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Win as a Team or Fail as Individuals: Cooperation
and Non-Cooperation in the Climate Tax Game*
Elmar Hillebrand†Marten Hillebrand‡
March 20, 2023
Abstract
This paper studies the strategic interaction of arbitrarily many heterogeneous
regions in a macroeconomic growth model of climate change. Regions differ with
respect to production technologies, productivity and factor endowments, stocks of
fossil fuels, and climate damages. They trade on friction-less international markets
for capital and fossil fuels and choose climate tax policies under different scenar-
ios of cooperation and non-cooperation. We derive closed form solutions of optimal
policies for both the non-cooperative equilibrium where regions internalize only do-
mestic climate damages and the efficient solution internalizing global climate dam-
ages. We extend all these results to cases with partial cooperation where regions
form coalitions and choose policies to maximize welfare of their coalition. Finally,
we study how transfers can incentivize regions to cooperate and determine a class
of optimal transfer schemes under which all regions are better-off under full co-
operation relative to the non-cooperative outcome. Numerical simulations based
on calibrated parameter values are used to illustrate and quantify our theoretical
results under different coalition scenarios.
JEL classification: C63, E61, H21, H23, Q54
Keywords: Climate Change; Multi-region dynamic general equilibrium model; Nash-
equilibrium; Cooperative solution; Coalitions; Optimal climate policy.
*Acknowledgements. We would like to thank Markus Epp, Bård Harstad, Marius Jäger, Oliver Land-
mann, and Klaus Wälde as well as participants of various research seminars and conferences for helpful
suggestions and comments.
†EEFA Research Institute, Muenster, Germany, email: e.hillebrand@eefa.de
‡Department of Economic Theory, Albert-Ludwigs University Freiburg, Rempartstrasse 10-16, 79085
Freiburg im Breisgau, Germany, email: marten.hillebrand@vwl.uni-freiburg.de (corresponding author)
1 Introduction
Climate change is a global threat requiring a coordinated effort of all nations to sig-
nificantly and permanently reduce their emissions from burning fossil fuels. Several
climate summits accompanied by intense negotiations held over that past years and
even decades suggest that there is wide agreement on this necessity. Yet, a political
solution that all countries have committed to has not been reached. A major obstacle to
implementing climate policies at the global level might be that nations or regions differ
substantially along many dimensions such as their state of economic development, re-
liance on fossil fuels, and the climate damages they are projected to suffer. Therefore,
the incentives for implementing climate polices vary considerably across regions. Past
climate agreements have also seen the formation of coalitions being an integral part
of negotiations. Hence, it is important to understand how the formation of coalitions
determines the outcome of the political process and which policies remain feasible.
A theoretical analysis of optimal climate policies and their successful implementation
must therefore be based on a framework which incorporates three key features. First,
decisions on climate policies are taken by politically autonomous regions acting in their
own self-interest rather than for the common good. Second, these regions differ sub-
stantially along key economic and other dimensions and, therefore, have different in-
centives for choosing and implementing a particular climate policy. Third, countries
may not be willing to fully cooperate but instead form coalitions of regions with com-
mon political interests.
The present paper studies the existence and form of optimal climate policies in a model
with heterogeneous regions that incorporates all of the previous features. Our anal-
ysis draws on the multi-region framework and results developed in our earlier work
Hillebrand & Hillebrand (2019, 2022). In these papers, we focused on optimal climate
policies under full cooperation. The present paper extends this approach to a general
non-cooperative setup in which the fully cooperative scenario emerges as a special case.
Within this setup, we address the following specific questions. First, which climate
policies emerge in a purely non-cooperative setting and how do they differ from the
fully cooperative solution? Second, which policies emerge with partial cooperation and
the formation of coalitions? Third, how can regions be incentivized to cooperate by
means of side payments and how can such transfer policies be characterized?
Answering these and a number of related questions is the general contribution of this
paper. Specifically, we derive closed form solutions of optimal climate policies for both
the fully non-cooperative and the fully cooperative case and extend these results to
scenarios with partial cooperation where regions form coalitions. All these results take
a very simple and intuitive form. Third, we provide a complete characterization of
transfer schemes which redistribute the gains from cooperation such that each region
1
has an incentive to cooperate. This defines a range of admissible transfers that forms
the basis for successful climate negotiations.
Our paper contributes to a large and growing literature studying the climate problem
from an economic perspective. Modern representatives of this field adopt the framework
of dynamic general equilibrium theory which is the standard approach in macroeco-
nomics.1Based on this approach, the paper closest to ours is Hambel, Kraft & Schwartz
(2021), henceforth HKS who derive optimal climate policies corresponding to abatement
efforts in a fully non-cooperative setup. A main contribution of our work relative to HKS
and others is that we allow for regions to trade on international markets for capital and
fossil fuels not permitted in HKS. Using the implicit definition of the social cost of car-
bon from Nordhaus (2014), HKS also obtain a measure of the implied regional emissions
taxes. In this paper, we use a different concept of the social cost of carbon derived from
an explicitly defined planning problem in line with Golosov et al. (2014) and Hillebrand
& Hillebrand (2019). This permits to derive the optimal climate tax in closed form and
admits a straightforward economic interpretation.
Our work also contributes to the game theoretic literature on climate change advanced
by Harstad (2016, 2012) and Battaglini & Harstad (2016). These papers usually focus
on game theoretic aspects and, therefore, employ highly stylized models of both the
macroeconomy and the climate system. As a consequence, the cost of climate change
and damages are specified directly in a somewhat ad-hoc fashion rather than being
derived as an endogenous outcome of the interaction between the economic production
sector and the climate system. Relative to this approach, we maintain the full-fledged
growth framework of Hillebrand & Hillebrand (2019) featuring an explicit description of
the production process and the climate system. Despite this more detailed specification,
we retain the virtue of being able to derive analytical results including closed form
solutions of optimal climate policies under different levels of cooperation among regions.
A final strand of research studies scenarios of cooperation and non-cooperation based
on the RICE model developed in Nordhaus & Yang (1996). Setting aside the concep-
tual problems facing the RICE model and the derivation of its solution (cf. Denning &
Emmerling (2017) or Hillebrand & Hillebrand (2019)), all these studies are confined to
purely numerical results and analytical expressions can not be derived.2In addition,
the RICE-framework severely restricts or even excludes trade among countries.
The paper is organized as follows. Section 2 introduces the model. Section 3 derives the
decentralized equilibrium under arbitrary climate policies. Section 4 studies the non-
cooperative solution which is compared to the efficient solution under full cooperation
in Section 5. Section 6 studies the case with partial cooperation under formation of
1Examples are Golosov et al. (2014), Hassler & Krusell (2012), or Gerlagh & Liski (2018).
2Models in this category include Bosetto et al. (2003), Eyckmans & Tulkens (2003), Eyckmans & Finus
(2003), or Finus et al. (2014).
2
coalitions. Section 7 presents results from a numerical simulation study. Section 8
concludes. Mathematical proofs and computational details are placed in the appendix.
2 The Model
2.1 World economy
The world economy is divided into L≥1 autonomous regions, indexed by ℓ∈L:=
{1,...,L}. Time evolves in discrete periods t∈{0,1,2,...}. All variables determined prior
to the initial period t=0 are treated as given parameters in the following setup. We
express the regional dependence of variables by a superscript ℓand identify summa-
tion of this variable over all regions by a bar superscript. For example, Xℓ
twill denote
fossil fuel consumption of region ℓin period tsuch that Xt:=Pℓ∈LXℓ
tdenotes global
emissions in period t. The major building blocks of the model are the production sector,
the climate model, and the consumption sector which are now explained in detail.
2.2 Production sector
Final output
In each region ℓ∈La single representative firm produces a homogeneous final output
commodity Yℓ
tusing capital Kℓ
tand fossil fuels Xℓ
tas inputs. The production technology
is of the general form
Yℓ
t=(1−Dℓ
t)Fℓ
t(Kℓ
t,Xℓ
t).(1)
Here, Dℓ
t∈[0,1] denotes endogenous climate damage which is further specified below.
Capital input Kℓ
tis rented in the international capital market at price rt. Input factor
Xℓ
tsubsumes different kinds of fossil fuels such as coal, oil, etc. which are purchased
in the global resource market at price vt. Time-dependence of the production function
Fℓ
tcaptures both regional population growth as well exogenous technological progress
of labor and energy efficiency.
