in electricity production and heating, we do have cheap and often competitive substitutes (wind, solar and hydro), and it is much easier to capture carbon emissions from bigger coal plants than from e.g. cars.

## Panel discussion

Kevin O’Rourke questioned the political feasibility of international coordination on a common climate policy. While climate policies might be optimal from the perspective of global GDP, there are important distributional consequences across countries, and politicians care about the maximization of national GDP. Martin Ellison also raised concerns about the political feasibility of the carbon tax. There is evidence for substantial fossil fuel subsidies, which indicates strong political opposition to a carbon tax. Hans-Werner Sinn worried that a constant carbon tax rate cannot be optimal if there are tipping points. Ingmar Schumacher argued that one advantage of a cap and trade system over a carbon tax might be that it does not require countries to settle on a price at the beginning, especially if all countries’ emissions stay below the cap initially.

Relating to a crucial model component, Andrea Ichino was wondering why the concave relationship between carbon and temperature, and the convex one between temperature and damage compensate each other to a linear relationship between carbon and damage, and how robust this assumption is. Alluding to the recent Volkswagen emission scandal, Johannes van Biesebroeck pointed out that measuring emissions is hard, so the advantage of a tax could be that it circumvents the measurement of emissions; however, as Hans-Werner Sinn stressed, it requires instead the measurement of inputs. It was also debated which role nuclear energy could play as an alternative to fossil fuel.

In response to the comments and questions, the authors stressed that their formula is not only nice because it is a simple one, but that it is actually very robust and easy to adjust, for example to non-logarithmic utility. One could even incorporate tipping points into the model; climate scientists have, however, not yet gathered enough evidence about the existence of tipping points. In contrast to their model, integrated assessment models are central planner models, and hence not very useful in thinking about policy. The authors stressed that the optimal tax rate is sensitive to γ, i.e. the elasticity of output net of damages with respect to the CO 2 concentration, and scientists should work on obtaining more reliable estimates of this parameter. Regarding the political feasibility of a carbon tax, indeed damages vary by countries, which surely matters in negotiations. Richer models that allow for cross-country heterogeneity are a next step in the research agenda.

If spillovers in green technologies exist, then these would imply optimal positive subsidies for green technologies, but this is motivated simply by the positive spillovers, not by being green itself, as long as the technologies do not replace coal. One should surely expect opposition against a carbon tax from industries relying on coal. Indeed, it is suspicious that these industries do not oppose subsidies of green technologies more: it likely indicates that these subsidies are not enough to replace coal by green technology.

2
The proportionality of the outflow to temperature is a linear approximation.
3
See Schwartz et al. (2014) . The value 3.7 is, however, not undisputed. Otto et al. (2013) use a value of 3.44 in their calculations.
4
Natural scientists attach a different meaning to the word equilibrium than economists. A translation to the language of economics would be steady state.
5
See IPCC (2013) Chapter 6, Box 6.1.
7
For a discussion, see Hassler and Krusell (2012) .
8
Krusell and Smith (2015a ) builds an integrated assessment model that focuses entirely on damages at a disaggregated level.
9
In contrast, in our discussion of top-down damages above we referred to regression estimates as percentages of output without reference to the given temperature.
10
11
Cases where this damage function is not a good approximation might, thus, include cases with stronger non-linearities in damages than those Nordhaus assumes. One case is that where global temperature appears with a power higher than 2; another is that with kinks. However, there is no consensus on such features, let alone at what level of carbon concentration they would occur.
12
Figure 2 shows the range of yearly average temperatures over the period for each country and the averages early and late in the sample.
13
It may also be that the mapping from emissions to carbon concentration becomes more convex. That will not by itself case our damage function to be a poor approximation, but it will complicate the optimal carbon tax calculations, as we will show below.
14
A path breaking study was Haldane (1905) . See Sherwood and Huber (2010) for a more recent study related to climate change.
15
This approach is also followed in the most recent work by Krusell and Smith (2015a) .
16
For evidence that optimizing saving at an aggregate level, compared to simple alternatives such as that entertained by Solow, is to be preferred, see Krusell and Smith (2015b ).
17
Population growth can be important in this context for some purposes but not, typically, for some key aspects of policy analysis, such as for the calculation of an optimal carbon tax.
18
For a discussion, see e.g., Hassler et al. (2012) .
