DEPARTMENT OF ECONOMICS
OxCarre
Oxford Centre for the Analysis of Resource Rich Economies
Manor Road Building, Manor Road, Oxford OX1 3UQ
Tel: +44(0)1865 281281 Fax: +44(0)1865 271094
[email protected] www.oxcarre.ox.ac.uk
OxCarre Research Paper 149
_
Non-Cooperative and Cooperative
Responses to Climate Catastrophes
in the Global Economy:
A North-South Perspective
Frederick van der Ploeg
OxCarre
&
Aart de Zeeuw
Tilburg University
Direct tel: +44(0) 1865 281281
E-mail: [email protected]
Non-Cooperative and Cooperative Responses to
Climate Catastrophes in the Global Economy:
A North-South Perspective*
Frederick van der Ploeg**
University of Oxford, United Kingdom
Aart de Zeeuw***
Tilburg University, the Netherlands
Abstract
The global response to a catastrophic shock to productivity which becomes
more imminent with global warming is to have carbon taxes to curb the risk of
a calamity and to accumulate precautionary capital to facilitate smoothing of
consumption. Our multi-region model of growth and climate change indicates
that without international lump-sum transfers the cooperative global response
to such stochastic tipping points requires converging carbon taxes for
developing and developed regions. Non-cooperative responses lead to a bit
more precautionary saving and lower diverging carbon taxes. Precautionary
capital suffers less from international free-rider problems than the carbon
taxes. We illustrate the various outcomes with a calibrated North-South model
of the global economy.
Key words: global warming, tipping point, precautionary capital, growth, risk
avoidance, carbon tax, free riding, international cooperation, asymmetries.
JEL codes: D81, H20, O40, Q31, Q38.
This draft: 11 December 2014
___________________
*
Van der Ploeg is also affiliated with the VU University Amsterdam and grateful for support
from the ERC Advanced Grant ‘Political Economy of Green Paradoxes’ (FP7-IDEAS-ERC
Grant No. 269788) and the BP funded Oxford Centre for the Analysis of Resource Rich
Economies. De Zeeuw is grateful for support from the European Commission under the 7th
Framework Programme (Socioeconomic Sciences and Humanities - SSH.2013.2.1-1 – Grant
Agreement No. 613420).
**
OXCARRE, Department of Economics, University of Oxford, Oxford OX1 3 UQ, U.K.,
+44-1865-281285, [email protected] .
***
CentER and TSC, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands,
+31-13-4662065, [email protected] .
1. Introduction
The standard recipe for the fight against global warming is to have a global
carbon tax which has to be set equal to the present value of all future marginal
damages arising from emitting one ton of carbon (e.g., Tol, 2002; Nordhaus,
2008; Stern, 2007; Golosov et al., 2014). Although these integrated assessment
studies acknowledge catastrophic non-marginal damages from global
warming, they typically allow for them by having convex damages. Here we
focus at the consequences for climate policy of a pending non-marginal
calamity which becomes more imminent with global warming (cf., Cai et al.,
2012; Lemoine and Traeger, 2014; Lontzek et al., 2014).1 Policy makers need
to react in two ways to such catastrophes (van der Ploeg and de Zeeuw, 2014):
global carbon taxation or a global emissions market to curb the risk of climate
calamities; and precautionary capital accumulation (ensured with a capital
subsidy if the market fails to internalize the need for precautionary capital) to
cope with the inevitable downward drop in consumption after the calamity.2
It is well known that international pollution control is subject to substantial
free-rider problems. If the stock of pollution causes damage, non-cooperative
differential game theory indicates that lack of international policy cooperation
leads to excessively large pollution stocks (van der Ploeg and de Zeeuw,
1992). Such international free-rider problems occur with vengeance in the
fight against global warming. Although multi-country versions of climate
assessment models such as RICE have been used to highlight the different
incentives to fight global warming in the different blocks of countries (e.g.,
Nordhaus and Boyer, 2000; Hassler and Krusell, 2012), dynamic game theory
1
These large, abrupt and persistent changes in the climate system are called regime shifts in
the ecological literature and a point where such a regime shift occurs is called a tipping point
(e.g. Biggs et al., 2012). Scientists have indeed given more prominence to the idea that climate
policy should focus at the small risk of abrupt and often irreversible climate disasters and
tipping points at high temperatures rather than at smooth global warming damages at low and
moderate temperatures (e.g., Lenton and Ciscar, 2013; Kopits et al., 2013; Pindyck, 2013).
2
Smulders et al. (2014) also emphasize the need for precautionary saving to deal with an
impending disaster, but their analysis uses a constant hazard rate whereas our analysis
highlights temperature-dependent hazard rates for climate policy.
1
is typically not used to assess the costs of free riding on international
negotiations. Furthermore, most of these studies focus only on carbon stocks
as international common bads and ignore other spill-over effects to do with
international trade in goods and services, migration and capital flows,
insurance and intertemporal trade.
Our contribution is to reinvestigate non-cooperative and cooperative responses
to the prospect of non-marginal climate catastrophes leading to a sudden and
irreversible drop in total factor productivity.3 The only connection between
regions is that carbon emissions in each region affect the common stock of
carbon in the atmosphere or global warming and thus the hazard of climate
tipping. We thus also abstract from international and intertemporal trade.4 We
do allow for two crucial asymmetries by, on the one hand, distinguishing
between a developed region with a high initial capital stock and a developing
region with a low initial capital stock, and, on the other hand, allowing the size
of the climate disaster to be bigger for the developing region than for the
developed region. Here a broad measure of capital, including human capital
and institutional quality, is taken.
Although international lump-sum transfers can in principle ensure smoothing
of consumption and a common carbon tax across the globe (Chichilniski and
Heal, 1994), this is hard to achieve in international negotiations. Our approach
focuses on internalizing the transboundary externalities of carbon emissions
that increase the hazard of climate change affecting everyone. Our main focus
is thus on cooperation in the setting of carbon taxes in the absence of
international transfers. Non-cooperative outcomes assume, in addition to such
transfers being infeasible, that transboundary externalities are not internalized.
Our approach is a novel type of differential game where the strategic
3
This may arise from, for example, flooding of cities, sudden increased occurrence of storms
and droughts, abrupt desertification of agricultural land, or reversal of the Gulf Stream. Other
