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Armon Rezai and Duncan K. Foley and Lance Taylor
Global Warming and Economic Externalities
Article (Accepted for Publication)
(Refereed)
Original Citation:
Rezai, Armon and Foley, Duncan K. and Taylor, Lance (2010) Global Warming and Economic
Externalities. Economic Theory. ISSN 0938-2259
This version is available at:
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Available in ePubWU: March 2011
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Global Warming and Economic Externalities∗
Armon Rezai†Duncan K. Foley‡Lance Taylor‡
December 6, 2010
Abstract
Despite worldwide policy efforts such as the Kyoto Protocol, the emis-
sion of greenhouse gases (GHG) remains a negative externality. Eco-
nomic equilibrium paths in the presence of such an uncorrected exter-
nality are inefficient; as a consequence there is no real economic oppor-
tunity cost to correcting this externality by mitigating global warming.
Mitigation investment using resources diverted from conventional in-
vestments can raise the economic well-being of both current and future
generations. The economic literature on GHG emissions misleadingly
focuses attention on the intergenerational equity aspects of mitigation
by using a hybrid constrained optimal path as the "business-as-usual"
benchmark. We calibrate a simple Keynes-Ramsey growth model to
illustrate the significant potential Pareto-improvement from mitiga-
tion investment, and to explain the equilibrium concept appropriate to
modeling an uncorrected negative externality.
Keywords: Global warming; Growth with negative externalities; Op-
timal economic growth
JEL Classification System: D62, O41, Q54
∗We received helpful comments from Larry Karp and an anonymous referee. Support
from the Schwartz Center for Economic Policy Analysis is gratefully acknowledged.
†Department of Socioeconomics, Vienna University of Economics and Business, Nord-
bergstr. 15/4, 1090 Vienna; email: arezai@wu.ac.at
‡Department of Economics, New School for Social Research, 6 East 16th Street, New
York, NY 10003
1 Introduction
Much discussion of the economics of global warming emphasizes the is-
sue of trade-offs in well-being between present and future generations
(Nordhaus, 2008; Nordhaus and Boyer, 2000; Stern, 2007). Specifi-
cally, is it socially beneficial for present and near future generations
to sacrifice their own consumption to mitigate global warming for the
benefit of generations yet to come?
In this paper we argue that the intergenerational distribution aspects
of climate policy are relevant only when the externality has been cor-
rected, and concern the distribution of welfare gains, not costs. If
global warming is a negative externality, standard welfare analysis
shows that all generations can benefit from its mitigation. Current
generations can direct less of their foregone consumption to physical
capital formation and more toward mitigation, thereby maintaining
their own levels of welfare while bequeathing a better mix of conven-
tional capital and stock of greenhouse gases (GHG) in the atmosphere
to the future.
We illustrate this point by solving a business as usual economic
growth model calibrated to current data for the intertemporal alloca-
tion of capital by a representative agent with an uncorrected external-
ity and comparing the results to a solution in which the externality is
corrected. The results show that the correction can represent a Pareto
improvement from an inefficient to an efficient growth path with higher
consumption levels and lower environmental damage. There are world
political efforts to implement institutions which enforce the social cost
of carbon emissions on individual agents, most notably the Kyoto Pro-
tocol which created a carbon market and is international law since
2005. Our contribution provides theoretical justification for such and
further measures; the existing framework only succeeds in partially
internalizing the externality with the emissions restrictions set in gen-
erous manners and the majority of emissions not being subject to any
caps.
In much of the literature this simple observation is obscured be-
cause the optimal path is compared to a reference path along which
the externality is partially corrected. This reference path maximizes
the present discounted value of the felicity of per-capita consump-
tion subject to the constraint that mitigation expenditure is equal to
zero. This constrained optimum implicitly includes the marginal so-
cial cost of emissions in the representative agent’s production and in-
vestment decisions, thus partially internalizing the externality. Com-
paring this solution to the true optimum incorrectly directs attention
toward ”inter-generation trade-offs” because typically the constrained
optimum shows higher per capita consumption for several decades.
While policy economists have persisted in presenting the global warm-
1
ing problem primarily as an issue of intergenerational equity, many
scholars emphasize the public-good nature of the problem, which leads
to the perspective developed in the present paper (Chichilnisky, 1994;
Chichilnisky and Heal, 1994; Chichilnisky and Sheeran, 2009).
When, on the other hand, the optimal and business as usual paths
are compared directly, the first-order effect of optimal mitigation is a
potential increase in per capita consumption in every time period. In-
tergenerational equity enters into the problem only as a second-order
effect as the optimal program distributes the potential gains from cor-
recting the externality across generations in accordance with the rep-
resentative agent’s preference for consumption smoothing. The com-
parison of the optimal and constrained optimal paths thus leads to
an upward biased estimate of the economic costs of mitigating global
warming.
This symposium includes contributions on other important aspects
of the global warming problem. The most related to our are those
by Karp and Zhang (2011), who discuss the effectiveness of different
policy instruments, Ostrom (2011), who discusses means of introducing
cost transparency at different governmental levels, and Chipman and
Tian (2011), who investigate the assumptions under which markets for
pollution rights lead to Pareto improvements. Burniaux and Oliveira
Martins (2011) aim at identifying the sensitivity of ”carbon leakage”
to key parameters in a general equilibrium model of climate policy.
Dutta and Radner (2011) and Lecocq and Hourcade (2011) discuss
further roadblocks in the way of efficient international climate policy
in a multi-region framework. The remaining contributions (Asheim,
Mitra, and Tungodden, 2011; Chichilnisky, 2011; Figuieres and Tiball,
2011; Lauwers, 2011) group around the ethical foundations of criteria
for sustainability and their application to economics.
2 The Global Warming Problem
Human (industrial) production entails emissions of GHG. Given sci-
entific evidence like the results presented in the 4th report of the In-
tergovernmental Panel on Climate Change (IPCC), such emissions im-
pact the world climate negatively. An increase in the concentration of
GHG is projected to increase the mean atmospheric temperature im-
plying a higher frequency of disasters and natural catastrophes (such
as droughts, floods, and heat waves), higher mortality rates, and a sig-
nificant loss of biodiversity. These consequences have economic costs,
the most apparent being a loss in the productive capacity of the world
economy.1
1The worldwide economic implications of climate change are hard to quantify. This
task becomes more difficult, the higher the assumed stock of carbon in the atmosphere.
2
The world climate is affected by the use of capital which produces a
negative externality in the form of emissions. Uninternalized externali-
ties lead to inefficiencies since economic agents do not perceive the true
cost of their actions and do not equalize (social) marginal costs and
benefits. In our case the representative agent is over-emitting GHG,
since she perceives the marginal cost of doing so (to her as an indi-
vidual) is zero. Under the perfect foresight assumption, she is able to
correctly predict the path of GHG (mainly CO2) concentrations given
her (and everybody else’s) consumption, production, and investment
choices. Although she is aware of the collective consequences of her
actions, she thinks her individual contribution to the overall result is
negligible. Consequently, she will not reduce her production-related
emissions, either through producing less or investing in mitigation, be-
cause she knows that nobody else will do so (as they believe their
actions to be insignificant, too). All agents end up choosing the same
inefficient allocation. This point was made in Foley (2009). Such
socially sub-optimal outcomes are well known from simple strategic
games, the most prominent being the ”Prisoners’ Dilemma”.
