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Optimal Climate Policies in a Dynamic Multi-Country Equilibrium Model *

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Abstract

This paper develops a dynamic general equilibrium model with an arbitrary number of different regions to study the economic consequences of climate change under alternative climate policies. Regions differ with respect to their state of economic development, factor endowments, and climate damages and trade on global markets for capital, output, and exhaustible resources. Our main result derives an optimal climate policy consisting of an emissions tax and a transfer policy. The optimal tax can be determined explicitly in our framework and is independent of any weights attached to the interests of different countries. Such weights only determine optimal transfers which distribute tax revenues across countries. We infer that the real political issue is not the tax policy required to reduce global warming but rather how the burden of climate change should be shared via transfer payments between different countries. We propose a simple transfer policy which induces a Pareto improvement relative to the Laissez faire solution. A calibrated example quantifies Pareto-improving transfers between rich and poor countries. JEL classification: E61, H21, H23, Q43, Q54
Optimal Climate Policies in a Dynamic
Multi-Country Equilibrium Model
Elmar HillebrandMarten Hillebrand
November 8, 2018
Final version
Abstract
This paper develops a dynamic general equilibrium model with an arbitrary num-
ber of different regions to study the economic consequences of climate change under
alternative climate policies. Regions differ with respect to their state of economic
development, factor endowments, and climate damages and trade on global mar-
kets for capital, output, and exhaustible resources. Our main result derives an
optimal climate policy consisting of an emissions tax and a transfer policy. The
optimal tax can be determined explicitly in our framework and is independent of
any weights attached to the interests of different countries. Such weights only de-
termine optimal transfers which distribute tax revenues across countries. We infer
that the real political issue is not the tax policy required to reduce global warm-
ing but rather how the burden of climate change should be shared via transfer
payments between different countries. We propose a simple transfer policy which
induces a Pareto improvement relative to the Laissez faire solution. A calibrated
example quantifies Pareto-improving transfers between rich and poor countries.
JEL classification: E61, H21, H23, Q43, Q54
Keywords: Multi-region dynamic general equilibrium model; Climate change; Optimal
climate tax; Optimal transfer policy; Pareto improvement.
First version: Dec. 2015, this version: November 8, 2018.
Acknowledgements. We would like to thank Martin Barbie, Inge van den Bijgaart, Reyer Gerlagh,
Philipp Harms, Tomoo Kikuchi, Carles Ma˜o, Oliver Saffran, Hagen Schwerin, and Klaus W¨alde for
fruitful discussions and seminar participants at various conferences including the 2016 Annual Meeting
of the Verein f¨ur Socialpolitik in Augsburg and the 2017 European Meeting of the Econometric Society
in Lisbon for their comments. We would also like to thank our two referees and the editor, Laura
Veldkamp whose comments helped us to substantially improve the paper.
EEFA Research Institute, Muenster, Germany, email: e.hillebrand@eefa.de
Department of Economics, Goethe University Frankfurt, Theodor-W.-Adorno-Platz 3, 60323 Frank-
furt am Main, Germany, email: Hillebrand@em.uni-frankfurt.de (corresponding author).
Introduction
On December 12, 2015, 195 countries joined the Paris Agreement to strengthen the
global response to climate change. Its central aim is to keep the increase in temperature
relative to pre-industrial level below two degrees Celsius until the end of the century.
Individual commitments how to achieve this goal, however, are non-binding and involve
voluntary climate policies chosen on the national level, so called ’Nationally Determined
Contributions’. This outcome reflects the complexity of any negotiations about joint
climate policies which are shaped by the bargaining power of individual countries and
the trade-offs between their political interests.
These observations suggests that climate change is inherently an economic problem and
understanding the incentives for individual regions to implement a given climate policy
is key for the success of any climate agreement. Conceptually, this calls for a theoretical
framework which incorporates the trade-off between the interests of different regions
and permits to analyze the effects of alternative climate policies at the regional level. In
this paper, we develop such a multi-region model and derive an optimal climate policy
that each region has an incentive to implement.
To successfully combat climate change, each country must contribute to the common
goal by reducing its emissions. A major obstacle to determine these individual con-
tributions is regional heterogeneity. For instance, regions differ considerably in their
dependence on fossil fuels and the mix of technologies they use to produce energy in-
puts. Thus, reducing emissions is more costly for some regions than for others. In
addition, various other differences such as state of economic development, future growth
prospects, or vulnerability to climate damages play an important role in climate negoti-
ations. This raises a first question that we will address in this paper: How much should
each region contribute to the common objective of reducing emissions?
In market economies, any equilibrium allocation is the outcome of decentralized decisions
made by firms and consumers who respond to incentives and taxes set by governments.
One way to reduce emissions is to levy a tax on emissions. Thus, a second questions
to be addressed is: Which tax policy should each region adopt to achieve its desired
reduction in emissions?
For a climate agreement to be successful, each region must have an incentive to imple-
ment the proposed policy. Thus, a third question to be answered is: How can climate
policies be designed such that each region has an incentive to join the climate agreement?
The present paper addresses these questions in a multi-region framework which incor-
porates several sources of regional heterogeneity which play a crucial role in climate
negotiations. Our model builds on the single-region framework in Golosov et al. (2014)
to which we add several new dimensions. First, we adopt a multi-region structure fea-
turing an arbitrary number of different regions. This is clearly required for our analysis.
1
Second, we devise a different model of the production process which distinguishing ex-
plicitly the resource stage at which fossil fuels are extracted and the energy stage at
which they are used to produce energy such as electricity or heat. This distinction is
important because empirically regions differ considerably not only in their dependence
on fossil fuels but also in the mix of technologies to produce energy outputs. Modelling
the energy stage explicitly allows us to capture these differences and study how climate
policies induce transitions from dirty to clean technologies in each region. Finally, we
allow for climate damages to differ across regions. This is another important source
of heterogeneity, notably because poorer countries tend to be more vulnerable to cli-
mate change. The economic part is complemented by a climate model describing how
emissions evolve in the atmosphere and damage the economy.
With these features, our model falls into the class of integrated assessment models which
incorporate the full interactions between climate variables and the economic production
process. In the literature, a large class of these models is based on the DICE framework
pioneered by Nordhaus (1977) and its multi-region extension, the RICE model developed
in Nordhaus & Yang (1996) and refined in Nordhaus & Boyer (2000). A typical feature
of these models stressed in Hassler et al. (2016) is that solutions are derived as planning
problems without explicit market structures and prices. Thus, these models make only
limited use of dynamic general equilibrium theory which confines the class of policies that
can be analyzed. In addition, the RICE-framework in conjunction with the employed
methodology entails strong restrictions on trade between regions.
The model developed in Golosov et al. (2014) takes full advantage of dynamic general
equilibrium theory with explicitly defined markets and price formation.1Hassler &
Krusell (2012) provide an extension to a multi-region framework which distinguishes oil-
producing and oil-consuming countries. To preserve analytical tractability, they impose
strong restrictions on trade between regions which are only allowed to trade oil which
is the only fossil fuel.2A major difference of our model to Hassler & Krusell (2012) is
that we allow for trade between regions and intertemporal borrowing and lending on an
international capital market. In fact, this assumption will be key for our results.
The general contribution of our analysis to the literature are the answers to the questions
posed above. First, we show that there is a unique efficient allocation which determines
the optimal level of emissions for each region. Second, we show that this efficient allo-
cation can be implemented by a uniform tax on emissions for which we derive a closed
form solution. Third, we devise a transfer scheme which distributes the revenue from
taxation such that each region has an incentive to implement the optimal emissions tax.
1Recent models based on the same paradigm are Barrage (2017), Gerlagh & Liski (2016, 2018), or
Rezai & van der Ploeg (2015, 2016) who extend the model in Golosov et al. (2014) in various directions.
2A related class of models studies climate policies in a two-region setting, e.g., Daubanes & Grimaud
(2010), Bretschger & Suphaphiphat (2014), or van den Bijgaart (2017). The model from Hassler &
Krusell (2012) is further extended in Hassler et al. (2017) to include directed technical change.
2
A crucial feature of our model to obtain these results is that the efficiency problem of
determining an optimal emissions tax can strictly be separated from the distributional
issue how global tax revenue should be shared via transfers. This separability result
requires a standard restriction on consumer preferences combined with borrowing and
lending between regions on a frictionless capital market. Static versions of this result
first proved by Bergstrom & Cornes (1983) are well-known in the public goods literature.
To the best of our knowledge, we are the first to extend it to a dynamic setting and
apply it to the climate problem. This is the methodological contribution of our paper.
The bargaining power of regions in our model is represented by a weighting scheme which
aggregates welfare in each region to a single utility index. An important consequence of
separability is that the optimal emissions tax does not depend on this aggregation. Thus,
determining the optimal carbon tax involves no trade-off between political interests and
regions could directly agree on this policy. Based on this finding, our analysis suggests
that the major political issue is how tax revenue should be shared via transfer payments.
Only the choice of a transfer scheme should therefore be the subject of negotiations.
To determine transfers in accordance with incentive constraints, our final theoretical
result proposes a simple transfer policy under which each region is strictly better off
relative to the Laissez faire scenario where no measures against climate change are taken.
The proposed policy thus satisfies the property of individual rationality discussed, e.g.,
in Eyckmans & Tulkens (2003)) which seems a minimal requirement for the optimal tax
policy to be implemented by each region. We present a calibrated numerical example
to quantify the range and size of Pareto-improving transfers between rich and poor
countries. The results show that rich countries can afford to transfer initially 1.6% and
subsequently up to 2.3% of their GDP to poor countries and would still benefit from a
global agreement to implement the optimal tax policy.
The paper is organized as follows. Section 1 introduces the model. The decentralized
equilibrium solution under different climate policies is studied in Section 2. Section
3 studies optimal allocations obtained as solutions to a planning problem. Optimal
climate policies which implement the optimal solution as an equilibrium allocation are
studied in Section 4. A calibrated example presented in Section 5 quantifies our results.
Extensions of our basic framework and robustness of our results are discussed in Section
6. Section 7 concludes, mathematical proofs are placed in the appendix.
1 The Model
1.1 World economy
The world economy is divided into L1 regions, indexed by L:= {1,...,L}.
Each region Lpursues its own interests and takes autonomous political decisions.
3
Although each region typically represent unions or groups of different countries, we will
nevertheless refer to region as a country. Regions are geographically or institutionally
separated, which imposes certain restrictions on trade between them.
The production process in each region Ldecomposes into three stages. The final sec-
tor produces a consumable output commodity based on a set of inputs including energy
goods and services. The second stage consists of a collection of energy sectors which
produces these goods and services based either on renewable or exhaustible resources.
The third stage is represented by the resource sectors which extract the domestic stock
of exhaustible resources.
These different production stages together with a global climate model and a description
of the consumption sector in each region constitute the main building blocks of our model
which will be described in detail in the following sections.
1.2 Production sectors
Time is discrete and indexed by t∈ {0,1,2,...}. There are I+ 1 production sectors in
each region Lwhich are identified by the index iI0:= {0,1,...,I}. Sector i= 0
is the final sector while iI:= {1,...,I}identifies the different energy sectors. Each
sector iI0consist of a single representative firm which employs labor N
i,t 0 and
capital K
i,t 0 as production factors in period t.