catastrophes such as a sudden eruption of methane from the permafrost can be considered too,
but we will focus at sudden drops in total factor productivity.
2
interactions take place via the carbon emissions of each region affecting the
common hazard of a climate catastrophe.5
Our core results are as follows. First, if international transfers are feasible,
international cooperation gives the first best with international consumption
smoothing and a common carbon tax throughout the globe to curb global
emissions and the probability of a regime shift. Furthermore, additional
precautionary capital is accumulated before the calamity to be better prepared
for when the calamity strikes and not suffer a too big a blow to consumption.
Second, if international transfers are infeasible, which seems to be more
realistic as we do not observe consumption smoothing across the rich and poor
parts of the globe, international cooperation can still aim for internalizing the
transboundary externalities of carbon emissions. Less developed regions give
higher priority to consumption, so have a lower carbon tax in the short run
than the developed regions of the world economy. Since the climate calamities
hit developing regions more, they have to engage more in precautionary
capital accumulation. Third, in the absence of any international cooperation,
there will be on average a lower carbon tax leading to more global warming
and a more imminent climate catastrophe. There is an absence of free-rider
problems in precautionary capital accumulation, but the bigger hazards of a
calamity under non-cooperation necessitate more precautionary capital
accumulation. The real cost of non-cooperative climate policies is that the
hazard of a climate calamity is brought forward. To the extent that this is an
irreversible catastrophe, the costs will be significant.
Section 2 presents our multi-region model of growth and development with
tipping points and abundant fossil fuel. Section 3 discusses what the outcomes
are after tipping has occurred, both with and without international lump-sum
transfers. Section 4 derives the outcomes under international policy
4
Although this can lead to international consumption smoothing, empirical evidence shows
that consumption is less correlated than output across countries (Backus et al., 1992).
3
cooperation if transfers are unavailable and compares these with the first-best
responses with transfers. Section 5 derives the non-cooperative outcomes for
climate policy and shows that these are less ambitious than under international
policy cooperation. Section 6 compares the cooperative and non-cooperative
outcomes with the first best and business-as-usual scenarios using a calibrated
North-South model of the global economy. Section 7 concludes.
2. A multi-region growth model with climate tipping
Consider a continuous-time Ramsey model of growth and global warming for
the global economy. We distinguish regions i = 1,..., n which are unconnected
by international trade, migration or capital flows. Subscripts i denote the
different regions. The only thing that connects these countries is a common
concern for global warming for the planet. This concern stems from a higher
stock of greenhouse gases bringing forward the expected date of a climate
catastrophe. In the calibration we consider two possible asymmetries between
a developing and a developed region (i = 2), namely different stages of
economic development, proxied by different levels of initial capital stocks,
K1 (0)  K 2 (0)  0, and different levels of damages to total factor productivity
as developing regions suffer relatively more from global warming, denoted by
 2   1  0 (see below), where region 1 denotes the developed and region 2
the developing region.
Fossil fuel Ei is an input into the production process and has constant marginal
cost d > 0. We assume that fossil fuel is in abundant supply (think of coal or
shale gas).6 The capital stock is denoted by Ki for region i. We assume that
capital and fossil fuel are cooperative factors of production. Total factor
5
Before-tip strategic interactions affecting the hazard of being removed from office have been
analysed in the context of dynamic resource games before (van der Ploeg, 2012).
6
In the background there may also a carbon-free imperfect substitute for fossil fuel, renewable
energy, which is produced with constant marginal cost. However, we suppress this and assume
this is optimized as in van der Ploeg and de Zeeuw (2014).
4
productivity is A before the regime shift and drops to (1   i ) A  A afterwards,
where 0   i  1 is the size of the climate disaster in region i. Utility is denoted
by U, consumption by Ci, the production function by AF ( Ki , Ei ) before and
(1   i ) AF ( K i , Ei ) after the tip, the depreciation rate of capital by δ > 0 and
the uniform rate of time preference by ρ > 0. For simplicity, we abstract from
population growth and technical progress.
The use of fossil fuels (measured in GtC) in each of the regions leads to
emissions of carbon dioxide. We denote the fraction of carbon emissions that
does not return quickly to the surface of the earth by ψ > 0. About a fifth of
emissions remain in the atmosphere for thousands of years (Golosov et al.,
2014; Gerlagh and Liski, 2012), but we suppose that all of the stock of
atmospheric carbon P decays at the rate γ > 0 (roughly 1/300) eventually and
returns to the surface of the earth. Fossil fuel use in all regions contribute to an
increase in the stock P and thus to global warming and a higher risk of climate
tipping. To formalize this, we suppose that a bigger stock of carbon P
increases the probability of climate change, hence h(t )  H  P(t )  with
H '( P)  0 . The size of the potential drop in total factor productivity is thus
known but it is not known when the climate regime shift will take place. With
global warming the expected duration before the regime shift occurs, 1/ H ( P),
falls with time, so failing climate policy makes the shock to productivity more
imminent. The conditional chance of the tip occurring at time T is
Pr[T  (t , t  t ) | T  (0, t )]
, so h(t )t is the probability that the tip
t 0
t
h(t )  lim
takes place in the infinitesimally small interval of time between t and t + Δt,
given that it has not occurred before t.
Although we do not allow for international lump-sum transfers when we
derive our cooperative and non-cooperative responses to the threat of climate
catastrophe, the first best does allow for such transfers (typically, from the
5
developing to the developed regions). Such transfers ensure that consumption
will be smoothed and lead to a uniform carbon tax across the globe. These
transfers may be desirable from a global welfare perspective but they are hard
if not impossible to realise in international negotiations. We therefore abstract
from them when we consider our second-best cooperative and non-cooperative
responses to the threat of abrupt climate calamities.
Social welfare in each of the regions is defined as the expected present
discounted value of the utility of the consumption:
(1)
   t