Given the inefficiency of over-accumulation of GHG stock in the at-
mosphere as a result of capital stock accumulation, the world economy
is not operating at the intertemporal production possibility frontier
(PPF). Future generations would appreciate lower stocks of CO2which
implies that current generations should accumulate less conventional
capital and consume more (of it) today. There is no intergenerational
trade-off despite the fact that such a trade-off is posited in most of
the global warming related economic publications. The mutual gains
can be illustrated by moving the economy from a point inside the PPF
to its boundary. This movement to an efficient equilibrium can be
achieved by cost transparency (which amounts to increasing the cost
of emitting to its true value).
Creating the correct price signal for GHG emissions (by whatever
means, including cap-and-trade permits, Pigouvian taxes, or direct
regulation) is sufficient to internalize the negative externality of global
warming. As a result our agent will start to invest into mitigation.
These mitigation costs, however, are small compared with the gain of
obviating GHG emissions. As is shown below, averting climate change
can represent a non-trivial Pareto improvement.
3 The Model
The model used here for analyzing the economic aspects of global
warming is a standard Ramsey-Cass-Koopmans model of the economy
The report of the IPCC (2007) presents convincing evidence for the negative relationship
between GDP and global warming. Tol (2009) gives an optimistic review.
3
extended to included GHG. In order to maximize the comparability of
our results with models in the literature based on neoclassical growth
theory, the economy only produces one good using a Cobb-Douglas
production function, F[K, L], in conventional capital and effective la-
bor. Effective labor consists of the exogenously given growth paths of
population, N, and Harrod-neutral technical change, B, (which can
be translated into Hicks-neutral technical progress given the Cobb-
Douglas technology), according to L=BN . The state equation for
conventional capital, K, is in its standard form with capital increasing
due to investment, I, and decreasing due to (exponential) depreciation
at rate δ.
Following Nordhaus and Boyer (2000), the stock of GHG in the at-
mosphere, CD (for carbon dioxide, CO2, measured in parts per million
volume, ppmv), enters the model as an additional state variable. Its dy-
namics depend on usable output, Y, and are governed by production-
related emissions, G[Y], mitigation efforts, M[m]Y, and (exponential)
depreciation at rate .mis the share of usable output invested in
mitigation. Mitigation efforts, M[m]Y, are linear in usable output
similar to the abatement cost function, Λ[.], used in Nordhaus and
Boyer (2000) and Nordhaus (2008).
In the interests of keeping the model as parsimonious as possible,
temperature dynamics are omitted and CD in excess of pre-industrial
levels lowers productive capacity directly via what we term a damage
function, Z[CD]. There are no sinks and no time lags. Emissions fully
affect output immediately and directly, Y=Z[CD]F[K, L]. Mitiga-
tion can take the form of removing existing CD from the atmosphere
or by preventing current emissions. Mitigation does not alter carbon
emissions intensity permanently.
Formally, the representative agent allocates shares of output to
consumption, c, and investment, s, of which a certain output share, m,
is invested into mitigation, in order to maximize utility, measured as
the discounted present value of the felicity of per capita consumption
over time. Let consumption C[t] = (1 −s[t])Y[t−1] and world output
Y[t] = Z[CD[t]]F[K[t], L[t]], then total utility is
U[C[t], t] =
T
X
t=1
1
(1 + ρ)(t−1) UC[t]
N[t]
These choices are subject to initial values K[0] and CD[0] and the
following state equations:
K[t+ 1] = (1 −δ)K[t]+(s[t+ 1] −m[t+ 1])Y[t]
CD[t+ 1] = (1 −)CD[t] + G[Y[t]] −M[m[t+ 1]]Y[t]
4
With λ[t] as the shadow price of capital and µ[t] the shadow price
of CO2in the atmosphere, both expressed in terms of undiscounted
felicity in the period t, and c[t] = C[t]
N[t]as per capita consumption in
period t, the adjoined Lagrangian for the above problem is
L[K, CD, λ, µ, t] =
T
X
t=1
1
(1 + ρ)(t−1)
(U[c[t]] + λ[t](K[t]−(1 −δ)K[t−1] −(s[t]−m[t])Y[t−1]) +
µ[t](CD[t]−(1 −)CD[t−1] −G[Y[t−1]] + M[m[t−1]]Y[t]))
(1)
Note that µ[t]<0, as CD affects production negatively. The
(negative of the) current dollar price of carbon emissions is given by
χ[t] = µ[t]
λ[t].
3.1 The optimal case (OPT)
For optimality the following first-order conditions have to hold, which
simultaneously represent a social competitive equilibrium under the as-
sumption that institutions (such as an optimal carbon tax, a universal
and optimal cap-and-trade system, or optimal direct regulation) exist
to impose the social costs of emission on producers and consumers:
∂λL= 0 ⇔K[t] = (1 −δ)K[t−1] + (s[t]−m[t])Y[t−1] (2)
∂µL= 0 ⇔CD[t] = (1 −)CD[t−1]+
G[Y[t−1]] −M[m[t]]Y[t−1] (3)
∂KL= 0 ⇔λ[t] = λ[t+ 1]
1 + ρ
(1 −δ+rK(1 −m[t+ 1] + (G0[Y[t]] −M[m[t+ 1]])χ[t+ 1]))
(4)
∂CDL= 0 ⇔µ[t] = λ[t]χ[t] = λ[t+ 1]
1 + ρ(rCD(1 −m[t+ 1])+
(1 −+rCD(G0[Y[t]] −M[m[1 + t]])) χ[1 + t]) (5)
∂sL= 0 ⇔λ[t] = c[t]U0[c[t]]
C[t](6)
∂mL= 0 ⇔χ[t] = −1
M0[m[t]] (7)
The first two equations are simply the state equations for K[t]
and CD[t]. With rK= (1 −Z[CD])FK[K, L] the marginal product
of capital, the next equation tells us that in an optimal program the
5
current value of capital must be equal to its marginal benefit, which is
the discounted value of its net marginal product factoring in mitigation
costs and net emissions resulting from a larger capital stock. Since
the time path of λ[t] reports the shadow price of capital, its change
times the discount factor yields the marginal rates of intertemporal
substitution which equal the real interest rate j[t] = λ[t]
λ[t+1] (1 + ρ)
yields the time path of the real interest rate, j[t].
With rCD =−Z0[CD]Y
Z[CD] the marginal product of CD, the same has
to hold in the fourth equation for CD: The price of CO2must be equal
to the discounted value of its net marginal product, again, factoring in
mitigation positively and net emissions resulting from higher output
negatively. This equation thus reflects the assumption that an effective
system of optimal pricing of emissions is in effect when we calculate
the OPT path. The last two equations are the Euler equations and
establish optimality with regard to the choice variables, s[t] and m[t].