Final sector
Sector i= 0 in region Lproduces final output in period tusing the technology
Y
t= (1 D
t)Q
0,tF0(K
0,t, N
0,t,(E
i,t)iI).(1)
Here, (E
i,t)iIis as collection of energy inputs used in addition to labor and capital in
production. The term Q
0,t >0 in (1) is an exogenous, possibly time- and region-specific
productivity parameter which is diminished by damages due to climate change. The
latter is measured by a damage index D
t[0,1[ which will be a function of total CO2
concentration in the atmosphere to be specified below.
Energy sectors
Energy is supplied by sectors iI. Their outputs should be broadly interpreted as
energy goods like electricity and heat or services like transportation. We distinguish
’exhaustible’ and ’renewable’ energy sectors depending on whether they base their pro-
duction on an exhaustible resource like coal, oil, and natural gas or a renewable resource
like wind, water, and solar energy.
Let IxIdenote the subset of exhaustible energy sectors. Each such sector iIxis
uniquely identified by the underlying resource on which production is based (like ’coal’
used for ’coal-fired power generation’ or ’oil’ used to provide ’fuel-based transportation
services’). The amount of exhaustible resource iIxused in region Lat time tis
4
denoted by X
i,t 0. Exhaustible resources are typically an essential input to production
in the respective sector and generate emissions proportional to their usage in production.
Energy sectors thus represent the production stage at which emissions are potentially
generated. Sectors which employ renewable sources do not cause emissions.3
With the previous distinction, the technology used by an exhaustible energy sector iIx
to produce energy output in period ttakes the form
Eℓ,s
i,t =Q
i,tFi(K
i,t, N
i,t, X
i,t) (2)
while production in a renewable energy sector iI\Ixin period tis given by
Eℓ,s
i,t =Q
i,tFi(K
i,t, N
i,t).(3)
Similar to (1), both specifications (2) and (3) allow for time- and region-specific produc-
tivity Q
i,t. In general, a higher productivity Q
i,t > Q
i,t may reflect a more developed
technology in region relative to or the fact that conditions to produce energy of type
iIare more favorable in region than in due to geographic conditions, etc.4
Technology and productivity
Denoting by Qt:= (Q
i,t)(ℓ,i)L×I0the world productivity vector in period t0, we
assume that the evolution of the sequence (Qt)t0is determined exogenously. The
remainder imposes the following standard restrictions on production technologies (1),
(2), and (3). The Inada condition ensures that each factor is employed in production.
Assumption 1
Each production function Fi:Rni
+R+,iI0is linear homogeneous, strictly
increasing, concave, and C2on Rni
++. The first partial derivatives satisfy the Inada-
condition limzmց0zmFi(z1,...,zni) = ∞ ∀z= (z1,...,zni)Rni
++ and m= 1,...,ni.
Resource sectors
Resource sectors are uniquely identified by the energy sector iIxwhich uses this
resource in production. In each region L, there exists a single firm which extracts
resources of type iIxand supplies them to the global resource market. The amount
of resource iextracted and supplied in period tis denoted Xℓ,s
i,t 0 (to be distinguished
from the amount X
i,t demanded by energy sector iIxin that region). Resource
firms face constant per unit extraction costs ci0 and take the initial resource stock
R
i,00 as a given parameter.5Feasible extraction plans are thus non-negative sequences
3This abstracts from emissions generated from using renewable resources like biomass, etc. which
are negligible relative to emissions from fossil fuels.
4For example, a solar energy plant located in the Sahara seems likely to produce more electricity
output than an identical plant located in a northern European region like Norway while the opposite
holds in the case with hydroelectric power generation.
5The assumption of constant extraction costs is a compromise between falling costs due to techno-
logical progress assumed in Golosov et al. (2014) and extraction costs which increase with the scarcity
of the resource as in Acemoglu, Aghion, Bursztyn & Hemous (2012) and also discussed in Hotelling
(1931). Our model should be amendable to extensions in either direction.
5
(Xℓ,s
i,t )t0which respect the feasibility constraint
X
t=0
Xℓ,s
i,t R
i,0.(4)
To avoid trivialities, we impose the initial condition PLR
i,0>0, i.e., initial world
resources are strictly positive for all iIx. It may, however, be the case that R
i,0= 0
in which case region does not own any resources of type i.
1.3 Climate model
Emissions of CO2are generated by using (’burning’) exhaustible resources like coal, oil,
and gas to produce energy. Thus, emission occur at the energy stage in production. The
amount of CO2generated by using one unit of exhaustible resource iIxis physically
determined by its carbon-content ζi0. In particular, ζi= 0 if the resource does no
generate emissions, like uranium in the case of nuclear energy production.6
Summing the different types of exhaustible resource inputs weighted by their respective
carbon content over all regions one obtains the total emissions of CO2in period tas
Zt:= X
LX
iIx
ζiX
i,t.(5)
For the following analysis, we will adopt the climate model from Golosov et al. (2014),
henceforth GHKT. However, none of our results depend on this particular specification.7
Assume that the climate state in period tconsists of permanent and non-permanent CO2
in the atmosphere, denoted as St= (S1,t, S2,t ). Given the sequence of emissions {Zt}t0
determined by (5), the climate state evolves as
S1,t =S1,t1+φLZt(6a)
S2,t = (1 φ)S2,t1+ (1 φL)φ0Zt(6b)
Specification (6) assumes that a share 0 φL<1 of emissions become permanent CO2.
Out of the remaining emissions, a share φ0becomes non-permanent CO2which decays
at constant rate 0 < φ < 1 while the remaining share 1 φ0leaves the atmosphere (see
GHKT for details). Total concentration of CO2at time tis thus given by
St=S1,t +S2,t.(7)
6Our specification abstracts from emissions which occur at the resource stage when resources are
extracted. While empirically such emissions certainly play a role, especially for uranium, they seem
quantitatively negligible compared to emissions occurring at the energy stage on which we focus.
7We show this formally in Section 6 where we adopt an alternative climate model from Gerlagh &
Liski (2018).
6
Climate damages and temperature in period tdepend exclusively on St. Denoting by
¯
S > 0 the pre-industrial level of CO2in the atmosphere, climate damage in region is
determined by a differentiable, strictly increasing function D: [ ¯
S, [[0,1[,
D
t=D(St).(8)
A specific functional form which will be used below is
D(S) = 1 exp{−γ(S¯
S)}, γ>0 (9)
which corresponds to the choice in GHKT.8Regional differences in climate damage thus
enter via region specific parameters γ,L. In particular, the climate problem would
be economically irrelevant if γ0.
1.4 Consumption sector
The consumption sector in region Lconsists of a single representative household
which supplies labor and capital to the production process and decides about consump-
tion and capital formation taking factor prices as given. In addition, the consumer is
entitled to receive all profits from domestic firms and transfers from the government.
Let K
0denote initial capital in t= 0 and Nℓ,s
t>0 the labor supplied in period t.
The sequence (Ns
t)t0of world labor supply Ns
t:= (Nℓ,s
t)Lis exogenously given in our
model. The household’s preferences over non-negative consumption sequences (C
t)t0
are represented by a standard time-additive utility function
U((C
t)t0) =
X
t=0
βtu(C
t).(10)
The subsequent analysis imposes the following restrictions on U.
Assumption 2
The discount factor in (10) satisfies 0< β < 1while uis of the form
u(C) = (C1σ
1σfor σ > 0, σ 6= 1
log(C)for σ= 1.(11)
Assumption 2 will be key for the separability between efficiency and optimality derived in
Section 3 and to determine the optimal climate policy in Section 4. Functional form (11)
is precisely the class of utility functions consistent with balanced growth in our model
with exogenous labor supply, cf. King, Plosser & Rebelo (1988). Thus, Assumption 2 is
a standard restriction in the presence of exogenous productivity growth. It is also used
in Nordhaus & Yang (1996) and almost any model of climate change and includes the
logarithmic specification in GHKT or Hassler & Krusell (2012) as a special case.
8The general version of GHKT allows for γto be time- and state-dependent. Here, we assume that
it is constant, as they do in their numerical simulations, too.
7
1.5 Summary of the economy
The economy Eintroduced in the previous sections can be summarized by its regional
and sectoral structure L,I0,Ix, the production technologies (Qt)t0,(Fi)iI0,(ci)iIx,
consumer characteristics (Ns
t)t0, u, β, and climate parameters (ζi)iIx, φ, φ0, φL. In
addition, initial values for capital supply Ks
0= (K
0)L, exhaustible resource stocks
R0= (R
i,0)(ℓ,i)L×Ix, and the initial climate state S1= (S1,1, S2,1) are given.
2 Decentralized Solution
This section studies the decentralized equilibrium solution of the economy where all
producers and consumers behave optimally under perfect foresight and market clearing
on all markets. All equilibrium variables are determined for a given climate policy which
imposes a tax on emissions and distributes the tax revenue as transfers across regions.
2.1 Equilibrium prices
Unless stated otherwise, all prices in period tare denominated in units of time tcon-
sumption. As labor and energy outputs will be immobile across countries, their prices
will, in general, be region-specific. Denote by w
t>0 the wage and p
i,t >0 the price
per unit of energy type iIin region and period t. By contrast, capital and ex-
haustible resources are traded on international markets implying that their prices are
not country-specific. The (rental) price of capital in period tis denoted as rt>0 and
the world price of resource iIxas vi,t >0.
Conceptually, all transactions take place in t= 0 and the consumption good in this
period is chosen as the numeraire. Since the economy is deterministic, the price of time
tconsumption measured in units of consumption at time zero can be expressed as9
qt=
t
Y
s=1
r1
s(12)
for each t0 where q0= 1. In the following analysis, the price defined in (12) serves
as a discount factor which discounts payments in period tto period zero.
2.2 Climate policies
Aclimate policy consists of two parts. The first one is a Carbon Tax Policy (CTP) which
levies a proportional tax on emissions. Taxes can vary over time and across regions.
9In our deterministic model, this holds because 1/rt+1 is the price of a bond traded in period tthat
pays-off one unit of the consumption good at time t+ 1. The prices in (12) are thus the Arrow-Debreu
prices for this economy.
8
Definition 1
A Carbon Tax Policy (CTP) is a non-negative sequence τ= (τt)t0where τt= (τ
t)L
are the (region-specific) taxes to be paid per unit of CO2emitted in period t0.
Since emissions occur at the energy stage, taxes are paid by energy producers. Tax
revenues are redistributed as lump-sum transfers to consumers. A natural restriction
would be to assume that transfers equal tax revenue in each region. We will, however,
adopt a more general setting which does not require taxes and transfers to balance at
the national level but allows for transfer payments across countries. This leads to the
following concept of a transfer policy which determines the share of tax revenue received
by each region. Such a transfer policy constitutes the second part of a climate policy.
Definition 2
A transfer policy is a mapping θ:LR,7→ θsatisfying PLθ= 1 which
determines the share of total tax revenue received by region Lin each period.
The pair (τ, θ) will be called a climate policy. Let T
tdenote the transfers received by
consumers in region in period t. These transfers are determined by tax revenue and
the given transfer policy as
T
t=θX
kL
Tax revenue in region k
z }| {
τk
t·X
iIx
ζiXk
i,t
| {z }
Emissions in region k
(13)
If T
t> τ
tPiIxζiX
i,t, region receives a net transfer from the other countries and
contributes a net transfer otherwise. Note that the case θ<0 is not excluded in this
definition, in which case consumers in region are taxed to finance transfers received by
other countries. Thus, the previous specification also allows for international redistribu-
tion via lump-sum taxation. Moreover, the assumption that transfer shares are constant
over time is without loss of generality, as the behavior of consumers derived below will
exclusively depend on their lifetime transfer income defined as
T:=
X
t=0
qtT
t.(14)
Thus, defining total discounted tax revenue
T:=
X
t=0
qtX
kL
τk
tX
iIx
ζiXk
i,t,(15)
lifetime transfers satisfy
T=θT=θ
X
t=0
qtX
kL
τk
tX
iIx
ζiXk
i,t.(16)
and we simply could have defined θas the ratio T/T .
