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OxCarre Research Paper 123
_
Abandoning Fossil Fuel:
How Fast and How Much
A third way to climate policy
Armon Rezai*
Vienna University of Economics & Business
&
Frederick van der Ploeg
OxCarre
*OxCarre External Research Associate
Direct tel: +44(0) 1865 281281
E-mail: [email protected]
ABANDONING FOSSIL FUEL: HOW FAST AND HOW MUCH
A third way to climate policy
Armon Rezai* and Frederick van der Ploeg**†‡
Abstract
Climate change must deal with two market failures: global warming and
learning by doing in renewable use. A third way for climate policy is therefore
required consisting of an aggressive renewables subsidy in the near term and a
gradually rising carbon tax which falls in long run. Such a climate policy
ensures that more renewables are used relative to fossil fuel, the carbon-free
era is brought forward, more fossil fuel is locked up and global warming is
lower. Further, there is an intermediate phase where fossil fuel and renewables
are used alongside each other. The climate externality is more severe than the
learning-by-doing one. The optimal carbon tax is not a constant proportion of
world GDP.
Keywords: climate change, integrated assessment, Ramsey growth, carbon tax,
renewables subsidy, learning by doing, directed technical change,
multiplicative damages, additive damages
JEL codes: H21, Q51, Q54
Revised June 2014
*
Department of Socioeconomics, Vienna University of Economics and Business,
Welthandelsplatz 1, 1020 Vienna, Austria. Also affiliated with IIASA.
Email address: [email protected].
**
Department of Economics, Oxford University, Manor Road Building, Manor Road, Oxford
OX1 3UQ, U.K. Also affiliated with VU University Amsterdam.
Email address: [email protected].
‡
This version of the paper has benefited from helpful comments from Erik Ansink, Reyer
Gerlagh, John Hassler, Per Krusell and seminar participants at Oxford, Tilburg and the
Tinbergen Institute and audiences at Annecy, SURED 2014, Ascona, and WCERE 2014,
Istanbul. The first author is grateful for support from the OeNB Anniversary Fund grant on
Green Growth (no. 14373). The second is grateful for support from the ERC Advanced Grant
‘Political Economy of Green Paradoxes’ (FP7-IDEAS-ERC Grant No. 269788).
1
1. Introduction
Climate policy has to deal with two crucial market failures: the failure for
markets to price carbon to fully internalize all future damages arising from
burning another unit of carbon and the failure of markets to internalize the full
benefits of learning by doing in using renewables. To correct for these market
failures one must set a carbon tax equal to the social cost of carbon (SCC),
which corresponds to the present value of all future marginal global warming
damages from burning one extra unit carbon,1 and set a renewable subsidy
equal to the social benefit of learning by doing (SBL), i.e., the present value of
all future reductions in the cost of renewables from using one unit of renewable
energy now. Our objective is to use a fully calibrated integrated assessment
framework of Ramsey growth and global warming to gain insights into such a
third way to climate policy. Our answer takes account of exhaustible fossil
fuel, increasing efficiency of carbon-free alternatives, gradual and abrupt
transitions from fossil fuel to renewables, a temporary population boom, and
ongoing technical progress and structural change.
We show that the third way to climate policy consists, on the one hand, of an
aggressive and temporary renewable subsidy, and, on the other hand, a
gradually rising carbon tax to price out fossil fuel. This policy mix curbs fossil
fuel use and promotes substitution towards renewables, leaves more untapped
fossil fuel in the crust of the earth and brings forward the carbon-free era.
Initially only fossil fuel is used, then, as a result of endogenous technical
progress in renewables, fossil fuel and renewables are used alongside each
other, and, finally, the economy only renewables are used. Reductions in
climate damages are optimally traded off against welfare losses from less
economic activity. Our analysis highlights the following three features.
1
Instead, the optimal price of carbon can be found on an efficient emissions market or can
correspond to the shadow price of direct control legislation.
2
First, in contrast to Nordhaus (2008, 2014), Rezai et al. (2012) and Golosov et
al. (2014), we analyse not only the optimal carbon tax but also the optimal
transition times for introducing renewables alongside fossil fuel and
abandoning fossil fuel altogether and the amount of untapped fossil fuel to be
left in the crust of the earth.2 Our model is annual instead of decadal as earlier
integrated assessment models or semi-decadal as in Nordhaus (2014), because
this allows us to pinpoint much more accurately the transition times towards
the carbon-free era and such long time intervals can introduce significant
numerical biases (Cai et. al, 2012). Building on IEA data, we furthermore
suppose that extraction costs rise as remaining reserves falls and less accessible
fields are explored. This allows us for the first time to quantitatively assess the
important role of untapped fossil fuel and stranded assets in the fight against
global warming. We thus offer an estimate of the maximum cumulative carbon
emissions (the ‘carbon budget’).
In our analysis fossil fuel is exhaustible and extraction costs are stock
dependent, hence the price of fossil fuel contains two forward-looking
components: the scarcity rent (the present discounted value of all future
increases in extraction costs resulting from an extracting an extra unit of fossil
fuel) and the SCC. The optimal transition times consequently depend on
expectations about future developments such as learning by doing in using
renewable energy. In contrast, most integrated assessment models assume that
fossil fuel is abundant so its demand does not depend on expectations about the
price of future renewables and the optimal distribution of finite resources
across time and thus the transition simply occurs when the price of fossil fuel
inclusive of the carbon tax reaches the price of the renewable energy source.
Second, we suppose that the cost of renewable energy falls as cumulative use
increases (Arrow, 1962). Although technological progress, diffusion, and
adoption are studied in the economics of climate change, these studies typically
2
We derive our results based on a calibrated and richer version of the theoretical growth and
climate model analysed in van der Ploeg and Withagen (2014).
3
do not allow for exhaustible fossil fuel and stock-dependent fossil fuel
extraction costs, and do not offer a fully calibrated integrated assessment
model.3 If cost reductions from learning-by-doing externalities are not fully
internalized, a subsidy is required alongside the carbon tax. We compare
“laissez faire” with a global first-best scenario where both the climate and
learning-by-doing externalities are fully internalized. Since international
climate negotiations have failed miserably and there is room for national
renewable energy policies, we also consider a scenario where only learning-bydoing externalities are internalized and the climate externality is not. This leads
to faster extraction of fossil fuel and acceleration of global warming, since
fossil fuel owners fear that their resources will be worth less in the future. This
phenomenon has been coined the Green Paradox by Sinn (2008).
Third, we show that with CES production and utility the optimal carbon tax
first rises and then declines with world GDP. The result of Golosov et al.
(2014) that the optimal carbon tax is proportional to world GDP is thus not
robust;4 it depends on logarithmic utility, Cobb-Douglas production, full
depreciation of capital in each period, no capital need in fossil fuel extraction,
3
Tiang and Popp (2014) provide recent econometric evidence for significant learning by doing
effects in renewable energy generation, suggesting that each new wind power project in China
(with 60 GW capacity) leads to a unit cost reduction of 0.25%. De Zwaan et al. (2002) are the
first to address optimal climate policy in the face of learning by doing in renewables in
integrated assessment models. Popp (2004) studies endogenous technical progress in a fully
calibrated IAM; Popp et al. (2010) review the implications of technical innovation and
diffusion for the environment; and Goulder and Mathai (2000) study the implications in a
stylized model of climate change. Manne and Richels (2004) argue that endogenous technical
progress does not alter climate policy recommendations, specifically the transition timing.
Jouvet and Schumacher (2012) find the opposite and so does Popp (2004) who studies optimal
policies in an adapted version of DICE of Nordhaus (2008). Hübler et al. (2012) present a
multi-region IAM with endogenous growth and study region-specific welfare effects. Fisher
and Newell (2008) study optimal interaction of policy instruments in a calibrated model of
heterogeneous energy producers limited to the US energy sector. None of these studies
considers the “laissez faire” decentralized economy. Fossil fuel stocks are assumed abundant
and their extraction costless. Thus these studies cannot address how much fossil fuel to lock up
in the crust of the earth and the time of phasing in renewables does not depend on expectations
about future policies and energy prices. These are crucial features of our analysis. Tsur and
Zemel (2005) analyze growth and R&D in a model with scarce resources, but do not offer a
calibration and estimates of optimal climate policy. Table F.1 in appendix F provides a detailed
comparison of prominent IAMs.
4
This formula is also used in Hassler and Krusell (2012) and Gerlagh and Liski (2013).
4
and multiplicative, exponential climate damages in production. The
proportional carbon tax performs poorly if policy needs to address multiple
market failures and intergenerational inequality aversion differs from unity.
The proportional tax suggested by Golosov et al. (2014) rise too slowly relative
to the first best and is, therefore, insufficient in pricing fossil fuels out of the
market. Like the renewable subsidy in isolation, it causes more fossil fuel to be
extracted in the near term and leads to a Green Paradox effect.
Section 2 discusses a simple two-stock model of carbon accumulation in the
atmosphere and global mean temperature due to Golosov et al. (2014) and
discusses our benchmark specification of climate damages which are bigger at
higher temperatures than Nordhaus (2008, 2014) following recent suggestions
by Stern (2013) and Dietz and Stern (2014). We do not adopt the
approximation to damages used in Golosov et al. (2014), since this leads to
unrealistic low damages at higher temperatures. Section 3 formulates our
general equilibrium model of climate change and Ramsey growth. Section 4
uses our calibrated integrated assessment model to highlight the different
outcomes for the optimal carbon tax, the renewables subsidy, untapped fossil
fuel, the time it takes to phase in renewable energy and to reach the carbon-free
era, and welfare under the optimal, the constrained optimal and the “laissez
faire” scenarios. It also discusses the performance of the simple formula for the
carbon tax put forward by Golosov et al. (2014). Section 5 discusses the
sensitivity of climate policy. Section 6 concludes.
2. The carbon cycle, temperature and global warming damages
We use an annual version of the decadal model of the carbon cycle due to
Golosov et al. (2014) and based on Archer (2005) and Archer et al. (2009):
(1)
EtP1  EtP  L Ft , L  0.2,
(2)
EtT1  (1  )EtT  0 (1  L )Ft ,   0.002304, 0  0.393, E0T  699 GtC,
E0P  103 GtC,
5
where EtP is the part of the stock of carbon (GtC) that stays thousands of years
in the atmosphere, EtT the remaining part of the stock of atmospheric carbon
(GtC) that decays at rate  , and Ft the rate of fossil fuel use (GtC/decade).5
About 20% of carbon emissions stay up ‘forever’ and the remainder has a
mean life of about 300 years, so   1 (1 0.0228)1/10  0.002304, where
0.0228 is the parameter proposed for the decadal model in Golosov et al.
(2014). The parameter 0  0.393, is calibrated so that about half the carbon
impulse is removed after 30 years.
The equilibrium climate sensitivity (ECS) is the rise in global mean
temperature following a doubling of the total carbon stock in the atmosphere,
Et  EtP  EtT .
A typical estimate for the ECS is 3 (IPCC, 2007). To facilitate
comparison with Golosov et al. (2014), we ignore lags between the stock of
atmospheric carbon and global mean temperature:6
(3)
Tt   ln  Et / 596.4  / ln(2),   3,
Et  EtP  EtT ,
where 596.4 GtC (280 ppmv CO2) is the IPCC figure for the pre-industrial
stock of carbon.7 The evolution of fossil fuel reserves follows from the
depletion equation:
(4)
St 1  St  Ft ,
S0  4000 GtC,
where St denotes the stock of reserves at the start of period t.
Nordhaus (2008) has combined detailed micro estimates of the costs of global
warming to get aggregate macro costs of global warming of 1.7% of world
5
The three reservoirs used by Nordhaus (2008) and the linear representation used in Gerlagh
and Liski (2013) highlight the exchange of carbon with the deep oceans arising from the
acidification of the oceans limiting the capacity to absorb carbon. Our carbon cycle ignores
time-varying coefficients as in Bolin and Erikkson (1958). It also abstracts from diffusive
rather than advective transfers of heat to the oceans (Allen et al., 2009) which leads to longer
and greater warming (Bronselaer et al., 2013; Baldwin, 2014). Finally, it does not consider
positive feedback effects which are more severe at higher temperatures.
6
This lag lowers the SCC (see section 5.3 and appendix C).
7
To convert to GtCO2, we multiply by 44/12.
6
GDP when global warming is 2.5o C. This figure is used to calibrate the
fraction of production output that is left after damages from global warming:
Z (Tt ) 
(5)
1
1   1Tt
2
  3Tt  4
, so Z ( Et )  Z  ln  Et / 596.4  / ln(2)  ,
with ζ1 = 0.00284, ζ2 = 2, and ζ3= ζ4 = 0 and plotted as the solid line in fig. 1.8
Golosov et al. (2014) approximate this with the exponential net output function
Z ( Et )  exp  ( Et  581)  with   2.379 105 , which gives rise to the dotted
line in fig. 1. The fit to the Nordhaus (2008) damages is good for small degrees
of global warming from the present, but at higher degrees the fit is much worse
and too optimistic about the size of global warming damages.9
Net output after daamge (% of gross)
Figure 1: What is left of output after damages from global warming
100%
90%
80%
70%
60%
50%
0
1
2
3
4
5
6
Temperature change above pre-industrial (°C)
DICE-07
DICE-2013R
Golosov et al. (2014)
Nordhaus-Weitzman
Weitzman (2010) and Dietz and Stern (2014) argue that damages rise more
rapidly at higher levels of temperature than suggested by (5), but empirical
studies on the costs of global warming at higher temperatures are not available.
With the heroic assumptions that damages are 50% of world GDP at 6o C and
8
The damage function resulting from the DICE-07 model is almost distinguishable (up to
temperatures of 7 Celsius) from that of the DICE-2013R model (see
http://www.econ.yale.edu/~nordhaus/homepage/Web-DICE-2013-April.htm )
9
The fit deviates by less than 1%-point only between 2°C and 4°C warming. Z(Et) = 1.0435 
0.0001 Et is a much better fit, also at higher temperature and carbon stocks (R 2 = 0.9994).
7
99% at 12.5o C, Ackerman and Stanton (2012) recalibrate (5) with ζ1 =
0.00245, ζ2 = 2, ζ3 = 5.021 x 10-6, and ζ4 = 6.76. The extra term in the
denominator captures potentially catastrophic losses at high temperatures.10
The dashed line in fig. 1 plots this recalibrated net output function. Although
the calibration of Nordhaus (2008) can be empirically justified for moderate
degrees of global warming, the revised calibration is more appropriate for
higher temperatures and matches it very closely with deviations of less than
0.5%-point up to 3°C of warming.
3. An integrated assessment model of Ramsey growth and climate change
The social planner's objective is to maximize utilitarian social welfare
 (Ct / Lt )11/  1 
(1   ) LtU t (Ct / Lt )  (1   ) Lt 
.
11 / 
t 0
t 0