They tell us, first, that marginal utility of consumption per capita in
period thas to be equal to cost of capital (which is measured in per
capita utils per unit of capital). Through equation (4) the marginal
utility of consumption per capita is equal to the per capita marginal
benefits of accumulating more over the remaining time horizon; sec-
ond, that marginal cost of mitigating has to be equal to the marginal
future benefit of doing so. The capital (usually dollar) price of carbon
emissions is fixed by the cost of the marginal emission reduction. This
thought will be taken up later. Given the Euler equations and the
co-state equations, one can derive Ramsey-Keynes rule equivalents of
the system.
3.2 The business-as-usual case (BAU)
We model the business-as-usual case as an equilibrium of the economy
in which global warming is the outcome of a negative externality. A
state variable is an externality when it has a real impact on the ob-
jective function or constraints, but no institutions exist to enforce the
social price on individual agent decisions involving it. Each agent as-
sumes that her decisions will not affect the path of the externality, but
when all agents make the same decisions the path of the externality
changes. On a perfect-foresight equilibrium path with an uncorrected
externality, each agent is assumed to correctly forecast the path of the
externality, but ignores the effect of her decisions on the path of the
externality. Thus on the equilibrium path with CD as an uncorrected
externality the typical agent solves the above maximization problem
expecting a certain time path of for the external CD[t]. The correct
forecasting assumption amounts to the side condition that expected
CDe[t] = CD[t], where CD[t] is the path of the externality correspond-
6
ing to the representative agent’s chosen decisions. The difference be-
tween the equilibrium path with an uncorrected externality and the
optimal path is the fact that the typical agent does not adjust her
controls to take account of their effect on the externality. She is not
aware of the true social cost of emitting. No social institutions exist to
provide the correct price signal to steer the economy. As a result, the
socially competitive equilibrium and the optimum diverge. Also, the
social bid price and social ask prices for the externality are not equal.
It is possible to express the second-best equilibrium path with an
uncorrected externality through the Lagrangian first-order conditions.
The first-order conditions with respect to the shadow prices return
the real laws of motion of the system, which must be obeyed. The
representative agent, however, ignores the effect of her decisions on
the external state variables, which corresponds to setting the shadow
price on these variables equal to zero in the first-order conditions with
respect to the non-external state and choice variables. The first-order
condition with respect to the external state variable then plays no
active role in the solution, but does keep track of the real social value
of the externality in terms of its shadow-price µ. The BAU path solves
these modified first-order conditions:
∂λLBAU = 0 ⇔K[t] = (1 −δ)K[t−1] + s[t]Y[t−1] (8)
∂µLBAU = 0 ⇔CD[t] = (1 −)CD[t−1] + G[Y[t−1]] (9)
∂KLBAU|µ[t]=0 = 0 ⇔λ[t] = λ[t+ 1]
1 + ρ(1 −δ+rK) (10)
∂CDLBAU|µ[t]=0 = 0 ⇔µ[t] = λ[t]χ[t] = 0 (11)
∂mLBAU|µ[t]=0 = 0 ⇔m[t] = 0 (12)
∂sLBAU|µ[t]=0 = 0 ⇔λ[t] = c[t]U0[c[t]]
C[t](13)
It is important to notice the subtle differences between these equa-
tions and the fully optimal path equations. The state equations (8)
and (9) are the same on the optimal and BAU paths. But on the BAU
path the controls are optimized without taking account of the exter-
nality, since µ[t] = 0 in the first-order condition. As a result m[t] = 0.
s[t] remains unaltered due to the specific form of the model (s[t] is
the total share of income saved). Likewise, equation (10) determines
the shadow price of the non-external state variables with the shadow
price of the externality, µ[t] = 0. In the absence of the externality, the
Euler equation for capital and the optimal savings decision reduce to
the usual Ramsey-Keynes rule.
There are two types of misallocation on the BAU path. First,
because there is no market price for carbon emissions, the typical agent
7
allocates too little (zero) resources to mitigation. Second, the typical
agent invests too much in conventional capital because she ignores the
impact of increasing output on increasing climate damage.
In our calculations we keep track of how m[t] and µ[t] would evolve
according to their first-best first-order conditions given the other, second-
best variables.
∂mL= 0 ⇔χ[t] = M0[m[t]]
∂CDL= 0 ⇔µ[t] = λ[t]χ[t] = λ[t+ 1]
1 + ρ(rCD(1 −m[t+ 1])+
(1 −+rCD(G0[Y[t]] −M[m[1 + t]])) χ[1 + t])
3.3 The constrained optimal case (COPT)
Researchers of the economic consequences of global warming (most
prominently and recently Nordhaus and Boyer, 2000; Nordhaus, 2008)
analyze an optimal path under the constraint that no mitigation is
undertaken. Because this type of path implicitly partially internalizes
the externality, it seems to us that it does not represent ”business-as-
usual”, and in the remainder of this paper we call this type of path
constrained optimal.
How exactly the constrained optimal path can be derived as an equi-
librium within the representative-agent perfect foresight methodology
is somewhat mysterious. Fully rational agents with perfect foresight
acting with complete markets adopt the first best solution presented
above as the optimal path. When markets are incomplete and there is
no price signal for the marginal social value of the externality, the equi-
librium is the BAU path described in the last section. The constrained
optimal (COPT) path, on the other hand, represents an inconsistent
mixture of assumptions about the representative agent’s information.
On the one hand, the representative agent on this type of path correctly
estimates the marginal social cost of emissions in making her consump-
tion, investment, and production decisions. On the other hand, she
seems to ignore the availability of mitigation technologies, despite this
understanding of the marginal social cost of emissions. This divergence
results in a difference between the marginal social value and marginal
social cost of mitigation. The agents in this mixed scenario perceive
the marginal social cost of emitting as zero, the only price that justi-
fies no mitigation. At the same time, however, the agent is confronted
with the true carbon price in her decision on how much output to
consume and how much to re-invest for capital formation. While this
inconsistency within the perfect foresight framework is corrected for
8
in the business-as-usual case above, we also solve for the constrained
optimal case given its importance in the economic literature on global
warming.
The first-order conditions of the constrained optimal case are the
special case of the fully optimal with m[t] = 0.
∂λLCOPT = 0 ⇔K[t] = (1 −δ)K[t−1] + s[t]Y[t−1] (14)
∂µLCOPT = 0 ⇔CD[t] = (1 −)CD[t−1] + G[Y[t−1]] (15)
∂KLCOPT = 0 ⇔λ[t] = λ[t+ 1]
1 + ρ(1 −δ+rK(1 + G0[Y[t]]χ[t+ 1]))
(16)
∂CDLCOPT = 0 ⇔µ[t] = λ[t]χ[t] = λ[t+ 1]
1 + ρ
(rCD + (1 −+rCDG0[Y[t]]) χ[1 + t]) (17)
∂sLCOPT = 0 ⇔λ[t] = c[t]U0[c[t]]
C[t](18)
Notable changes include the altered state equation for CD[t] in
which carbon dioxide concentration can only be changed through emis-
sions and dissipation and the two modified co-state equations. The
change in the marginal benefit of capital from the optimal to the con-
strained optimal path depends on the functional form of M[m[t]]. For
any meaningful mitigation function, the marginal benefit of capital
will be lower in the constrained optimal case. As a result, less capital
will be accumulated. This is intuitive as a reduction in Kis the only
means by which CD[t] increases can be counteracted with mitigation
constrained to zero. Note that while agents are deprived of the choice
to mitigate, they still fully respond to changes in the price of carbon
in their accumulation decisions. The same logic applies to the price of
CD[t]. Lower marginal benefit will lead to higher CO2concentration,
as CD is a ”bad”.