Wi  E   e U (Ci (t ))dt  , i  1,.., n.
0

Capital accumulation in each of the regions is given by
(2) Ki (t )  Ai (t ) F  Ki (t ), Ei (t )   dEi (t )  Ci (t )   K i (t ), K i (0)  K i 0 , i  1,..n,
with total factor productivity in each of the regions given by
(3)
Ai (t )  A, 0  t  T ,
Ai (t )  (1   i ) A  A, t  T , i  1,.., n,
where the tipping point T is driven by the hazard rate H ( P) with the
accumulation of the atmospheric stock of carbon given by
(4)
 n

P(t )     Ei (t )    P(t ), P(0)  P0 .
 i 1

We focus on and compare the following outcomes for the global economy:
1. Cooperative responses when global welfare is maximized.
2. Non-cooperative responses when each region maximizes its own
welfare taking the actions of the other region at any point of time as
given (Nash equilibrium).
3. Business-as-usual scenario where in contrast to the previous three
outcomes the regions behave in a naïve fashion and do not take
account of threat of catastrophic climate catastrophes.
6
None of these outcomes allow for international transfers and international
consumption smoothing. If one does and allows for international cooperation
one would obtain the first best (see appendix 1). These cooperative and noncooperative outcomes are second best and focus only on the transboundary
externalities of carbon emissions raising the hazard of climate change. Section
6 illustrates these outcomes numerically with a stylized calibrated two-region
model of growth and development for the global economy.
3. After-tip outcomes
After the climate catastrophe has occurred, each region i independently solves
a standard Ramsey growth problem with total factor productivity (1   i ) A. It
does not matter whether regions cooperate or not after the tip, but it does
matter whether international transfers are available.
The maximum levels of output net of energy costs and capital depreciation are
Y ( Ki ,  i )  Max  (1   i ) AF ( Ki , Ei )  dEi   Ki  ,
(5)
E
YKi  (1   i ) AFKi    0, Y i   AFi  0, i  1,.., n.
Fossil fuel use increases with the capital stock Ki and decreases in the size of
the disaster i and the price, so YdKi  0 and Yd i  0.
The Hamilton-Jacobi-Bellman (HJB) equations in the value function VA are
(6)
V A ( Ki ,  i )  Max U (Ci )  VKA ( Ki ,  i ) Y ( Ki ,  i )  Ci  , i  1,.., n.
i
Ci
where superscript A denotes after-tip values. The optimality condition for
consumption implies that marginal utility of consumption should equal the
marginal value of capital, U '(Ci )  VKAi ( Ki ,  i ), which gives
(7) C  C ( K i ,  i ), C
A
A
Ki

VKAi Ki
U "(Ci )
 0, C i 
A
7
VKAi i
U "(Ci )
 0, t  T , i  1,.., n.
Consumption increases with the capital stock given that the value function is
concave in capital. A bigger disaster i boosts the marginal value of capital
VKAi which requires a boost to marginal utility of consumption U '(Ci ) and thus
a bigger fall in consumption.
Differentiating (6) with respect to Ki and using (7) yields
(8)
Y ( Ki ,  i )  C A ( Ki ,  i )  CKAi ( Ki ,  i )   YKi ( K i ,  i )    C A ( K i ,  i ),
  U '(Ci ) / CiU "(Ci )  0,
i  1,.., n,
where  is the uniform elasticity of intertemporal substitution. Relative risk
aversion and intergenerational inequality aversion equal 1/. Equation (8)
yields the Keynes-Ramsey rule and the dynamics of capital:
(9)
Ci (t )   YKi  K i (t ),  i     Ci (t ),
K i (t )  Y  K i (t ),  i   Ci (t ), K i (T )  K iT , t  T , i  1,..n.
We denote the steady state with a bar across the variable. The steady-state
capital stocks KiA ( i ) follow from the modified golden rules YKi ( KiA ,  i )   ,
i  1,..., n, and are low if the region-specific disaster is large and the discount
rate is high. As far as the transient phase is concerned, the capital stock is
predetermined at time T, but the rate of consumption jumps down at the time
of
the
tip
to
place
the
economy
on
the
stable
manifold,
Ci (T )  C A  K i (T ),  i  , i  1,.., n. The optimal path along the stable
manifold is Ci (t )  C A  Ki (t ),  i  , t  T , i  1,.., n. Rearranging (6) gives
V ( Ki ,  i ) 
A
(10)
U  C A ( K i ,  i )   U '  C A ( K i ,  i )  Y ( K i ,  i )  C A ( K i ,  i ) 

i  1,.., n,
where we use (cf. van der Ploeg and de Zeeuw, 2014):
8
,

 K (t ) 
K A ( i )
C ( K i (t ),  i )  Y  K ( i ),  i   Ai
,


z
 0,

i
i
A
K
(

)
Y
K
(

),



i 

i
i
A
(7)
zi 
A

2

1
2
 2  4 YK K ( K A ( i ),  i )CiA    0, i  1,..n.
i i
4. Before-tip cooperative outcomes
The question is how the prospect of a climate regime shift affects the optimal
growth paths before this shift occurs. Since the hazard rate H(P) of a climate
disaster depends on the stock of atmospheric carbon P, the value functions
Vi B ( Ki , K j i , P), i  1,.., n, where the superscript B indicates before-tip values,
are functions of the capital stocks K1 ,.., K n , and the global carbon stock P.
Here we analyse the cooperative outcome where the regions internalize the
global warming externality and postpone treatment of the non-cooperative
equilibrium to section 5.
4.1. International cooperation: no international transfers
With no transfers, Si  0, i  1,.., n, the world social planner maximizes
expected utilitarian global social welfare taking account of how an imminent
stochastic tipping point affects the optimal growth path. It thus solves:
(11)
 T  t  n

 n

max E   e  U  Ci (t )   dt e  T V A  K i (T ),  i  
Ci , Ei
 i 1

 i 1

0
subject to (2)-(5) and the after-tip value function (10). Optimal control
problems with an endogenous hazard rate can be solved via a HJB equation
including a term that captures the expected capitalized losses from a climate
disaster (cf. Polasky et al., 2011; van der Ploeg and de Zeeuw, 2014). If the
regions cooperate, we find that we can write the global value function in the
n
separable form C ( K1 ,.., K n , P)   Vi C ( K i , P), where the superscript C
i 1
9
denotes the cooperative before-calamity outcome. We denote V A ( K i ,  i ) from
(11) by Vi A ( K i ). Hence, the HJB equation is
n
  Vi C ( K i , P) 
(12)
i 1
 U (C )  H ( P) V
n
Max
C1 ,..,Cn , E1 ,.., En
i
i 1
C
i