19
For details on the conditions under which the result obtains exactly, see Golosov et al. (2014) . The authors also show remarkable robustness of the formula to departures from the assumptions they state. Extensions and closely related settings include van der Ploeg and Withagen (2014 ), Rezai and van der Ploeg (2014) , Anderson et al. (2014) , Li et al. (2014) , Gerlagh and Liski (2012) , and Traeger (2015) .
20
The planner’s optimal choice comes from a condition that requires the marginal product of labour in each sector to be the same when one takes into account that a unit of labour in the coal sector adds damages to the consumption sector. It is thus very straightforward to derive.
21
The first term represents the duration of the part of emissions that is assumed to remain ‘forever’ in the atmosphere. As the subjective discount rate approaches zero, this term approaches infinity. However, the assumption that a share $ϕL$ of emissions remain in the atmosphere for a horizon that from an economic point of view is infinite does become less reasonable with a discount rate close to zero. In this case, the fact that over tens of thousands of years, also this share of emissions is slowly absorbed by the oceans starts to matter. Then a depreciation rate capturing this very slow process should be added to the denominator of the first term. For typical calibrations of subjective discount rates, say larger than 0.1% per year, however, the effect of this slow depreciation is quantitatively negligible.
22
Often, the carbon tax is expressed per mass unit of CO 2 , i.e., including the oxygen.
23
This is an exact result that involves a specific degree of risk aversion (that given by a logarithmic function) together with the exponential damage function.
24
See, for example, Lenton et al. (2008) or, for a more popular scientific description, Levitan (2013) .
25
Note that a higher climate sensitivity would imply that the time until the steady state is reached for a given emission scenario increases. This means that more time for developing techniques for adaptation and carbon capture is allowed.
26
BP (2015) reports global oil reserves to 2398Gt. Using a carbon content of 0.775 this is 196GtC. The same source reports natural gas reserves to 187.1 trillion m 3 . Using a carbon content of 0.574 kg/m 3 , this is 107 GtC. At current extraction rates, both these stocks would last approximately 50 years.
27
BP (2015) reports 891Gt of coal. Using a carbon content of 0.716, this corresponds to 638GtC.
28
Rogner estimates coal reserves to 3400Gtoe, which is approximately 3500GtC.
29
In 2015, the tax was 1.12 SEK per kg CO 2 corresponding to 4.11 SEK per kg carbon since 1 kg carbon produces 3.67 kg CO 2 when burned.
30
We use a carbon content of 0.846 for oil and 0.716 for coal.
31
Pigou (1920) was first to show how taxes can solve market failures caused by externalities.
32
It should be said that an increasing number of consumers, especially in rich countries such as Sweden, appear to be willing to ‘internalize’ the externalities by imposing restrictions on their own behaviour such as reducing their fossil-fuel demands. However, this group is, and arguably will be for the foreseeable future, negligible viewed from the global perspective.
33
Strictly speaking, the damage elasticity is constant, implying that the marginal cost curve is not flat but slightly downward-sloping (through the effect of emissions on the level of GDP). However, the slope is small enough to be negligible.
34
In principle, one could think of a European Emission Trading Central Bank with the responsibility to stabilize the price of emission rights at $τ∗.$ A simpler implementation of the optimal policy would, however, be to have a tax, which would not have to vary as energy demand and thus emission values vary with the level of economic activity.
35
36
An example of such an inefficiency is that the Swedish government in 2014 sold unused emission rights to Merril Lynch at a price of 0.03 SEK/kg CO 2 at the same time as it taxed Swedish emission at the rate 1.08 SEK/kg CO 2 . Unfortunately, we believe that this example is not an outlier.
37
The carbon was introduced in 1991 at 0.25 SEK per kg CO 2 . Since then the tax rate has more than quadraupled but various reductions for industry have also been introduced ( Ministry of Sustainable Development, Sweden, 2005) .
38
The total energy bill in the economy is on the order of magnitude of 5% of GDP.
39
Throughout the discussion in this paper, we take green energy to mean other energy sources than fossil fuel. Of course, many non-fossil energy sources have other negative side effects, often on the environment, and the label ‘green’ can be questioned from this perspective. For example, a full evaluation of nuclear power would clearly require a separate analysis.
40
See also McGlade and Ekins (2015) who estimate the optimal use of fossil fuels under the restriction that global warming be limited to 2 °C. They conclude that there should be no reduction in oil consumption until 2050 but radical reductions in coal use.