In our calculations below we retain the (for this program) super-
fluous optimality condition for m[t] in order to see what mitigation
effort would be under an optimal scenario (at that point in time of the
program).
∂mL= 0 ⇔χ[t] = −1
M0[m[t]]
3.4 Basic Logic
The basic logic and qualitative features of the three cases can be seen
even without specifying functional forms, calibrating the model to re-
flect current economic values, and solving it over long time horizons.
9
On the OPT equilibrium path agents will invest enough resources
in mitigation to compensate emissions to the point where marginal cost
of doing so is equal to the benefit in output due to less environmental
damages. This level is defined by the damage function which is key
to the quantitative outcomes of the model. The carbon price will be
defined by mitigation efforts. It will be equal to the cost of marginal
mitigation efforts.
On the BAU equilibrium path agents are not only deprived of the
mitigation instrument, but also see themselves incapable of affecting
the stock of CO2. Their decisions are taken solely with respect to
maximizing their intertemporal utility and consumption. The carbon
dynamics drive the system. Emissions from rapid capital accumulation
will drive up carbon dioxide levels to the point where environmental
damage chokes off further accumulation due to the falling profit rate.
On the COPT equilibrium path agents are deprived of the miti-
gation instrument. In order to move the economy to a steady state
equilibrium, the savings decisions and the capital stock will have to
do all of the adjustment. It is important to note that in the COPT
case (as well as in the BAU case), the climate dynamics dominate the
outcome. A steady state can only be achieved when emissions equal to
the dissipation of existing carbon stock in the atmosphere. This sets
the level of admissible capital stock and the appropriate savings rate.
Given the logic of the three model cases, it is trivial, but nonetheless
important, to note that the overall utility will be the greatest on the
OPT path, followed by the COPT and BAU paths. Growth in capital
stock will be highest on the OPT, followed by the BAU. Investment will
be lower in the COPT than in the BAU scenario as agents are aware of
the deleterious effects of their accumulation decisions and, hence, more
cautious. Climate catastrophes – meaning high equilibrium levels of
CO2– are certain in the BAU and very likely in the COPT case. Over
the range of all realistic mitigation functions, CO2levels will stabilize
at low or moderate levels in the OPT case.
These characteristics are given by the model structure and are,
thus, independent of the functional forms and parameter values. We
hope that in this light the secondary relevance of much of the current
debate on discounting factors will be apparent.
Fast convergence to the steady state implies that the steady state
results will drive much of the model’s behavior and its transition dy-
namics. As can be seen below, the optimal path reaches its steady
state within 10 decades. Nordhaus (2008) and Stern (2007) do not
devote much attention to the steady states of their models and the
implications of steady state values on the transition dynamics.
10
4 Functional Forms
There are several choices that need to be made about the forms the
production, damage, mitigation, and emissions functions take. The
functional forms we use in the simulations reported here are as follows:
Utility has its traditional iso-elastic manifestation U[c[t]] = c[t]1−η
1−η, or
U[c[t]] = Log[c[t]] when η= 1. Potential output is a Cobb-Douglas
production function, F[K[t], L[t]] = AK [t]αL[t]1−α. Carbon-related
damages are measured on a scale between 0 and 1 with zero damage
at the pre-industrial level of 280 ppmv and complete output loss at a
CDMax = 780ppmv, with the damage function Z[CD[t]] = 1−CD[t]−280
CDMax−280 1
γγ
.
This functional form implies that even at current CO2levels of 380
ppmv a certain fraction of potential output is lost due to environmen-
tal degradation. Emissions, G[Y[t]] = βY [t], are linear in output at
a constant carbon intensity of production. The mitigation function,
M[m[t]] = ζ1−e−νm[t]
ν, where ζis a scaling parameter and 0 < ν is
a semi-elasticity reflecting diminishing returns to m, which converts
the unitless proportion of output devoted to mitigation, m, into CO2
reduction per $ spent on mitigation, that is ppmv/$. In our specifica-
tion of the mitigation function, we diverge from the other studies in
assuming positive mitigation costs even at very low mitigation efforts
(∂mM[m[t]]|m[t]=0 =ζ6= 0). From the first-order condition for m[t],
M0[m[t]] equals the carbon price χ[t] in the first best solution.
Population growth follows Nordhaus (2008) and UN projections in
assuming that world population will rise from currently 6500 to 8600
million over the next 10 to 20 decades, and then stabilize at this level.
Labor productivity is assumed to start at a yearly growth rate of 2%
and to flatten out at 3 times its current value after 30 decades.
Given these functional forms, the adjoined Lagrangian (with L[t] =
B[t]N[t]) is:
L[K[t],CD[t], λ[t], µ[t], t] =
T
X
t=1
1
(1 + ρ)(t−1)
(1−s[t])Z[CD[t−1]]F[K[t−1],L[t−1]]
N[t](1−η)
1−η+λ[t](K[t]−(1 −δ)K[t−1]
−(s[t]−m[t])Z[CD[t−1]]F[K[t−1], L[t−1]])
+µ[t]CD[t]−(1 −)CD[t−1] −β−ζ1−e−νm[t]
ν
Z[CD[t−1]]F[K[t−1], L[t−1]]))
11
5 Calibration
Given the assumptions of functional forms, the actual functions need to
be calibrated to match economic and physical realities. All parameters
are geared towards a decadal time interval. In the benchmark case
Lagrangian, the discounting factor, ρ= 0.1. With output measured in
units current (2000-2010) $ trillion, initial capital stock is assumed to
be K0= 200. CO2is measured as parts per million volume (ppmv).
Initial CO2concentration CD0= 380. Capital decays radioactively
with δ= 0.7.
Currently, 7 Gt carbon are burnt per year. This corresponds to
an increase in CD of 3.37 ppmv. With a yearly world output of $60
trillion this implies a carbon dioxide emission intensity β=3.37
60 =
0.0561 ppmv
$trillion . As the actual increase in atmospheric carbon dioxide
concentration is only about 2 ppmv, dissipation is 1.37 ppmv. This
yields the yearly dissipation rate =1.37
380 = 0.0036.
The marginal product of capital in the Cobb-Douglas production
function αis set at 0.35 in line with standard economic research. Total
factor productivity is calibrated to match current world output of $60
trillion. The elasticity parameter in the utility function ηis set at 2,
in our baseline simulations.