(6)

t


t
Here Lt is the size of the exogenous world population, Ct aggregate
consumption, U the instantaneous CES utility function,  > 0 the rate of pure
time preference and  > 1 the elasticity of intertemporal substitution. The
ethics of climate policy depend on how much weight is given to future
generations and on how small intergenerational inequality aversion is or how
easy it is to substitute current for future consumption per head. The most
ambitious climate policies result if society has a low rate of time preference
and little intergenerational inequality aversion (low , high ).
Output at time t is produced with three inputs: manmade capital Kt, labour, Lt,
and energy. Types of energy are renewables Rt, (e.g., solar or wind energy),
and fossil fuels (oil, natural gas and coal), Ft. The production function H(.) has
constant returns to scale, is concave and satisfies the Inada conditions.
Renewables are subject to learning, so their unit production cost b(Bt) falls with
10
We capture catastrophic losses at high levels of atmospheric carbon, but abstract from
positive feedback and uncertain climate catastrophes that occur once temperature exceeds
certain thresholds (e.g., Lemoine and Traeger, 2014; van der Ploeg and de Zeeuw, 2013).
8
cumulated past production Bt and thus b < 0. Fossil fuel extraction cost is
G(St )Ft , with St remaining reserves. Extraction is harder as less accessible
fields have to be explored, so G '  0. What is left of production after covering
the cost of resource use is allocated to consumption Ct , investments Kt 1  Kt ,
and depreciation  Kt with a constant depreciation rate :
(7)
Kt 1  (1   ) Kt  Z ( Et ) H ( Kt , Lt , Ft  Rt )  G(St ) Ft  b( Bt ) Rt  Ct ,
where damages follow from (5). The initial capital stock K0 is given. The
development of atmospheric carbon follows from (1) and (2), that of fossil fuel
stock reserves from (4), and that of knowledge for renewable use from
(8)
Bt 1  Bt  Rt ,
B0  0.
Current technological options favour fossil energy; complete decarbonisation
requires substantial reductions in the cost of renewables versus that of fossil
fuel. Apart from carbon taxes, technological progress is an important factor in
determining the optimal combination of fossil and renewable energy sources
(Acemoglu et al., 2012; Mattauch et al., 2012). We thus capture learning and
lock-in effects by making the cost of renewables a decreasing function of past
cumulated renewable energy production, b '  0 with Bt   Rs . 11
s 0
t
Proposition 1: The social optimum maximizes (6) and satisfies (1)-(8), the
Euler equation for consumption growth

(9)
Ct 1 / Lt 1  1  rt 1 

 , rt 1  Z t 1 H Kt 1   ,
Ct / Lt
 1  
and the efficiency conditions for energy use
(10a) Z ( Et ) H Ft  Rt ( Kt , Lt , Ft  Rt )  G( St )  tS  tE , Ft  0, c.s.,
11
We prefer learning-by-doing over other specifications of endogenous technical change, such
as investment in R&D in Bovenberg and Smulders (1996) and Acemoglu et al. (2012), due to
the availability of empirically validated learning curves.
9
(10b) Z ( Et ) H Ft  Rt ( Kt , Lt , Ft  Rt )  b( Bt )  tB , Rt  0, c.s. ,
where the scarcity rent , the SCC and the social benefit of learning by doing in
final goods units (the SBL) are, respectively, given by
(11) tS   G '(St 1s )Ft 1s t s ,
s 0