41
Empirical evidence on the size of (specific and general) spillovers is hard to come by; Dechezleprêtre et al. (2013) is an exception, arguing that green spillovers are stronger than dirty ones.
42
Formally, Pigou-based analysis rests on marginal conditions. With non-convexities, one can imagine multiple local optima and, hence, a role for additional policies in order to select globally among multiple equilibria or steady states.
43
See, e.g., Laibson (1997) .
44
However, the low return on safe government bonds is hard to reconcile with subjective discount rates substantially above zero.
45
See Arrow et al. (2014) for a discussion on whether declining discount rates should be used in policy analysis.
46
In general, the Pigouvian principle only holds exactly when the climate externality is the sole source of inefficiency. The existence of other distortionary taxes, for example on labour, implies that the optimal tax may deviate from the Pigouvian principle. Intuitively, if the carbon tax makes distortions on the labour market worse, the optimal carbon tax should be lower than otherwise. Our assessment, however, is that these other second-best considerations are of minor importance. See Bovenberg and de Mooij (1994 ) for a static analysis and Barrage (2014) and Schmitt (2014) for a dynamic one.
47
A comprehensive coverage of economic policy in the area of environment is Sterner and Coria (2011) . They also briefly discuss politics. Other comprehensive studies include Goulder and Parry (2008) and Aldy et al. (2009) .
48
The present paper was written before the conclusion of the Paris negotiations, where a global agreement was reached. This agreement was not a concrete agreement on quotas (or taxes) but rather on ‘intent’ and hence it is hard to classify. We tentatively view the agreement as a result precisely of moving away from the quantity focus, but a full discussion of the interpretation of the agreement would require a discussion that is outside the scope of the present analysis.
49
The road toll example can be made to liken the case of climate change even more if one imagined that the boroughs were represented by the following leaders: Barack Obama (or Donald Trump?), Vladimir Putin, Xi Yinping, … It is not hard to understand why an agreement on ride quotas would not work in this more elaborate example. Despite the heterogeneity of these borough leaders, however, we do think that there would be a chance of an agreement of a unit toll price even between such leaders.
50
Other articles that also look at in how far the results in Golosov et al. (2014) are robust are Jensen and Traeger (2014) , Traeger (2015) and Gerlagh and Liski (2014) who look at uncertainty, Hassler and Krusell (2012) who have a multi-regional set-up, Gerlagh and Liski (2012) and Iverson (2012) who look at non-constant discounting.
51
This would increase initial temperature and one wonders how this result would be augmented if one takes climate thresholds into account.
52
This is based on the variable rtfpna, which is real TFP at 2005 constant national prices. We obtain a similar result for their variable ctfp, which also accounts for differences in terms of trade.
53
This discount rate is not exactly equal to zero as it adjusts for the impact of catastrophes and major disasters.
54
The elasticity of intergenerational inequality aversion, let us call it θ , is based upon a utility function such as $u(c)=(c1−θ−1)/(1−θ)$ . Thus, for increasing θ we have an increasing aversion to intergenerational inequality. The logarithmic case is the one where θ = 1. As noted in Dasgupta (2008) , θ reflects the maximum sacrifice one generation would be willing to undertake in order to transfer income to another generation.
55
There is another important key parameter which is the damage elasticity. In particular it is assumed to be exogenously given. However, there exists a wealth of literature discussing the possibility of optimal adaptation measures that could reduce this damage elasticity. We will not discuss this one here further.
56
See also Meyer and Meyer (2005) for further discussions.
57
What should be added is that Nordhaus (2014) nicely shows that the optimal carbon tax is very much insensitive to the mix between the discount rate and the elasticity of intertemporal inequality aversion, as long as this mix is based upon the Ramsey rule.

## REFERENCES

Acemoglu
D.
Aghion
P.
Bursztyn
L.
Hémous
D.
(
2012
). ‘
The environment and directed technical change
’,
American Economic Review
,
102
,
131
66
.
Aldy
J.E
Krupnick
A.J
Newell
R.G
Parry
I.W.
Pizer
W.A
(
2009
). ‘
Designing climate mitigation policy
’,
Journal of Economic Literature
,
48
,
903
34
.
Anderson
E
Brock
W
Hansen
L.P.
Sanstad
A.H
(
2014)
. ‘Robust analytical and computational explorations of coupled economic-climate models with carbon-climate response’, RDCEP Working Paper No. 13-05.
Archer
D.