The damage function Z[CD[t]] also takes an iso-elastic form. This
allows us to combine the apparent global warming optimism of economists
towards the low damages of global warming at low carbon dioxide con-
centration with the serious warnings of climate scientists about severe
output loss at high carbon dioxide concentration (which is set to 780
ppmv in our model). We set γ= 0.5, which is at the higher end of
potential damages (Barker, 2008). The calibration points usually cited
are the IPCC (2008) predictions of an increase to a doubling of pre-
industrial concentrations (about 280 ppmv) leading to a temperature
rise of 3◦C and an increase of 4◦C leading to a potential output loss
of 1 −5% of current output. Nordhaus (2008) assumes that current
damages to the world economy are 0.15% of output. This corresponds
to γ= 0.3. The results for γ= 0.3 are reported in the sensitivity anal-
ysis section below. We deviate from Nordhaus as his damage function
is lacking any reasonable upper limit on temperature and ultimately
CO2concentration. Rezai (2010) shows that our parametric form of
a damage function lies consistently above Nordhaus’ for γ= 0.5 and
below it with γ= 0.3 up to 500 ppmv (which is the relevant range for
the optimal program). It is worth mentioning that the asymptotic be-
havior of the damage function for high concentrations has little impact
on the OPT path as long as its shape for low concentrations is close to
the mentioned calibration points. This holds for our range of damages.
Figure 1 plots the damage function for γ= 0.5 and a band of 0.7≥
γ≥0.3, around which a sensitivity analysis is carried out below. It also
12
includes the assumptions on environmental damages from Nordhaus
(2008) as a dark gray area up to CD = 580. Note that while our
assumptions on γregarding damages can be regarded as high, so are
the assumptions on mitigation costs with the carbon price at $160 (per
t of C).
Figure 1: Damage Function with γ= 0.5
The parameter ζrepresents the marginal reduction in CO2concen-
tration (ppmv) per $T from spending a small amount on mitigation
when mitigation is zero. If it costs $160 to remove one tonne of C at
present (carbon markets suggest level between $75 and $125), when ef-
fectively m= 0, then to reduce CO2concentration by 1 ppmv through
removing 2.07 Gt C from emissions would cost (2.07)(0.160$trillion) =
0.331$T, so we set ζ= 3ppmv
$T . Using this specification the carbon price
is directly linked to and anchored by marginal mitigation efforts. Note
that the assumption of a lower current carbon price increases ζand
makes the mitigation function more effective.
Figure 2: Mitigation Function for ζ= 3 and ν= 6
The table below provides an overview over the parameter assump-
tion.
13
Function Parameter Value
L[.] ρ0.1
K0200
CD0380
δ0.7
0.036
U[.] η2
F[.] α0.35
A28.4
Z[.] γ0.5
CDMax 780
G[.] β0.056
M[.] ζ3
ν6
Table 1: Overview of the parameter values used in the numerical simulation
6 Computational Implementation
The above systems of equations can be reduced to 4 laws of motion, 2
for the state variables and 2 for the co-state variables by substituting
the optimal expressions for the controls into the other equations and
forming thus the maximized Lagrangian. These 4 difference equations,
of which some form a subsystem in the COPT and BAU case, have
to hold for t= 1,..., T . In addition, initial conditions on the two
state variables and terminal transversality conditions on the co-state
variables have to hold. This yields 4(T+ 1) conditions to determine
4(T+ 1) variables.
In order to solve this set of equations, we make use of the pro-
gram Mathematica and its root finding command. In this process the
specification of initial search parameters is crucial. Generally, Mathe-
matica proves to be quite agile in finding the equilibrium path even if
state and co-state variables are persistently shifting on a steady growth
path. Specifying variables in logarithmic forms some times facilitates
the search routine in this case.
7 Growth Paths
The above first-order conditions are sufficient as well as necessary for
a global maximum as the maximized Hamiltonian is concave in K
and CD for any given λand µ(that is, the objective function is quasi-
concave and the constraints convex in Kand CD). The programs below
are set up as finite-horizon problems with the additional requirement
of the transversality conditions λ[T]K[T] = 0 and µ[T]CD[T] = 0 with
a fixed Tat 60 decades. These terminal conditions guarantee that the
14
calculated OPT path is a valid optimum of the primal problem over
the finite time horizon. We choose the time horizon sufficiently long
such that the paths to approximate the ”steady-state” in the middle of
the time horizon. This characteristic is known as the turnpike property
which assures that finite horizon programs mimic their infinite horizon
twins for sufficiently long horizons (Samuelson, 1965). The paths be-
low effectively reach their steady state within 30 decades. Solving over
60 periods becomes sufficiently close to the infinite horizon problem.
Note that it would be possible to change the transversality end-point
conditions to require, for example, some minimal capital and maxi-
mal CO2stocks at the end of the program, but it is not very easy to
see how to choose those levels, or, in fact, to use this method except
in the finite time horizon case. (It is not really correct to force the
path to converge to the steady state at any finite time, for example.)
We also emphasize the illustrative character of the simulations below,
since long-run projections of any economic growth model are subject
to high uncertainties in key parameters and the possible appearance of
intervening factors not included in the model.
Figure 3 is a comparison of the optimal, constrained optimal and
business-as-usual paths for these parameters, in terms of world per
capita consumption, the damage from global warming, the implied
price of carbon, and the CO2concentration. The finite-horizon pro-
gram approaches its steady state very quickly, as to be expected from
the turnpike theorem.
On the unconstrained optimal path capital accumulation combined
with ”small” mitigation efforts of around 1 −2% of GDP enables sus-
tainably rising output and consumption levels. Mitigation efforts are
front-loaded, meaning that most of the mitigation is done in the first
few decades, such that only current CO2emissions have to be mitigated
in later periods. The carbon price stabilizes around $180/t which is
close to zero mitigation carbon cost of $160. In fact, atmospheric car-
bon concentration decreases almost to its pre-industrial level.
On the BAU path agents lack the correct price signals to cor-
rect the negative externality. This leads to inefficiencies in several
respects. Capital rises at a rate similar to the optimal case during the
first 100 years although no mitigation can be carried out; the impli-
cation of this over-accumulation is rapidly rising CO2concentration
and environmental damage. Since the capital accumulation equation
for λ[t]BAU is independent of any carbon related costs (or their price
signals), accumulation continues until damages are so high that fur-
ther accumulation cannot occur due to the declining productivity of
capital and labor inputs. Output and consumption are bound to de-
cline due to ever higher carbon concentration and damages until the
marginal product of capital has fallen sufficiently to approach a sta-
15
ble equilibrium. Equilibrium output and consumption per capita are
almost 20% below current levels and less than 25% of the OPT levels
Figure 3: The optimal equilibrium path, OPT, is plotted in pale-gray, the BAU equilibrium in
gray, and the constrained equilibrium, COPT, in dark-gray. The carbon price on the optimal path
is about $200/t, and the damage on the optimal path is less than 1% of potential output. The
carbon price is the social marginal value of foregoing the emissions from 1t of carbon. For OPT
and COPT the carbon price is effective in economic decisions, but not in BAU (and is plotted as
a dashed line). For OPT the mitigation percentage is the proportion of world product devoted
to mitigation. For COPT and BAU the mitigation percentage is the investment called for by the
imputed carbon price (and is plotted as a dashed line), while the actual mitigation is zero.