9
2.3 Producer behavior
Firms in each sector iI0choose their productions plans to maximize the discounted
stream of current and future profits. As there is no intertemporal linkage between these
decisions, the decision problems of final and energy sectors can be formulated and solved
on a period-by period basis. This, however, is not possible in the case of resource sectors
which solve intertemporally dependent problems.
Final sector
Given damage-adjusted productivity and factor prices for labor, capital, and the list of
energy prices p
t= (p
i,t)iI0, the final sector in region solves the following decision
problem in period t0:
max
(K,N,E1,...,EI)R2+I
+(1 D
t)Q
0,tF0(K, N, (Ei)iI)w
tNrtKX
iI
p
i,tEi(17)
Under Assumption 1, any solution to (17) satisfies the following first order conditions
which equate prices and marginal products of each production factor:
(1 D
t)Q
0,tKF0(K
0,t, N
0,t,(E
i,t)iI) = rt(18a)
(1 D
t)Q
0,tNF0(K
0,t, N
0,t,(E
i,t)iI) = w
t(18b)
(1 D
t)Q
0,tEiF0(K
0,t, N
0,t,(E
i,t)iI) = p
i,t iI.(18c)
Energy sectors
A renewable energy sector iI\Ixin region Ltakes sector specific productivity,
factor prices for labor and capital, and the domestic energy price p
i,t >0 as given and
solves the following decision problem in each period t0:
max
(K,N )R2
+p
i,tQ
i,tFi(K, N )w
tNrtK.(19)
A solution to (19) is characterized by the following first order conditions
p
i,tQ
i,tKFiK
i,t, N
i,t=rt(20a)
p
i,tQ
i,tNFiK
i,t, N
i,t=w
t.(20b)
An exhaustible energy sector iIxtakes, in addition, the resource price vi,t >0 and
the tax τ
t0 per unit of CO2as given parameters in the decision in period t. As each
unit of resource igenerates ζiunits of CO2, the decision problem in period treads:
max
(K,N,X )R3
+p
i,tQ
i,tFi(K, N , X)w
tNrtK(vi,t +τ
tζi)X.(21)
Clearly, the solution to (21) becomes independent of τ
tif ζi= 0, i.e., the firm employs
a clean technology. The first order conditions associated with (21) are given by:
p
i,tQ
i,tKFiK
i,t, N
i,t, X
i,t=rt(22a)
p
i,tQ
i,tNFiK
i,t, N
i,t, X
i,t=w
t(22b)
p
i,tQ
i,tXFiK
i,t, N
i,t, X
i,t=vi,t +ζiτ
t.(22c)
10
Resource sectors
The resource sector iIxin region Lchooses a non-negative extraction sequence
(Xℓ,s
i,t )t0which satisfies the resource constraint (4). Resources extracted in period t
are supplied to the global resource market. Given sequences of resource prices (vi,t)t0,
discount factors (qt)t0defined by (12) and constant extraction costs ci0, the resource
sector maximizes the discounted sum of future profits. The decision problem reads
max
(Xℓ,s
i,t )t0
X
t=0
qt(vi,t ci)Xℓ,s
i,t |Xℓ,s
i,t 0t0,(4) holds.(23)
To avoid trivialities, assume R
i,0>0. Then, the existence of optimal extraction plans
determined as solutions to problem (23) can be characterized as follows.
Lemma 1
If R
i,0>0, the solution to (23) satisfies the following:
(i) An interior solution (X
t)t00exists if and only if resource prices satisfy
vi,t ci= (vi,0ci)/qt0t0.(24)
(ii) If condition (24) is satisfied, there are two cases:
(a) If vi,0=ci, any sequence (X
t)t0satisfying P
t=0 X
tR
i,0is a solution.
(b) If vi,0> ci, any sequence (X
t)t0satisfying P
t=0 X
t=R
i,0is a solution.
Condition (24) is a version of the classical Hotelling rule (cf. Hotelling (1931)) under
which net resource prices must grow at the rate of interest for resource firms to be
indifferent between extracting resources in different periods. Hassler & Krusell (2012)
also derive a version of the Hotelling rule in their multi-region framework. One also
observes from Lemma 1 (ii) that only in case (a) where vi,t =cifor all t0 may it be
optimal not to exhaust the entire stock of resources.
In either case, (24) permits maximum profits Π
i:= P
t=0 qt(vi,t ci)X
tto be written as
Π
i= (vi,0ci)R
i,0.(25)
Intuitively, the discounted profit stream (25) of resource sector iIxis the excess value
of the initial stock of resources valued at time-zero prices net of extraction costs. Also
note that given an optimal extraction plan (Xℓ,s
i,t ) determined as a solution to (23), the
period profit of resource sector iIxin region Lis given by
Π
i,t = (vi,t ci)Xℓ,s
i,t 0.(26)
In general, however, the quantity in (26) will be indeterminate at equilibrium due to
the multiplicity of solutions to (23).
11
Equilibrium profits
All firms in region are owned by domestic consumers who are entitled to receive all
profits. A direct consequence of Assumption 1 and the first order conditions derived in
(18), (20), and (22) is that profits in final production and all energy sectors are zero.
Thus, by (25) the total lifetime profit income of consumers in region Lis
Π=X
iIx
Π
i=X
iIx
(vi,0ci)R
i,0.(27)
2.4 Consumer behavior
Budget constraints
In each period t0, consumers in region Lreceive labor income w
tNℓ,s
t>0, the
return rton their current net asset holdings K
t, profit income Π
t0, and transfers T
t.
Their choices of current consumption C
t0 and investment K
t+1 satisfy the period
budget constraint
C
t+K
t+1 =rtK
t+w
tNℓ,s
t+T
t+ Π
tt0.(28)
At the individual level, capital investment may be negative10 but must satisfy the No-
Ponzi game condition
lim
t→∞ qtK
t+1 0 (29)
requiring consumers to ultimately repay any outstanding debt. Using (28) and (29) one
can recursively eliminate investment to obtain the consumer’s lifetime budget constraint
X
t=0
qtC
tW+T.(30)
Here, Tdenotes lifetime transfer income defined in (14) and
W:= r0K
0+ Π+
X
t=0
qtw
tNℓ,s
t(31)
is the consumer’s lifetime non-transfer income determined by initial asset holdings K
0,
lifetime profit income Πdetermined by (27), and lifetime labor income.
Optimal consumption plans
The consumer chooses a consumption sequence (C
t)t0to maximize lifetime utility (10)
10This assumption can be justified by assuming that an international bond market coexists with the
capital market on which borrowing and lending takes place. The consumer in period tchooses capital
investment ˜
K
t+1 0 and bond purchases B
t+1 subject to the budget constraint (28). Market clearing
requires Kt+1 =PL˜
K
t+1 on the capital market and PLB
t+1 = 0 for the bond market. Setting
K
t+:= ˜
K
t+1 +B
t+1 then implies the equilibrium conditions derived below.
12
subject to her lifetime budget constraint. The decision problem reads:
max
(C
t)t0(
X
t=0
βtu(C
t)C
t0t0,(30) holds).(32)
Equation (30) shows that existence of a solution to (32) requires the solvency condition
T>W(33)
imposing a lower bound on transfers which becomes relevant with taxation (θ<0).
Standard (variational) arguments imply that any solution (C
t)t0to (32) must satisfy
the Euler equations
rt+1βu(C
t+1) = u(C
t)t0.(34)
and the lifetime budget constraint (30) holds with equality. The latter is equivalent to
the transversality condition
lim
t→∞ qtK
t+1 = 0.(35)
In fact, the restriction imposed by Assumption 2 allows us to characterize the unique
solution to (32) in the following lemma.
Lemma 2
Let Assumption 2 hold and the solvency condition (33) be satisfied. Then, problem (32)
has a unique solution (C
t)t0given by
C
t=(βt/qt)1
σW+T
P
s=0 qs(βs/qs)1
σ
t0.(36)
Optimal consumption behavior defined by (36) thus satisfies the permanent income
hypothesis by consuming a fraction of lifetime income including transfers in each period.
Note that this behavior requires the possibility of unconstrained borrowing and lending
in a frictionless capital market.
2.5 Market clearing
Restrictions on trade
Trade between countries occurs on global markets for capital, final output, and ex-
haustible resources of each type iIx. All these goods can freely be exported without
additional costs. As there is no sign restriction on capital investment at the individual
level, consumers can also take loans permitting intertemporal borrowing and lending of
final output between regions. By contrast, labor supply is immobile and can only be
employed in domestic production sectors. Likewise, energy goods and services can only
13
be used in domestic final good production.11 Thus, there are domestic markets for labor
and energy of all types iIin each region L.
Domestic markets
Market clearing on the domestic labor market in region in period t0 requires
X
iI0
N
i,t
!
=Nℓ,s
t.(37)
Since energy is non-tradable across countries, energy demanded in final production must
coincide with domestic energy production in each region. The market clearing condition
for energy type iIin region Lwhich must hold in each period tis therefore simply
E
i,t
!
=Eℓ,s
i,t .(38)
International markets
Let Kt:= PLK
tbe the aggregate stock of productive capital supplied to production
in period t. Market clearing on the world capital market in period trequires
X
LX
iI0
K
i,t
!
=Kt.(39)
Since period profit income (26) is, in general, indeterminate at equilibrium, so is con-
sumers’ individual net capital position K
t,Lin the period budget constraint (28).
The market clearing condition for exhaustible resource iIxin period treads
X
L
Xℓ,s
i,t
!
=X
L
X
i,t.(40)
If X
i,t < Xℓ,s
i,t region is a net exporter of resource iIxin period tand a net importer
otherwise. Summing (4) over all countries and using (40), the allocation of resources
across production sectors must satisfy the world resource constraint
X
t=0 X
L
X
i,t Ri,0(41)
where Ri,0:= PLR
i,0is the total initial stock of resource i. An immediate consequence
of Lemma 1 is that the equilibrium amount of resources Xℓ,s
i,t supplied by region and
period profits (26) will, in general, be indeterminate. As resources extracted in different
countries are perfect substitutes, however, the equilibrium extraction plans can always
be chosen compatible with the resource constraint (4) in each region.
Finally, summing the consumers’ period budget constraints (28) over all regions and
exploiting the first order and zero profit conditions for all sectors iI0in conjunction
11In fact, this is one reason why we disentangle the energy stage and the resource stage in our model,
compared to GHKT.
14
with (26) and the market clearing conditions (37), (38), (39), and (40) together with
(13) one obtains the market clearing condition for final output in period tas
Kt+1 +X
L
C
t+X
iIx
ciX
L
X
i,t =X
L
Y
t.(42)
Here, Kt+1 is productive capital formed in period tand supplied to production in t+ 1.
2.6 Equilibrium
For purposes of a compact notation, we employ the following vector notation for the
variables introduced in the previous sections for each t:
Qt:= (Q
i,t)(ℓ,i)L×I0Ns
t:= (Nℓ,s
t)LCt:= (C
t)L
Yt:= (Y
t)LEt:= (E
i,t)(ℓ,i)L×IXt:= (X
i,t)(ℓ,i)L×Ix
Nt:= (N
i,t)(ℓ,i)L×I0Kt:= (K
i,t)(ℓ,i)L×I0St:= (St,1, St,2)
wt:= (w
t)L,pt:= (p
i,t)(ℓ,i)L×Ivt:= (vi,t)iIx.