(
2005
). ‘
The fate of fossil fuel CO 2 in geologic time
’,
Journal of Geophysical Research
,
110
,
C09S05, doi:10.1029/2004JC002625
.
Archer
D
Eby
M
Brovkin
V
Ridgwell
A
Cao
L
Mikolajewicz
U
Caldeira
K
Matsumoto
K
Munhoven
G
Montenegro
A.
Tokos
K
(
2009
). ‘
Atmospheric lifetime of fossil fuel carbon dioxide
’,
Annual Review of Earth and Planetary Sciences
,
37
,
117
34
.
Arrow
K.J.
Cline
W.R.
Maler
K.G.
Munasinghe
M.
Squitieri
R.
Stiglitz
J.E.
(
1996
).
Intertemporal Equity, Discounting, and Economic Efficiency
,
Cambridge University Press
,
Cambridge, UK, New York and Melbourne
.
Arrow
K.J.
Cropper
M.L.
Gollier
C.
Groom
B.
Heal
G.M.
Newell
R.G.
Nordhaus
W.D.
Pindyck
R.S.
Pizer
W.A.
Portney
P.R.
Sterner
T.
Tol
R.S.J.
Weitzman
M.L.
(
2013
). ‘How should benefits and costs be discounted in an intergenerational context? The views of an expert panel’, Technical Report, Resources for the Future, RFF DP 12-53.
Arrow
K.J.
Cropper
M.L.
Gollier
C.
Groom
B.
Heal
G.M.
Newell
R.G.
Nordhaus
W.D.
Pindyck
R.S.
Pizer
W.A.
Portney
P.R.
Sterner
T.
Tol
R.S.J.
Weitzman
M.L.
(
2014
). ‘
Should governments use a declining discount rate in project analysis?’
,
Review of Environmental Economics and Policy
,
8
,
145
63
.
Barrage
L.
(
2014
). ‘Optimal dynamic carbon taxes in a climate-economy model with distortionary fiscal policy’, Working Paper, University of Maryland.
Bluedorn
J.C
Valentinyi
A.
Vlassopoulos
M
(
2010
). ‘The long-lived effects of historic climate on the wealth of nations’, mimeo, Southampton University.
Boden
T.
Marland
G.
Andres
B.
(
2011
).
National CO 2 emissions from fossil-fuel burning, cement manufacture, and gas flaring
’,
Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory
.
Bovenberg
A.L.
de Mooij
R.A
(
1994
). ‘
Environmental levies and distortionary taxation
’,
American Economic Review
,
84
,
1085
9
.
BP
(
2015
). BP Statistical Review of World Energy June 2015, http://www.bp.com/ .
Campbell
J.Y.
Mankiw
N.G.
(
1989
). ‘Consumption, income and interest rates: reinterpreting the time series evidence’ , in
NBER Macroeconomics Annual 1989
, Volume
4
,
MIT Press
, pp.
185
246
.
Ciscar
J.C
Feyen
L
Soria
A
Lavalle
C
Raes
F
Perry
M
Nemry
F
Demirel
H
Rozsai
M
Dosio
A
Donatelli
M
Srivastava
A
Fumagalli
D
Niemeyer
S
Shrestha
S
Ciaian
P
Himics
M
Van Doorslaer
B
Barrios
S
Ibáñez
N
Forzieri
G
Rojas
R
Bianchi
A
Dowling
P
Camia
A
Libertà
G
San Miguel-Ayanz
J
de Rigo
D
Caudullo
G
Barredo
J.I
Paci
D
Pycroft
J
Saveyn
B
Van Regemorter
D
Revesz
T
Vandyck
T
Vrontisi
Z
Baranzelli
C
Vandecasteele
I
Batista e Silva
F
Ibarreta
D
(
2014
). ‘Climate impacts in Europe. The JRC PESETA II project’, European Commission.
Ciscar
J.C
Iglesias
A
Feyen
L
Szabo
L
Van Regemorter
D
Amelung
B
Nicholls
R
Watkiss
P
Christensen
O
Dankers
R
Garrote
L
Goodess
D
Hunt
A
Moreno
A
Richards
J
Soria
A
(
2011
). ‘
Physical and economic consequences of climate change in Europe
’,
Proceedings of the National Academy of Sciences
,
108
,
2678
83
.
Cooper
R.N.
(
1998
).
‘Toward a real global warming treaty’
,
Foreign Affairs
,
77
,
66
79
.