Restrictions on technology
We impose the following standard restrictions on the production function Fℓ
tin (1). The
left-side Inada condition (2) ensures that each factor will be employed at equilibrium.
Assumption 1
The function
Fℓ
t:R2
+−→ R+
is strictly increasing, strictly concave, and continuously
differentiable on
R2
++
. The partial derivatives satisfy the boundary condition
lim
ziց0
∂Fℓ
t(z1,z2)
∂zi
= ∞
for both
i=1,2
and all
z=(z1,z2)∈R2
++.(2)
3
Extraction of fossil fuels
Region ℓ∈Lpossesses an initial stock of fossil fuels Rℓ
0≥0 which can be extracted at
constant unit costs cx≥0. Extraction of fossil fuels takes place in a resource sector
operated by a single firm which chooses an extraction sequence (Xℓ,s
t)t≥0with Xℓ,s
t≥0
denoting the extraction of fossil fuels in period t. Feasible extraction plans satisfy
∞
X
t=0
Xℓ,s
t≤Rℓ
0.(3)
Extracted resources Xℓ,s
tare supplied to the global resource market in each period t. To
avoid trivialities, we impose the initial condition Pℓ∈LRℓ
0>0, i.e., initial world resources
of fossil fuels are strictly positive.
2.3 Climate model
Emissions
Emissions of CO2are generated by using fossil fuels in production. Measuring fossil
fuel inputs Xℓ
tin (1) directly in units of CO2, global emission in period tare given by
Xt:=X
ℓ∈L
Xℓ
t.(4)
Emissions prior to t=0 are given and we assume that X−t=0 for t>0 sufficiently large
reflecting the fact that emissions before the industrial age were uniformly zero.
Atmospheric level of carbon
Climate damages in period tare determined by atmospheric carbon concentration St
relative to the pre-industrial level that depends on past aggregate emissions, i.e.,
St=
∞
X
n=0
δnXt−n.(5)
The non-negative sequence (δn)n≥0in (5) determines the evolution and persistence of
emissions in the atmosphere. The specification (5) encompasses various climate models
in the literature including Golosov et al. (2014) or Gerlagh & Liski (2018).
Climate damages
Climate damages in region ℓ∈Lat time tdepend exclusively on carbon concentration
Stgiven by (5) and are determined by the damage function Dℓ:R+−→ [0,1[,
Dℓ
t=Dℓ(St) :=1−exp{−γℓSt},γℓ>0.(6)
Regional differences in climate damages are captured by the region-specific parameter
γℓ,ℓ∈L. The exponential form (6) is also widely used in the literature, cf. again Golosov
et al. (2014) or Gerlagh & Liski (2018). Golosov et al. (2014) show that this specification
approximates well the damage function of the DICE-model ((Nordhaus & Sztorc 2013))
in the empirically relevant range.
4
2.4 Consumption sector
Representative consumer
The consumption sector in region ℓ∈Lconsists of a single representative household who
supplies capital Kℓ,s
tto the global capital market in each period t. Initial capital Kℓ,s
0
in period t=0 is taken as given in the decision. In addition, the consumer is entitled to
receive all profits from domestic firms and transfers from the government.
Consumer preferences
The household’s preferences over non-negative consumption sequences (Cℓ
t)t≥0are rep-
resented by a standard time-additive utility function
U((Cℓ
t)t≥0)=
∞
X
t=0
βtu(Cℓ
t) where u(C)=(C1−σ
1−σfor σ>0,σ6= 1
log(C) for σ=1.(7)
The discount factor satisfies 0 <β<1. The previous specification is widely used in
models of climate change. It is key for the separability between efficiency and optimality
exploited in Hillebrand & Hillebrand (2019) to derive an optimal climate policy.
2.5 Summary of the economy
The economy Eintroduced in the previous sections can be summarized by its regional
structure, production parameters, climate model, and consumer parameters. Formally,
E=DL,³(Fℓ
t)t≥0´ℓ∈L,cx,(δn)n≥0,(γℓ)ℓ∈L,β,σE.(8)
In addition, the initial distribution of capital (Kℓ,s
0)ℓ∈Land initial stocks of fossil fuels
(Rℓ
0)ℓ∈Lare given as well as aggregate emissions (X−t)t>0prior to t=0.
3 Decentralized Equilibrium
The decentralized equilibrium reconciles optimal behavior of consumers and producers
in each region with market clearing on regional and global markets. The equilibrium is
determined for a given climate policy chosen by each region to be specified next.
3.1 Climate policy
Climate tax
Each region ℓ∈Llevies a tax τℓ
ton regional emissions in period tto be paid by the
final sector. The tax sequence (τℓ
t)t≥0is the first building block of a climate policy.
5
This sequence may consist of given exogenous numbers or can be generated by a time-
invariant rule that depends on endogenous variables.
Transfers
All tax revenue is returned to consumers as a lump sum transfer. As for the distribution
of transfers across regions, we distinguish the cooperative and the non-cooperative case.
In the non-cooperative case, each region simply rebates its tax revenue to domestic
consumers implying that consumers in region receive a transfer in period tequal to
Tℓ
t=τℓ
tXℓ
tfor each ℓ∈L.(9)
In the cooperative case, regions pool their tax revenue and agree on a transfer policy
θ:L−→ R,ℓ7→ θℓsatisfying Pℓ∈Lθℓ=1 which determines the transfer share θℓreceived
by region ℓ. Transfers received by consumers in region ℓat time tthen follow as
Tℓ
t=θℓX
k∈L
τk
tXk
tfor each ℓ∈L.(10)
The assumption of time-invariant transfers shares imposes no restriction since con-
sumer behavior depends exclusively on lifetime transfers defined below. Any time-
dependent distribution of transfers is therefore equivalent to a time-invariant transfer
policy in the sense that it implies the same consumption behavior.
3.2 Producer behavior
Final sectors
The representative firm in the final sector chooses non-negative factor inputs in period
tto maximize its period-profit, taking climate damages and prices for capital, labor,
and fossil fuels as given. The latter includes the tax on emissions. The formal problem
determining profits Πℓ
tin period treads:
Πℓ
t=max
Kℓ,Xℓ≥0n(1−Dℓ
t)Fℓ
t(Kℓ,Xℓ)−rtKℓ−(vt+τℓ
t)Xℓo.(11)
Profit maximizing factor demand (Kℓ
t,Xℓ
t) solves the standard first order conditions3
(1−Dℓ
t)∂KFℓ
t(Kℓ
t,Xℓ
t)=rt(12a)
(1−Dℓ
t)∂XFℓ
t(Kℓ
t,Xℓ
t)=vt+τℓ
t.(12b)
Profits determined by (11) are fully transferred to consumers in region ℓ.
Resource sector
Unlike the final sector, the resource sector solves an intertemporal decision problem
3We denote by ∂KF(K,X) :=∂F(K,X)
∂Kand ∂XF(K,X) :=∂F(K,X)
∂Xthe partial derivatives of a function F.