( K i , P)  Vi A ( K i ) 
 C
 n
 
C
  ViKi ( K i , P )  AF ( K i , Ei )  dEi  Ci   K i   ViP ( K i , P )   E j   P  
i 1 
 j 1
 

n
with the optimality conditions
n
(13)
AFEi ( K i , Ei )  d   iC ,  iC  
U '(Ci )  ViKBi ,
 ViPC
i 1
ViKBi
, i  1,...., n,
where τiC is region i's cooperative social cost of carbon.
With τiC as the additional cost of fossil fuel input, we define YiB as maximum
output net of fossil fuel costs and capital depreciation:
(14)
Yi B ( K i , iC )  Max  AF ( K i , Ei )  (d   iC ) Ei   K i  , i  1,...., n.
Ei
Differentiating (12) with respect to Ki and P, using (13)-(14), yields a set of
differential equations for the first-order derivatives of Vi B as functions of time
(the Pontryagin conditions). This leads to (omitting the dependence on time t):
(15a)
ViKCi  YiKBi ( Ki , iC )    H ( P)  ViKCi  H ( P)ViKAi ( Ki ),
(15b) ViPC       H ( P)ViPC  H '( P) Vi C  Vi A ( Ki )  , i  1,...., n.
From the first part of (15) using (13), we get the Keynes-Ramsey rule:
ViKAi ( Ki ) 
(16) C   Y ( Ki , )      C ,   H ( P) 
 1 , i  1,...., n,
B
 U '(Ci ) 
C
i
where  iC
B
Ki
C
i
C
i
C
i
C
i
is the precautionary return on capital accumulation and
  U '/ CU "  0 is the constant elasticity of intertemporal substitution. The
10
growth rate of consumption is thus proportional to the marginal net product of
capital plus the precautionary return minus the pure rate of time preference.
The precautionary return on capital is proportional to the hazard of a climate
calamity and through this channel increases with global warming and the stock
of atmospheric carbon. This precautionary return, if necessary forced upon the
market by a capital subsidy, induces precautionary capital accumulation and
softens the blow to consumption when the calamity strikes.
Using (14) and (13), we get the dynamics for the social costs of carbon:
(17)
 n

C
A
  V j ( K j , P)  V j ( K j ) 
 , i  1,.., n,
 iC  riC iC  H '( P)  j 1
U '(CiC )






where riC  YiKBi ( Ki , iC )    H ( P)  iC . We thus have that the social costs of
carbon under international cooperation are the present discounted value of
expected non-marginal damages from a calamity to all regions together:
 V
n
(18)
 
 iC (t )   H '( P)  e 
s
r
riC
( s ) ds j 1
C
j
( K j ( s ), P( s ))  V jA ( K j ( s ))
U '(CiC ( s ))
t
ds,
i  1,.., n,
The relevant discount rate is the sum of the interest rate plus the rate of
atmospheric decay, the hazard of a climate calamity and the precautionary
return. Hence, the optimal carbon tax is large if the drops in future welfare
from climate calamities and the marginal hazard are large. The convexity of
the hazard pushes up the carbon tax; the level of the hazard depresses it (via
the higher discount rate).
Due to the infeasibility of international transfers, it is not optimal to equalize
carbon taxes across the globe. Poorer regions have lower levels of
consumption and therefore have a higher marginal utility of consumption.
They also have a lower capital stock and thus a higher marginal product of
11
capital and employ a higher discount rate. As can be seen from (18), both of
these effects imply that it is optimal from a global perspective for poorer
regions to have a lower carbon tax than richer regions in the short run but the
carbon taxes will converge throughout the globe in the long run. In as far as
poorer countries suffer more from the impact of climate calamities than richer
countries, (16) indicates that the precautionary return or the required capital
subsidy and thus the degree of precautionary capital accumulation must be
higher for poorer countries. This offsets somewhat the downward effect on the
socially optimal carbon taxes to be set for poorer countries.
The equations for the pre-tip accumulation of capital and carbon are:
K i C  Yi B ( K i C , iC )   iCYiBi ( K i C , iC )  CiC , K i C (0)  K i 0 , i  1,...., n,
(19)
n
P C    Y jB C ( K j C , Cj )   P C , P C (0)  P0
j 1
0  t  T.
j
The second terms in the right-hand side of the capital dynamics, the first part
of (19), are the lump-sum rebates of the tax revenues if the social costs of
carbon τ are implemented as a carbon tax.
The steady state and transient pre-tip dynamics under international cooperation
follow from solving the saddle-point system (16), (17) and (19) given the
value functions evaluated from (12) and (7), where
predetermined variables and
C1 ,.., Cn ,1 ,.., n 
K1 ,.., K n , P
are the
the non-predetermined
variables.
5. Non-cooperative before-tip climate policies
Carbon emissions increases the atmospheric carbon stock irrespective of
whether they emanate from the developed or developing regions of the global
economy. This therefore creates a transboundary externality between the
regions. Since the carbon externality is the only connection between the
12
various regions, we use the value functions Vi N ( Ki , P; Ei ), i  1,.., n, where
Ei corresponds to E j , j  i . Hence, regions only react to their own capital
stock taking the emissions of all the other regions as given. However, even
though under cooperation this is justified due to the separable nature of the
value functions, this is not necessarily so under non-cooperation. There may
thus be other non-cooperative Nash equilibrium outcomes where each region
does directly react to the capital stocks of the other regions, but we abstract
from these in this paper.7 Of course, in our non-cooperative Nash equilibrium
with restricted information sets (denoted by the superscript N) the value
functions will in equilibrium depend on all the capital stocks and the carbon
stock in the usual way, since the rival regions’ emission levels depend on the
rival regions’ capital stocks and carbon tax (which depend on their capital
stock and the carbon stock).
If the regions of the globe do not cooperate, the HJB equations become
Vi N ( K i , P; Ei )  Max U (Ci )
Ci , Ei
V ( K i , P; E i )  AF ( K i , Ei )  dEi  Ci   K i 
N
iKi
(20)
 n