Dasgupta
P.
(
2008
).
‘Discounting climate change’
,
Journal of Risk and Uncertainty
,
37
(
2–3
),
141
69
.
Dechezleprêtre
A
Martin
R.
Mohnen
M
(
2013
). ‘Knowledge spillovers from clean and dirty technologies: a patent citation analysis’, Grantham Research Institute Working Paper.
Dell
M
Jones
B.
Olken
B
(
2012
). ‘
Temperature shocks and economic growth: evidence from the last half century
’,
American Economic Journal: Macroeconomics
,
4
,
66
95
.
Desmet
K.
Rossi-Hansberg
E
(
2015
). ‘
On the spatial economic impact of global warming
’,
Journal of Urban Economics
,
88
,
16
37
.
Fischer
C.
Morgenstern
R.D.
(
2006
).
‘Carbon abatement costs: why the wide range of estimates?’
,
The Energy Journal
,
27
,
73
86
.
Frederick
S.
Loewenstein
G.
O’donoghue
T.
(
2002
).
‘Time discounting and time preference: a critical review’
,
Journal of Economic Literature
,
40
(
2
),
351
401
.
Fredriksson
P.G.
Millimet
D.L.
(
2002
).
‘Strategic interaction and the determination of environmental policy across US states’
,
Journal of Urban Economics
,
51
(
1
),
101
22
.
Gars
J.
(
2012
). ‘Essays on the macroeconomics of climate change’, Ph.D thesis, Institute of International Economic Studies, Stockholm University.
Gerlagh
R.
Liski
M.
(
2012
). ‘Carbon prices for the next thousand years’, Technical Report, CESifo Group Munich.
Gerlagh
R.
Liski
M.
(
2014
). ‘Carbon prices for the next hundred years’, Technical Report, CESifo Group Munich.
Green
K.P
Hayward
S.F.
Hassett
K.A
(
2007
). ‘ Climate change: caps vs. taxes’ ,
AEI Outlook Series
,
American Enterprise Institute for Public Policy Research
,
Washington, DC
.
Golosov
M
Hassler
J
Krusell
P.
Tsyvinski
A
(
2014
). ‘
Optimal taxes on fossil fuel in general equilibrium
’,
Econometrica
,
82
,
41
88
.
Goulder
L.H.
(
1995
).
‘Effects of carbon taxes in an economy with prior tax distortions: an intertemporal general equilibrium analysis’
,
Journal of Environmental economics and Management
,
29
(
3
),
271
97
.
Goulder
L.H.
Parry
I.W
(
2008
). ‘
Instrument choice in environmental policy
’,
Review of Environmental Economics and Policy
,
2
,
152
74
.
Goulder
L.H.
Hafstead
M.A.
(
2013
).
‘ Tax reform and environmental policy: options for recycling revenue from a tax on carbon dioxide’
,
Resources for the Future DP
,
13
31
.
Haldane
J.S.
(
1905
). ‘
The influence of high air temperatures: no. 1
.’,
The Journal of Hygiene
,
5
,
494
513
.
Hansjürgens
B.
(ed.). (
2005
).
Emissions Trading for Climate Policy: US and European Perspectives
.
Cambridge University Press
.
Harari
M.
La Ferrara
E
(
2012
). ‘Conflict, climate and cells: a disaggregated analysis’, IGIER Working Paper 461.
Harrington
W.
Morgenstern
R.D.
(
2004
).
Choosing Environmental Policy: Comparing Instruments and Outcomes in the United States and Europe
.
Resources for the Future
.
Hassler
J.
Krusell
P
(
2012
). ‘
Economics and climate change: integrated assessment in a multi-region world
’,
Journal of the European Economic Association
,
10
,
974
1000
.
Hassler
J
Krusell
P.
Olovsson
C
(
2012
). ‘Directed, energy-saving technical change’, Working Paper.
Hassler
J.
Krusell
P.
Nycander
J.
(
2016
). ‘Climate Policy’, Economic Policy .
Helland
E.
(
1998
).
‘Environmental protection in the federalist system: the political economy of NPDES inspections’
,
Economic Inquiry
,
36
(
2
),
305
19
.
Interagency Working Group on Social Cost of Carbon, United States Government
(
2015
). Technical Update of the Social Cost of Carbon for Regulatory Impact Analysis under Executive Order 12866 (May 2013, Revised July 2015). https://www.whitehouse.gov/sites/default/files/omb/inforeg/scc-tsd-final-july-2015.pdf .