(12) tB   b '(Bt 1s )Rt 1s t s  and
s 0

(13) tE  s0 L  0 (1  L )(1   )s  t s Z '( Et 1s ) H ( Kt 1s , Lt 1s , Ft 1s  Rt 1s ),
with the compound discount factors given by t s   (1  rt 1s ' )1, s  0.
s '0
s
Proof: see appendix B.
The Euler equation (9) indicates that the growth in consumption per capita rises
with the social return on capital (rt+1) and falls with the rate of time preference,
especially if intergenerational inequality aversion (1/) is small.
Equation (10a) implies that, if fossil fuel is used, its marginal product should
equal total marginal cost, which is the sum of current extraction cost G(St) the
scarcity rent  tS and the SCC  tE . If fossil fuel is not used, its marginal product
is below marginal cost. Equation (10b) states that, if the renewable is used, its
marginal product must equal its current cost b(Bt) minus the SBL  tB .
Equation (11) states that the scarcity rent of keeping an extra unit of fossil fuel
unexploited is the present discounted value of all future reductions in fossil fuel
extraction costs. It follows from the Hotelling rule which states that the return
on extracting an extra unit of fossil, selling it and getting a return on it, the rate
of interest  rt 1tS  minus the increase in future extraction cost  G '(St 1 )Ft 1  ,
must equal the expected capital gain from keeping an extra unit of fossil fuel in
the earth tS1  tS  . Equation (12) indicates that the SBL equals the present
discounted value of all future learning-by-doing reductions in the cost of
renewable energy.
10
Equation (13) states that the SCC is given by the present discounted value of all
future marginal global warming damages from burning an additional unit of
carbon, taking due account of that part stays in the atmosphere for ever and the
rest gradually decays at a rate corresponding to roughly 1/300 per year.
Proposition 2: If the utility function is logarithmic, the production function is
Cobb-Douglas, global warming damages are
Z ( Et )  exp  ( Et  581)  ,
depreciation of physical capital is 100% every period and energy production
does not require capital input, the SCC (13) per Giga ton of carbon becomes
(13)
 1   

 1  
tE , Golosov c.s.   
 L  
 0 (1   L )  Z ( Et ) H ( K t , Lt , Ft  Rt ).
   
  

Proof: see Golosov et al. (2014).
This result that the optimal SCC is proportional to world GDP requires bold
assumptions which are much less appropriate in an annual than in a decadal
model. The factor of proportionality is independent of intergenerational
inequality aversion and the factor shares in production; it increases if the social
rate of discount  is smaller, the permanent fraction of the atmospheric stock of
carbon L is larger, and the lifetime of the transient component of the
atmospheric stock of carbon 1/ is larger.
Decentralized market outcome
In a decentralized market economy firms choose inputs to maximize profits,
Z (Tt ) H ( Kt , Lt , Ft  Rt )  wt Lt  (rt 1   ) Kt ( pt   t )Ft  qt t  Rt , under perfect
competition taking the wage rate wt, the market interest rate rt+1, the market
price for fossil fuel pt, the specific tax t on carbon emissions, the market price
for renewable energy qt, the specific subsidy t on the use of renewable energy,
and
temperature
Tt
as
given.
Capital
accumulation
follows
from
Kt 1  (1   ) Kt  I t , where It is the investment rate at time t. Fossil fuel owners
also operate under perfect competition and maximize the present value of their
11

profits,
   p F  G(S ) F 
t
t t
t
t
t 0
with t   (1  r1s )1, t  0, subject to the
s 0
t
depletion equation (4), taking the market price of fossil fuel pt as given and
internalizing the effect of depletion on future extraction costs. Producers of
renewable energy also operate under perfect competition and maximize the