ViPN ( K i , P; E i )   E j   P 
 j 1


 H ( P ) Vi N ( K i , P; Ei )  Vi A ( K i )  , i  1,...., n.
The Nash equilibrium characterized by (20) implies that each region focuses
on their own growth path, taking emissions of the other regions as given and
ignoring the transboundary externality.
The non-cooperative Nash equilibrium optimality conditions for problem (19)
become:
(13)
7
U '(Ci )  V ,
N
iKi
ViPN
AFEi ( K i , Ei )  d   ,    N , i  1,...., n,
ViKi
N
i
N
i
More general information sets may yield other Nash equilibria (Basar and Olsder, 1982). It is
13
where τiN is region i's non-cooperative social cost of carbon. With τiN as the
additional cost of fossil fuel input, we define Yi B ( K i , iN ), i  1,...., n, as the
maximum output net of fossil fuel costs and capital depreciation (see (14)).
As before, differentiating (20) with respect to Ki and P yields the Pontryagin
conditions for the non-cooperative outcome:
(15a) ViKNi  YiKBi ( Ki , iN )    H ( P)  ViKNi  H ( P)ViKAi ( Ki ),
(15b) ViPN       H ( P)ViPN  H '( P) Vi N  Vi A ( Ki )   ..., i  1,...., n.
As other regions’ fossil fuel use, Ei, does not depend on Ki, (15b) is
unaffected by these cross terms. Ei does depend on i and this explains why
the value functions Vi in (15a) and (15b) depend on i. Since we focus at an
open-loop Nash equilibrium in feedback representation which takes the time
paths of fossil fuel use and emissions paths of the other regions as given, we
do not need extra terms in (15a) and (15b) to allow for the effects of i and
Ki and P via Ei on Vi. This is why the value functions for each region
simply depend on their own capital stock and the stock of atmospheric carbon.
The conditions (15) are the same as (15) for the cooperative case except for
the social costs of carbon. If the regions cooperate, they internalize the
externality of putting the other regions to higher risk from increasing the stock
of atmospheric carbon and thus employ a higher social cost of carbon as can
be seen by comparing the expression for  iC in (13) with that for  iN in (13).
From the first part of (17) and (15), we get the Keynes-Ramsey rules
ViKAi ( Ki ) 
(16) C   Y ( Ki , )      C ,   H ( P) 
 1 , i  1,...., n,
B
 U '(Ci ) 
N
i
B
Ki
N
i
N
i
N
i
not easy to characterize these other equilibria.
14
N
i
which are the same as the Keynes-Ramsey rules under international
cooperation (16) except that the non-cooperative taxes determine energy
demand and the level of net output. Comparing (16) and (16), we see that in
contrast to the expressions for the carbon taxes there is no non-cooperative
bias in the expressions for the precautionary returns on capital accumulation.
The reason is that the international externality plays out via global warming. In
general equilibrium, however, the non-cooperative level of global warming
will be higher. The non-cooperative precautionary returns or capital subsidies
will therefore be higher than under international cooperation in general
equilibrium. The precautionary returns are also affected by the percentage
drop in consumption after the tip, but that should not differ too much for
cooperative and non-cooperative outcomes.
Using (15) and (13), we get the dynamics of the social costs of carbon:
Vi N ( Ki , P; E i )  Vi A ( Ki ) 
,
U '(CiN )


 iN  ri N iN  H '( P ) 
(17)
where ri N  YiKBi ( Ki , iN )    H ( P)  iN . Hence, the non-cooperative social
costs of carbon for each region are the present discounted values of expected
non-marginal climate damages to this region:
(18)
 (t )   H '( P) 
N
i

t
s
ri
e r

N
( s ) ds
Vi N ( K i ( s ), P( s ); Ei ( s ))  Vi A ( K i ( s ))
ds,
U '(CiN ( s ))
i  1,.., n,
Hence, comparing (18) with (18), we establish that for the symmetric noncooperative outcome the expressions for the non-cooperative optimal carbon
taxes are half that of those for the cooperative carbon taxes. Of course, as the
rate of consumption and capital and carbon stocks differ for the noncooperative and cooperative outcomes, this does not necessarily mean that the
carbon taxes are half those in the non-cooperative outcome. For example, the
carbon stock and global warming will be higher if countries do not cooperate.
15
Hence, to the extent that the hazard function is convex this puts some upward
pressure on the non-cooperative carbon taxes. On the other hand, the higher
damages in the absence of cooperation depress rates of consumption and boost
marginal utilities of consumption, thus pushing non-cooperative carbon taxes
further downwards. How these and other effects play out will be manifest in
the numerical illustrations for the global economy (see section 6).
The equations for the accumulation of capital and greenhouse gases are:
K i N  Yi B ( K i N , iN )   iN YiBi ( K i N , iN )  CiN , K i N (0)  K i 0 , i  1,...., n,
(19)
n
P N    Y jB N ( K j N , Nj )   P N , P N (0)  P0 ,
j
j 1
which are the same as (19) except that the social costs of carbon and thus the
rates of output and fossil fuel use are set to their non-cooperative levels.
The steady state and transient pre-tip non-cooperative dynamics follow from
the saddle-point system (16), (17) and (19), with the same predetermined and
non-predetermined variables as in section 4.1.
Comparison of steady states
The steady states follow from the modified golden rules of capital
accumulation YKBi ( KiB , i )    i , i  1,.., n, with
(21)
ViKAi ( KiC )

 1 , i  1,...., n,
 U '(C )

i C  H ( P C ) 
C
i
n
 iC 
(22)
 H '( P C ) V jC ( K Cj , P C )  V jA ( K Cj ) 
j 1
     H ( P C )  U '(CiC )
n

 H '( P C ) U (C Cj )  V jA ( K Cj ) 
j 1
   H ( P )       H ( P C )  U '(CiC )
C
in the cooperative case, and
16
, i  1,...., n,
i 
N
(22)