IPCC (Intergovernmental Panel on Climate Change)
(
2007a
). ‘ Climate change 2007: the physical science basis’ , in
S.
Solomon
Qin
D.
Manning
M.
Chen
Z.
Marquis
M.
Averyt
K.B.
Tignor
M.
Miller
H.L.
Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change.
IPCC
(
2007b
). ‘Climate change 2007: impacts, adaptation and vulnerability’, in M. Parry, O. Canziani, J. Palutikof, P. van der Linden, C. Hanson (eds.), Contribution of Working Group II to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change .
IPCC
. (
2013
). ‘Climate Change 2013: The Physical Science Basis’, in T.F. Stocker, D. Qin, G.-K. Plattner, M. Tignor, S.K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex and P.M. Midgley (eds.), Working Group I Contribution to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change .
Iverson
T.
(
2012
). ‘Optimal carbon taxes with non-constant time preference’, Technical Report, MPRA Paper No. 49588 2012.
Iverson
T.
(
2014
). ‘Optimal carbon taxes with non-constant time preference’, Working Paper, Colorado State University.
Jensen
S.
Traeger
C.P.
(
2014
).
‘Optimal climate change mitigation under long-term growth uncertainty: stochastic integrated assessment and analytic findings’
,
European Economic Review
,
69
,
104
25
.
Kallbekken
S.
Kroll
S.
Cherry
T.L.
, (
2011
).
‘Do you not like Pigou, or do you not understand him? Tax aversion and revenue recycling in the lab’,
Journal of Environmental Economics and Management
,
62
(
1
),
53
64
.
Krusell
P.
Smith
A.A
Jr.
(
2014
). ‘Regional damages from temperature fluctuations’, Working Paper.
Krusell
P.
Smith
A.A
Jr.
(
2015a
). ‘Climate change around the world’, Working Paper.
Krusell
P.
Smith
A.A
Jr.
(
2015b
). ‘
Is Piketty’s’ second law of capitalism’ fundamental?
’,
Journal of Political Economy
,
123
,
725
48
.
Laibson
D.
(
1997
). ‘
Golden eggs and hyperbolic discounting
’,
The Quarterly Journal of Economics
,
112
,
443
77
.
Lenton
T.M
Held
H
Kriegler
E
Hall
J.W
Lucht
W
Rahmstorf
S
Schnellnhuber
H.J
(
2008
). ‘
Tipping elements in the Earth’s climate system
’,
Proceedings of the National Academy of Sciences
,
105
,
1786
93
.
Levitan
D.
(
2013
). ‘
Quick-change planet: do global climate tipping points exist?
’,
Scientific American
,
25
.
Li
X
Narajabad
B.
Temzelides
T
(
2014
). ‘Robust dynamic optimal taxation and environmental externalities’, CESifo Working Paper Series 4562.
McGlade
C.
Ekins
P
(
2015
). ‘
The geographical distribution of fossil fuels unused when limiting global warming to 2 degrees Celsius
’,
Nature
,
517
,
187
90
.
Mendelsohn
R
Nordhaus
W.H.
Shaw
D
(
1994
). ‘
The impact of global warming on agriculture: a Ricardian analysis
’,
American Economic Review
,
84
,
753
71
.
Meyer
D.J.
Meyer
J.
(
2005
).
‘Relative risk aversion: what do we know?’
,
Journal of Risk and Uncertainty
,
31
(
3
),
243
62
.
Ministry of Sustainable Development, Sweden
(
2005
). ‘Sweden’s fourth national communication on climate change’, DS 2005:55, Government of Sweden.
Nordhaus
W.D.
(
1993
)
‘Optimal greenhouse-gas reductions and tax policy in the “DICE” model’
,
The American Economic Review
,
83
(
2
),
313
7
.
Nordhaus
W.D.
(
2007
). ‘
To tax or not to tax: alternative approaches to slowing global warming
’,
Review of Environmental Economics and Policy
,
1
,
26
44
.
Nordhaus
W.
(
2014
).
‘Estimates of the social cost of carbon: concepts and results from the DICE-2013R Model and alternative approaches’
,
Journal of the Association of Environmental and Resource Economists
,
1
(
1/2
),
273
312
.
Nordhaus
W.D.
Sztorc
P
(
2013
). ‘DICE 2013R: introduction and user’s manual’, http://aida.wss.yale.edu/˜nordhaus/homepage/ .