16
despite significant technical progress and population growth. The gray
dashed lines in the carbon price and mitigation graphs depict the im-
plied carbon price and the mitigation called for by this price. Implicit
carbon price and mitigation efforts on the BAU are fifty times higher
than their optimal counterparts. The inefficiency of the BAU can also
be seen in the equilibrium saving rates. While higher capital stock
implies higher saving to compensate the lower marginal product, the
equilibrium BAU saving rate with its lower capital stock is higher in
our simulations due to the (unnecessary) high carbon concentration.
The constrained optimal path does slightly better than the BAU
path. Although the agents in the COPT also are confined to zero
mitigation, they conceive the correct price signals and run up GHG in
the atmosphere much more cautiously than in the BAU scenario. In
fact, overall capital stock is decreasing on this path as current levels are
suboptimally high, implying too many carbon emissions. As carbon
concentration increases, so must the marginal product of capital which
can only be achieved with a lower capital stock. Equilibrium GHG
concentration entails a high carbon price which, again, calls for high
mitigation efforts. Equilibrium carbon concentration and price are
lower than on the BAU path. The carbon price is about twenty times
the optimal level.
While the quantitative results of our simulations are dependent on
the specific parameter assumptions, it is important to point out that
the qualitative results are independent of these assumptions. Espe-
cially the finding that moving from the inefficient BAU path to the
efficient OPT path through mitigation constitutes a Pareto improve-
ment should be noted. This result implies that there is no cost to
mitigation, but there are significant gains from doing so (in our simu-
lations up to 400% of GDP). Higher or lower discounting rates will not
alter this. In light of the magnitude of the avail, we argue that ques-
tions of uncertainty and intergenerational equity which are discussed
further below can be considered of second-order importance.
The high carbon prices and low, often negative real interest rates
implied by these simulation results also underline the important method-
ological point that the present discounted value of future costs and
benefits are conditional on some particular consumption path for the
representative agent. On the BAU path simulated here, for example,
the implicit real interest rate is negative for a significant part of the
transition path and below the discount rate on average over the whole
path. The real interest rate is negative because the representative agent
would pay a high price to move consumption from the early part of the
BAU path, where capital and labor are highly productive, to the later
part of the path, where productivity has been destroyed by the climate
catastrophe. Even though the time horizon of the simulations is very
long, the present discounted value of carbon emissions on the actual
17
path is, as a result, quite substantial. Much of the existing literature
on global warming ignores the dependence of present discounted values
on the consumption path, and calculates present discounted values of
future costs and benefits using steady-state values of real interest rates.
Since steady-state real interest rates are positive, discounting at these
interest rates results in very low present discounted values even of very
significant future damage, if the damage is far enough in the future
(see Chichilnisky and Eisenberger, 2010). The consistent present dis-
counted values using real interest rates implied by equilibrium paths
are much higher.
8 Intergenerational Equity
An important aspect of the current climate change debate in the eco-
nomic literature centers on the problem of intergenerational equity.
This focus on generational equity arises primarily from a failure to
appreciate that the business-as-usual path with an uncorrected exter-
nality is inefficient. In particular, it is a consequence of the mistaken
use of the COPT path rather than the real BAU path as the benchmark
with which the OPT path is compared. The most striking difference
between the COPT and OPT paths is the generational distribution of
consumption. As we have explained above, the COPT path is not a
theoretically relevant benchmark, because it represents an inconsistent
mixture of partial internalization of the global warming externality and
a failure to divert resources from conventional investment to mitiga-
tion.2The use of the COPT path as the benchmark comparison leads
to the misleading impression that the problem of correcting the global
warming externality is primarily an issue of intergenerational equity,
how much to sacrifice the consumption of current generations to protect
the environment for future generations. In an optimal growth frame-
work the resolution of this trade-off depends on the discount factor,
ρ, and the degree of social preference for consumption smoothing ex-
pressed in the elasticity of felicity with respect to consumption, η. But
with the correct BAU benchmark the correction of the global warm-
ing externality can provide an intergenerational Pareto improvement,
raising the per capita consumption and felicities of every generation.
2Shiell and Lyssenko (2008) offers a clear explanation of the logic supporting this point,
in the context of a two-state variable model very similar to the present one. This paper also
presents an ingenious alternative method to ours of computing approximate BAU paths.
Shiell and Lyssenko also find that the asymptotic behavior of COPT and BAU paths is
very similar. Because they do not focus on the initial transient levels of consumption on
the OPT and BAU paths, however, their discussion does not bring out the critical role
of the COPT path in suggesting that an intergenerational distributional tradeoff is at the
center of global-warming policy evaluations.
18
The parameters ρand ηinfluence the distribution of this intergenera-
tional gain, which is a second-order consideration.3In fact, with our
benchmark values of ρ=.1/decade and η= 2, the OPT path exhibits
slightly lower per capita consumption than BAU in the first decade.
One way to understand this fact is that the representative agent in
the BAU equilibrium values current consumption too highly relative
to investment in mitigation; when she grasps the full social marginal
value of mitigation she prefers to reduce her consumption slightly in
the first decade because of the high rate of return of this investment
to the consumption of future generations. If the representative agent
put more weight on intergenerational smoothing of consumption, for
example, if η= 3, the corresponding OPT path would dominate the
BAU path for η= 2, thus yielding higher utility for any value of ηand
ρand demonstrating the inefficiency of the BAU equilibrium.
In Figure 4 below we plot the initial decades of four paths to under-
line these points. The COPT, and BAU paths relative to the OPT path
all of which are the same as those plotted in Figure 3 and calculated
with the benchmark parameters, in particular with η= 2. The OPT’
path has the same parameters for the technical side of the model, but
sets η= 3, to emphasize the inefficiency of the BAU path. The figure
shows that the use of the COPT equilibrium as the benchmark dis-
torts the perception of the economic issues involved in global warming
policy by incorrectly suggesting that correction of the global warming
externality will depress per capita world consumption significantly for
several decades.
Figure 4: Comparisons of the consumption paths for the business-as-usual, constrained optimal,
and the altered optimal pale-gray OPT’ (with η= 3) cases. All normalized to the optimal
consumption stream.
Given the inefficiency of over-accumulation of GHG stock in the at-
mosphere as a result of capital stock accumulation, the world economy
is not operating at the intertemporal production possibility frontier
(PPF). Future generations would appreciate lower stocks of CO2which
3On the relation between ρand ηand intergenerational equity, see Arrow (2007),
who emphasizes the public good nature of global warming, without, however, noting the
inefficiency of the BAU path.
19
implies that current generations should accumulate less conventional
capital and consume more (of it) today. There is no intergenerational
trade-off as is posited in most of the global warming related economic
literature. The mutual gains are illustrated by moving the economy
from an inefficient point inside the PPF (the BAU path) to its bound-
ary (the OPT’ path with η= 3).