6
involving a stream of future profits. To discount these payments to period t=0, we
define the time t-discount factor (Arrow-Debreu price) as
qt:=
t
Y
n=1
r−1
nfor each t=0,1,2,... where q0=1.(13)
The resource firm’s discounted profit stream in t=0 generated by an optimal extraction
sequence is then determined as
Πℓ
x:=max
(Xℓ,s
t)t≥0(∞
X
t=0
qt(vt−cx)Xℓ,s
t¯¯¯(3) holds, Xℓ,s
t≥0 for all t≥0).(14)
Applying standard arguments (cf. Hillebrand & Hillebrand (2019)), the existence of an
optimal extraction sequence requires the Hotelling rule
vt−cx=rt(vt−1−cx) for all t=1,2,3,... (15)
under which maximum profits (11) are given by
Πℓ
x=(v0−cx)Rℓ
0.(16)
3.3 Consumer behavior
In each period t, the representative consumer in region ℓreceives factor income from
supplying capital Kℓ,s
tformed in the previous period t−1. In addition, the consumer
collects profits Πℓ
tfrom the final sector and Πℓ
x,t=(vt−cx)Xℓ,s
tfrom the domestic resource
sector as well as transfers Tℓ
tfrom the government. Consumption Cℓ
tand newly formed
capital Kℓ,s
t+1satisfy the period budget constraint
Kℓ,s
t+1=Πℓ
t+rtKℓ,s
t+(vt−cx)Xℓ,s
t+Tℓ
t−Cℓ
t.(17)
We interpret Kℓ,s
t+1as the consumer’s net asset position and, therefore, do not exclude
negative values. To exclude Ponzi-schemes, however, we impose the usual condition
limt→∞ qtKℓ,s
t+1≥0. Using this and (16) and defining lifetime transfer income Tℓ:=
P∞
t=0qtTℓ
tand the discounted stream of final profits Πℓ:=P∞
t=0qtΠℓ
twe can solve (17)
forward to obtain the lifetime budget constraint
∞
X
t=0
qtCℓ
t≤Πℓ+r0Kℓ,s
0+(v0−cx)Rℓ
0+Tℓ.(18)
The consumer chooses consumption (Cℓ
t)t≥0subject to (18) to maximize lifetime utility
Udefined in (7). Optimality is determined by equality of (18) and the Euler equation
Cℓ
t+1=Cℓ
t(βrt+1)1
σfor all t=0,1,2,... (19)
7
3.4 Market clearing and equilibrium
Market clearing
In each period t, capital and fossil fuels are traded on international markets. The
market clearing conditions for period tread:
X
ℓ∈L
Kℓ
t
!
=X
ℓ∈L
Kℓ,s
tand X
ℓ∈L
Xℓ
t
!
=X
ℓ∈L
Xℓ,s
t.(20)
Since the Hotelling rule (15) makes resource firms indifferent between the timing of
extraction, the extraction sequence (Xℓ,s
t)t≥0will, in general, be indeterminate at equi-
librium. Due to (17), the same will be true of regional capital supply (Kℓ,s
t+1)t≥0. However,
regional consumption of fossil fuels (Xℓ
t)t≥0is uniquely determined at equilibrium and
satisfies the world resource constraint
∞
X
t=0X
ℓ∈L
Xℓ
t=
∞
X
t=0
Xt≤R0:=X
ℓ∈L
Rℓ
0.(21)
Writing profits (11) as Πℓ
t=Yℓ
t−rtKℓ
t−(vt+τℓ
t)Xℓ
tand exploiting the definition of trans-
fers (9) or (10) we can sum the consumer’s budget constraint (17) over all regions ℓand
combine the result with the market clearing conditions (20) to obtain the evolution of
world capital as
X
ℓ∈L
Kℓ
t+1=X
ℓ∈L
Yℓ
t−cxX
ℓ∈L
Xℓ
t−X
ℓ∈L
Cℓ
t.(22)
Equation (22) can also be interpreted as a market clearing condition for the world com-
modity market.
Finally, for purposes of a compact definition of equilibrium we combine (4), (5), and (6)
to obtain regional climate damages as a function of past regional emissions given by
Dℓ
t=1−expÃ−γℓ∞
X
n=0
δnX
ℓ∈L
Xℓ
t−n!for all ℓ∈L.(23)
Definition of equilibrium
We are now in a position to formally define a decentralized equilibrium for this economy.
Definition 1
An equilibrium of
E
consists of climate taxes and transfers
((τℓ
t,Tℓ
t)ℓ∈L)t≥0
, an allocation
A∗=((Kℓ∗
t,Xℓ∗
t,Cℓ∗
t)ℓ∈L)t≥0
, and prices
P∗=(r∗
t,v∗
t)t≥0
such that for all
t=0,1,2,...
:
(i) Factor inputs
(Kℓ∗
t,Xℓ∗
t)
solve conditions (12) for given prices
(r∗
t,v∗
t)
, taxes
τℓ
t
,
and damages
Dℓ
t
determined by (23) for each region
ℓ∈L
.
(ii) Prices
P∗
satisfy the Hotelling rule (15) and global fossil fuel consumption is con-
sistent with the world resource constraint (21).
8
(iii) Taxes and transfers and emissions are consistent with (9) or (10).
(iv) Regional consumption
(Cℓ∗
t)t≥0
satisfies the Euler equation (19) and constraint
(18) with equality with discount factors
(q∗
t)t≥0
determined by (13).
(v) Initial capital
(Kℓ
0)ℓ∈L
satisfies the capital market clearing condition (20) for
t=0
and (22) holds for all
t
with output determined by (1) and damages by (23).
Properties of equilibrium
The form of the utility function (7) and a frictionless capital market imply a world-
consumption distribution that is constant over time. Thus, each region acquires a con-
stant share of world consumption in each period. We state this result formally in the
next lemma. The proof is analogous to Hillebrand & Hillebrand (2019) and is omitted.
Lemma 1
Equilibrium consumption of each region
ℓ∈L
is a constant fraction
µℓ∈]0,1[
of world
consumption
C∗
t:=Pℓ∈LCℓ∗
t
in each period
t
, i.e.,
Cℓ∗
t=µℓC∗
t
where
µℓ=
Πℓ+r0Kℓ,s
0+(v0−cx)Rℓ
0+Tℓ
Pk∈L³Πk+r0Kk,s
0+(v0−cx)Rk
0+Tk´.(24)
4 The Non-Cooperative Equilibrium
This section considers the non-cooperative equilibrium in which each region ℓsets its
climate tax policy τℓ=(τℓ
t)t≥0to maximize domestic welfare, taking as given the deci-
sions of other regions. In this non-cooperative scenario, transfers are determined by
(9) excluding net transfers between regions. The regionally optimal tax policy can be
derived in two steps. In the first step, each region chooses a regionally optimal alloca-
tion taking emissions from other regions as well as global prices for capital and fossil
fuels as given. The second step then determines regional climate taxes such that the
regionally optimal allocation materializes as a decentralized equilibrium.
4.1 Regional planning problem
Constraints
The planner in region ℓtakes local initial stocks of capital Kℓ,s
0and fossil fuels Rℓ
0≥0
as well as aggregate emissions (X−t)t≥1prior to t=0 as fixed parameters. Likewise,
the sequence (X−ℓ
t)t≥0of future emissions X−ℓ
t:=Pk∈L\{ℓ}Xk
tfrom other regions as well
as international prices P=(rt,vt)t≥0are taken as given. We assume that these prices
satisfy the Hotelling rule (15) as they will at equilibrium and define (qt)t≥0by (13).
9
A crucial deviation from the decentralized solution is that the planner in region ℓtakes
into account the impact of its own emissions and those of other regions on local climate
damage Dℓ
tdetermined by (23). Using the previous notation, we write damages as
Dℓ
t=1−expn−γℓ∞
X
n=0
δn³Xℓ
t−n+X−ℓ
t−n´o for ℓ∈L.(25)
The regional planner chooses factor inputs to the production technology (1) in each pe-
riod t. The decision also involves the accumulation of capital and extraction of fossil
fuels. However, these variables are not necessarily fully employed in domestic produc-
tion but can also be traded on global capital and resource markets at the given prices.
As before, a superscript ’s’ signifies variables supplied to these markets. The regional
stock of capital then evolves according to the resource condition
Kℓ,s
t+1=(1−Dℓ
t)Fℓ
t(Kℓ
t,Xℓ
t)+rt¡Kℓ,s
t−Kℓ
t¢+vt¡Xℓ,s
t−Xℓ
t¢−cxXℓ,s
t−Cℓ
t.(26)
It follows from (3) and the Hotelling rule (15) that the extraction sequence (Xℓ,s
t)t≥0cho-
sen by region ℓgenerates total discounted revenue given again by (16). Using this and
imposing again the No-Ponzi condition limT→∞ Kℓ,s
T+1qT≥0, one can solve (26) forward
to obtain the lifetime budget constraint
∞
X
t=0
qt³rtKℓ
t+vtXℓ
t+Cℓ
t−(1−Dℓ
t)Fℓ
t(Kℓ
t,Xℓ
t)´≤r0Kℓ,s
0+(v0−cx)Rℓ
0.(27)
This condition somewhat parallels the consumer’s condition (18). The main difference is
that the planer chooses inputs to final production directly and, more importantly, takes
into account the impact of fossil fuel inputs Xℓ
ton domestic climate damages via (25).