OECD
. (
2013
). ‘Climate and carbon: aligning prices and policies’, OECD Environment Policy Papers, No. 1.
OMB
(
2003
). ‘Circular A-4: regulatory analysis’, Technical Report, Office of Management and Budget, Washington, DC: Executive Office of the President. http://www.whitehouse.gov/omb/circulars/ .
Otto
A
Otto
F.E.L
Allen
M.R
Boucher
O
Church
J
Hegerl
G
Forster
P.M
Gillett
N.P
Gregory
J
Johnson
G.C
Knutti
R
Lohmann
U
Lewis
N
Marotzke
J
Stevens
B
Myhre
G
Shindell
D
(
2013
). ‘
Energy budget constraints on climate response’
,
Nature Geoscience
,
6
,
415
6
.
Pigou
A.C.
(
1920
).
The Economics of Welfare
,
Macmillan
,
London
. [Database]
Rezai
A.
van der Ploeg
F
(
2014
). ‘Intergenerational inequality aversion, growth and the role of damages: occam’s rule for the global carbon tax’, Discussion Paper 10292, CEPR, London.
Rezai
A.
van der Ploeg
F
(
2015
).
‘Intergenerational inequality aversion, growth and the role of damages: occams rule for the global carbon tax’
,
Journal of the Association of Environmental and Resource Economics
,
3
,
493
522
.
Rogner
H.H.
(
1997
). ‘
An assessment of world hydrocarbon resources
’,
Annual Review of Energy and the Environment
,
22
,
217
62
.
Schmitt
A.
(
2014
). ‘Beyond Pigou: climate change mitigation, policy making and distortions’, Ph.D thesis, Institute of International Economic Studies, Stockholm University.
Schwartz
S
Charlson
R
Kahn
R
Rodhe
H
(
2014
). ‘
Earth’s climate sensitivity: apparent inconsistencies in recent assessments’, Earth’s
Future
,
2
,
601
5
.
Sherwood
S.C.
Huber
M
(
2010
). ‘
An adaptability limit to climate change due to heat stress
’,
Proceedings of the National Academy of Sciences
,
107
,
9552
5
.
Sinn
H.W.
(
2008
). ‘
Public policies against global warming: a supply side approach
’,
International Tax and Public Finance
,
15
,
360
94
.
Sinn
H.W.
(
2012
).
The Green Paradox. A Supply Side Approach to Global Warming
,
MIT Press
,
Cambridge, MA
.
Stern
N.
(
2007
).
The Economics of Climate Change: The Stern Review
,
Cambridge University Press
.
Sterner
T.
Coria
J
(
2011
).
Policy Instruments for Environmental and Natural Resource Management
,
Resources for the Future Press
,
Washington, DC
.
Tol
R.S.J.
(
2009
). ‘
The economic effects of climate change
’,
Journal of Economic Perspectives
,
23
,
29
51
.
Traeger
C.
(
2015
). ‘Closed-form integrated assessment and uncertainty’, Cesifo Working Paper Series 22.
van den Bijgaart
I.
Gerlagh
R.
Liski
M.
(
2016
).
‘A simple formula for the social cost of carbon’
,
Journal of Environmental Economics and Management
,
77
,
75
94
.
van der Ploeg
F.
Withagen
C
(
2014
). ‘
Growth, renewables, and the optimal carbon tax
’,
International Economic Review
,
55
,
283
311
.
Weitzman
M.
(
2009
). ‘
On modeling and interpreting the economics of catastrophic climate change
’,
Review of Economics and Statistics
,
91
,
1
19
.
Weitzman
M.
(
2012
). ‘
GHG targets as insurance against catastrophic climate damages
’,
Journal of Economic Public Theory
,
14
,
221
44
.
Weitzman
M.
(
2014
). ‘
Can negotiating a uniform carbon price help to internalize the global warming externality?
’,
Journal of the Association of Environmental and Resource Economists
,
1
,
29
49
.

## Author notes

1Paper presented at the 30th Anniversary Panel of Economic Policy , Luxembourg, October 16–17, 2015. We are particularly grateful for comments and suggestions offered by Nicola Fuchs-Schündeln, Timo Goeschl, and Ingmar Schumacher, and for very useful feedback from many other participants at the meeting.
The Managing Editor in charge of the paper was Nicola Fuchs-Schündeln.