 H '( P N ) Vi N ( K iN , P N )  Vi A ( K iN ) 
     H ( P N )  U '(CiN )
 H '( P N ) U (CiN )  Vi A ( K iN ) 
   H ( P N )       H ( P N )  U '(CiN )
, i  1,...., n,
in the non-cooperative Nash equilibrium with
CiN  Yi B ( K iN , i N )   i N YiBi ( K iN , i N ), i  1,...., n,
(23)
PN  


n
 Y (K
B
i 1
i
i
N
i
, i N ).
This is only a target steady state, because after the tip the system will move to
the after-tip steady states K iA . The precautionary returns i push up the steady
state capital stocks KiB to be prepared for a possible shock but the social costs
of carbon (  i N or  iC ) push these down again in order to reduce the risk. The
steady-state social costs of carbon (  i N or  iC ) depend on the gaps between the
before-tip and after-tip values, all evaluated at the targeted steady states. If
regions do not cooperate, only the own gap is taken into account. If regions
cooperate, also the gaps of the other regions are taken into account. Ignoring
general equilibrium effects, the cooperative costs of carbon are n times the
average of the non-cooperative costs of carbon but the precautionary returns
are the same in the two cases. However, with higher carbon taxes the stock of
atmospheric carbon is kept lower so that the risk of a climate disaster is lower
and less precautionary saving is needed. Therefore, it is to be expected that the
precautionary returns are lower in the cooperative case than in the Nash
equilibrium. In the simulations in the next section we quantify these effects.
6. Illustrative calculations for a North-South model of the global economy
Here we calculate numerically the outcomes derived in sections 3-5 for a
stylized North-South model of the global economy of which the calibration is
17
based on van der Ploeg and de Zeeuw (2014) and discussed in appendix 2.
Region 1 is the developed region (the “North”) and starts out with an initial
capital stock which is 9 times larger than that of region 2 (the “South”). We
assume the same total factor productivity and a relatively high capital share.
This captures that institutions themselves evolve over time as the economy
develops and institutional quality is therefore subsumed in the capital stocks of
each region. Human capital to the extent that it develops by investments is also
included in this broad measure of capital. The other key asymmetry is that we
assume catastrophic drops of 10 and 30 percent in total factor productivity for,
respectively, the North and South, i.e., 1 = 0.1 and 2 = 0.3.
We suppose that at the initial carbon stock, P0 = 826 GtC (or 388 ppm by vol.
CO2), the hazard rate is H(826) = 0.02 and that it rises linearly to 0.033 if the
carbon stock is 1652 GtC. With a climate sensitivity of 3, doubling of the
carbon stock thus induces an additional 3 degrees Celsius and a drop in the
average time it takes for the tip to occur from 50 to 30 years.8 This gives
H ( P)  0.02  1.614  10 5 ( P  826). 9
We first discuss the steady-state outcomes and then the transient paths of the
cooperative, non-cooperative and business-as-usual (BAU) outcomes.
6.1. Steady-state outcomes
The business-as-usual and the after-tip scenarios with and without
international transfers have zero carbon taxes and no precautionary savings
(  1   2  1   2  0 ) and give rise to the steady states reported in table 1. As
a result of the tip we see that capital, economic activity and consumption are
lower, especially in the South where the calamity strikes hardest. As a result,
there will be less carbon emissions and thus the carbon stock will be
considerable lower after the tip (1414 instead of 1992 GtC). Consequently,
8
We use 3ln( Pt / 596.4) / ln(2) for the temperature compared to pre-industrial temperatures.
18
global warming will be less after the tip (3.7 instead of 5.2 degrees Celsius).
As a comparison, we also give the results in case the calamity hits all parts of
the world in the same way (final column).
Table 1: Naïve and after-tip steady states
Business
as usual
After tip
 1  0.1,  2  0.3
After tip
 1   2  0.2
K1 ($ T)
226.8
192.3
159.8
K 2 ($ T)
226.8
129.6
159.8
C1 ($ T)
34.03
28.85
23.98
C2 ($ T)
34.03
19.45
23.98
45.37
38.46
31.97
GDP 2 ($ T)
45.37
25.93
31.97
P (GtC)
Temperature
(degrees Celsius)
1992
1414
1404
5.22
3.74
3.71
_______
GDP1 ($ T)
_______
The after-tip stable manifolds in the asymmetric case are 3.0K10.43 and
2.4 K 0.43 , respectively. These are used to evaluate the after-tip value functions,
which are needed to calculate the cooperative and non-cooperative pre-tip
steady states reported in table 2. We make the following observations.
First, the first column of table 2 shows that international cooperation
introduces precautionary returns of about 0.46% per annum and carbon taxes
of around $80 per ton of carbon emitted. These precautionary returns and
carbon taxes do not differ much for South and North despite the absence of
international transfers and the different impact of the climate calamity on the
9
If the carbon stock quadruples to 3304 GtC and global warming increases with an additional
6 degrees Celsius, the hazard rate increases to 6 percent per annum and an expected time for
the tip to occur of 16.7 years.
19
two regions. Of course, these policies differ in the transient period when the
South is still catching up with the North (see section 6.2).
Table 2: Pre-tip target steady states
 1  0.1,  2  0.3
 1   2  0.2
Cooperative
Noncooperative
Cooperative
Noncooperative
K1 ($ T)
249.6
252.7
284.1
287.9
K 2 ($ T)
330.9
336.4
284.1
287.9
C1 ($ T)
34.27
34.33
34.53
34.57
C2 ($ T)
34.62
34.63
34.53
34.57
46.33
46.88
48.32
48.76
GDP 2 ($ T)
50.69
51.09
48.32
48.76
P (GtC)
Temperature
(degrees Celsius)
1836
2005
1849
2007
4.87
5.25
4.90
5.25
1 (%/year)
0.46
1.50
0.96
0.98
 2 (%/year)
0.47
1.54
0.96
0.98
 1 ($/tC)
80.0
14.8
74.4
33.9
 2 ($/tC)
81.7
58.5
74.4
33.9
_______
GDP1 ($ T)
_______
Second, fossil fuel emissions, economic activity and consumption under
cooperation are in the long run higher in the South due to the bigger impact of
the climate calamity demanding more precautionary capital accumulation.
Column 3 indicates that this is not the case if the impact is the same across the
world. Consumption in the South is lower in the short run, because it needs to
catch up much more and because precautionary capital is accumulated to
prepare for the calamity (see section 6.2.). Steady-state carbon taxes are a little
20
higher in the South because steady-state consumption in the South is a little
higher due to the higher precautionary saving, and thus marginal utility of
consumption is a little lower.
Third, the non-cooperative outcome reported in the second column indicates
that carbon taxes are much lower than under international cooperation,
especially in the North: about 15$ in the North and $60 in South compared
with about $80 per ton of emitted carbon under cooperation. As a result, the
carbon stock ends up higher (2005 instead of 1836 GtC) and there will be
more global warming which brings forward the expected date of the climate
calamity. The precautionary returns are almost triple those under cooperation,