The regional planning problem
Using (25) and (27) the planning problem of region ℓ∈Lcan now be stated as follows.
max
(Kℓ
t,Xℓ
t,Cℓ
t)t≥0nU((Cℓ
t)t≥0)¯¯¯(25) and (27) hold, Kℓ
t,Xℓ
t,Cℓ
t≥0 for all t=0,1,2,...o.(28)
Clearly, the solution to (28) depends on the given emissions (X−ℓ
t)t≥0of other regions
and prices P=(rt,vt)t≥0. Crucially, the planner is a ’price-taker’ on international mar-
kets and does not take into account the impact of his decision on the determination of
these prices as equilibrium outcomes.
Solution to the problem
Problem (28) is a constrained optimization problem that can be solved by standard
Lagrangean methods. The details can be found in Section A.1 in the Appendix. This
way, on can derive the following optimality conditions which characterize the solution
in addition to the technological and resource constraints. First, consumption satisfies
the Euler equation (19) and (27) holds with equality. Second, the marginal product of
capital in period tequals its return rtsuch that (12a) holds. Finally, resource input in
10
period tearns a marginal product equal to its price vtplus the total discounted marginal
damage in production:
(1−Dℓ
t)∂XFℓ
t(Kℓ
t,Xℓ
t)=vt+
∞
X
n=0
βn³Cℓ
t+n/Cℓ
t´−σδnγℓYℓ
t+n.(29)
Intuitively, the damage-related term in (29) consists of three factors. First, a discount
factor βn¡Cℓ
t+n/Cℓ
t¢−σserving to measure damages in t+nin units of time tconsump-
tion. Second, the term δnwhich measures the quantity of emissions at time tthat are
still in the atmosphere at time t+n. Third, the marginal loss in domestic output Yℓ
t+nat
time t+ncaused by an additional unit of carbon in the atmosphere. Summation of these
factors over all periods t,t+1,t+2,... then gives the total domestic damage generated
by one additional unit of emissions in period t.
For later reference, we state the previous result formally in the next lemma.
Lemma 2
Let prices
P=(rt,vt)t≥0
and emissions
(X−ℓ
t)t≥0
of other regions be given. If the regional
allocation
(Kℓ
t,Xℓ
t,Cℓ
t)t≥0
solves (12a), (19), and (29) with
Dℓ
t
determined by (25) for all
t=0,1,2,...
and satisfies (27) with equality, then it is a solution to the problem (28).
4.2 Non-cooperative solution as a Nash equilibrium
The market maker
In a non-cooperative equilibrium the decisions of all regions ℓ∈Ldetermined as a so-
lution to (28) are mutually compatible in the sense that emissions of all other regions
are correctly anticipated and market clearing on global capital and resource markets
obtains in each period t. To formalize this idea, we embed the previous structure into
the setup of a non-cooperative game and invoke the formal definition of a Nash equilib-
rium. Since the regional planning problem involves not only the decision variables from
other regions, but also the prices of capital and fossil fuels, we introduce an additional
player ℓ=0 that will be referred to as a ’market-maker’. This player chooses prices
P=(vt,rt)t≥0as his strategy subject to the Hotelling rule (15). Since the Hotelling rule
determines all future prices (vt)t>0from the initial choice v0and capital returns and
choosing the sequence (rt)t≥1is equivalent to choosing the AD-prices (qt)t≥0, we can
identify the choice of player ℓ=0 with the strategy (v0,r0,(qt)t≥0).
Decision problem of the market maker
We now set up the market maker’s decision problem such that market clearing obtains
along the Nash equilibrium defined below. Suppose player ℓ=0 bases his choice of
prices on the ’mismatch’ between demand and supply on international markets deter-
mined by the allocations chosen by players. To this end, the resource price v0is chosen
11
to maximize the value of excess demand in the resource market.4Formally, the price is
a solution to the linear problem
max
v0((v0−cx)µ∞
X
t=0X
ℓ∈L
Xℓ
t−R0¶¯¯¯cx≤v0≤vmax
0).(30a)
Here, vmax
0>0 is some arbitrary upper bound chosen sufficiently large to exceed any
possible equilibrium value vnc
0defined below. Thus, the market maker sets the resource
price to its minimal value v0=cxif the total demand for resources P∞
t=0Pℓ∈LXℓ
tsummed
over all regions and periods is lower than the available resource stock R0. Conversely,
if resource demand exceeds this stock, the resource price is set to its maximum value
v0=vmax
0. It follows that problem (30a) has an interior solution cx<v0<vmax
0(in fact,
infinitely many) if and only if the resource constraint (21) holds with equality such that
total demand exhausts the global stock.
In a similar fashion, the market maker determines r0and the values qtfor t=0,1,2,...
based on the excess demand for capital in t=0 and in the consumption good market in
period t, respectively. Formally, these problems read
max
r0(r0X
ℓ∈L³Kℓ
0−Kℓ,s
0´¯¯¯0≤r0≤rmax
0)(30b)
and, for each t=0,1,2,...
max
qt(qtX
ℓ∈L³Cℓ
t+cxXℓ
t+Kℓ
t+1−(1−Dℓ
t)Fℓ
t(Kℓ
t,Xℓ
t)´¯¯¯0≤qt≤qmax
t).(30c)
Again, the upper bounds rmax
0and qmax
tare chosen sufficiently large and the initial
distribution of capital (Kℓ,s
0)ℓ∈Lentering (30b) is given and climate damage Dℓ
tin (30c)
determined by (25) for each ℓ. Linearity of each of the underlying problems then implies
that existence of interior solutions to (30b) and (30c) is equivalent to the capital market
clearing condition in (20) for t=0 and (22) for each t=0,1,2,...5
Definition of non-cooperative equilibrium
Using L0:={0}∪Las the set of players we can now define a Nash equilibrium as follows.
Definition 2
A non-cooperative (Nash) equilibrium of
E
is an allocation
Anc =((Kℓ,nc
t,Xℓ,nc
t,Cℓ,nc
t)ℓ∈L)t≥0
and prices
Pnc =(rnc
t,vnc
t)t≥0
such that the following holds:
(i) The regional allocation
(Kℓ,nc
t,Xℓ,nc
t,Cℓ,nc
t)t≥0
chosen by player
ℓ∈L
solves the
planning problem (28) given prices
Pnc
and emissions
(X−ℓ,nc
t)t≥0
of other regions.
4This idea is used in finite-dimensional Walrasian economies to construct a price correspondence of
which equilibrium prices obtain as fixed points. Details of this approach can be found, e.g., in Mas-Colell,
Whinston & Green (1995, Section 17.C).
5Since the choices of r0,v0, and (qt)t≥0are independent, one could also determined them simultane-
ously by maximizing the sum of (30a), (30b), and (30c) summed over all t=0,1,2,...
12
(ii) Prices
Pnc
chosen by player
ℓ=0
satisfy the Hotelling rule (15) and
(vnc
0,rnc
0)
and
qnc
t
defined by (13) are solutions to the decision problems (30a)-(30c) for all
t≥0
.
It is worth noting that (21) may or may not bind along the non-cooperative equilibrium.
If it does, fossil fuels carry a scarcity rent implying v0>cxand, by (15), vt>cxfor all t.
On the other hand, if P∞
t=0Pℓ∈LXℓ,nc
t<R0, the solution to problem (30a) satisfies v0=cx
and implies vt=cxfor all tby (15). In this case, fossil fuels are abundant and there is
no scarcity rent.
4.3 Time-consistency of the non-cooperative solution
In this section we show that the non-cooperative equilibrium is time-consistent in the
sense that in each future period t=N≥1, each player if permitted to re-optimize will
stick to his original strategy provided everybody else does. In other words, the Nash
equilibrium of the non-cooperative game defined above satisfies the one-shot deviation
principle, i.e., no player can increase their pay-off by singularly deviating from the equi-
librium strategy in any period. Hence, the non-cooperative equilibrium from Definition
2 is time-consistent or also subgame perfect.
To show this formally, define the wealth of the consumer in region ℓ∈Lat the beginning
of period t=0,1,2,... as
Wℓ
t:=rtKℓ,s
t+(vt−cx)Rℓ
t(31)
where Rℓ
tis the regional resource stock at the beginning of period tdetermined recur-
sively as
Rℓ
t+1=Rℓ
t−Xℓ,s
tfor all t=0,1,2,... (32)
Intuitively, wealth Wℓ
tconsist of capital income and the value of the current resource
stock net of extraction costs in period t. Note that initial wealth Wℓ
0appears on the
right-hand side of the lifetime budget constraint (27).