Consumption in the COPT case lies above consumption in the op-
timal case in the first three decades as agents who are aware of the
deleterious effects of global warming wisely choose to accumulate less
and consume more. This positive difference in the first few decades
forms the basis for the intergenerational equity discussion since the
choice of the appropriate program now depends on its parameteriza-
tion (most importantly the discount factor and the elasticity of felicity
with respect to per capita consumption). Nordhaus has placed much
emphasis on the dominance of baseline consumption over optimal con-
sumption in the first few decades. This implies that current gener-
ations would attain lower utility levels if they started investing into
mitigation. The consumption paths of the optimal and business-as-
usual scenarios virtually move together in the first few decades with
the BAU dominating the OPT path for the first two decades. The
OPT’ path, however, shows that this effect is due to the representa-
tive agent preferring to transfer some of the gain from global-warming
mitigation to future generations given the very high rate of return to
mitigation investment given the initial conditions.
Figure 5 illustrates these equilibrium allocations in terms of PPFs
and indifference curves. The curve closest to the origin represents
all attainable allocations with µ= 0 (the BAU PPF). The dark line
next to it is the COPT PPF and represents all supportable allocations
with m= 0. The internalization of the externality in the absence of
mitigation allows a welfare maximizing COPT allocation with higher
present and future consumption levels compared to BAU. The out-
most curve represents what is commonly understood as a PPF: all
technologically feasible, unconstrained first-best allocations. It is ap-
parent that there are many allocation along the OPT PPF which yield
higher utility than COPT and BAU. The preference parameters ρand
ηpin down the welfare-maximizing one. In our results they are such
that the optimal intertemporal allocation (OPT) features the lowest
present consumption level. As figure 5 demonstrates, the gains from
moving from the BAU to the OPT allocation are much greater than
from the COPT to the OPT. In particular, the set of allocations with
non-decreasing consumption in either period is greater in the former
than in the latter. This is due to the fact that the high saving rate
in the BAU allocation provides more room to reshuffle the investment
portfolio toward green investment in the form of mitigation. While the
OPT’ allocation does not lie on the highest indifference curve and is
20
suboptimal, it distributes the gains from moving from the BAU to the
OPT PPF such that consumption in neither period falls.
Figure 5: Comparisons of the equilibrium allocations for the business-as-usual, constrained opti-
mal, the optimal and the altered optimal OPT’ scenarios. Given the parameter values for ηand ρ),
OPT has the lowest present consumption level. OPT’ distributes the gains suboptimally, but such
that consumption levels rise in both periods.
9 Parameter Sensitivity
In order to gauge the effect of parametric changes on the quantitative
results of our simulations, this section explores sensitivity analysis for
some model parameters. Generally, consumption smoothing dominates
most of the adjustment in parametric changes. Due to spatial limita-
tions, only the selected parameters are reported here. A complete set
can be obtained from the authors. Table 2 presents an overview over
the effects of parametric changes (rows) on selected variables (columns)
relative to their values with standard parameters (which are reported
in the bottom row). The sensitivity table reports lower and upper
bounds for each parameter.
For example, the first table in table 2 captures the effects on the
OPT path. The first two rows report changes in the discount rate of
which the standard value is 0.1. The first row has ρ= 0.05 and the
second ρ= 0.15. The first column reports the changes on consumption
per capita. A lower time preference increases consumption by 0.5%
at the median and the 20% quantile compared to the consumption
stream under standard parameter assumptions. Consumption rises
by 0.6% above the corresponding standard consumption level at the
80% quantile. An increase in the discount rate to ρ= 0.15 decreases
consumption by 0.6% at the 20% quantile and by 0.7% at the median
and 80% quantile. The next column reports effects of the changes on
the carbon price. The change in pure time preference has virtually no
21
Table 2: Effects of Parametric Changes on Time Paths of Selected Variables Relative to their Standard Value
α→0.35, β →0.056, γ →0.5, δ →0.7, →0.036, ζ →3., η →2, ν →6, ρ →0.1,A→28.4,CDMax →780
OPT Cons per cap ($/yr) Carbon price ($/t) Mitigation/Y (%) CO2 (ppmv) K ($trillions) Interest Rate (%/yr)
Q20 Median Q80 Q20 Median Q80 Q20 Median Q80 Q20 Median Q80 Q20 Median Q80 Q20 Median Q80
ρ0.05 1.005 1.005 1.006 1.1.1.1.003 1.003 1.003 0.989 0.993 0.993 1.099 1.104 1.104 0.539 0.604 0.762
0.15 0.992 0.993 0.993 1.1.1.0.997 0.997 0.997 1.007 1.008 1.011 0.911 0.911 0.914 1.209 1.38 1.439
γ0.3 0.97 0.971 0.971 0.998 0.998 0.999 0.981 0.982 0.992 1.229 1.244 1.307 0.969 0.97 0.97 0.999 1.1.002
0.7 1.086 1.086 1.086 1.1.001 1.001 0.997 1.007 1.007 0.949 0.979 0.982 1.086 1.086 1.086 0.999 1.1.
CDMax 580 1.061 1.061 1.061 1.1.001 1.001 0.998 1.005 1.005 0.966 0.986 0.988 1.061 1.061 1.061 1.1.1.
980 0.959 0.984 0.984 1.1.1.0.998 0.998 1.004 1.017 1.019 1.027 0.969 0.984 0.984 1.1.1.011
ν3 1.001 1.001 1.001 0.945 0.945 0.946 0.972 0.973 0.973 0.997 0.999 0.999 1.001 1.001 1.001 1.1.1.
12 0.998 0.998 0.998 1.13 1.132 1.132 1.061 1.062 1.062 1.002 1.003 1.006 0.998 0.998 0.998 1.1.1.001
ζ1.5 0.964 0.967 0.967 2.261 2.262 2.263 2.119 2.119 2.139 1.024 1.029 1.064 0.961 0.965 0.965 1.1.1.005
6 1.015 1.015 1.016 0.472 0.473 0.473 0.482 0.487 0.487 0.972 0.989 0.991 1.016 1.016 1.017 0.998 1.1.