(43)
All variables defined in (43) take values in the appropriate positive orthant of Rn.
Definition of equilibrium
Let the climate tax policy τand transfer policy θdefined as above be given. The
following definition of equilibrium is standard. Here and in the remainder we denote
equilibrium variables by a * superscript.
Definition 3
Given tax policy τand transfer policy θ, an equilibrium of Eis an allocation A=
C
t, K
t+1,Y
t,E
t,K
t,N
t,X
t,S
tt0and prices P=r
t,w
t,p
t,v
tt0such that:
(i) The allocation is consistent with the production technologies (1), (2), and (3) and
the market clearing conditions/resource constraints (37), (38), (39), (41), and (42).
(ii) Producers behave optimally, i.e., equations (18), (22), and (20) hold for all t0.
Profits of resource firms are given by (25) while resource prices evolve as in (24).
(iii) Consumers behave optimally as described in Lemma 2 with profit incomes deter-
mined by (27) and transfers satisfying (16) and the solvency condition (33).
(iv) Climate variables evolve according to (6) with emissions given by (5) and climate
damages in (1) determined by (7) and (8).
Properties of equilibrium
If we want to emphasize the dependence of the equilibrium allocation on policy variables,
we will write A(τ, θ), etc. A special case of Definition 3 is the Laissez faire equilibrium
15
with no taxation, i.e., τ0. The induced equilibrium allocation ALF := A(0, θ) is in-
dependent of θand constitutes an important benchmark in the subsequent discussion.12
It is clear that this solution will, in general not constitute a Pareto optimal outcome
due to the climate externality in production.
The following results establish additional properties of equilibrium that follow from our
restrictions on technologies and preferences. As before, we denote by Ri,0=PLR
i,0
to be the total initial stock of resource iIx.
Lemma 3
Under Assumptions 1 and 2, any equilibrium allocation has the following properties:
(i) The allocation is interior, i.e., A0.
(ii) Consumption C
tis a constant share of world consumption ¯
C
t:= PLC
t, i.e.,
C
t=µ¯
C
t(44)
for all Land t0where µ>0and PLµ= 1.
(iii) Prices and extraction of each resource iIxsatisfy the following:
(a) If Ri,0=, then vi,0=ciimplying vi,t ci.
(b) If Ri,0<, then limt→∞(vi,t +ζiτ
t) = .
(c) If Ri,0>P
t=0 PLX
i,t, then vi,0=ciand limt→∞(vi,t +ζiτ
t) = .
Lemma 3 (ii) shows that consumption in each region moves in lock-step with world con-
sumption over time. This property is due to consumer utility restricted by Assumption
2 and will play a key role to characterize how policy affects equilibrium allocations.
The result in (iii) shows that an infinite stock of resources drives resource prices down
to extraction costs. Hence, there is no scarcity rent on that resource and profits are
zero.13 Conversely, if the resource stock is finite, gross resource prices (including taxes)
must converge to infinity which requires vi,0> ciin the absence of taxation. Moreover,
(iii) shows that each resource is completely exhausted unless its scarcity rent is zero
and taxes on emissions grow infinitely large. Thus, aggressive taxation is needed to
prevent resources from being completely exhausted. In particular, all resources are
12Our subsequent discussion and notation presume that each policy considered induces a unique
equilibrium. A general proof of existence/uniqueness of equilibrium is beyond the scope of this paper.
The recursive structure of equilibria and their computation is discussed in detail in Hillebrand &
Hillebrand (2018) suggesting that the uniqueness assumption is justified.
13An infinite resource stock is typically justified by assuming the existence of a backstop technology
which provides an equivalent substitute for the resource in the future. Such a backstop technology is
implicitly assumed in GHKT for coal.Hillebrand & Hillebrand (2018) provide a critical discussion of
this asssumption and show that it is key to obtain the quantitative predictions in GHKT.
16
fully exhausted at the Laissez faire equilibrium. See Hillebrand & Hillebrand (2018) for
further discussion and application of these results in a numerical study of the model.
The final result of this section disentangles the separate impact of the tax policy τ
and the transfer policy θon the equilibrium allocation A(τ, θ). Essentially, it shows
that transfers only determine consumption in each region as a constant share of world
consumption but are irrelevant for all other equilibrium variables. This result will play
a major role in Section 4 when we study climate policies which implement the social
optimum as an equilibrium allocation.
Proposition 1
Given a climate policy (τ, θ), let A=C
t, K
t+1,Y
t,E
t,K
t,N
t,X
t,S
tt0be the
induced equilibrium allocation and Pthe equilibrium price system. For each t0, let
¯
C
t=PLC
tdenote aggregate equilibrium consumption. Then, the following holds:
(i) The tax policy τdetermines the aggregate equilibrium allocation
¯
A=¯
C
t, K
t+1,Y
t,E
t,K
t,N
t,X
t,S
tt0(45)
and the price system Pwhich are both independent of the transfer policy θ.
(ii) The transfer policy θonly affects the distribution of aggregate consumption across
regions, i.e., the consumption shares µin (44) which take the form
µ=W+θT
PkLWk+T.(46)
with non-transfer incomes Wdetermined by (31) and tax revenue Tby (15).
The aggregate allocation ¯
Ain (45) determines world consumption but does not specify
its distribution across regions. Apart from that, it contains the same variables as the
equilibrium allocation A. The equilibrium distribution of consumption determined by
(46) corresponds to the relative sizes of consumers’ lifetime incomes including transfers.
Only this equilibrium quantity depends on the transfer policy. Note that consumption
shares become independent of θalong the Laissez faire equilibrium where T= 0.
3 Optimal Solution
In this section we determine an optimal allocation as the solution to a planning problem
(PP) which maximizes consumer utility subject to the feasibility constraints imposed
by technology, resources, and climate change. The major difference to the decentralized
solution is that the planning problem incorporates the climate externality and the link
between emissions, climate damage, and productivity in final good production.
17
In a multi-region world, there is no unique choice of the planner’s objective function
which must necessarily incorporate the trade-offs between the interests of different coun-
tries. The standard approach in the literature (see, eg., Nordhaus & Yang (1996)) also
adopted here aggregates utilities in different countries based on a weighting scheme
which assigns a certain weight to the utility of consumers in each region. Such a weight-
ing scheme is essentially equivalent to choosing certain minimum utility levels for all
countries 6= 1 and then maximizing utility of region 1 as is done in Eyckmans &
Tulkens (2003) to obtain a Pareto-optimal allocation.
A major advantage of our restrictions on preferences in Assumption 2 is that it gives
rise to a separation result which permits to compute an optimal allocation in two steps.
First, we determine an efficient allocation which maximizes utility of a fictitious world
representative consumer. This efficient solution completely specifies the optimal climate
path and the entire allocation of production factors and resources across countries to-
gether with aggregate world consumption. Most importantly, the efficient solution is
independent of the employed weighting scheme. In a second step, we determine an op-
timal distribution of world consumption across different countries to obtain an optimal
allocation which maximizes a weighted utility index reflecting the trade-off between the
interests of different countries. This separability between efficiency and optimal distri-
bution will be the key to determine an optimal climate policy in the next chapter. In
fact, this separation result has been well-known in the literature on public goods. A
finite-dimensional version was first proved by Bergstrom & Cornes (1983).
3.1 Optimal allocations
Feasible allocations
Consider a planner who chooses a feasible world allocation subject to the restrictions
imposed by technology, factor mobility, and resource constraints. Formally, using the
notation introduced in the previous section, the planner takes the sequences of pro-
ductivity (Qt)t0and labor supply (Ns
t)t0as given. In addition, initial world capital
K0>0, the initial world stock Ri,0>0 of each exhaustible resource iIx, and the
initial climate state S1= (S1,1, S2,1) are given as well. While final output, capi-
tal, and exhaustible resources can freely be allocated across countries, labor and energy
outputs can only be used within each region. Thus, the planner faces essentially the
same restrictions as producers and consumers in the decentralized solution including
constraints (37), (38), (39), and (42) when allocating labor, capital, and output. Fur-
ther, the given initial stock of world resources imposes the restrictions (41) on the use
of exhaustible resource iIxin production.14 This leads to the following definition of
14As resources extracted in different countries are perfect substitutes, the solution to the planning
problem does not determine where these resources are extracted. However, any allocation of exhaustible
resources (X
i,t)(ℓ,i)L×Ix)t0satisfying the feasibility constraint (41) can always be chosen compatible
18
a feasible allocation.
Definition 4
(i) A feasible allocation is a sequence A=Ct, Kt+1,Yt,Et,Kt,Nt,Xt,Stt0which
satisfies (1), (2), (3), (5), (6), (7), (37), (38), (39), (41), and (42) for all t0.
(ii) The set of feasible allocations of the economy Eis denoted A.
In particular, equilibrium allocations studied in the previous section are feasible, i.e.,
AA.
Objective function
The distribution of consumption across countries necessarily induces a trade-off between
the utility levels attained by consumers in different countries. To incorporate this trade-
off, assume that the planner uses a weighting scheme corresponding to a list of utility
weights ω= (ω)Lwhere ω0 represents the weight attached to consumer utility in
region in the planner’s decision. Formally, we have
Definition 5
A utility weighting scheme is a map ω:LR+,7→ ωwhich satisfies PLω= 1.
In what follows, let ∆ := {(x1,...,xL)RL|PL
=1 x= 1}denote the unit-simplex in
RL. Then, the set of all possible weighting schemes can be identified with ∆+:= ∆RL
+.
Each weighting scheme ω+defines the following weighted utility index
V((Ct)t0;ω) :=
X
t=0
βtX
L
ωu(C
t) (47)
which depends on world consumption (Ct)t0where Ct= (C
t)Lfor all t0.
Weighted Planning Problem
Given ω+, use (47) to define the following Weighted Planning Problem (WPP)
max
AV((Ct)t0;ω)A= (Ct, Kt+1,Yt,Et,Kt,Nt,Xt,St)t0A.(48)
Assuming it exists and is unique, denote the solution to (48) as
Aopt =Copt
t, Kopt
t+1,Yopt
t,Eopt
t,Kopt
t,Nopt
t,Xopt
t,Sopt
t)t0.(49)
It is clear that, in general, the solution to (48) will depend on the weighting scheme ω.
Thus, we will write Aopt(ω) as a way of emphasizing this dependence. It is also clear
that for any weighting scheme ω+the solution Aopt(ω) to (48) is a Pareto-optimum
on the set of feasible allocations A.
with the individual resource constraints (4) in each region.
19
3.2 Efficient aggregate allocations
Feasible aggregate allocations
Consider now a modified planning problem which faces the same restrictions as before
but does not specify the distribution of consumption across different countries. As
before, denote aggregate world consumption in period tby ¯
Ct0 and write the resource
constraint (42) as
Kt+1 +¯
Ct+X
LX
iIx
ciX
i,t =X
L
Y
t.(50)
Replacing (42) by (50) leads to the following definition of a feasible aggregate allocation
which specifies aggregate world consumption but not its distribution across countries.
Apart from that, it involves the same variables as a feasible allocation defined above.