Combining definition (31) and (32) with the period-budget constraint (17), the evolution
of the wealth sequence (Wℓ
t)t≥0is determined recursively by the equation
Wℓ
t+1=rt+1³Wℓ
t+(1−Dℓ
t)Fℓ
t(Kℓ
t,Xℓ
t)−rtKℓ
t−vtXℓ
t−Cℓ
t´for all t=0,1,2,... (33)
with the initial value Wℓ
0determined by (31) from the given initial values Kℓ,s
0and Rℓ
0.
Solving (33) forward one obtains initial wealth at the beginning of time t=N>0 as:
qNWℓ
N=
N−1
X
t=0
qt³(1−Dℓ
t)Fℓ
t(Kℓ
t,Xℓ
t)−Cℓ
t−rtKℓ
t−vtXℓ
t´+Wℓ
0.(34)
Observe that although the consumer’s net capital position Kℓ,s
tand regional extraction
Xℓ,s
tare, in general, indeterminate at equilibrium, the consumer’s wealth position is
13
uniquely determined recursively by (34).
Consider now the behavior of the consumer in region ℓ∈Lin some period t=N≥1.
Let previous wealth Wℓ
N−1and the strategies of all regions (Kℓ
t,Xℓ
t,Cℓ
t)0≤t<Nas well as
the sequence (rt,vt)0≤t<Nchosen by the market maker prior to period t=Nbe given.
In particular, aggregate emissions (Xt)t<Nprior to period t=Nare given. Suppose in
period t=Neach player ℓ∈Lchooses an updated strategy (Cℓ
t,Xℓ
t,Kℓ
t)t≥Nto maximize
the remaining utility
UN((Cℓ
t)t≥N)=
∞
X
t=N
βtu(Cℓ
t) (35)
subject to the updated time Nlifetime budget constraint
∞
X
t=N
qt,N³Cℓ
t+rtKℓ
t+vtXℓ
t−(1−Dℓ
t)Fℓ
t(Kℓ
t,Xℓ
t)´≤Wℓ
N(36)
where qt,N:=qt/qN=Qt
n=N+1r−1
nand Wℓ
Ndetermined by (34) or, equivalently, by (33)
setting t=N−1. The time-Nre-optimization problem of player ℓ∈Lthen reads:
max
(Cℓ
t,Xℓ
t,Kℓ
t)t≥NnUN((Cℓ
t)t≥N)|Cℓ
t,Kℓ
t,Xℓ
tfor all ≥0,t≥N,(36) holdso.(37)
The following lemma describes the properties of solutions to (37). The proof parallels
the one of Lemma 2 and is therefore omitted.
Lemma 3
If the sequence
(Cℓ
t,Xℓ
t,Kℓ
t)t≥N
satisfies (12a), (19), and (29) with
Dℓ
t
determined by (25)
for all
t≥N
as well as (36) with equality, then it is a solution to problem (37).
In a similar vein, suppose the market maker ℓ=0 is permitted to update his strategy as
well by choosing values (rt,vt)t≥Nsubject to the Hotelling rule (15). Clearly, this choice
is equivalent to choosing values vNand rNand qt,Nfor all t>N. Suppose these choices
are made to maximize the values (vN−cx)Pℓ∈L¡Rℓ
N−P∞
t=NXℓ
t¢,rN¡Pℓ∈LKℓ
N−Ks
N¢, and
qt,NPℓ∈L¡Cℓ
t+rtKℓ
t+vtXℓ
t−(1−Dℓ
t)Fℓ
t(Kℓ
t,Xℓ
t)¢for all t≥N, respectively. Here, Ks
N:=
Pℓ∈L¡(1−Dℓ
N−1)Fℓ
N−1(Kℓ
N−1,Xℓ
N−1)−Cℓ
N−1−cxXℓ
N−1´is world capital supply in period N.
Combining this behavior with that of players ℓ∈L, we can define a time NNash equi-
librium as a list of continuation strategies (Cℓ
t,Xℓ
t,Kℓ
t)t≥Nfor each ℓ∈Land (rt,vt)t≥N
for ℓ=0 which solve the updated decision problems at time Ngiven previous decisions.
Now suppose until time N−1, all players followed the strategies prescribed by the non-
cooperative equilibrium. Formally, each player ℓ∈Lchose (Kℓ,nc
t,Xℓ,nc
t,Cℓ,nc
t)0≤t<Nas
their strategy and the market maker ℓ=0 chose (rnc
t,vnc
t)0≤t<N. Then, the continuation
strategies (Kℓ,nc
t,Xℓ,nc
t,Cℓ,nc
t)t≥Nfor each player ℓ∈Lare clearly feasible, provided the
market maker chooses (rnc
t,vnc
t)t≥N. Moreover, these strategies satisfy the optimality
conditions from Lemma 3, implying that each player will find it advantageous to stick
to these strategies, provided everybody else does. This establishes the following main
result of this section.
14
Proposition 1
Let
N≥0
be arbitrary and suppose all players
ℓ∈L0
followed the strategies of the
non-cooperative equilibrium up to time
t=N−1
. Then, the continuation strategies
(Kℓ,nc
t,Xℓ,nc
t,Cℓ,nc
t)t≥N
for each player
ℓ∈L
and
(rnc
t,vnc
t)t≥N
for player
ℓ=0
constitute a
non-cooperative (Nash) equilibrium at stage
t=N
.
We remark that a stronger version of this result would be a Markov-perfect equilibrium
where all strategies are generated by time-invariant decision rules defined on a suitable
state space. This concept is widely used in game-theoretic studies of the climate prob-
lem, see, e.g., (Harstad 2016). In our setting, however, with time-varying production
functions (1) due to population growth, technological progress, etc., it is in general not
possible to obtain such a time-invariant structure. Therefore, the subgame-perfectness
demonstrated in this section appears to be the strongest result possible in our setting.
4.4 Regionally optimal climate policy
Implementing the non-cooperative solution
Returning to the decentralized equilibrium discussed in Section 3, we seek to imple-
ment the non-cooperative solution as a decentralized equilibrium. This requires de-
termining the climate tax policy (τℓ
t)t≥0for each region ℓsuch that the equilibrium
allocation and price system satisfy A∗=Anc and P∗=Pnc. Formally, this can be accom-
plished by ensuring the optimality conditions determining the non-cooperative solution
coincide with the equilibrium conditions from Definition 1.
Regionally optimal climate tax
Along the non-cooperative equilibrium, the regional lifetime budget constraint (27)
holds with equality. Using the definition of profits in (11) and the form of transfers
(9), we obtain
(1−Dℓ
t)Fℓ
t(Kℓ
t,Xℓ
t)−rtKℓ
t−vtXℓ
t=Πℓ
t+Tℓ
t.(38)
Using this in (27) we obtain the consumer’s lifetime budget constraint (18) with equality
as required. Furthermore, invoking Lemma 2, we see that the Euler equations (19) are
automatically satisfied and so is the optimality condition (12a) with respect to capital
input. The price sequence Pnc satisfies the Hotelling rule (15) by assumption. Finally,
comparing the optimality conditions (12b) and (29) wit respect to fossil fuel inputs, we
can see that implementing the regionally optimal solution to (28) requires choosing the
regional climate tax according to the rule
τℓ
t=
∞
X
n=0
βn³Cℓ
t+n/Cℓ
t´−σδnγℓYℓ
t+n.(39)
Hence, if taxes are determined by the tax rule (39), all conditions determining the de-
centralized and non-cooperative solution coincide. We state the previous result formally
in the following theorem.
15
Theorem 1
Suppose each region
ℓ∈L
chooses a climate tax policy
(τℓ
t)t≥0
according to the rule
(39) and consumers in region
ℓ
receive transfers determined by (9). Then, the induced
equilibrium allocation and prices satisfy
A∗=Anc
and
P∗=Pnc
.