present value of their profits,
  q  b( B ) R  , taking the market price of
t
t
t
t
t 0
renewable energy qt and the stock of accumulated knowledge about using
renewable energy Bt as given. Households maximize utility (6) subject to their
budget constraint AtH1  (1  rt 1 ) AtH  wt Lt  Tt  Ct , where AtH denotes household
assets and Tt lump-sum transfers from the government at time t. The
government budget constraint is  t Ft  t Bt  Tt , where we abstract from
government consumption and investment. Without loss of generality, the
government balances its books and hands back revenue from carbon taxes
minus cost of renewable subsidies as lump-sum transfers. Asset market
equilibrium requires that
AtH  Kt
and the final goods market equilibrium
condition is Z (Tt ) H ( Kt , Lt , Ft  Rt )  Ct  It  G(St ) Ft  b( Bt ) Rt , so the market
must produce enough final goods to satisfy demand for consumption and
investment goods and cover fossil fuel extraction costs and production costs of
renewables.
Proposition 3: The decentralized market outcome satisfies (1)-(8), the Euler
equation (9), the energy producers’ optimality conditions
(10a) pt  G(St )  tS , Ft  0, c.s.,
(10b) qt  b( Bt ), Rt  0, c.s. ,
where the scarcity rent  tS is (11), and the final good firms’ optimality
conditions wt  Z (Tt ) H Lt , rt 1    Z (Tt ) H Kt , pt   t  qt   t  Z (Tt ) H Ft  Rt .
12
Proof: Follows directly from the optimality conditions for firms, fossil fuel
producers and households. 
Proposition 4: The social optimum is replicated in the decentralized market
economy if  t  tE and  t  tB , t  0, where these follow from (12) and (13).
Proof: Comparing conditions of proposition 2 with those of proposition 3. 
The first best is thus sustained in the market economy if the specific carbon tax
is set to the SCC, the specific subsidy on renewable subsidy is set to the SBL,
and net revenue is handed back in lump-sum fashion to households.
Definition 1: A decentralized market economy has the following outcomes.
I. First best or social optimum with  t  tE and t  tB , t  0.
II. Constrained first best with  t  0 and t  tB , t  0.
III. Constrained first best with  t  tE and t  0, t  0.
IV. “Laissez faire” or business as usual with  t  t  0, t  0. 
Policy scenario I leads to the social optimum and sets the specific carbon tax t
to the SCC (13) and the renewable subsidy t to the SBL (12). Scenario II
internalizes the benefits of learning to use the renewable but not the SCC, so
that the optimal renewable subsidy is given by t  tB and the carbon tax t is
set to zero (i.e., (13) is irrelevant). No international climate agreement can be
reached but national governments move ahead using subsidies (e.g. feed-in
tariffs) to stimulate cost reduction in the production of renewable energy.
Scenario III internalizes the SCC but not learning to use the renewable, so that
 t   tE and the optimal renewable subsidy is zero (i.e., (12) is irrelevant).
Scenario IV corresponds to “laissez faire” and corrects neither for climate nor
for learning by doing, so t = t = 0.
Given that initially renewable energy sources are not competitive, the market
economy starts with an initial phase of only fossil fuel use where fossil fuel
demand follows from setting its marginal product, Z (Tt ) H Ft , to the sum of
13
extraction cost, scarcity rent and carbon tax, G ( St )  tS   t . After some time,
the relative cost of renewable energy has fallen (due to the rising carbon tax
and technical progress) and extraction costs have risen sufficiently (as less
accessible fields have to be extracted) so that it becomes attractive to phase in
clean energy. During this intermediate phase of simultaneous use, demand
follows from Z (Tt ) H Ft  Rt  G ( St )  tS   t  b( Bt )  t . After some more time,
fossil fuel is phased out and the final carbon-free phase starts. Since fossil fuel
extraction costs become infinitely large as reserves are depleted, reserves will
not be fully exhausted and thus fossil fuel will be left untapped in the crust of
the earth at the end of the intermediate phase. Since fossil fuel and renewable
energy are perfect substitutes, simultaneous use is infeasible without learningby-doing or without renewable subsidy or carbon tax (except possibly for a
single period of time). Learning by doing introduces convexity in the
production cost of the renewable so an intermediate phase with simultaneous
use emerges. From that moment on the in-situ stock of fossil fuel remains
unchanged, but the carbon in the atmosphere gradually decays leaving
ultimately only the permanent component of the carbon stock. During the final
phase Z (Tt ) H Rt  b( Bt )  t , which shows that renewable use increases in
capital, the stock of renewable knowledge and the renewable subsidy but
decreases in global mean temperature.
We are interested in how the optimal timing of transitions from introducing the
renewable alongside fossil fuel and from phasing out fossil fuel altogether
differs across scenarios I-IV; for example, we wish to know by how much
carbon taxes and renewable subsidies bring forward the carbon-free era. These
transition times follow from the conditions that energy prices at these transition
times should not jump. The stock of untapped fossil fuel at the end of the
intermediate phase follows from the condition of indifference between using
fossil fuel and renewable energy and a vanishing scarcity rent:
14
(14)
G(St )   t  b( Bt )  t for 0  t  tCF ,
G(St )   t  b( Bt )  t , St  StCF , t  tCF ,
where tCF is the time at which the economy for the first time relies on using
only renewable energy. From (14) we see that the stock of untapped fossil fuel
clearly increases in the renewable subsidy and carbon tax.
4. Policy simulation and optimization
In our simulations time runs from 2010 till 2600 and is measured in years. The
functional forms and calibration of the carbon cycle, temperature module and
global warming damages have been discussed in section 2. The functional
forms and benchmark parameter values for the economic part of our model put
forward in section 3 are discussed in appendix D. Our benchmark parameters
assume low extraction costs and an initially high cost for renewable energy
generation which biases our outcomes toward fossil fuel use. The reported
simulations use the CES production function (with elasticity of substitution
between energy and the capital-labour aggregate equal to  = 0.5).
Apart from I-IV table 1 gives two further scenarios in which the carbon tax
(13) is implemented by either (V) choosing the renewable subsidy optimally or
(VI) ignoring learning by doing in renewable use. The results for these
scenarios are reported in fig. 2 and tables 2 and 3.
Table 1: Scenarios
Carbon tax, τt
0
Proportional to GDP (13)
(I) first-best
(II) only subsidy
(V) proportional carbon tax and
optimal renewable subsidy
(III) only carbon tax
(IV) “laissez faire“
(VI) only proportional carbon tax
Renewable
subsidy, t
from (13)
from (12)
0
15
Figure 2: Benchmark policy simulations
Consumption, Ct
Output after Damage, Yt
300
$trillions / yr
$trillions / yr
200
150
100
50
250
200
150
100
50
0
0
2010 2060 2110 2160 2210 2260 2310
2010 2060 2110 2160 2210 2260 2310
Mean Global Temperature, Tt
6
°C (above pre-industrial)
$trillions (2010)
Capital Stock, Kt
600
400
200
0
5
4
3
2
1
0
2010 2060 2110 2160 2210 2260 2310
2010 2060 2110 2160 2210 2260 2310
Renewable Energy Use, Rt
25
40
20
30
GtC / yr
GtC / yr
Fossil Fuel Use, Ft
15
10
20
10
5
0
0
2010
2060
2110
2160
2210
2010
800
500
600
400
400
200
2110
2160
2210
Renewable Energy Subsidy, νt
$ / tC
$ / tC
Social Cost of Carbon, τt
2060
300
200
100
0
0
2010 2060 2110 2160 2210 2260 2310
2010 2060 2110 2160 2210 2260 2310
Key: laissez faire (
), carbon tax only (
), proportional carbon tax only (
),
renewable subsidy only (
), first best (
), prop. tax & optimal subsidy (
)
16
Figure 2: Benchmark policy simulations (continued)
Hotelling Rent, ϴst
Cumulative Emissions
3000
400
2500
300
$ / tC
GtC
2000
1500
1000
200
100
500
0
0
2010
2060
2110
2160
2210
2010 2060 2110 2160 2210 2260 2310
Key: laissez faire (
), carbon tax only (
), proportional carbon tax only (
),
renewable subsidy only (
), first best (
), prop. tax & optimal subsidy (
)
4.1. How to quickly de-carbonize and leave more fossil fuel untapped
Under the first-best scenario I (blue, solid) consumption, GDP and the capital
stock are monotonically increasing. The transition to renewable energy takes
place smoothly as soon as 2035 and fossil energy is phased out completely by
2040. Over this period 320 GtC are burnt, so most of the 4000 GtC of fossil
fuel reserves are abandoned. This leads to a maximum increase in temperature
of 2.1°C or a maximum atmospheric carbon stock of 970 GtC (from (3)) which
is close to the maximum of a trillion tons of carbon used in Allen et al. (2009).
This rapid and unambiguous first-best transformation towards a carbon-free
economy is achieved through the implementation of a carbon tax and a
renewable subsidy policy. Both follow an inverted U-shaped time profile. The
global carbon tax is set to the SCC which starts at 109 $/tC or 30 $/tCO2 and
reaches a maximum of 409 $/tC or 112 $/tCO2 in 2240, long after the
transition to renewable energy has occurred. The renewable subsidy starts at
175 $/tC or 47$/tCO2 and quickly rises to 387 $/tC or 105 $/tCO2 in 2038 and
rapidly falls to zero as all learning has occurred by the end of this century.
The optimal policy mix, therefore, combines a quick and aggressive subsidy to
phase in renewable energy quite early and a carbon tax which gradually rises
(and falls) to price fossil energy out of the market once renewable energy
sources are competitive.
17
Table 2: Transition times and carbon budget
I
II
III
IV
Social optimum
Carbon tax only
Renewable subsidy only
Laissez faire
Only fossil fuel Simultaneous use Renewable Only Carbon used
2010-2035
2035-2040
2040 –
320 GtC
2010-2080
x
2080 –
770 GtC
2010-2075
2075-2085
2085 –
1080 GtC
2010-2175
x
2175 –
2500 GtC
In the “laissez faire” scenario IV (orange, dashed) both externalities remain
uncorrected. As a result the economy uses much more fossil fuel, 2500 GtC in
total, so much less fossil fuel is left in the crust of the earth. Global mean
temperature increases by a maximum of 5.1°C matching recent IPCC and IEA
estimates for business as usual. The transition to renewable energy occurs
much later, in 2175, and abruptly. The reason is that climate benefits of
renewable energy and learning by doing are not taken into account; hence the
potential cost reductions are not recognized and take longer to materialize.
Table 3 indicates that these inefficiencies (see section 5.2 for more detail)
cause a substantial welfare loss relative to the first best of almost 6 times
today’s world GDP. 12
Table 3: Social costs of carbon, renewable subsidies, and welfare losses
I
II
III
IV
Social optimum
Carbon tax only
Renewable subsidy only
Laissez faire
Welfare
Loss
(% of GDP)
0%
-48%
-95%
-598%
Maximum
carbon tax τ
($/tC)
410 $/GtC
690 $ / GtC
Maximum
renewable
subsidy ($/tC)
390 $/GtC
460 $/GtC
max T
(°C)
2.1 °C
3.0 °C
3.5 °C
5.1 °C
Failing to reach an international climate agreement on carbon taxation and
implementing only a renewable subsidy to encourage learning by doing up to a
rather higher level of 460 $/GtC or 125 $/tCO2 is insufficient to bring forward
the transition to renewable energy and to avoid severe climate change. This