BAU Cons per cap ($/yr) Carbon price ($/t) Mitigation/Y (%) CO2 (ppmv) K ($trillions) Interest Rate (%/yr)
Q20 Median Q80 Q20 Median Q80 Q20 Median Q80 Q20 Median Q80 Q20 Median Q80 Q20 Median Q80
ρ0.05 0.965 0.972 0.977 1.093 1.096 1.129 1.024 1.025 1.033 1.002 1.002 1.004 1.062 1.068 1.088 0.504 0.51 0.879
0.15 1.015 1.025 1.028 0.848 0.913 0.915 0.963 0.976 0.976 0.994 0.998 0.998 0.923 0.94 0.945 1.099 1.467 1.477
γ0.3 1.019 1.027 1.044 0.978 1.197 1.201 0.995 1.048 1.049 1.03 1.032 1.035 0.989 1.026 1.043 0.822 1.005 1.036
0.7 0.939 0.955 0.971 0.855 0.858 0.91 0.958 0.959 0.974 0.944 0.95 0.951 0.949 0.956 1.0.568 0.987 0.997
CDMax 580 0.718 0.755 0.76 1.107 1.121 1.226 1.027 1.03 1.049 0.756 0.757 0.833 0.736 0.755 0.898 0.293 1.1.007
980 1.172 1.235 1.301 0.7 0.894 0.901 0.92 0.97 0.972 1.02 1.231 1.234 0.988 1.235 1.255 0.244 0.998 1.107
COPT Cons per cap ($/yr) Carbon price ($/t) Mitigation/Y (%) CO2 (ppmv) K ($trillions) Interest Rate (%/yr)
Q20 Median Q80 Q20 Median Q80 Q20 Median Q80 Q20 Median Q80 Q20 Median Q80 Q20 Median Q80
ρ0.05 0.98 1.019 1.042 1.122 1.137 1.15 1.032 1.036 1.039 0.969 0.985 0.991 0.725 0.803 0.837 −0.209 0.345 0.482
0.15 0.969 0.991 1.015 0.894 0.905 0.91 0.969 0.972 0.974 1.005 1.009 1.023 1.086 1.092 1.142 1.476 1.617 2.173
γ0.3 1.063 1.083 1.117 1.063 1.166 1.181 1.017 1.043 1.047 1.041 1.049 1.059 0.711 0.716 0.922 0.481 1.012 1.069
0.7 0.896 0.925 0.93 0.901 0.91 0.975 0.97 0.974 0.993 0.941 0.944 0.961 0.947 1.096 1.11 0.848 0.993 1.235
CDMax 580 0.683 0.765 0.787 1.11 1.158 1.172 1.029 1.042 1.044 0.764 0.776 0.872 0.467 0.54 0.55 0.212 1.024 1.51
980 1.199 1.228 1.298 0.814 0.892 0.915 0.942 0.968 0.976 1.072 1.201 1.226 1.474 1.496 1.597 0.8 0.976 1.541
22
effect on the carbon price. The third column reports the effects of
parametric changes on mitigation efforts, the fourth on atmospheric
carbon concentration, and so on. The third and fourth row the changes
in the severity of damage, γ, the fifth and the sixth changes in the upper
bound of atmospheric changes in CD, and so on.
The table does not report parameter changes in νand ζfor the
BAU and COPT paths, since these parameters concern the mitigation
function and there is no mitigation available in these scenarios. Vary-
ing the discount factor has very different effects in the three scenarios.
In the optimal case higher time preference yields higher consumption
today and higher CO2 concentration and damage in the future. This
is the standard result which most of the intergenerational literature
rests on. In the BAU the higher consumption induced by higher time
preference leads to less capital accumulation and less carbon emissions
which implies less carbon damage later on. This lowers the extent
of the climate crisis. A lower valuation of the future, thus, leads to
(marginally) higher consumption today and tomorrow as the external-
ity is mitigated. In the constrained optimal case the increase in ρleads
to the standard intertemporal reallocation. A higher consumption level
is sustained by higher accumulation with higher carbon concentrations
early on. The shift also implies that the marginal rates shift. In the
future a higher damage level has to be compensated with higher main-
tenance investments. Consumption is, consequently, lowered.
A reduction in γto 0.3 leads to lower environmental damages. In
the optimal case capital, output, and consumption can rise, while mit-
igation efforts fall. For marginal rates to equalize, CO2concentration
and damages rise. In the constrained optimal case lower damages,
again, lead to higher output and consumption and higher CO2concen-
trations. Damages and capital stock, however, fall. A milder damage
function allows higher GHG concentrations in the atmosphere although
these are not run up to the same damage level as before. Less dam-
age requires less capital stock for the same level of consumption and
less maintenance investment. The resources freed increase consump-
tion even further. In the BAU case a milder damage function implies a
milder climate crisis. More capital and CD can be accumulated. In the
BAU capital accumulation is only slowed down by falling profitability
due to environmental damages. A softer damage function allows a
higher level of consumption to be obtained before and after a climate
catastrophe. The climate crisis itself is more pronounced as can be
seen in the steep fall of output and the reversal of the interest rate to
almost −6%. World output and consumption increase while CO2is
run up, but fall more drastically thereafter.
In the optimal case agents are able to fully internalize the exter-
nality. As a result the effects of increases in the maximal permissible
CO2level are negligible. In the BAU the increase of CDMax increases
23
output and consumption considerably. Generally the transition to the
steady state occurs much smoother which is due to the relative decrease
of carbon emissions to carbon dissipation at higher CO2levels. This
allows a higher capital stock and higher economic activity. The same
rationale holds true for the constrained optimal case. Capital stock,
output and consumption increase together with CD, while damages
decrease.
The mitigation function is only relevant in the optimal case and
consists of two parameters. ζmeasures the cost of mitigation as it ties
marginal mitigation to the carbon price. νmeasures the effectiveness
of mitigation for different levels of mitigation investment, m. An in-
crease in the cost structure of mitigation leads to a higher carbon price
and higher mitigation costs, with slight increases in CD and environ-
mental damages to offset some of the cost increase. A decrease in the
effectiveness of mitigation leads, again, to increased mitigation costs
and a higher carbon price. This movement is counteracted by a slight
increase in CO2and environmental damage. Note that front-loading
of mitigation efforts is not a robust result and can change depend-
ing on parametric assumptions regarding the mitigation and damage
functions. Convergence to full mitigation levels remains robust at 6
decades.
10 Conclusions
Human emissions of carbon dioxide into the atmosphere are a negative
externality; despite world political efforts such as the Kyoto Protocol
and its carbon market, the majority of individual producers do not take
into account the impact of their emissions on atmospheric CO2, but
atmospheric concentration of CO2affects their (or their descendants’)
collective well-being through the deleterious effects of global warming.
Such a negative externality leads to a market failure and an inefficient
allocation of resources. In the case of global warming the inefficiency
takes the form of over-accumulation of capital and under-investment
in mitigation. The correction of this market failure through the imple-
mentation of institutions which enforce cost transparency represents a
Pareto improvement: the current generation invests less, spending the
retrenchment on consumption and mitigation such that future gener-
ations enjoy higher output, higher consumption combined with lower
GHG concentration.
Our simulations show that the gains from such a movement to the
intertemporal production possibility frontier are large. While this re-
allocation will imply significant changes of our life-style and the tech-
nology used, the mitigation costs to return to pre-industrial carbon
dioxide levels are, however, relatively small with their peak at around
24
2% of world product.
The question of how such cost transparency can be achieved and
which mechanisms to use are important, but cannot be answered in our
framework.4This is also true for the Pareto improvement question
of who needs to compensate whom. Given our representative agent
assumption, such a realistic question is clearly unanswerable. These
problems will need to be solved by the world political system.5
If global warming is a real economic problem, there is no economic
cost to correct it. In principle, the costs of reducing emissions in the
current generation can be shifted to the future generations who will
benefit from a cooler planet by reducing conventional investment.
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