By Lemma 1, equilibrium consumption satisfies Cℓ
t=µℓCtpermitting to replace the
discount factor in (39) by aggregate consumption. This yields (39) in equivalent form
τℓ
t=
∞
X
n=0
βn³Ct+n/Ct´−σδnγℓYℓ
t+n.(40)
Finally, along a balanced growth path where regional output and consumption grow at
constant and identical rate g, (39) reduces to
τℓ
t=γℓYℓ
t
∞
X
n=0³β(1+g)1−σ´nδn.(41)
The formula (41) is an extension of the main result in Golosov et al. (2014) who derive
the globally optimal climate tax in closed form and show that it can be expressed as a
constant fraction of output determined by climate parameters and the discount factor.
5 Equilibrium under Full Cooperation
5.1 The efficient solution
The global planning problem
This section compares the previously derived non-cooperative solution to the globally
efficient solution that obtains under full cooperation between regions. Intuitively, the
non-cooperative solution is what the world is doing while the cooperative solution is
what it should be doing to combat climate change.
Hillebrand & Hillebrand (2019) show that the efficient solution maximizes utility of a
world representative consumer on the set of feasible allocations determined by techno-
logical constraints and the climate system. Since the present framework differs from
the one used in Hillebrand & Hillebrand (2019), we briefly restate this problem here.
Consider a global planner choosing an aggregate allocation A=((Kℓ
t,Xℓ
t)ℓ∈L,Ct)t≥0of
production factors for each region along with aggregate global consumption. The term
’aggregate’ signified by the bar notation reflects the fact that the allocation Aspecifies
only aggregate consumption, but no its distribution across regions. Note however that
this aggregation only refers to consumption while production inputs are specified sepa-
rately for all regions.
The planner takes into account the impact of emissions on climate damage via (23), the
global resource constraint (21), and the feasibility constraint
Ct≤X
ℓ∈L³(1−Dℓ
t)Fℓ
t(Kℓ
t,Xℓ
t)−cxXℓ
t−Kℓ
t+1´for all t=0,1,2,... (42)
16
Furthermore, the initial capital allocation (Kℓ
0)ℓ∈Lis subject to the initial condition
X
ℓ∈L
Kℓ
0≤Ks
0:=X
ℓ∈L
Kℓ,s
0.(43)
Based on these constraints, the global planner maximizes utility of a fictitious world
representative consumer who consumes aggregate consumption Ctin each period t.
Formally, the global planning problem can be stated as:
max
((Kℓ
t,Xℓ
t)ℓ∈L,Ct)t≥0nU((Ct)t≥0)¯¯¯(21),(23),(42),(43) hold, Kℓ
t,Xℓ
t,Ct≥0 for all t≥0,ℓ∈Lo.
(44)
We denote the solution to (44) by Aeff =((Kℓ,eff
t,Xℓ,eff
t)ℓ∈L,Ceff
t)t≥0and refer to it as the
efficient aggregate allocation. The following lemma characterizes the efficient solution
formally. The proof can be found in Section A.2 in Appendix A.
Lemma 4
If the aggregate allocation
A=((Kℓ
t,Xℓ
t)ℓ∈L,Ct)t≥0
satisfies the feasibility conditions (42)
and (43) with equality for all
t=0,1,2,...
, the intratemporal efficiency conditions
(1−Dℓ
t)∂KFℓ
t(Kℓ
t,Xℓ
t)=(1−Dk
t)∂KFk
t(Kk
t,Xk
t) (45a)
(1−Dℓ
t)∂XFℓ
t(Kℓ
t,Xℓ
t)=(1−Dk
t)∂XFk
t(Kk
t,Xk
t) (45b)
for all
k,ℓ∈L
, the intertemporal efficiency conditions
βC−σ
t+1
C−σ
t
(1−Dℓ
t+1)∂KFℓ
t+1(Kℓ
t+1,Xℓ
t+1)=1 (45c)
βC−σ
t+1
C−σ
t³(1−Dℓ
t+1)∂XFℓ
t+1(Kℓ
t+1,Xℓ
t+1)−cx−ˆ
τt+1´=(1−Dℓ
t)∂XFℓ
t(Kℓ
t,Xℓ
t)−cx−ˆ
τt(45d)
for all
ℓ∈L
with damages determined by (23) and
ˆ
τt:=
∞
X
n=0
βn³Ct+n/Ct´−σδnX
k∈L
γkYk
t+n
for all
t=0,1,2,... (46)
as well as the transversality condition
limT→∞ βTC−σ
TPℓ∈LKℓ
T+1=0
and the resource
constraint (21), then it is a solution to (44).
Intuitively, the efficient solution equates marginal products across all regions in each
period and also intertemporally. Equation (45a) implicitly determines a (shadow) capi-
tal return ˆ
rtwhich ensures an efficient world capital allocation. This return can be used
in (45c) ensuring intertemporally efficient consumption and capital accumulation. Fur-
thermore, condition (45b) allocates fossil fuel usage efficiently to equate marginal prod-
ucts across world regions. This property can be used to define a shadow resource price
ˆ
vt:=(1 −Dℓ
t)∂XFℓ
t(Kℓ
t,Xℓ
t)−ˆ
τtcorresponding to the marginal product of the resource
in production net of the total discounted damage captured by ˆ
τt. With this definition,
equation (45d) corresponds to the Hotelling rule (15) ensuring efficient extraction of the
resource over time. In particular, if the resource constraint (21) is non-binding at the
optimal solution, then ˆ
vt=cx, as shown in the appendix.
17
5.2 Implementing the efficient solution
We now explore conditions under which the decentralized allocation A∗is efficient.
Formally, the equilibrium allocation is efficient if and only if for all t=0,1,2,...
Kℓ∗
t=Kℓ,eff
tand Xℓ∗
t=Xℓ,eff
tfor all ℓ∈Land X
ℓ∈L
C∗
t=Ceff
t.(47)
Climate tax policies under which (47) holds will be termed efficient and denoted by
(τeff
t)t≥0. Note that such a policy will necessarily be uniform across all regions. We now
show that the efficient climate tax policy takes the following explicit form
τeff
t:=
∞
X
n=0
βn³Ceff
t+n/Ceff
t´−σδnX
k∈L
γkYk,eff
t+nfor all t=0,1,2,.... (48)
A similar result is derived in Hillebrand & Hillebrand (2019) in a slightly different
setup. To prove this claim, we must show that the equations from Lemma 4 generating
the efficient allocation are satisfied at equilibrium if taxes are chosen based on (48).
This result holds independently of how transfers (Tℓ
t)t≥0are determined.
The final sector’s optimality condition (12a) implies equalization of capital returns (45a).
Further, under uniform taxation, optimality condition (12b) implies that marginal prod-
ucts of fossil fuels equalize as required by (45b). Further, the Euler equation (19) can be
aggregated and combining it with (12a) to replace the return on capital by its marginal
product yields the aggregate Euler equation (45c).6Further, solving (12b) for vtand
substituting the result into the Hotelling rule (15) using the form of taxes (48) and
replacing again the capital return rt+1by its marginal product based on (12a) gives
(45d). At equilibrium, the capital market clearing condition (20) for t=0, the world
resource constraint (21), and the commodity market equilibrium condition (22) hold di-
rectly. Finally, using the market clearing condition (20) of the capital market individual
transversality conditions can be aggregated based on (45c) to obtain the aggregate ver-
sion in Lemma 4.7
We state the previous result formally in the following theorem.
Theorem 2
Suppose all regions choose uniform climate taxes given by (48). Then, the equilibrium
allocation
A∗=((Kℓ∗
t,Xℓ∗
t,Cℓ∗
t)ℓ∈L)t≥0
is efficient, i.e., (47) holds for all
t=0,1,2,...
6We remark that this aggregation property of the Euler equation is key for the separability result from
Hillebrand & Hillebrand (2019) on which the present section is based. It requires the form of preferences
(7) and equalization of capital returns via an international capital market.
7For this, one can use the (arbitrary but given) sequence of transfers (Tℓ
t)t≥0and assume an arbitrary
extraction sequence (Xℓ,s
t)t≥0consistent with (3) and satisfying Pℓ∈LXℓ,s
t=Pℓ∈LXℓ
tfor all t. Then, one
can use (17) to recursively obtain the sequence (Kℓ,s
t+1) satisfying the capital market clearing condition
in (20) and the transversality condition limt→∞ Kℓ,s
t+1βtu′(Cℓ,∗
t)/u′(Cℓ,∗
0)=limt→∞ Kℓ,s
t+1βtu′(C∗
t)/u′(C∗
0)=0.