12
Stern (2007) expresses cost of inaction in annuity terms; Nordhaus (2008) in terms of
today’s consumption.
18
“renewable subsidy only” case III performs better than “laissez faire” in terms
of welfare and climate outcomes but temperature rises by as much as 3.5°C
above pre-industrial levels and 1080 GtC are used in total (a bit less than half
of that under “laissez faire” case but still 3 to 4 times the optimal carbon
budget). Subsidizing renewable energy induces simultaneous use of renewable
along with fossil energy sources as late as 2075 with a complete transition by
2085; this indicates the power of the carbon tax in quickening the uptake of
renewables. Not internalizing the climate externality, however, still lowers
welfare by 95% compared to the first best. There are Green Paradox effects
(Sinn, 2008) as fossil fuel use is, compared with under “laissez faire”,
increased in anticipation of fossil fuel being made less competitive relative to
the renewable energy.13 Fossil fuel deposits are depleted more quickly to avoid
them becoming obsolete.
Establishing carbon markets, taxing emissions, or direct control but ignoring
the learning externality (see the orange, solid lines) is politically perhaps less
likely, but it is more effective in avoiding global warming and increasing
welfare than a subsidy for renewable energy. In scenario III the carbon tax rises
to a much higher level of 690 $/tC or 188 $/tCO2 than the first-best carbon tax.
A carbon tax ends fossil fuel use not until 2080 but does this abruptly as the
learning externality is still not recognized. This policy limits global warming to
3.0°C and carbon use to 770 GtC. The inefficiency of the learning externality
seems to matter less in that not internalizing it only lowers welfare by 48%.
The comparison of welfare indicators across partial policy scenarios gives
further credence to the assessment of Stern (2007) that climate change is the
biggest externality our planet faces and that policy makers should prioritize
climate negotiations over renewable policy. If such negotiations fail due to
insurmountable free-riding problems with international treaties, (national)
13
Under Leontief production technology, there are no Green Paradox effects as the energy
demand is a fixed proportion of output.
19
renewable subsidies are yet a relatively efficient way to avert the most severe
aspects of global warming.
4.2. Time paths for the market price of fossil fuel and the renewable
Market prices for both types of fossil and renewable energy are depicted in
fig. 3. Regular lines depict prices of fuels in use, faint lines prices of fuels not
in use. The price of fossil energy consists of the sum of marginal extraction
cost and the Hotelling rent plus any carbon tax (see equation (10a)). The
market price of renewable energy is set to its production cost minus any
learning subsidy (see equation (10b)). Initially prices are rising in all scenarios
and only on these rising sections are fossil fuels used. A renewable subsidy
enables a period of simultaneous use (indicated by boxed sections) and
smoothes the transition to the carbon-free era.
Figure 3: The market price of fossil and renewable energy ($/tC)
1000
1000
800
800
$ / tC
1200
$ / tC
1200
600
600
400
400
200
200
0
2010 2060 2110 2160 2210 2260 2310
Key left: laissez faire (
), carbon tax only (
right: renewable subsidy only (
0
2010 2060 2110 2160 2210 2260 2310
)
), first best (
). Boxes indicate simultaneous use.
First, consider the “laissez faire” scenario IV (orange, dashed, left panel) where
the carbon tax and the subsidy are set to zero. The market cost of renewable
energy is above the market price of fossil energy and constant in the absence of
any production and/or subsidy. Fossil fuel is in use initially and its price rises
due to increasing extraction costs and the increasing Hotelling rents. As
extraction costs approach the back-stop price, the fossil fuel era draws to an
end and the Hotelling rent falls. This fall in the Hotelling rent mitigates rising
20
extraction costs and prolongs the era of fossil fuel use. In its final period, the
scarcity value reaches zero and extraction costs equal renewable unit costs.
After this switch point, the cost of fossil energy, consisting now only of the
extraction cost component, remains constant (coloured in faint orange as not in
use). The cost of producing renewable energy falls quickly and approaches its
lower floor due to learning by doing. This scenario is clearly sub-optimal, since
renewable energy is too expensive for much too long relative to fossil fuel.
Next, consider the case where only the climate externality is internalized using
a carbon tax, scenario II (orange, solid, left panel). Adding the SCC to the price
of fossil energy increases its initial price and its growth rate significantly. It
surpasses the market price of renewable energy much earlier but the transition
to the carbon-free era is still abrupt with a price spike in 2080 as the learningby-doing externality is still not recognized.
If instead of the carbon tax, a subsidy for renewable energy is introduced,
scenario III (dark blue, dashed, right panel), the market price of energy falls
below its “laissez faire” level since fossil fuel owners anticipate that their
resources will be worth less in the future. Lower market prices stimulate higher
fossil fuel use (of up to 20%), faster extraction of fossil fuel, and acceleration
of global warming (the Green Paradox). The subsidy lowers the market price of
renewable energy to allow for simultaneous use (indicated by boxed sections).
Once renewable energy is brought into use, its price rises with that of fossil
energy. The market price of renewable energy starts declining only as the
transition to carbon-free energy is complete.
To implement the first best, scenario I (dark blue, solid, right panel), a carbon
tax needs to be added to the renewable subsidy.14 The carbon tax increases the
14
If the SCC is taken account of, the price of fossil energy continues to rise even as no fossil
energy is produced (see the faint solid orange and dark blue lines). The SCC rises initially,
since decay is limited and consumption is increasing. This yields a smaller marginal utility of
consumption and thus a higher SCC (see (13)). However, after some point of time decay of
atmospheric carbon dominates the decrease in marginal utility of consumption and the SCC –
and with it the market price of fossil energy – start to fall. If the fall is sufficiently large, fossil
21
price if fossil energy beyond its “laissez faire” level. This lowers the required
subsidy for renewable energy, brings forward the switch points to simultaneous
use and to full renewable energy use, and precludes the Green Paradox effect.
4.3. “Laissez faire” overinvests in dirty and underinvests in clean capital
How does “laissez faire”, scenario IV, differ from the first best? Initially,
output, consumption and capital accumulation take place at very similar levels.
However, the impacts of the climate and learning externalities are large enough
to drastically change accumulation paths as temperatures rise. This is also
reflected in the substantial welfare loss of about 6 times initial GDP. “Laissez
faire” leads to inefficient allocation of resources, since the economy overvalues
the returns to conventional capital accumulation and undervalues investments
in green energy sources. Failure to cooperate induces excessive fossil fuel
extraction and capital accumulation leading to high global warming damages.
This inefficient use of resources lowers welfare because it keeps consumption
low in early periods of the program to allow for capital accumulation and
consumption low in future periods due to high global warming damages.
Damages under “laissez faire” are large enough to lower factor returns
sufficiently to induce decumulation of capital and a fall in consumption. From
2140-2190 the capital stock falls by 25% from a peak of $410 trillion to a
trough $310 trillion, consumption drops by 17% from a peak of $107 trillion to
a trough of $88 trillion. This climate crisis is ended as extraction costs rise
above the cost of renewable energy. At this point all of energy use is sourced
from renewables and the climate and economy recover from previous excessive
use of fossil fuels. The inefficiency of this scenario is also reflected in the high
share of GDP expended on energy which rises from around 7% in 2010 to 10%
in 2110, compared to 6.5% and 6%, respectively, under first best. Once the
economy switches to the renewable and stocks of atmospheric carbon recede,
energy becomes competitive again. As fig. 3 indicates, the time horizon that we consider is too
short and the learning-based reduction in renewable costs too large to make such re-switching
optimal. A carbon tax proportional to GDP or consumption does not allow for re-switching.
22
the return to capital and the interest rate increase. This leads to a resumption of
global growth. As only a fraction of carbon dissipates, welfare remains below
the social optimum even in steady state.
Close inspection demonstrates that consumption in the social optimum is the
lowest for some initial periods. This implies that internalization of the climate
externality has to be phased in more slowly which is the case with higher
intergenerational inequality aversion (lower ).15
4.4. The optimal carbon tax versus aggregate consumption or world GDP
To examine whether the linear formula for the optimal carbon tax (13) put
forward by Golosov et al. (2014) holds up in our more general and detailed
integrated assessment model of Ramsey growth and climate change, fig. 4 plots
the ratio of the optimal carbon tax to both world GDP and aggregate
consumption. We immediately see that the optimal carbon tax (dark blue line)
is not well described by a constant proportion of world GDP or aggregate
consumption. The general pattern is that during the initial phases of fossil fuel
use the social cost of carbon rises as a proportion of world GDP as more
carbon emissions raise marginal damages of global warming and during the
carbon-free phases the SCC falls as a proportion of world GDP as a significant
part of the stock of carbon in the atmosphere is gradually returned to the
surface of the oceans and the earth. The required rise and fall of the SCC is
substantial and non-linear due to rapid rises in productivity and population.
The dotted lines in fig. 2 provide further details on the time profiles of key
variables under scenarios V and VI which set climate policy according to (13).
This proportional tax internalizes part of the SCC and improves welfare
relative to no policy. A carbon tax proportional to GDP increases the price of
15
Alternatively, the social optimum might have to be abandoned and instead one has to devise
second-best intergenerational compensation schemes for policy to increase consumption in all
periods. Karp and Rezai (2014) and Karp (2013) discuss the identification of generations in
Ramsey models, intergenerational discounting, and the various effects of environmental policy
in an OLG setting.
23
fossil fuel and encourages a transition to renewable energy earlier on.
However, the proportional tax starts at a too low level and rises too slowly (and
even falls in scenario VI) compared to first best. Still, in this sub-optimal