Definition 6
(i) A feasible aggregate allocation is a sequence ¯
A=¯
Ct, Kt+1,Yt,Et,Kt,Nt,Xt,Stt0
which satisfies (1), (2), (3), (5), (6), (7), (37), (38), (39), (41), (50) for all t0.
(ii) The set of feasible aggregate allocations of the economy Eis denoted ¯
A.
In particular, the aggregate equilibrium allocation in (45) is feasible, i.e., ¯
A¯
A.
Aggregate planning problem
The following modified planning problem maximizes utility of a fictitious world repre-
sentative consumer who consumes ¯
Ctin period tand has the same utility function as
consumers in each region. This leads to the following Aggregate Planning Problem
(APP):
max
¯
A
X
t=0
βtu(¯
Ct)¯
A= ( ¯
Ct, Kt+1,Yt,Et,Kt,Nt,Xt,St)t0¯
A.(51)
A solution to (51) will be denoted
¯
Aeff = ( ¯
Ceff
t, Keff
t+1,Yeff
t,Eeff
t,Keff
t,Neff
t,Xeff
t,Seff
t)t0(52)
and referred to as an efficient aggregate allocation. An efficient solution ¯
Aeff specifies
the entire allocation of production factors and resources across countries but leaves
undetermined the distribution of aggregate consumption across countries. It is therefore
independent of any weights attached to the interests of different countries and constitutes
the main ingredient to our separation result.
3.3 Optimal distribution of consumption
To prepare the main result of this section, let a weighting scheme ω+be arbitrary
but fixed and consider the static problem of distributing a given level ¯
C > 0 of aggre-
gate consumption across countries in an arbitrary period. The decision variable is a
20
consumption distribution µ= (µ)Lwhere µ0 is the share of ¯
Cgiven to region
. Since PLµ= 1, any feasible consumption distribution satisfies µ+defined
as above. An optimal consumption distribution µopt = (µℓ,opt)Lcan be determined as
the solution to the problem
max
µ=(µ)L(X
L
ωu(µ¯
C)µ0L,X
L
µ1).(53)
The following lemma establishes that (53) has a unique solution which can be computed
explicitly and which, crucially, is independent of ¯
C. The result in (ii) establishes an
essential equivalence between the choice of a weighting scheme ωand a consumption
distribution µwhich will be exploited in Section 4.3.
Lemma 4
Let ube of the form (11) from Assumption 2. Then, the following holds:
(i) For each weighting scheme ω= (ω)L+, there exists a unique consumption
distribution µopt(ω) = (µℓ,opt(ω))Lwhich solves (53) taking the form
µℓ,opt(ω) = (ω)1
σ
PkL(ωk)1
σ
, L.(54)
(ii) For any consumption distribution µ= (µ)L+there exists a weighting
scheme ω+which rationalizes µin the sense that µ=µopt(ω)solves (53).
3.4 From efficiency to optimality
The main result of this section shows that the solution (49) to the WPP (48) can be
obtained from the efficient allocation (52) by distributing aggregate consumption opti-
mally across countries. Moreover, for any weighting scheme ω, the optimal distribution
is time-invariant and can be determined as the solution to (53). Using Lemma 4 allows
us to state this result in the following theorem.
Theorem 1
Let Assumptions 1 and 2 be satisfied and define the efficient allocation Aeff as in (52).
Given a weighting scheme ω, let µopt(ω) = (µℓ,opt (ω))Lsolve (53). Then, the allocation
A= (µopt(ω)¯
Ceff
t, Keff
t+1,Yeff
t,Eeff
t,Keff
t,Neff
t,Xeff
t,Seff
t)t0(55)
solves the WPP (48), i.e., A=Aopt(ω).
In words, Theorem 1 says that a solution to the WPP can simply be obtained from
the efficient solution (52) by distribution aggregate consumption in an optimal fashion
21
determined by the weighting scheme ω. In particular, the entire allocation of production
factors, resources, and emissions is completely determined by the efficient solution.
As its major implication, the previous result allows to compute a unique efficient allo-
cation of production factors, resources, emissions, climate damages, etc. which is com-
pletely independent of the weights that the interests of different countries receive in the
decision. The weighting scheme is therefore irrelevant for answering the question what
the optimal climate path is and where and how emissions should be reduced. Intuitively,
there exists a unique allocation of factors and resources across regions to produce an
efficient world consumption sequence which incorporates the climate externality. The
weighting scheme becomes relevant only to determine how this efficiently produced world
consumption sequence should be distributed across regions. An optimal consumption
distribution can be computed using (54) once a suitable weighting scheme has been
chosen and, by Lemma 4 (ii) is in fact equivalent to such a choice.
3.5 Computing the efficient allocation
The previous results show that the social optimum is essentially characterized by the
efficient solution (52). Adopting a standard infinite-dimensional Lagrangian approach,
it is now straightforward to obtain explicit conditions which completely characterize this
solution. Detailed computations can be found in Section A.7 in the appendix. The main
findings are as follows.
Social cost of carbon
The total costs Λtof emitting one additional unit of CO2in period t(measured in units
of time tconsumption) correspond to the discounted sum of all future marginal climate
damages in all regions caused by this emission. Formally,
Λt=
X
n=0
βnu(¯
Ct+n)
u(¯
Ct)
| {z }
discount factor
φL+ (1 φL)φ0(1 φ)n
| {z }
climate parameters X
L
D(St+n)Q
0,tF0K
0,t, N
0,t,(E
i,t)iI
| {z }
marginal damage in region
.
(56)
Equation (56) is a multi-region version of the result in GHKT. Note that Λtis indepen-
dent of and iand depends on the structural parameters of the model and endogenous
model variables in a complicated way, unless stronger restrictions similar to those in
GHKT are imposed.15 The term (56) is the key quantity to incorporate the climate
externality into the (shadow) price of exhaustible resources.
Efficiency conditions
The remaining optimality conditions essentially ensure intratemporal and intertemporal
15Climate costs Λtare independent of because future climate damages are discounted by the same
discount factors in each region. For a different setup with region-specific discount rates see Eyckmans
& Tulkens (2003).
22
efficiency in production and resource extraction in each period t0. Denote by
ˆp
i,t = (1 D
t)Q
0,tEiF0K
0,t, N
0,t,E
t.(57)
the time tshadow price of energy type iIin region L(measured in units of time
tconsumption units). Using (57), marginal products of capital are equalized across all
countries and sectors, i.e., for all ℓ, ℓLand all iI:16
(1 D(St))Q
0,tKF0K
0,t, N
0,t,E
t= ˆp
i,tQ
i,tKFi(K
i,t, N
i,t, X
i,t) (58)
Second, in each region Lmarginal products of labor are equalized across all sectors
(although not across countries due to labor immobility), i.e., for all iI
(1 D(St))Q
0,tNF0K
0,t, N
0,t,(E
i,t)iI= ˆp
i,tQ
i,tNFi(K
i,t, N
i,t, X
i,t).(59)
Third, for all iIx, the constraint (41) is binding and resource extraction is intratem-
porally efficient in each period t0, i.e., for all L:
ˆp
i,tQ
i,tXFi(K
i,t, N
i,t, X
i,t)ζiΛt= ˆvi,t.(60)
Equation (60) defines the true shadow price of resources as the marginal product in
production minus the cost of emissions defined in (56). Compared to the laissez-faire
equilibrium allocation, the social planner includes a wedge between the marginal product
of dirty resource iand its (shadow) price which accounts for the externality cost of an
additional unit of emissions. This is the key difference to the equilibrium condition (22c)
along the Laissez faire equilibrium which fails to take this cost into account. If D0,
however, the efficient solution coincides with the aggregate equilibrium allocation (45).
Intertemporal efficiency of final good allocation (consumption vs. capital formation) is
ensured by the standard Euler equation which holds for all Land t1:
u(¯
Ct1) = βu(¯
Ct)(1 D(St))Q
0,tKF0K
0,t, N
0,t,(E
i,t)iI.(61)
Condition (61) in conjunction with (58) also equates (implicit) capital returns across
countries in each period.
Defining the shadow price of resource extraction as in (60), intertemporally efficient
extraction of resource iIxis ensured by the condition:
ˆvi,t ci=βu(¯
Ct+1)
u(¯
Ct)ˆvi,t+1 ci.(62)
Finally, standard arguments also require the transversality condition
lim
t→∞ βtu(¯
Ct)Kt+1 = 0.(63)
The following proposition summarizes the result of this section. The proof is given in
Section A.7 in the appendix.
16 Here and in the sequel, we incur a slight abuse of notation by including X
i,t as a ’dummy’ argument
of Fieven if i /Ixto save some notation.
23
Proposition 2
Let Assumptions 1 and 2 hold. Then, any feasible aggregate allocation ¯
A¯
Awhich
satisfies conditions (56)-(62) for all t0as well as (63) is efficient, i.e., solves (51).
4 Optimal Climate Policies
This section determines a climate policy which implements the ω-optimal allocation (49)
as an equilibrium allocation in the sense of Definition 3. Such a policy will be referred
to as an optimal climate policy. The properties of equilibria stated in Proposition 1 and
the separation result from Section 3 allow us to derive the optimal policy in two steps.
In the first step, an efficient climate tax policy τeff is computed which implements the
efficient allocation (52) as an aggregate equilibrium allocation defined as in (45), i.e.,
¯
A(τeff ) = ¯
Aeff . In the second step, an optimal transfer scheme θopt(ω) is computed
based on a given weighting scheme ωwhich together with τeff implements the ω-optimal
allocation as an equilibrium allocation, i.e., A(τeff , θopt(ω)) = Aopt(ω). Evidently, the
latter requires the choice of a weighting scheme ω. We offer a simple idea how such a
scheme could be chosen such that each region has an incentive to adopt the optimal
climate tax policy.
4.1 The optimal climate tax policy
Using the efficient allocation (52), define the climate tax policy τeff = (τeff
t)t0as
τeff
t=
X
n=0
βnu(¯
Ceff
t+n)
u(¯
Ceff
t)φL+(1φL)φ0(1φ)nX
L
D(Seff
t+n)Q
0,tF0Kℓ,eff
0,t , Nℓ,eff
0,t ,(Eℓ.eff
i,t )iI.
(64)
Equation (64) is a classical example of a Pigovian tax which equates taxes to the total
discounted marginal damage caused by each unit of CO2determined by (56). Under
this policy, emissions taxes τ
tτeff
tare uniform across dirty sectors in all countries and
incorporate the total damage from emitting one unit of CO2in period t. The following
result shows that this policy implements the efficient solution (52) as an aggregate
equilibrium allocation defined as in (45). Recall from Proposition 1 that this allocation
is independent of the transfer policy. Thus, the CTP defined by (64) will be referred to
as the efficient tax policy.
Theorem 2
Let Assumptions 1 and 2 hold and define the climate tax policy τeff as in (64). Then, the
induced aggregate equilibrium allocation defined in (45) is efficient, i.e., ¯
A(τeff ) = ¯
Aeff .
24
With the specific functional forms (9) and (11) for climate damage and consumer utility,
the efficient tax formula (64) takes the more specific form
τeff
t=
X
n=0
βn¯
Ceff
t+n
¯
Ceff
tσφL+ (1 φL)φ0(1 φ)nX
L
γYℓ,eff
t+n.(65)
In particular, efficient taxes are zero if climate damages are absent, i.e., γ0.