Aggregating this condition over all regions using the capital market clearing condition in (20) then gives
the result.
18
The distribution (µℓ)ℓ∈Lof aggregate consumption (Ceff
t)t≥0depends on whether there
are transfers between regions. If there is no re-distribution, transfers are determined by
(9) and consumption shares follow from equation (24). In a cooperative setup, however,
transfers can be used to incentivize regions to implement the globally rather than the
regionally optimal tax. This issue will be studied next.
5.3 Incentive-compatible transfer policies
Welfare gain under cooperation
Denote the consumption distribution along the non-cooperative equilibrium by (µℓ,nc)ℓ∈L
and define aggregate consumption as Cnc
t:=Pℓ∈LCℓ,nc
t. The aggregate allocation Anc =
((Kℓ,nc
t,Xℓ,nc
t)ℓ∈L,Cnc
t)t≥0associated with the non-cooperative equilibrium satisfies all
the constraints in (44). Therefore, the implied total welfare is less than at the efficient
equilibrium, i.e.,8
Unc :=U((Cnc
t)t≥0)<U((Ceff
t)t≥0)=:Ueff.(49)
We now explore whether this efficiency gain can be distributed such that each regionis
better-off nder efficient taxation relative to the non-cooperative equilibrium.
Pareto-improving consumption shares
To induce a Pareto-improvement under efficient taxation relative to the non-cooperative
equilibrium, each region ℓ∈Lmust receive a consumption share µℓsuch that
U((µℓCeff
t)t≥0)≥U((µℓ,ncCnc
t)t≥0).(50)
Exploiting the form of utility (7), we can determine explicit lower bounds µℓ
crit for this
share such that (50) holds whenever µℓ≥µℓ
crit. These shares are characterized in the
following result. The proof can be found in Section A.3 in Appendix A.
Lemma 5
Under efficient taxation of all regions, suppose each region
ℓ∈L
receives a consumption
share
µℓ≥µℓ
crit :=
µℓ,nc ³Unc/Ueff´1
1−σ
if
σ>0,σ6= 1
µℓ,nc ·e−(1−β)(Ueff−Unc)
if
σ=1.
(51)
Then, the induced disaggregated efficient allocation
A=((Kℓ,eff
t,Xℓ,eff
t,µℓCeff
t))t≥0
Pareto
improves the non-cooperative allocation
Anc =((Kℓ,nc
t,Xℓ,nc
t,Cℓ,nc
t))t≥0
.
8The strict inequality can be inferred from Theorems 1 and 2 by observing that taxes along the non-
cooperative equilibrium are determined by (39) rather than by (48). Therefore, ¯
Anc cannot be efficient.
19
Noting that µℓ
crit <µℓ,nc for all ℓ∈L, condition (51) holds in particular if µℓ=µℓ,nc, i.e.,
each region attains the same consumption share as in the non-cooperative equilibrium.
Pareto-improving transfer shares
We now explore how transfers between regions can be designed such that (51) holds.
For this purpose, we assume that regions pool their tax revenue and agree on a trans-
fer policy (θℓ)ℓ∈Ldetermining transfer payments as in (10). Let Aeff be the efficient
allocation and define the induced equilibrium prices Peff =(reff
t,veff
t)t≥0by setting
reff
t:=(1−Dℓ,eff
t)∂KFℓ
t(Kℓ,eff
t,Xℓ,eff
t) and veff
t:=(1−Dℓ,eff
t)∂XFℓ
t(Kℓ,eff
t,Xℓ,eff
t) (52)
for t=0,1,2,... with climate damages Dℓ,eff
tdetermined by (23). Note that the quan-
tities in (52) are well-defined., i.e., independent of the regional index ℓdue to (45a)
and (45b) and satisfy the Hotelling rule (15) due to (45d). Denote by (qeff
t)t≥0the in-
duced Arrow-Debreu prices defined by (13) and let Teff
t:=Pℓ∈Lτeff
tXℓ,eff
tdenote total tax
revenue and Πℓ,eff
t:=Yℓ,eff
t−reff
tKℓ,eff
t−veff
tXℓ,eff
tfinal profits in region ℓat time t. As
before, write the induced lifetime profit incomes as Πℓ,eff :=P∞
t=0qeff
tΠℓ,eff
tand total tax
revenue Teff :=P∞
t=0qeff
tTeff
t. Then, we can use the result from Lemma 1 and condition
(24) to establish the following one-to-one relation between consumption shares (µℓ)ℓ∈L
and transfer shares (θℓ)ℓ∈Lalong the efficient equilibrium:
µℓ=
Πℓ,eff +reff
0Kℓ,s
0+(veff
0−cx)Rℓ
0+θℓTeff
Πeff +reff
0Ks
0+(veff
0−cx)R0+Teff .(53)
Solving (53) for θℓand combining it with the result from Lemma 5 gives the following
complete characterization of incentive-compatible transfer policies inducing a Pareto-
improvement over the non-cooperative solution.
Theorem 3
Define
(µℓ,crit)ℓ∈L
as in Lemma 5. Suppose each region
ℓ∈L
chooses efficient taxes
determined by (61) and receives a constant share
θℓ
of global tax revenue satisfying
θℓ≥θℓ
crit :=µℓ,crit +
µℓ,crit ³Πeff +reff
0Ks
0+(veff
0−cx)R0´−³Πℓ,eff +reff
0Kℓ,s
0+(veff
0−cx)Rℓ
0´
Teff .
Then, all regions are better-off relative to the non-cooperative equilibrium.
Observe that the minimal transfer shares defined in Theorem 3 satisfy
X
ℓ∈L
θℓ
crit =X
ℓ∈L
µℓ,crit −¡Pℓ∈Lµℓ,crit −1¢³Πeff +reff
0Ks
0+(veff
0−cx)R0´
Teff <1.(54)
Thus, the set of transfer policies (θℓ)ℓ∈Lsatisfying the conditions of Theorem 3 is non-
empty.
20
6 Partial Cooperation and Coalitions
6.1 A cooperative setup
The previous sections compared the allocation under full cooperation by all members of
Lto the case where there is no cooperation at all and each region solves its own planning
problem (28). Clearly, there are many intermediate scenarios where some regions join
forces and coordinate their climate policies by forming coalitions.
Coalitions and aggregation
To describe a scenario with coalitions formally, let P(L) be the power set of Lconsisting
of all subsets of L. Any non-empty subset L′∈P(L) of Lwill be referred to as a coalition.
The case L′=Lis called the grand coalition consisting of all regions. Let L′∈P(L) be
an arbitrary coalition. In what follows, we index variables referring to the coalition as
a whole by a superscript L′. For variables referring to individual members, we continue
to use the regional index ℓ∈L′. For example, RL′
0:=Pℓ∈L′Rℓ
0is the initial stock of fossil
fuels owned by the coalition L′while (Xℓ
t)t≥0denotes fossil fuel consumption in member
region ℓ∈L′. Based on this convention, a coalitional variable xL′obtains as the sum of
the list of individual variables (xℓ)ℓ∈L′, i.e., xL′=Pℓ∈L′xℓ.
The coalitional planning problem
Suppose the members of coalition L′join forces by aggregating their resource stocks and
capital endowments. Summing (27) over all regions in L′one obtains the constraint
∞
X
t=0
qtÃX
ℓ∈L′³(1−Dℓ
t)Fℓ
t(Kℓ
t,Xℓ
t)−rtKℓ
t−vtXℓ
t´−CL′
t!+r0KL′,s
0+(v0−cx)RL′
0≥0.(55)
Defining the complement −L′:=L\L′, regional climate damages (23) can be written as
Dℓ
t=1−expµ−γℓ∞
X
n=0
δn³XL′
t−n+X−L′
t−n´¶for ℓ∈L′(56)
Assume that coalition L′maximizes the total utility attained by the aggregated con-
sumption stream (CL′
t)≥0. This approach is based on the separation principle established
in Hillebrand & Hillebrand (2019) permitting to first determine an efficient solution by
maximizing aggregate utility and then distributing consumption among coalition mem-
bers based on some suitable weighting scheme. The distribution of consumption among
coalition members corresponds to an imputation in the cooperative game defined below.