scenario a Green Paradox effect emerges around 2080 arising from the
anticipation of the imminent phasing in of renewables. The welfare loss of such
an inefficient policy is equivalent to 155% of current GDP. Once the
proportional carbon tax is supplemented with an optimal renewable subsidy,
compensating the non-optimal climate policy, the social optimum is matched
quite closely and the welfare loss is less than 1% of initial GDP.
Figure 4: The SCC as fraction of aggregate world GDP and consumption
(i) Social Cost of Carbon / Output
(ii) Social Cost of Carbon /
Consumption
4
5
1/$
1/$
3
2
4
3
2
1
2010 2060 2110 2160 2210 2260 2310
Key: social optimum (
), carbon tax only (
proportional tax & optimal subsidy (
1
2010 2060 2110 2160 2210 2260 2310
),
), proportional carbon tax only (
)
5. Robustness of the social cost of carbon
Fig. 5 shows the sensitivity of the social optimum to some other key
parameters and to the proportional tax rule proposed by Golosov et al. (2014)
and table 4 give the corresponding transition times and carbon budgets.
Our general finding of an inverted-U time profile for the social cost of carbon
is robust. The exact timing and magnitude depends on specific parameters.
Introducing a time lag in the temperature response to carbon increases as in
Gerlagh and Liski (2013), lowers the SCC and slows the transition to carbonfree sources of energy.
24
Table 4: Transition times and carbon budget (extended)
Scenario
Fossil fuel Simultaneous use
Renewable Only
Carbon used
Baseline
Temperature lag
2010–2035
2010–2055
2035–2040
2055–2060
2040 –
2060 –
320 GtC o
600 GtC o
Nordhaus damage
2010–2105
2105–2110
2110 –
845 GtC o
 = 0 (Leontief)
2010–2035
2035–2040
2040 –
325 GtC o
2010–2040
2040–2045
2045 –
340 GtC o
2010–2105
x
2105 –
1,340 GtC o
proportional tax &
optimal subsidy (V)
prop. tax only (VI)
Climate policy is also less ambitious if damages are lower and less convex as
in Nordhaus (2008), which implies that more carbon is burnt and much less
fossil fuel is left abandoned in the crust of the earth. Lowering the
substitutability between the capital-labour aggregate and energy in the CES
production function (  = 0) leads to higher (fossil and renewable) energy use,
which causes higher stocks of atmospheric carbon and a higher SCC as the
destructive and costly energy production cannot be substituted. Climate policy
is less ambitious with a proportional carbon tax set according to (13) where the
presence of an optimal subsidy reduces the amount of carbon burnt to mimic
the first-best closely.
Figure 5: Sensitivity analysis for the time paths of the optimal SCC
$/tC
Social Cost of Carbon, τt
Renewable Energy Subsidy, νt
500
500
400
400
300
300
200
200
100
100
0
2010
2210
CES = 0
$
/
t
C
Key:
2110
0
2010
2310
Baseline
2060
Nordhaus Damage
Social Cost of Carbon, τt
2110
2160
2210
Temperature lag
500
400
300
200
100
0
The subsidy for renewable energy use and the optimal carbon tax are
2010
2060
2110
2160
2210
2260
2310
complements. The subsidy policy increases in cases in which the carbon tax
25
(and with it the market price of fossil energy) falls: lower damages and less
responsive temperature.
We also conducted further robustness checks: A much lower social rate of
discount ( = 0) leads to a much more ambitious climate policy with a much
higher social cost of carbon, earlier phasing in of renewables and more fossil
fuel left in situ. A higher elasticity of intertemporal substitution implies less
intergenerational inequality aversion. For example, with zero intergenerational
inequality aversion (   instead of  = ½), this implies in a growing
economy that the carbon tax hurts earlier generations much more than later
generations. The social planner is relatively more concerned with fighting
global warming than with avoiding big differences in consumption of different
generations. Climate policy is also more aggressive with a higher equilibrium
climate sensitivity ( = 6). Starting with half the initial capital stock (K0 = 100)
or increasing population and productivity growth (A(∞) = 6 and L(∞) = 10.6)
hardly affect the SCC. The increase in the SCC under lower discounting and
intergenerational inequality aversion and a more sensitive climate allows for
less aggressive renewable subsidies because of instrument complementarity.
6. Conclusion
Our main findings based on an integrated assessment model of climate change
and Ramsey growth are that it is important to not only internalize the climate
externality but also learning by doing in using renewables. Pricing carbon and
subsidizing renewable use curbs fossil fuel use and promotes substitution away
from fossil fuel towards renewables, increases untapped fossil fuel, and brings
forward the carbon-free era. Our benchmark results give rise to a global carbon
tax which rises from about 110 $/tC (or 30 $/tCO2) initially to 190 $/tC (52
$/tCO2) in 2050 and a renewable subsidy which rises from 175 $/tC
(48 $/tCO2) initially to 380 $/tC (104 $/tCO2) in 2040 and then falls quickly to
zero. The optimal policy mix, therefore, consists of an aggressive subsidy
26
making renewable energy competitive early on and a gradually rising carbon
tax pricing fossil energy out of the market. The total amount of carbon burnt is
320 GtC which is much less than the 2,500 GtC under “laissez faire”.
Consequently, the social optimum manages to limit the maximum temperature
to 2.1 °C instead of 5.1 °C under “laissez faire”. The welfare loss without
policy is almost 6 times today’s world GDP. Climate policy becomes less
ambitious as the social cost of carbon is higher, fossil fuel is abandoned less
quickly and more carbon is used in total if the discount rate is higher,
intergenerational inequality aversion is weaker, and the equilibrium climate
sensitivity is lower. Less substitutability between energy and the capital-labour
aggregate leads to less energy use especially in capital-scarce economies and
less global warming. Hence, the required global carbon tax is lower.
If international agreements on cooperation in climate change mitigation cannot
be reached, governments can move forward unilaterally by introducing
subsidies to renewable energy. Such policy internalizes the learning externality
and accelerates fossil fuel extraction and global warming as fossil fuel owners
fear that their resources will be worth less in the future. The level and the
duration of the subsidy policy increases compared to the social optimum to
compensate for the missing carbon tax. This limits global warming to 3.5 °C
and the loss in welfare to 95% relative to the social optimum. If a carbon tax is
introduced instead of the subsidy, the welfare loss reduces to 50% of today’s
world GDP. The comparison of welfare indicators across partial policy
scenarios gives further credence to the assessment of Stern (2007) that climate
change is the biggest externality our planet faces and leads us to conclude that
policy makers should prioritize climate negotiations over renewable policy. If
such negotiations fail, due to insurmountable free-riding problems associated
with international treaties, (national) renewable subsidies are yet a relatively
efficient instrument to avert the most severe aspects of global warming.
Renewable subsidies in isolation, however, are insufficient to combat climate
change. Making total factor and energy productivities also endogenous (using
27
the empirical estimates of the determinants of growth rates given in Hassler et
al., 2011) allows further substitution possibilities between energy and the
capital-labour aggregate in the longer run and justifies a more ambitious
climate policy. R&D subsidies should be set such that the price of renewable
energy follows the (rising) price of fossil energy and need to be complemented
by a carbon tax in order to avoid excessive extraction associated with the
Green Paradox.
A recent interagency working group (2010) suggests that US institutions use a
social cost of carbon of initially $80/tC (22 $/tCO2) rising to 165 $/tC
(45 $/tCO2) in 2050 in project appraisal based on a discount rate of 3% per
annum. A discount rate of 2.5% per annum would imply an initial social cost of
carbon of 129 $/tC (35 $/tCO2) rising to 238 $/tC (65 $/tCO2) in 2050, which
in line with our estimates.16 These figures are typically based on existing
integrated assessment models which are often very elaborate and in which
consumers rarely maximize utility as in the Ramsey model. Hence, it is not
always clear what the underlying assumptions and the crucial parameters
deriving the results are. Golosov et al. (2014) offer a tractable fully consistent
general equilibrium model of climate change and Ramsey growth, but employ
unrealistically low damages at higher temperatures and need to make some
very bold assumptions to ensure that both aggregate consumption and the
carbon tax are a fixed proportion of world GDP. Much too much carbon is
burnt and the welfare losses are substantial with this proportional carbon tax,
especially if there is no policy in place for encouraging learning by doing in
renewable production. In the absence of an optimal renewable subsidy, the
proportional tax starts at a too low level and rises too slowly and causes
extraction rates to rise, as predicted by the Green Paradox.
16
Such models of climate change yield estimates of the social cost of carbon starting from 5 to
35 $ per ton of carbon in 2010 and rising to $16 to $50 per ton in 2050 (e.g., the DICE, PAGE
and FUND models of Nordhaus (2008), Alberth and Hope (2007) and Tol (2002),
respectively), but the Stern Review obtains much higher estimates of the social cost of carbon
with a much lower discount rate (Stern, 2007).
28
Our model of Ramsey growth and climate change has more realistic damages
and finds that the optimal carbon tax is a hump-shaped function of world GDP.
Our analysis also pays careful attention to how fast and how much fossil fuel
should be abandoned and how quickly and how much renewables should be
phased in. Our results suggest a ‘third way’ in climate policy which consists of
a quick and aggressive path of upfront renewable subsidies to stimulate use of
renewables and enjoy the fruits of learning by doing, combining the logic of
direct technical change and kick-starting green innovation developed in
Acemoglu et al. (2012) and Mattauch et al. (2012) with a gradually rising
carbon tax as advocated in most integrated assessment studies including
Nordhaus (2008, 2014), Stern (2007) and Dietz and Stern (2014).
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I
Appendix A: A simple formula for the social cost of carbon
Golosov et al. (2014) use a decadal Cobb-Douglas production function with
negative exponential damages:
(A1)
Yt  Z ( Et ) AK t  Ft   exp   (2.13 Et  581)  AK t  Ft  ,
  2.379  10 5 ,
where Kt is the capital stock at the start of period t, A is the calibrated total
factor productivity (including the contribution of fixed factors such as labour
and land),  is the share of capital in value added, and  is the share of energy
in value added. Apart from (1), (2) and (A1), Golosov et al. (2014) assume
logarithmic utility, 100% depreciation of manmade capital each period, zero
fossil fuel extraction costs and thus full exhaustion of initial fossil fuel
reserves. With these bold assumptions one can show that the social cost of
carbon or the optimal carbon tax is a constant fraction of world GDP:

 tGolosov c.s.  1000 
L
1  (1   )
1


 0.004758

(1  L )0
0.00748
Y 

Y,
1  t
1
1  t
1  (1   )(1   ) 
1  0.9772(1   ) 
1  (1   )
where  > 0 is the rate of time preference (see (13)). Decadal world GDP is
630 US$ trillion in 2010, so that using a discount rate of 10% per decade (or
0.96% per year),  = 0.1, we get for 2010 a SCC of 75 US$/tC (or 20 $/tCO2).
This estimate is lower if society is less patient. The beauty of this expression is
that no detailed integrated assessment model of growth and climate change is
needed. The carbon tax follows directly from the rate of time preference ,
world GDP and some technical damage and carbon cycle parameters. The
formula breaks down if intergenerational inequality aversion or factor
substitution differs from unity, extraction costs are non-zero (especially if they
are stock dependent) and the bad fit of the exponential approximation of (5)
plays up.
II
Appendix B: Proof of proposition 1
The adjoined Lagrangian for our model reads:
L




t 0

t 0


t 0
S
B

(1   )t  LU
t t (Ct / Lt )  t (St 1  St  Ft )  t (Bt 1  Bt  Rt )


(1   )t  tPE ( EtP1  EtP  L Ft )  tTE EtT1  (1   ) EtT  0 (1  L ) Ft 


(1   )t t  Kt 1  (1   ) Kt  Z ( EtP  EtT ) H ( Kt , Lt , Ft  Rt )  G(St ) Ft  b( Bt ) Rt  Ct ,
where tS denotes the shadow value of in-situ fossil fuel, tB the shadow value
of learning by doing, tPE and tTE the shadow disvalue of the permanent and
transient stocks of atmospheric carbon, and t the shadow value of manmade
capital. Necessary conditions for a social optimum are:
(A2a) U '(Ct / Lt )  (Ct / Lt )1/  t ,
(A2b) Zt H Ft  Rt  G(St )  tS  L tPE  0 (1  L )tTE  / t , Ft  0, c.s.,
(A2c) Zt H Ft  Rt  b( Bt )  tB / t , Rt  0, c.s.,
(A2d) (1    Zt 1HK )t 1  (1   )t ,
t 1
(A2e) tS1  (1   )tS  G '(St 1 )Ft 1t 1,
(A2f) tB1  (1   )tB  b '(Bt 1 )Rt 1t 1,
PE
(A2g) tPE
 Z '(EtP1  EtT1 )Ht 1t 1,
1  (1   )t
TE
(A2h) (1  )tTE
 Z '( EtP1  EtT1 ) Ht 1t 1.
1  (1   )t
Equations (A2a) and (A2d) yield the Euler equation (9). The Kuhn-Tucker
conditions (A2b) and (A2c) can be rewritten as (10a) and (10b) after defining
E
PE
TE
tS  tS / t , tB  tB / t and t  L t  0 (1  L )t  / t in final good
units. Equations (A2e) and (A2d) yield the Hotelling rule
(A3)
tS1  (1  rt 1 )tS  G '(St 1 )Ft 1 .
Forward summation over time of (A3) gives (11). (A2f) and (A2d) yield
III
tB1  (1  rt 1 )tB  b '(Bt 1 )Rt 1 .
(A4)
Forward summation over time of (A4) yields (12). Defining tPE  tPE / t and
tTE  tTE / t in final good units we use (A2g), (A2h) and (A2a) to get:
PE
(A5a) tPE
 Z '(EtP1  EtT1 )Ht 1,
1  (1  rt 1 )t
TE
(A5b) (1  )tTE
 Z '(EtP1  EtT1 )Ht 1.
1  (1  rt 1 )t
Solving (A5a) and (A5b) it can be shown that tE  LtPE  0 (1  L )tTE is
given by (13).
Appendix C: A temperature lag
Suppose a lag between atmospheric stock of carbon and temperature:


Tt 1  (1  T )Tt  T  ln ( EtP  EtT ) / 581 ,
(3)
where T is the speed at which a higher stock of atmospheric carbon gets
translated into a higher global mean temperature. Since it takes on average
about 70 years, we set T = 1.70. With this additional equation the Lagrangian
function becomes:
 (1   )  L U (C / L )   (S  S  F )   (B  B  R ) 
 (1   )   T  (1   )T    ln  ( E  E ) / 581


 (1   )   ( E  E   F )   E  (1   ) E   (1   ) F 


 (1   )   K  (1   ) K  Z (T ) H ( K , L , F  R )  G ( S ) F  b( B ) R  C .

L
t
t
t 0

t
T
t
t 0

t
t 0
t
t
t
t
t 1
PE
t
t 0

t
t 1
P
t 1
T
P
t
S
t
t 1
t
T
t
B
t
t
P
t
TE
t
L t
t
t
t 1
t
t
T
t
T
t 1
t
t
T
t
t
t
0
t
L
t
t
t
t
t
where tT is the marginal disvalue of global warming. Necessary conditions for
a social optimum are (11a)-(11f) as before and:
PE
(11g) tPE
 T ( EtP1  EtT1 )1 tT1,
1  (1   )t
TE
(11h) (1  )tTE
 T ( EtP1  EtT1 )1 tT1, and
1  (1   )t
IV
(11i) (1  T )tT1  (1   )tT  Z '(Tt 1 )Ht 1t 1.
We now get as before (12)-(15). The dynamics of the permanent and transient
components of the SCC become:
(16)
PE
tPE
  ( EtP1  EtT1 ) 1tT1 ,
1  (1  rt 1 ) t
TE
(1   )tTE
  ( EtP1  EtT1 ) 1tT1 ,
1  (1  rt 1 ) t
where tT  tT / t . Solving (16) yields the SCC as the present discounted
value of all future marginal damages from global warming:
(A6)
tC   s 0  L  0 (1   L )(1   ) s   t  s ( EtP s 1  EtT s 1 ) 1tT s 1 .

Solving (11i) together with (11d) yields:
(A7)
(1  T )tT1  (1  rt 1 )tT  Z '(Tt 1 ) Ht 1.
This equation yields the marginal cost of global warming:
(A8)
tT  s0 (1  T )s t s Z '(Tt s 1 )Ht s 1 .

Upon substituting (A8) into (A6) we get the SCC.
Relationship to Golosov et al. (2014)
Golosov et al. (2014) assume that there is no temperature lag. Following
Gerlagh and Liski (2013) we do allow for such a lag. Under this set of
assumptions, (A8) shows that the marginal cost of global warming at the social
optimum is proportional to global GDP (cf. equation (6)):
T
5  1  
(A8) t  2.379 10 
   T

 Z (Tt ) H ( Kt , Lt , Ft  Rt ).

Upon substitution of (A8) into (A6), we get:
 1   
s
P
T
1
tC  2.379  105 
  s 0   L  0 (1   L )(1   ) t  s ( Et  s 1  Et  s 1 ) Z (Tt  s 1 ) H t  s 1 .




T 


This expression does not simplify to a simple expression depending only on
V
current global GDP. However, if we use a dynamic reduced-form temperature
module with a distributed lag between carbon stock and damages,
Zt  Z (Et ),
(A9)
Et 1  (1  T )Et  T (EtP1  EtT1 ),
we have the Lagrangian function
 (1   )  L U (C / L )   (S  S  F )   ( B  B  R ) 
 (1   )    E  (1   ) E   ( E  E )


  (1   )   ( E  E   F )   E  (1   ) E   (1   ) F 


 (1   )   K  (1   ) K  Z ( E ) H ( K , L , F  R )  G ( S ) F  b( B ) R  C .

L
t
t
t 0

t
E
t
t 0

t

t 0
t
t
T
P
t 1
P
t
t 1
t
S
t
t
t 1
PE
t
t 0
t
t
t 1
t 1
t
t
T
t 1
TE
t
t
B
t
t
P
t 1
T
L t
t
t
T
t 1
t
t
T
t
t
t
0
t
L
t
t
t
t
t
and thus we get:
PE
(11g) tPE
 T tE ),
1  (1   )(t
TE
(11h) (1  )tTE
 T tE ),
1  (1   )(t
(11i) (1  T )tE1  (1   )tE  Z '(Et 1 )Ht 1t 1.
It thus follows that
(A10) tC   s 0  L  0 (1   L )(1   ) s  t  stE s 1 ,

(A11) tE   (1  T )s t s Z '(Et s1 )Ht s1 .
s 0

Under the assumptions made by Golosov et al. (2014), we get
(A11) tE  2.379  105 
1  
 Z ( Et ) H t
   T 
and thus the following simple expression for the SCC:
(A10)
 1 
tC  2.379  10 5 
   T

  1   
 1  
 
 L  
 0 (1   L )  Z ( Et ) H ( K t , Lt , Ft  Rt ).
   