so there is more precautionary saving leading to higher long-run capital stocks.
Finally, if the impact of the climate calamity is the same for North and South,
the last two columns indicate that the non-cooperative carbon taxes are a bit
less than half of the cooperative taxes but that long-run non-cooperative and
cooperative precautionary returns do not vary much. This is a result of the
non-cooperative bias mainly manifesting itself in the carbon taxes, not in the
precautionary returns.
6.2. Transient dynamics
Of course, just comparing the steady-state effects of our prudent outcomes for
the two regions with business as usual can be very misleading, especially in
the presence of asymmetries. For example, the prudent pre-tip steady states
reported in table 2 are not affected by the different stages of development of
the North and the South ( K1 (0)  K 2 (0) ). This additional form of asymmetry
has a strong impact on the transient dynamics. To illustrate this, figure 1
reports for our benchmark  1  0.1,  2  0.3 , where the South gets hit more by
climate catastrophe than the North, the cooperative (solid lines), non-
21
cooperative (dashed lines) and business-as-usual (dotted lines) dynamic
simulations for the North (in blue) and the South (in red).10
Figure 1: Pre-tip simulations (  1  0.1,  2  0.3 )
10
The outcomes are calculated by log-linearizing the relevant saddle-point system and solving
22
for the dynamic trajectories using spectral decomposition.
23
We make the following observations.
24
First, the South has to catch up with the North so that consumption is in the
beginning substantially lower in the South. If tipping does not occur for a long
time, the consumption levels converge close together in all cases. In the long
run, the business-as-usual, cooperative and non-cooperative outcomes lead to
approximately the same levels of consumption.
Second, the capital stock and economic activity in the South start low but if
tipping does not occur for a long time, the capital stock in the South will
eventually be larger than the capital stock in the North. After all, the South has
to prepare for a larger climate catastrophe. Both capital stocks are always
higher than in the business-as-usual outcome if tipping can occur. In the noncooperative outcome, the capital stocks in both the North and the South are
slightly larger than under cooperation. The reason is that the hazard rate is a
bit higher under non-cooperation, since the stock of atmospheric carbon and
risk of catastrophe are a bit higher, and thus precautionary saving must rise.
Third, the carbon taxes are, of course, higher in case of cooperation than in
case the North and South regions fail to cooperate. It is interesting to note that
the carbon taxes under cooperation in the North and the South differ a lot in
the beginning but converge in the long run, if tipping does not occur for a long
time. The carbon taxes in the non-cooperative outcome, however, start at the
same level and diverge.
Finally, convergence of the atmospheric carbon stock and global mean
temperature is very slow. It is clear, however, that the atmospheric carbon
stock and the use of fossil fuel are the highest when business is as usual and
the lowest when the North and the South cooperate.
7. Conclusion
Future costs of global warming may to a large extent result from climate
tipping points (Lenton and Ciscar, 2013). The catastrophic damages as such
25
are an important driver of the carbon tax. Another important driver is how
much more imminent the tipping point becomes with global warming. The
most striking result in analysing these tipping points is that both precautionary
capital accumulation and a carbon tax are needed. Precautionary saving may
be picked up by the market but if not, a capital subsidy is needed. More capital
is required to be prepared for the shock and to smooth consumption over time
and less fossil fuel use is required to curb the risk of a tipping point.
With full international cooperation and the developed region transferring funds
to the developing region to smooth consumption across the globe, the first best
will result. This will lead to the same carbon tax for all parts of the global
economy. This first-best outcome is highly unrealistic as such international
transfers are, in practice, never used to fully smooth consumption across the
globe. However, without such transfers countries may be willing to aim for the
second-best global optimum where the transboundary externalities of carbon
emissions leading to a higher carbon stock and therefore a higher hazard of a
climate catastrophe are internalized. In this case carbon taxes will be higher in
the developed than in the developing parts of the world economy, as poorer
countries need to catch up with the rich countries, but will converge in the
long run if tipping does not occur for a long time. If countries do not
coordinate and cooperate on their climate policies, the outcome will be third
best. In this case, carbon taxes are lower, of course, and also diverge instead of
converge in the long run. As a consequence, fossil fuel use and thus the carbon
stock and the hazard rate are higher, and consequently global warming will be
more severe. It follows that precautionary saving is higher if the countries do
not cooperate, even though there are no direct externalities with respect to the
accumulation of the capital stocks. Since poorer countries suffer relatively
more from climate calamities, they need to engage more in precautionary
capital accumulation.
We envisage four future directions of research. First, following van der Ploeg
and de Zeeuw (2014) one can allow for more conventional gradual, marginal
26
global warming damages in addition to the damages resulting from stochastic
tipping points. Second, one can allow for catastrophic shocks to the carbon
cycle instead of total factor productivity as in van der Ploeg (2014). Third, one
can allow for the scarcity of fossil fuel and the scarcity rents that this implies.
Engström and Gars (2014) have investigated similar issues in the tractable
discrete-time aggregate model of global growth and climate change developed
by Golosov et al. (2014). It is important to see how this plays out in a multiregion world.11 Fourth, our two-region model of the global economy is highly
stylized and designed for illustrative purposes. With a greater fragmentation of
regions the non-cooperative biases in carbon taxes will be higher. It is also
important to study how the various cooperative and non-cooperative models
play out with international trade and capital mobility. Finally, one might
investigate the international political economy of dealing with tipping points
that affect different parts of the globe differently.
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29
Appendix 1: First best
To achieve the global first best, international transfers Si , i  1,.., n, are
n
needed. These transfers must obey the budget constraint
S
i 1
i
 0.
After tip
The
after-tip
HJB
equation
in
the
after-tip
value
function
 A ( K1 ,.., K n ,  1 ,.. n ) becomes
 A ( K1 ,.., K n ,  1 ,.. n ) 
(A1)
 n