In general, expression (65) can not be computed explicitly, as it involves the entire paths
of future output in each region and aggregate consumption. However, if the efficient
solution induces a balanced growth path on which output and consumption grow at
constant and identical rate g0, (65) takes the much simpler form
τeff
t= ¯τeff X
L
γYℓ,eff
t,¯τeff := φL
1β(1 + g)1σ+φ0
1φL
1β(1 + g)1σ(1 φ).(66)
Thus, on a balanced growth path, the optimal tax is a constant share ¯τeff of world output
weighted by the damage parameters γ. For logarithmic utility (σ= 1) and homogeneous
climate damages (γγ), equation (66) recovers precisely the tax-formula derived in
GHKT under a set of additional restrictions (log utility, Cobb-Douglas production, all
capital used in the final sector).17 None of these restrictions is required here if the
assumption of a balanced growth path is satisfied. The numerical simulations presented
in Hillebrand & Hillebrand (2018) and in Section 5 below show that the efficient solution
converges quickly to a balanced growth path suggesting that (65) is well-approximated
by (66) in applications of the model.
4.2 Optimal transfer policies
Under the efficient climate tax policy (64), the aggregate equilibrium allocation (45)
coincides with the efficient allocation (52). While this determines aggregate world con-
sumption together with an optimal allocation of production factors and climate variables
in each period, it leaves undetermined how world consumption is distributed across coun-
tries. The latter requires the choice of a weighting scheme ω+based on which the
optimal consumption distribution µopt(ω) can be determined by (53).
We now explore the existence of a transfer policy θunder which the induced equilibrium
allocation A(τeff , θ) coincides with the optimal allocation Aopt(ω) defined in (49) in
which each region attains the optimal consumption share µopt(ω). As before, let τeff be
the efficient tax policy which implements the efficient aggregate allocation ¯
Aeff and, by
Lemma 3 also determines the price system Peff supporting the efficient allocation. Let
Wℓ,eff denote the induced lifetime non-transfer income of consumers in region defined
17A general formula for the Social Cost of Carbon and its approximation along a balanced growth
path is derived in van den Bijgaart et al. (2016).
25
in (31), Weff := PLWℓ,eff aggregate non-transfer lifetime income and Teff the total
tax revenue defined as in (15). Given a weighting scheme ω+and consumption
shares µopt(ω) = (µℓ,opt(ω))Ldetermined by Lemma 4, consider the following transfer
policy θopt = (θℓ,opt)Ldefined for each Las
θℓ,opt(ω) = µℓ,opt(ω)Weff +Teff Wℓ,eff
Teff .(67)
Note that (67) determines consumer ’s lifetime cum-transfer income W+Tto be a
share µℓ,opt(ω) of world cum-transfer income Weff +Teff . The following result shows that
the transfer policy (67) together with τeff constitutes indeed an optimal climate policy.
Theorem 3
Let Assumptions 1 and 2 hold and define the climate tax policy τeff as in (64). Given any
weighting scheme ω+, define µopt(ω)by (53) and the transfer policy θopt(ω)by (67).
Then, the induced equilibrium allocation is ω-optimal, i.e., A(τeff , θopt(ω)) = Aopt(ω).
4.3 A Pareto-improving transfer policy
Applying the optimal transfer policy defined in (67) requires the choice of a particular
weighting scheme ω+or, equivalently, invoking Lemma 4 (ii), a desired consumption
distribution µ= (µ)L+. This raises the question how such a distribution can
and should be determined. Ideally, one might want to choose ωresp. µas an equal
weighting scheme based, e.g., on relative population sizes to ensure a fair world allocation
of consumption. In any quantitative study of the model, however, such a choice would
induce massive transfers unrelated to climate change but due to the fact that the world
income distribution is very unequal and strongly biased towards industrialized countries.
The analysis of this paper, however, is not about fairness and redistribution of world
income but how transfers can be determined such that each region has an incentive to
implement the optimal tax on emissions. For this reason, the present section offers an
alternative approach which chooses the consumption distribution based on the shares
that each region attains in the Laissez faire allocation. This target seems a natural choice
because the Laissez faire solution corresponds to the extreme case where all countries
agree not to take any measures against climate change. It is therefore a natural threat
point in any bargaining process about transfers.
Formally, let µLF = (µℓ,LF)Ldenote the consumption shares along the Laissez faire
equilibrium allocation ALF which are constant by Lemma 3 (ii). Using the same notation
as in the previous subsection, define the transfer policy θLF = (θℓ,LF )L∆ as
θℓ,LF := µℓ,LF Weff +Teff Wℓ,eff
Teff , L.(68)
26
Under transfer policy θLF, each region attains the same relative wealth and the same
share µℓ,LF of world consumption along the efficient equilibrium allocation as in the
Laissez faire allocation. The following main result shows that this policy makes each
country better-off, i.e., consumers in each region enjoy utility strictly higher than in the
Laissez faire allocation if they agree to jointly implement the efficient tax policy.18
Theorem 4
The equilibrium allocation A(τeff , θLF)Pareto-improves the laissez faire allocation ALF .
5 Quantitative results
This section provides a quantitative example which illustrates the previous theoretical
results.19 We consider the case with two regions (L= 2) which are broadly calibrated
to match rich (= 1) and poor (= 2) world regions represented by OECD and Non-
OECD countries. Details on these calibrations and the computation of equilibria in our
model can be found in Hillebrand & Hillebrand (2018).
5.1 Calibration
The simulation starts in t= 2015 and ends in t= 2215 with one period representing ten
years which is a standard choice in the literature. Initial world labor supply N1,s
0+N2,s
0is
normalized to one and distributed across regions based on empirical population shares.
Regional differences in productivity are captured by the parameters Q
0,t Q
0in (1)
which are constant and chosen to match the world income distribution between regions
and an initial world output of 700 trillion current US$ as in GHKT. Growth enters
our model via labor augmenting technological change due to which labor supply Nℓ,s
tin
each region grows at constant rate g. As argued in the previous section, population
growth can be incorporated in the discount factor βwhich is chosen as in GHKT. We
also assume a logarithmic utility function by setting σ= 1.
There are two energy sectors (I= 2) one of which (i= 1) produces dirty energy based
on fossil fuels and the other (i= 2) clean energy based on renewable resources. The
production functions in equations (1), (2), and (3) are specified as follows:
F0(K, N, E1, E2) = Kα0N1α0ν0[κ(E1)̺1
̺+ (1 κ)(E2)̺1
̺]
̺ν0
̺1(69a)
F1(K, N, X) = Xν1Kα1N1α1ν1, F2(K, N ) = Kα2N1α2.(69b)
18Clearly, this does not eliminate the free-riding problem that a single region may have an incentive
to deviate from the optimal policy. A more elaborate game-theoretic analysis of this problem within
the previous framework is beyond the scope of this paper but left for future research.
19The simulation data are available at http://www.marten-hillebrand.de/research
27
We choose the shares of capital α0and energy ν0in (69a) as in GHKT and set ̺= 3
to obtain a high substitutability between clean and dirty energy as in Acemoglue et
al. (2012) or Rezai & van der Ploeg (2015) who even assume the two energy types to
be perfect substitutes. Choosing κ= 0.5 yields a unit relative price between clean and
dirty energy also assumed in GHKT. The production elasticities α1, ν1,α2in (69b) are
chosen to match the empirical cost structure of energy sectors reported in Hillebrand &
Hillebrand (2018). Productivity parameters Q
i,t Q
iin energy production (2) and (3)
are constant and chosen to obtain an empirically plausible energy mix in each region
with clean energy acquiring a share of 22% in rich and 14% in poor countries in the
initial period (see Hillebrand & Hillebrand (2018) for details).
Exhaustible resources in our simulation model comprise all fossil fuels (coal, oil, and
natural gas). As GHKT, we assume that these resources are abundant and do not
have a scarcity rent by setting R1
0=. By (24) and Lemma 3(a), resource prices are
thus equal to extraction costs c1in each period. To obtain a plausible value for c1,
we combine the two estimates for coal and oil extraction from Hillebrand & Hillebrand
(2018) weighted by the empirical global resource shares. The same approach determines
the Carbon content ζ1of fossil fuels as a weighted average of coal, oil, and natural gas.
The parameters defining the climate model (6) and damage function (9) are chosen
as in GHKT and the initial climate state S1= (S1,1, S2,1) is chosen to match the
empirically observed CO2concentration of 850 Gigatons of carbon in t= 2015 with the
same shares of permanent and non-permanent carbon as in GHKT. Table 1 summarizes
our parameter set.
Consumer sector Final sector Energy sector 1 Energy sector 2 Climate model
N1,s
00.18 Q1
03.23 Q1
14.2 Q1
220 S1,1722
N2,s
00.82 Q2
00.65 Q2
112 Q2
240 S2,1110
K00.18 α00.3 α10.52 α20.75 φL0.2
g0.16 ν00.04 ν10.27 φ0.0228
β0.98510 ̺3c10.000071 φ00.393
σ1κ0.5 ζ10.5835 γ0.000053
¯
S581
Table 1: Parameter set.
5.2 Equilibrium dynamics
Our parametrization yields an optimal carbon tax equal to 34 $/t CO2in 2015, which is
in the range of optimal emissions taxes reported, e.g., in Nordhaus (2007) and Golosov
et al. (2014). Comparing the Laissez-faire and optimal solution, Figure 1 depicts the
predicted evolution of selected economic and climate variables over the next 200 years.
28
As one would expect, the optimal emissions tax curbs global emissions which stay flat
over the entire time window and decline to zero asymptotically. This limits the increase
in global temperature20 to a maximum of 2 degrees relative to pre-industrial level and
is thus in line with the two-degree target. Laissez-faire causes an exponential increase
in emissions and temperature exceeding the two-degree target within the next 50 years.
At the disaggregated level, GDP in both regions follows a balanced growth path under
optimal taxation with roughly constant growth rates while these growth rates decline
continually under Laissez-faire. Introducing the optimal tax comes at a slight initial
cost which is limited to at most 0.8% of GDP in both regions. After fifty years, the gain
in GDP relative to Laissez faire becomes positive and increases continually thereafter.
0
200
400
600
800
1000
2015
2035
2055
2075
2095
2115
2135
2155
2175
2195
2215
replacemen
GtC
optimallaissez faire
(a) Global emissions
10
2015
2035
2055
2075
2095
2115
2135
2155
2175
2195
2215
C
laissez faire optimal
(b) Global temperature
10
2015
2035
2055
2075
2095
2115
2135
2155
2175
2195
2215
Trn. $
rich,lf rich,opt poor,lf poor,opt
(c) GDP in both regions
-
5
0
5
10
15
20
2015
2035
2055
2075
2095
2115
2135
2155
2175
2195
2215
% of GDP
rich countries poor countries
(d) GDP loss/gain relative to Laissez-faire
Figure 1: Evolution under laissez faire and optimal taxation.
5.3 Pareto-improving transfers
The distribution of global tax revenue depends on the weighting scheme ω= (ω1, ω2)
which is fully determined by the weight ω1attached to rich countries. By Lemma 4
and (54), our choice σ= 1 permits to interpret ω1directly as the target consumption
share of rich countries along the optimal equilibrium. An important reference is the case
where ω1equals the consumption share along the laissez faire equilibrium. Denote this
20Following GHKT, we compute global temperature in period tas T EM Pt=λlog(St/¯
S)/log(2),
cf. their equation (25) on page 66.
29
choice by ω. By Theorem 3, choosing ω1=ωinduces a Pareto improvement relative to
laissez faire. To obtain the set of Pareto-improving weighting schemes, we compute two
critical values ω
min and ω
max. For ω1=ω
max, region 2 is exactly indifferent between the
optimal equilibrium and laissez faire while region 1 is exactly indifferent if ω1=ω
min.