   

VI
A lag between atmospheric carbon and damages (T > 0) thus pushes down the
SCC so our estimates of the optimal global carbon tax will be biased upwards.
Appendix D: Functional forms and calibration
Preferences
As stated in (8), we suppose a CES utility function. We set the elasticity of
intertemporal substitution to  = ½ and thus intergenerational inequality
aversion to 2. The rate of pure time preference  is 1% per year.
Cost of energy
We employ an extraction technology of the form G(S )  1 (S0 / S ) , where 1 and
2
 2 are positive constants. This specification implies that reserves will not be
fully be extracted; some fossil fuel remains untapped in the crust of the earth.
Extraction costs are calibrated to give an initial share of energy in GDP
between 6%-7% depending on the policy scenario. This translates to fossil
production costs of $350/tC ($35/barrel of oil), where we take one barrel of oil
to be equivalent to 1/10 ton of carbon, giving G( S0 )   1  0.35. The IEA
(2008) long-term cost curve for oil extraction gives a doubling to quadrupling
of the extraction cost of oil if another 1000 GtC are extracted. Since we are
considering all carbon-based energy sources (not only oil) which are more
abundant and cheaper to extract, we assume only a doubling of production
costs if a total 2000 GtC is extracted. With S0  4000 GtC,1 this gives  2  1. 2
This implies that we assume very low extraction costs and a high initial stock
of reserves which biases our findings toward using more fossil fuel longer.
1
Stocks of carbon-based energy sources are notoriously hard to estimate. IPCC (2007) assumes
in its A2- scenario that 7000 GtCO2 (with 3.66 tCO2 per tC this equals 1912 GtC) will be burnt
with a rising trend this century alone. We roughly double this number to get our estimate of
4000 GtC for initial fossil fuel reserves. Nordhaus (2008) assumes an upper limit for carbonbased fuel of 6000 GtC in the DICE-07.
2

Since G(2000) / G(4000)  (4000 / 2000) 2  2 2 and 2 2  2.
VII
Initial capital stock and depreciation rate
The initial capital stock is set to 200 (US$ trillion), which is taken from Rezai
et al. (2012). We set the depreciation rate  to be 0.1 per year.
Global production and global warming damages
Output before damages is

H t  (1   ) AKt ( AtL Lt )1


0    1. This

11/
 (
Ft  Rt

1
11/  11/
)


,   0, 0    1 and
is a constant-returns-to scale CES production function in energy
and a capital-labour composite with  the elasticity of substitution and  the
share the parameter for energy. The capital-labour composite is defined by a
constant-returns-to-scale Cobb-Douglas function with  the share of capital, A
total factor productivity and AtL the efficiency of labour. The two types of
energy are perfect substitutes in production. Damages are calibrated so that
they give the same level of global warming damages for the initial levels of
output and mean temperature. It is convenient to rewrite production before
1
11/
11/ 11/


 AKt ( AtL Lt )1 
 Ft  Rt 



. We set the
damages as H t  H 0 (1   ) 


H

H


0
0






share of capital to  = 0.35, the energy share parameter to  = 0.06, and the
elasticity of factor substitution   0.5 which we will refer to as the CES run.
Initial world GDP in 2010 is $63 trillion. Given A1L  1 , we calibrate A = 3.78
to yield initial output under “laissez faire”. The energy intensity of output  is
calibrated to an initial energy use of 9 GtC under “laissez faire”,   0.15.
Population growth and labour-augmenting technical progress
Population in 2010 (L1) is 6.5 billion people. Following Nordhaus (2008) and
UN projections population growth is given by Lt  8.6  2.1e0.35t . Population
growth starts at 1% per year and falls below 0.1% percent within six decades
VIII
and flattens out at 8.6 billion people. In the sensitivity analysis in section 4.3
we assume faster growth and a higher plateau to reflect more recent forecasts.
Without loss of generality the efficiency of labour AtL  3  2e0.2t starts out with
A1L  1 and
an initial Harrod-neutral rate of technical progress of 2% per year.
The efficiency of labour stabilizes at 3 times its current level.
Cost of the renewable and learning by doing
We model learning by doing with initial cost reductions and a lower limit for
the cost of the renewable, i.e.,
b( Bt )  1  2e 3Bt , 1 ,  2 , 3  0.
This
formulation differs from the usual power law definition of learning curves
(Manne and Richels, 2004) but allows us to better calibrate initial learning
rates (which can reach infinity for power law) and specify a lower limit for unit
cost. We calibrate unit cost of renewable energy to the percentage of GDP
necessary to generate all energy demand from renewables. Under a Leontief
technology, with   0 , energy demand is  Zt Ht . The cost of generating all
energy carbon free is  Zt Ht b / Zt Ht   bt . Nordhaus (2008) states that it costs
5.6% of GDP to decarbonise today’s economy in a model of back-stop
mitigation. We increase this cost significantly to 7-8% of GDP. To calibrate
the production cost of renewable energy, this cost estimate needs to be
combined with the cost of producing conventional energy which ranges
between 6%-7%. This gives
 b1  0.14
or with
  0.15
we get
b1  b(0)  1   2  0.94 or $940/tC. Through learning by doing this cost can be
reduced by 60% to a lower limit of 9% of GDP, so that b()  1  0.563 and
thus  2  0.375. In our simulations with   0, this lower limit falls to about 6%
of GDP due to substitution of energy through capital. We assume that
cumulative renewable production lowers unit cost at a falling rate and the
parameter 3 measures this speed of learning. We calibrate learning such that
costs would decrease slowly. We suppose a mere 1.25% reduction in cost if all
of world energy use would be supplied by renewable sources in the initial
IX
period, so that 3 = 0.00375. These assumptions imply at most a 14% reduction
for a doubling of cumulative production at the peak of learning-by-doing and
are very conservative: Manne and Richels (2004) assume a 20% cost reduction
(and consequently need to impose unrealistic constraints on renewable use due
to the strong curvature of the power law learning curve). Alberth and Hope
(2007) assume a cost reduction of 5%-25% for a doubling of cumulative
experience. Our calibration implies a shallow learning curve and, together with
the assumption with the calibration of abundant fossil fuel, biases the model
toward fossil fuel use.
Appendix E: Computational implementation (for online publication only)
In our simulations we solve the model for finite time and use the turnpike
property to approximate the infinite-horizon problem. All equilibrium paths
approach the steady state quickly such that the turnpike property renders
terminal conditions essentially unimportant. We allow for continuation stocks
to reduce the impact of the terminal condition on the transitions paths in the
early periods of the program. We use the computer program GAMS and its
optimization solver CONOPT3 to solve the model numerically. The social
planner solution, scenario I, in which the externality is taken into account fit
the program structure readily. To solve the “laissez-faire” equilibrium paths,
we adopt the iterative approach discussed in detail in Rezai (2011).
Conceptually, the aggregate economy is fragmented into N dynasties in the
externality scenarios. Each dynasty has 1/Nth of the initial endowments and
chooses consumption, investment and energy use in order to maximize the
discounted total utility of per capita consumption. (All production and cost
functions are homogeneous of degree 1 and invariant to N). The dynasties
understand the contribution of their own emissions to the climate change and
learning in renewable energy, but take carbon emissions and knowledge
generation of others as given. The climate and knowledge dynamics are
X
affected by the decisions of all dynasties. This constitutes the market failures.
Under “laissez faire” all dynasties essentially play a dynamic non-cooperative
game, which leads to a Nash equilibrium in which each agent forecasts the
paths of emissions and renewable generation correctly and all agents take the
same decisions.
With identical dynasties, equilibrium requires
RtExog  ( N 1) / N Rt .
Ft Exog  ( N 1) / N Ft
and
If N  1 the externalities are internalized and we obtain the
social optimum. As
N ,
we obtain the “laissez faire” outcome
characterized in section 2. For this outcome, the transition equations (cf.
equations (1), (2) and (8)) become
EtP1  EtP  L Ft Exog ,
EtT1  (1   ) EtT  0 (1   L ) Ft Exog ,
Bt 1  Bt  RtExog .
The numerical routine starts by setting the time path of emissions exogenous to
the planner’s optimization, 0 Ft Exog and 0 RtExog , at an informed guess. (The left
superscript denotes the index of iteration.) GAMS solves for the welfaremaximizing investment, consumption, and energy use choices conditional on
this level of exogenous emissions. These decisions define the time profile of
exogenous fossil and renewable energy use in the next iteration,
and
i 1
RtExog  i Rt .
i 1
Ft Exog  i Ft
The routine is repeated and Ft Exog and RtExog are updated until
the difference in the time profiles between iterations meets a pre-defined
stopping criterion. In the reported results iterations stop if the deviation at each
time period is at most 0.001%.
XI
Appendix F: Comparison of prominent IAMs (for online publication only)
Table F.1 compares the key properties of various IAMs: the DICE-2013R
model as discussed in the decadal version of Nordhaus (2008) and the semidecadal version of Nordhaus (2014), the FUND model as presented in Tol
(2002), the GHKT model as put forward in Golosov et al. (2014), the GL
model as developed by Gerlagh and Liski (2013), and the AABH model as set
out in Acemoglu et al. (2012), and finally the RP model as put forward in the
present paper. The recent Dietz and Stern (2014) extends the DICE model to
make the case for stricter climate policy by allowing for the endogeneity of
growth, the convexity of damages and climate risk, but still has decadal time
resolution and abundant fossil fuel and can thus not discuss issues to do with
how much fossil fuel to lock in the crust of the earth.
Table F.1: Comparing Integrated Assessment Models of Climate Change
Unit of time (in yrs)
DICE
FUND
GHKT
GL
AABH
RP
10 or 5
1
10
10
5
1
Scarce fossil resources






Stranded reserves






Explicit energy market






Endogenous technical
progress in renewables






Population growth






Productivity growth






Optimal growth






N. A.
C.-D.
C.-D.








Aggregate production
function (incl. energy)
Leontief
CES (not logarithmic)
utility

N. A.
100% depreciation


Climate damages
(at high temperature)
low
N. A.
very
low
very
low
Prod.
Health
Prod.
Prod. &
Utility
Type of damage
CES
CES
high
Utility
Prod.