Max   U (Ci )   KAi Y ( K i ,  i )  Ci  Si    A Si  , i  1,.., n,
Ci , Si
 i 1



where A is the Lagrange multiplier corresponding to the budget constraint.
The first-order conditions for the transfers and the rates of consumption are
(A2)
 KAi  U '(Ci )   A , i  1,.., n.
It follows from (A2) that by judicious setting of international transfers the
after-tip consumption rates are the same across the globe even if capital stocks
and catastrophe impacts differ, Ci (t )  C A , i  1,.., n, t  T . Hence,


(A3) YKi  Ki (t ),  i   YK1  K1 (t ),  1   Ki (t )  K1 (t ) ( 1 ,  i ), i  2,.., n, t  T .
A bigger adverse impact of the tip on the developing regions implies that its
capital stock is lower than in the developed region 1. This is clearly
unrealistic.
Before tip
The before-tip value HJB equation with international transfers is
30
n
  Vi B ( K i , P) 
i 1
n
 U (C )  H ( P) V
n
Max
C1 ,..,Cn , E1 ,.., En , S1 ,.., Sn
i 1
i
B
i

( K i , P )  Vi A ( K i ) 


  ViKBi ( K i , P)  AFi ( K i , Ei )  dEi  Ci   K i  Si 
(A4)
i 1
n 

 n


  ViPB ( K i , P )   E j   P    Si  ,
i 1 

 j 1


so that the optimality conditions for transfers are
(A5)
ViKBi  U '(Ci )   , i  1,.., n.
It follows from (A5) that the pre-tip rates of consumption are the same in all
regions, Ci (t )  C B , i  1,.., n, t  T . The social costs of carbon become
n
 iC  
 V jPB
j 1
U '(C B )
Precautionary
  C , i  1,...., n,
returns
are
and are the same across the globe.
identical
across
the
globe,
U '(C A ) 
 1   C , i  1,...., n, and thus social rates of interest are
B
U
'(
C
)


iC  H ( P) 
the same across the globe too, YKBi ( Ki , C ), i  1,...., n.
Using international lump-sum transfers achieves international consumption
smoothing despite differences in stages of economic development. Hence, the
first-best carbon taxes are the same across the globe too (cf. Chichilnisky and
Heal, 1994). In fact, the precautionary returns, social rates of interest and
capital stocks are the same throughout the globe.
31
Appendix 2: Functional forms and calibration
The utility functions U have a constant elasticity of intertemporal substitution
of σ = 0.5 and a pure rate of time preference of ρ = 0.014. The Cobb-Douglas
production functions F ( K i , Ei )  K i Ei  , i  1, 2, have a capital share of  =
0.3 and an energy share of  = 0.0623. The depreciation rate is δ = 0.05. We
calibrate to the business-as-usual (i.e., negligible carbon taxes and no
precautionary capital accumulation) outcome for the world economy for the
year 2010. Data sources are the BP Statistical Review and the World Bank
Development Indicators. The initial 2010 capital stocks are set to K1 (0)  180
and K 2 (0)  20 trillion US dollars and the 2010 level of world GDP is 63
trillion US dollars. We measure fossil fuel in GtC, so the emission-input ratio
equals one. We use a market price for fossil fuel of d = 504.3 US$ per ton of
carbon (or 9 US$ per million BTU). Global fossil fuel use in 2010 is 8.3 GtC

0.3
E (0)  K1 (0) 1 
1 0.0623


9
 2.02 , we get
(or 468.3 million GBTU). Using 1

E2 (0)  K 2 (0) 
E1 (0)  5.551 and E1 (0)  2.749 GtC in 2010. We thus have 42.1 and 20.9
trillion US dollars for GDP in the developed and developing part of the global
economy. The level of total factor productivity that matches these levels of
output and inputs in both regions is A = 8.5044.
The initial capital stocks of 180 and 20 trillion US dollars are below the
steady-state levels of 211 and 104 trillion US dollars to reflect that big parts of
the global economy, especially the developed region, are still catching up. The
fraction of carbon staying in the atmosphere is set to  = 0.5 and the rate of
decay of atmospheric carbon is set equal to  = 0.003. Table A1 summarizes
our calibration.
32
Table A1: Calibration of the two-country model with a tipping point
Variable/parameter
Pure rate of time preference, ρ
Elasticity of substitution, σ
Depreciation rate of capital, δ
Share of capital in value added, α
Share of fossil fuel in value added, β
Total factor productivity, A
Eventual climate shock, πi
Fraction of carbon staying up, ψ
Natural decay of carbon, γ
Cost of fossil fuel, d
2010 levels of GDP
2010 levels of capital, Ki0
2010 fossil fuel use, Ei0
Initial stock of carbon, P0
0.014
0.5
0.05
0.3
0.0623
8.5044
0.1, 0.3
0.5
0.003
9 US $/million BTU = 504.3 US $/tC
42.1 and 20.9 trillion US $
180 and 20 trillion US $
5.55 and 2.75 GtC
826 GtC = 388 ppm by vol. CO2
Optimal use of fossil-fuel use is Ei 
 Ai Fi
. Output net of fossil fuel costs
d i
1

      1 
1 
and capital depreciation is Y ( K i , i )  (1   )  Ai 
  Ki   Ki ,
  d   i  
i  1,.., n. The modified golden rules YKi ( Ki , i )    i , i  1,.., n, yields the preand after-tip steady-state capital stocks:
1
(A6)
1 
 1  
 
    

B

Ki   A 
,

B 
B 
      i   d   i  
1
1 
 1  

     
A
Ki  (1   i ) A 
, i  1,.., n.
   
      d  

Equation (7) gives:
33
(A7)
1
1  
1
zi    
 2  4 (    ) 
2
 1 
 2
 Ci A 
 K A   0.06457 ,
 i 
so C A ( K1 )  3.001K10.4303 and C A ( K 2 )  2.398K 2 0.4303 . These policy rules give
 
explicit expressions for the marginal value functions VKA ( Ki )  U ' Ci A
and
the value functions V A ( K i ,  i ) .
The pre-tip steady states follow from


Ci
B 
 Ci ( K i ) 
i  H ( P B ) 
B
A

 1 ,  i 


 i       i
Ci B   1   

d i  


 (C j B )(11/ )

 V A ( K j B ) 
j 1  1  1 / 
,
B
B
   H ( P )       H ( P ) 
n
1/
 H '( P B )(Ci B )1/  
n 
 B


B


K
,
P


 i


j1  



 

 d  j
     j
 


 B
 K j ,


where the summation in the expression for the carbon tax (the second steady
state) reduces to just the j = i term in the non-cooperative case.
34