Thus, any choice ω1[ω
min, ω
max] induces a Pareto-improvement. For each of the three
-3,0%
-1,0%
1,0%
3,0%
2015
2035
2055
2075
2095
2115
2135
2155
2175
2195
2215
% of domestic GDP
ω
min
ω
ω
max
(a) Rich countries
-5,0%
0,0%
5,0%
10,0%
2015
2035
2055
2075
2095
2115
2135
2155
2175
2195
2215
% of domestic GDP
ω
min ωω
max
(b) Poor countries
Figure 2: Pareto-improving transfers between rich and poor countries
cases ω1∈ {ω
min, ω, ω
max}, Figure 2 depicts the associated transfers received by each
region which are expressed as a percentage of regional GDP. For ω1=ω, rich countries
are entitled to receive about 63.4% of global tax revenue corresponding to a transfer of
roughly half a percentage point of their GDP in each period. Transfers received by poor
countries vary between 0.5% and 0.7% of domestic GDP. For ω1=ω
max, rich countries
receive 213.8% of global tax revenue which must partially be financed by consumers in
the poor region who would have to pay a lump-sum tax. This tax amounts to 1.5% of
GDP in the initial period and increase to up 2.1% in subsequent periods. Conversely,
for ω1=ω
min, poor countries receive a share of 382.7% of total tax revenue in which
case rich countries must levy a tax on their consumers. This tax amounts to initially
1.6% of GDP and increases to up to 2.3% in subsequent periods. These figures define a
range of possible transfer payments that both regions could agree on in negotiations on
the optimal climate policy.
In absolute terms, transfers in the initial period compute as follows. Annual GDP in rich
countries is 48.8 trillion U.S.$ and 21.4 trillion U.S.$ in poor countries. Global annual
tax revenue is about 283.7 billion U.S.$. For ω1=ω, rich countries receive an annual
transfer of about 179.9 billion U.S.$ and poor countries about 103.8 billion U.S.$. If
ω1=ω
max, poor countries must raise an annual tax revenue of 322.8 billion U.S.$ to
finance an annual transfer of 606.5 billion U.S.$ to rich countries. Finally, if ω1=ω
min,
poor countries receive an annual transfer of 1085.6 billion U.S.$ part of which is financed
by a tax on consumers in rich countries which amounts to 801.6 billion U.S.$.
30
6 Extensions and discussion
The present section discusses which of our previous restrictions on technologies, pref-
erences, and the climate model are crucial for our results and which ones are merely
convenient to work with and can be relaxed. For the sake of readability, some of the
underlying formal arguments are relegated to Section A.11 in the appendix.
6.1 Climate change affecting utility
An alternative approach to modeling the adverse effects of climate change is to assume
a direct impact on consumer utility. Models which adopt this idea include Acemoglu
et al. (2012), Barrage (2017), Gerlagh & Liski (2016, 2018), or Rezai & van der Ploeg
(2016). Qualitatively, all our main results continue to hold if we modify our period
utility function (11) to be of the form
u(C, S) = log Cv(S) (70)
where vis any differentiable, increasing, and concave function of carbon concentration
defined as in (7). Since both climate damage Dtand global temperature can be written
as functions of St, specification (70) also includes cases where utility depends negatively
on global temperature or damage as in Barrage (2017).
In the decentralized solution, consumers in each region now take the sequence (St)t0
as an additional exogenous parameter in their decision problem. The additive struc-
ture (70) then implies that the solution to this problem and, therefore, all equilibrium
conditions derived in Section 2 remain unchanged. By contrast, both the weighted
and aggregate planning problems (48) and (51) now incorporate the direct impact of
emissions on utility. Thus, the social costs of carbon now comprises both the damage to
production and consumer utility. Formally, Λt= Λprod
t+Λcons
twhere Λprod
tis determined
by (56) as before while Λcons
tcaptures the marginal disutility from emissions given by
Λcons
t=
X
n=0
βn¯
C1
t+n
¯
C1
tφL+ (1 φL)φ0(1 φ)nv(St+n)
¯
C1
t+n
.(71)
Note the similarity of the previous formula to the one derived in Barrage (2017), see her
equation (20). Economically, Λcons
trepresents the marginal welfare loss of a fictitious
representative world consumer who has the same utility (70) as households in each
region. In this regard, it is important to note that the separability result from Theorem
1 requires utility (70) to be logarithmic in consumption and breaks down under the
more general form (11) if σ6= 1. Defining the emissions tax as τeff
t= Λtfor all tthen
implements the efficient solution as an aggregate equilibrium allocation as before.
31
6.2 An alternative climate model
Our specification of the climate model is taken directly from GHKT. An alternative,
more recent specification is developed in Gerlagh & Liski (2018) which permits to de-
compose the size, delay, and persistence of emissions in the atmosphere. Using our
notation, the key equation of their climate model determines atmospheric CO2as
St=
X
τ=0
δτZtτ.(72)
Here, Ztdenotes total CO2-emissions defined as in (5) which are set to zero for tT
where Tis the beginning of the industrial revolution. Equation (72) is derived in Gerlagh
& Liski (2018) based on a deeper specification of the underlying climate model with
multiple layers which contains the three-layer specification used in the DICE/RICE
model as a special case. Thus, equation (72) also covers the climate model adopted in
Nordhaus & Boyer (2000). Also note that Gerlagh & Liski (2018) assume δ0= 0 in
which case emissions do not have an instantaneous effect on the climate state.
Using now (5), (8), and (72) as our climate model while maintaining all other previous
assumptions, the main results of the previous analysis continue to hold. In particular,
the efficient allocation can be computed exactly as before and permits to obtain an
optimal allocation via Theorem 3. The only major change is that the social cost of
carbon (56) defining the efficient tax τeff
t(64) now takes the form
Λt=
X
n=0
βnu(¯
Ct+n)
u(¯
Ct)δnX
L
D(St+n)Q
0,t+nF0K
0,t+n, N
0,t+n,(E
i,t+n)iI.(73)
As Gerlagh & Liski (2018) also assume an exponential damage function, their model
is included by our damage function (9) with region-specific damage parameters γ.21
Under this specification, the social cost of carbon (73) takes the simpler form
Λt=
X
n=0
βnu(¯
Ct+n)
u(¯
Ct)δnX
L
γY
t+n.(74)
One observes that due to (11) the SCC (74) is again proportional to damage weighted
output PLγY
ton a balanced growth path. Thus, the alternative climate model (72)
fully preserves our multi-region version (66) of the GHKT result.
6.3 Population growth
The previous framework formally assumed a stationary population in each region. It is
now straightforward to modify this restriction and include constant population growth
21In the notation of Gerlagh & Liski (2018), final output can be expressed as yt=Atω(st)f(kt, et, ly,t)
where total factor productivity is ω(st) = exp{− P
τ=1 θτztτ}with stbeing the history of emissions
before t. Function fdepends on capital, energy, and labor and Atis a productivity parameter.
32
in our model. To see this, denote total consumption in region by C
tas before and let
L
tdenote the population size corresponding to the number of consumers in region in
period t. Assume that the population grows at constant rate g
Lsuch that
L
t= (1 + g
L)L
t1(75)
with the initial population to normalized to L
0= 1. Adopting the same approach as in
Rezai & van der Ploeg (2015), define aggregate utility in region as
U((C
t)t0) =
X
t=0
L
tβtu(C
t/L
t) (76)
which replaces our earlier specification (10). Exploiting the functional form (11) and
using L
0= 1 and (75) permits to write utility (76) equivalently as
U((C
t)t0) =
X
t=0
(β(1 + g
L))tu(C
t) + ¯u
if σ= 1 and
U((C
t)t0) =
X
t=0
(β(1 + g
L)σ)tu(C
t) + ¯u
if σ6= 1. Here, ¯uis a constant determined by exogenous parameters that has no economic
implications. Thus, population growth merely transforms the original discount factor
βto ˆ
β:= β(1 + g
L)σ. Since our results require all regions to share the same discount
factor, the model is compatible with constant population growth as long as growth rates
are uniform across regions, i.e., g
LgLand the transformed discount factor satisfies
ˆ
β < 1. In this case, all our previous results continue to hold without change.
6.4 Redistribution via non-lump sum transfers
The revenue from taxation of emissions is distributed to consumers in the form of lump-
sum transfers. In the literature, the assumption of lump-sum taxes is frequently chal-
lenged as unrealistic. In principle, similar arguments could be made against lump-sum
transfers, although they seem much less controversial.22 Thus, it is important to discuss
how our results change if lump-sum transfers are not available.
Alternative transfers could take the form of proportional subsidies on labor income,
capital income, or final consumption. Denoting the percentage subsidy on these incomes
as η
n,t,η
k,t η
c,t, respectively, the consumer’s period budget constraint (28) changes to
C
t(1 η
c,t) + K
t+1 = (1 + η
k,t)rtK
t+ (1 + η
n,t)w
tNℓ,s
t+ Π
t.(77)
22The assumption of lump sum transfers is also made in GHKT and in Hassler & Krusell (2012).
Even Barrage (2017), who studies how distortionary taxation affects the optimal climate tax assumes
that governments can make lump-sum transfers to consumers.
33
As before, denote by θthe share of total tax revenue Tt:= τtPLPiIxζiX
i,t received
by region . These revenues are fully distributed as subsidies to consumers such that
η
c,tC
t+η
k,trtK
t+η
n,tw
tNℓ,s
t=θTt.(78)
The consumer’s intertemporal optimality condition (34) now takes the form
rt+1(1 + η
k,t+1)βC
t+1
C
tσ1η
c,t
1η
c,t+1
= 1.(79)
Note that wage subsidies do not enter this condition. Thus, if all transfers are paid as
proportional subsidies η
n,t on labor income, this merely changes the consumers lifetime
income (31) to W=r0K
0+Π+P
t=0 qt(1+η
n,t)w
tNℓ,s
t. Thus, the structural properties
of equilibrium from Proposition 1 remain unchanged and by setting η
n,t := θTt
w
tNℓ,s
t
, an
optimal climate policy can be obtained exactly as before. Clearly, the key feature
required for this result is that labor supply is exogenous which is a standard assumption
made in most models of climate change (an exception is Barrage (2017)).23 In this case,
wage subsidies are non-distortionary and essentially equivalent to lump sum transfers.
One also observes from (79) that the distortionary effect of consumption subsidies van-
ishes if η
c,t η
c. Such a constant choice, however, will typically not satisfy condition
(78) for all t. If constant consumption subsidies are amended by non-distortionary wage
subsidies chosen to satisfy (78), Proposition 1 and all major results remain true.
Transfers paid either as capital subsidies or time varying consumption subsidies distort
the consumer’s intertemporal first order condition. In such cases, aggregation of this
constraint, which is key for the result from Proposition 1 is no longer possible. While
the first best optimal allocation could still be determined as before, implementing this
allocation as an equilibrium by separating efficiency from distribution is no longer pos-
sible, since transfers also affect the aggregate equilibrium allocation. To determine an
optimal climate policy based on distortionary transfers, one would most likely have to
employ a Ramsey framework as in Barrage (2017) which incorporates the response of
consumers to the proposed climate policy and maximizes a weighted utility index of the
form (47). We do not expect to preserve our separation result under such a modification.
In this sense, the existence of non-distortionary transfers is crucial for our main results.