Based on the previous idea, the coalitional planning problem can be stated as follows:
max
((Kℓ
t,Xℓ
t)ℓ∈L′,CL′
t)t≥0nU((CL′
t)t≥0)¯¯¯(55),(56) hold, Kℓ
t,Xℓ
t,CL′
t≥0 for all t=0,1,2,...o.(57)
As in the regional problem (28), emission (X−L′
t)t≥0of all other regions and global prices
P=(rt,vt)t≥0are taken as given in the decision (57). Further, the coalitional planer
21
takes into account the impact of emission of coalition members on damages (56) in each
member region ℓ∈L′.
Solution to the coalitional planning problem
It is again straightforward to solve the coalitional problem (57) using Lagrangean meth-
ods. The proof of the following result parallels the one of Lemma 2 and is omitted.
Lemma 6
Let prices
P=(rt,vt)t≥0
and emissions
(X−L′
t)t≥0
of non-coalition members be given. If
the allocation
((Kℓ
t,Xℓ
t)ℓ∈L′,CL′
t)t≥0
solves conditions (12a) and
(1−Dℓ
t)∂XFℓ
t(Kℓ
t,Xℓ
t)=vt+
∞
X
n=0
βn¡CL
t+n/CL
t¢−σδnX
k∈L′
γkYk
t+n(58)
for all
ℓ∈L′
with
Dℓ
t
determined by (56) and the Euler equation
CL′
t+1=CL′
t(βrt+1)1
σ(59)
for all
t=0,1,2,...
and satisfies (55) with equality, then it is a solution to problem (57).
Note that (58) corrects the marginal product of fossil fuel in production by the dis-
counted future climate damages of all coalition members.
6.2 Coalitional equilibrium
Classes of coalitions
Now let L={L1,...,LN}⊂P(L) be a partition of the set of all regions Linto 1 ≤N≤L
coalitions, i.e., ∪N
n=1Ln=Land Ln∩Lm= ; for all n6= mwhere n,m∈{1,...,N}. Following
Yi (1997), we call Lacoalition structure. Suppose the members of each coalition L′∈L
solve a coalitional planning problem of the form (57). The market maker ℓ=0 acts as
in the previous section to enforce market clearing. Consistency of coalitional decisions
leads to the following definition of a coalitional equilibrium.
Definition 3
Let
L⊂P(L)
be a coalition structure. A coalitional equilibrium is an allocation
¯
A=³³(Kℓ
t,Xℓ
t)ℓ∈L′,CL′
t´t≥0´L′∈L(60)
and a price system
P=(rt,vt)t≥0
such that the following holds:
(i) For each
L′∈L
, the allocation
((Kℓ
t,Xℓ
t)ℓ∈L′,CL′
t)t≥0
solves the coalitional planning
problem (57) given prices
P
and emissions
(X−L′
t)t≥0
of all other regions..
(ii) Prices
P
chosen by player
ℓ=0
satisfy the Hotelling rule (15) and
(v0,r0)
and
qt
defined by (13) are solutions to the decision problems (30a)-(30c) for all
t≥0
.
22
When we want to emphasize the dependence of coalitional equilibrium on the coali-
tion configuration, we write ¯
ALand PL. The bar-notation is used to emphasize that
the distribution of consumption within a coalition is not determined and subject to an
imputation to be bargained over by coalition members.
6.3 Decentralization of coalitional equilibrium
Based on the optimality condition (58), it is now straightforward to extend the results
from Theorem 1 to the following theorem describing the climate policy under which the
coalitional allocation is decentralized.
Theorem 4
Let
L⊂P(L)
be a coalition structure. For each coalition
L′∈L
, suppose all its mem-
bers
ℓ∈L′
set taxes according to the rule
τℓ
t=τL′
t:=
∞
X
n=0
βnÃCL′
t+n
CL′
t!−σ
δnX
k∈L′
γkYk
t+n
for all
ℓ∈L′.(61)
Then, the coalitional equilibrium is decentralized.
On a balanced growth path where aggregate consumption and output in region of the
coalition grow at constant rate g, the optimal tax formula (61) reduces again to
τℓ
t=τL′
t=X
k∈L′
γkYk
t
∞
X
n=0¡β(1+g)1−σ¢nδnfor all ℓ∈L′.(62)
Two special coalition structure are of particular interest. The first one is the coalition
L={{1},...,{L}} where each region acts individually. In this case, Definition 3 coincides
with that of a non-cooperative equilibrium from Definition 2 and so ¯
AL∼
=Anc.9
A second scenario is where L={L}corresponding to the grand coalition consisting of
all regions. For this case, the tax formula (61) coincides with the efficient solution (48).
Therefore, by virtue of Theorem 2 ¯
AL∼
=¯
Aeff, i.e., the induced allocation is efficient.
We state these two insights as the following and final main result.
Proposition 2
(i) If
L={{1},...,{L}}
, then
¯
AL∼
=Anc
.
(ii) If
L={L}
, then
¯
AL∼
=¯
Aeff
.
We remark that all results from Section 5.3 on optimal transfer policies designed to
induce cooperation among regions carry over to the case with partial cooperation. In
fact, such polcies can also be derived in closed form based on the same principles as in
the non-cooperative case.
9Due to the slightly different arrangement of terms in ¯
ALand Anc, we write ¯
AL∼
=Anc and ¯
AL∼
=¯
Aeff
below to mean that the two allocations are isomorphic rather than directly identical.
23
7 Quantitative results
7.1 Regional structure and coalition scenarios
This section illustrates and quantifies our theoretical results based on calibrated pa-
rameter values chosen to match selected empirical targets. Specifically, we distinguish
the non-cooperative and fully cooperative solution as well as two additional scenarios
with some key regions not joining the global climate agreement.
Regions
Setting L=10 we distinguish the following region displayed in Table 1. The number
of regions is small enough to allow for a compact presentation of the results and large
enough to distinguish the main players in past climate agreements.
Table 1: Regions in the simulation model
Region Label Index ℓRegion Label Index ℓ
United States USA 1 India IND 6
OECD Europe OEU 2 Russia RUS 7
Australia & New Zealand ANZ 3 Brazil BRA 8
Other High Income OHI 4 Developing countries DEV 9
China CHN 5 Low Income Countries LIC 10
Time structure
One time period tin our model represents ten years which is a standard choice in the
literature. The initial model period t=0 represents the years 2010-2019 and is referred
to as the baseline period 2015. Subsequent periods representing years 2020 −2029,
2030−2039, etc. are referred to by their midpoints 2025,2035, etc. Flow variables such
as production output or emissions are generally aggregated over the entire period while
stocks such as capital or atmospheric carbon usually refer to the beginning of the period.
Coalition scenarios
As for the structure of coalitions, we distinguish four scenarios detailed in the following
table. Scenario 1 is the efficient benchmark which is opposed to the completed absence
Table 2: Coalition structures in our simulation
Scenario Description Coalition structure
1 Full non-cooperation L={{ℓ}|ℓ∈L}={{1},{2},...,{10}}
2 Full cooperation (grand coalition) L={L}={{1,. ..,10}}
3 Grand coalition except US L={L\{1},{1}}
4 Grand coalition except China L={L\{5},{5}}
24
of cooperation in Scenario 2. Scenarios 3 and 4 correspond to the cases where either
the US or China decide not to cooperate. This allows us to evaluate and quantify the
relative importance of these players for a successful global climate agreement.
7.2 Functional forms and parameters
Production sector
Similar to Hassler, Krusell & Olovsson (2021) we assume that the production function
Fℓ
t:R2
+−→ R+in (1) is of the form
Fℓ
t(K,X)=·κ³Kα(Qℓ
N,tNℓ
t)1−α´ε−1
ε+(1−κ)(Qℓ
X,tX)ε−1
ε¸ε
ε−1,0<κ<1,0<α<1,ε>0.
(63)
Here, Nℓ
tand Qℓ
N,tdenote population size and labor efficiency in region ℓat time twhile
Qℓ
X,trepresents energy efficiency. Parameter εcontrols the elasticity of substitution
between fossil energy and the capital-labor aggregate.
Variables Qℓ
N,tand Qℓ
X,tcapture labor-augmenting and energy-augmenting technical
change, respectively. They both grow at constant exog