6.5 Relation to RICE and Negishi weights
It is important to contrast our approach of determining an optimal policy to the one
adopted in the RICE model as described in Nordhaus & Yang (1996), see also Nord-
haus & Boyer (2000). To determine an efficient allocation in this model, all endogenous
23From an empirical standpoint, the assumption of perfectly inelastic labor supply seems justified
whenever the length of each time period is sufficiently long such that short term business cycle fluctua-
tions are eliminated. This is certainly the case if one model period corresponds to ten years, a standard
parametrization used in Acemoglu et al. (2012), GHKT, and also in Section 5.
34
variables are determined by solving a planning problem of the type (48) with (initially)
constants weights. Unlike our approach which treats these weights as policy parame-
ters, however, the RICE algorithm determines the weights endogenously to make the
solution compatible with an (asymptotic) no-trade equilibrium. This is required by
the restrictions on regional trade in the RICE framework. While the existence of such
weights would ensure that the resulting allocation is Pareto-optimal and, therefore, can
be interpreted as a competitive equilibrium allocation due to Negishi’s theorem (Negishi
(1960)), the solution is then again perturbed by choosing time-varying weights, referred
to as ’modified Negishi-weights’. This perturbation is done in order to avoid excessive
reallocation of capital across regions but does not necessarily lead to a Pareto-optimal
allocation (see Denning & Emmerling (2017) for a discussion of these problems). Thus,
the analysis in RICE does not determine a socially optimal allocation in our sense for
arbitrary weights but the solution is - at best - Pareto optimal. Further, it is not pos-
sible to treat the welfare weights as policy parameters representing bargaining power,
population size, fairness considerations, etc. as we do. Instead, they are dictated by
economic restrictions and the employed methodology in the RICE framework.
6.6 Relation to single-region models
Disaggregated consumption in our model is fully described by aggregate consumption.
This raises the question whether the production side can be aggregated in a similar way
such that world output can be written as a function of aggregate factor inputs. It turns
out that the answer is negative unless technologies in all regions are identical. To see
this formally, consider an arbitrary period t. Dropping time index tfor convenience, let
damages Dand sector-specific productivities (Q
i)iI0in each region Lbe given.
Substitute (2) and (3) into (1) to obtain final output in region as
Y=F((K
i, N
i, X
i)iI0) := (1 D)Q
0F0(K
0, N
0,(Q
iFi(K
i, N
i, X
i))iI).
The derived production functions Fare linear-homogeneous. They are identical across
regions precisely when damages and productivities are identical, i.e., FFif and only
if DDand Q
iQifor all iI0. In this case, standard arguments can be used to
show that aggregate output Y:= PLYis determined by aggregate inputs of capital
Ki:= PLK
i, labor Ni:= PLN
i, and exhaustible resources Xi:= PLX
i, i.e.,
Y=F((Ki, Ni, Xi)iI0).
The previous argument breaks down as soon as climate damages or productivities in at
least one sector differ across regions. Thus, it is not possible to represent the production
side by an aggregate technology except for the trivial case where regions are essentially
identical. In all other cases, world output is determined by a technology that involves all
disaggregated production variables. In our calibrated example and also in our companion
35
paper Hillebrand & Hillebrand (2018), we show that differences in productivities are
crucial to match certain regional characteristics of the production process. Moreover,
differences in climate damages are known to be particularly important when regions
represent rich and poor countries, cf. Bretschger & Suphaphiphat (2014) and references
therein. For these reasons, our framework is not equivalent to a single region model but
permits to study many important questions which are due to regional heterogeneity and
can not be addressed in a single region model.
7 Conclusions
The problem of determining an optimal climate path and an efficient allocation of pro-
duction factors and exhaustible resources admits a unique solution which is completely
independent of the interests of different countries. This solution can be implemented as
a decentralized equilibrium by levying a uniform global tax on carbon emission which
can be computed (or approximated) in closed form. In principle, all countries should
agree on this policy.
The real issue in the political debate about climate change is therefore not how and
where emissions should be taxed, but rather how countries should share the tax revenue
via transfers. These transfers determine the world distribution of consumption or income
and provide a mechanism to compensate regions for climate damages. As the choice of
an optimal transfer policy induces a trade-off between the interests of different countries,
one might want to determine the transfer policy such that each region has an incentive
to implement the optimal tax policy. An example of a transfer policy which leads to a
Pareto-improvement relative to the Laissez faire allocation was devised in this paper.
The latter results mark only a first step towards a more elaborate model of the political
process which determines climate policies. In future research, we intend to model this
process as a (cooperative or non-cooperative) game between regions as in Dutta &
Radner (2006). Within the framework of this paper, Lwould be the set of players each
of which chooses a domestic emissions tax policy τas their strategy and receives utility
of domestic consumers as their pay-off. Transfers across regions then correspond to side
payments which can be used to incentivize each region to implement a certain strategy.
This raises the question whether there exists a transfer policy under which the optimal
climate tax policy derived in this paper can be obtained as the Nash equilibrium of this
non-cooperative game. One could also study cooperative versions of this game where
some regions join forces to combat climate change by forming coalitions.
In addition, the framework developed in this paper can be extended in various directions.
One such extension is to replace the deterministic setup by a stochastic environment
with random perturbations which allows to include various forms of uncertainty into the
model. A second extension is a setup with endogenous growth and directed technical
36
change as in Acemoglu et al. (2012). Both modifications were considered in GHKT and
we believe that our framework is also amendable to them.
It is also well-known in the literature that a uniform carbon tax across regions may fail
to be optimal if lump-sum transfers are not available, cf. Chichilnisky & Heal (1994) or
d’Autume et al. (2016) and also the discussion in Section 6.4. Including such additional
restrictions in our model to study how this affects the optimal climate policy is a final
goal of future research.
A Mathematical Appendix
A.1 Proof of Lemma 1
Using standard Lagrangian arguments, a non-negative sequence (X
t)t0is a solution to
(23) if P
t=0 X
tR
i,0and there exist non-negative Lagrangian variables (σ
t)t0and
λ0 such that ((X
t, σ
t)t0, λ) solve the first order and Kuhn-Tucker conditions
qt(vi,t ci) + σtλ= 0 t0 (A.1a)
σtXt= 0 t0 (A.1b)
λ
X
t=0
XtR
i,0= 0.(A.1c)
If X
t>0 for all t0, then, σ
t= 0 by (A.1b) and vi,t ciby (A.1a) for all t. Using
q0= 1 and qt/qt1=r1
tfor all t > 0 in (A.1a), resource prices must evolve as in (24).
The remaining assertions follow immediately.
A.2 Proof of Lemma 2
Under Assumption 2, (12) and (34) imply that the solution to (32) evolves as
C
t=C
t1(βrt)1
σ=C
0
t
Y
s=1
(βrs)1
σ=C
0βt
qt1
σ
t1.(A.2)
Using (A.2), the l.h.s. in the lifetime budget constraint (30) can be written as
X
t=0
qtC
t=
X
t=0
qtC
0βt/qt1
σ=C
0
X
t=0 βtqσ1
t1
σ.(A.3)
Using (A.3) in (30) - which holds with equality- gives
C
0=W+T
P
t=0 βtqσ1
t1
σ
.(A.4)
Using (A.4) in (A.2) yields (36).
37
A.3 Proof of Lemma 3
(i) This is a direct consequence of the equilibrium conditions (18), (20), (22) and the
boundary behavior of Fiand uimposed by Assumptions 1 and 2.
(ii) Set µ:= C
0/¯
C
0for L. By Lemma 2, the growth rates of each sequence
(C
t)t0are independent of and equal to the growth rates of aggregate consumption
(¯
C
t)t0. Induction then implies that (44) holds for all t0.
(iii) Let iIxbe arbitrary. If Ri,0=, there exists a region Lfor which R
i,0=.
As profits must be finite at equilibrium, (25) implies vi,0ci.
If Ri,0<, (41) implies limt→∞ X
i,t = 0 for all L. The boundary behavior from
Assumption 1 gives limt→∞ XFiK
i,t, N
i,t, X
i,t=and the claim therefore follows
from (22c).
If Ri,0>P
t=0 PLX
i,t, the same arguments employed to prove (b) give the second
result while the first one follows from Lemma 1 (ii).
A.4 Proof of Proposition 1
(i) Using the result from Lemma 2, aggregate consumption in period tsatisfies
¯
C
t=(βt/qt)1
σPLW+T
P
s=0 qs(βs/qs)1
σ
t0 (A.5)
and determines Kt+1 by (42). These and all other equations relevant to determine the
variables in ¯
Aand prices Pare independent of θ.
(ii) Using Lemma 2 and (14) in conjunction with (46) and (A.5) gives
C
t=(βt/qt)1
σW+θT
P
s=0 qs(βs/qs)1
σ
=µ¯
C
t
for all t0 and L, proving the claim.
A.5 Proof of Lemma 4
(i) Let ω+and ¯
C > 0 be given. It is clear that ω= 0 implies µ= 0 which is
implied by the solution (54). As we can always restate problem (54) as a maximization
problem involving only those µfor which ω>0, the remainder of the proof assumes
w.l.o.g. that ω0. The boundary behavior of udefined in (11) implies that any
solution to (53) is bounded away from zero by, say, µ= (µ)L0. This and the
constraint PLµ1 define a compact, convex subset of RL
++ on which the map
µ7→ PLωu(µ) is continuous and strictly concave ensuring that (53) has a unique
38
solution µopt. Standard Lagrangian type arguments imply the existence of a multiplier
λ > 0 such that the solution satisfies µ= (ω¯
C1σ)1
σfor all Land PLµ= 1.
Combining these conditions to eliminate ¯
C1σgives (54). For later reference, let a= 0
if σ6= 1 and a= 1 otherwise and use (11) to write the objective function in (53) as
X
L
ωu(µ¯
C) = a u(¯
C) + X
L
ωu(µ)!+ (1 σ)u(¯
C)X
L
ωu(µ).(A.6)
Using (A.6), the maximum value (53) can be expressed as
X
L
ωu(µℓ,opt ¯
C) = (1 σ)m(ω)u(¯
C)σ6= 1
m(ω) + u(¯
C)σ= 1 (A.7)
where m(ω) := PLωu(µℓ,opt(ω)). Note that (1 σ)m(ω)>0 whenever σ6= 1.
(ii) Let ˜µ= (˜µ)L+be arbitrary. Defining ˜ω:= (˜µ)σ(PkL(˜µk)σ)1for each
L, one verifies directly that ˜µ=µopt(˜ω) solves (53) under this weighting scheme.
A.6 Proof of Theorem 1
Let a weighting scheme ω+be arbitrary but fixed and µopt(ω) be the unique solution
to (53). Denote the efficient solution (52) to the APP (51) by ¯
Aeff = ( ¯
Ceff
t,Γeff
t)t0¯
A
and define A= (Ct,Γeff
t)t0Awhere Ct= (C
t)L:= µopt(ω)¯
Ceff
tas in the theorem.
To establish that Ais ω-optimal, i.e., solves (48), let A= (C
t,Γ
t)t0Abe any other
feasible allocation where Ct= (C
t)L,t0. We have to show that
V((C
t)t0;ω)V((Ct)t0;ω).(A.8)
Define aggregate consumption ( ¯
C
t)t0induced by (Ct)t0as ¯
C
t:= PLC
t,t0.
Then, ( ¯
C
t,Γt)t0¯
Aand, since ( ¯
Ceff
t,Γeff
t)t0solves the APP (51),
X
t=0
βtu(¯
C
t)
X
t=0
βtu(¯
Ceff
t).(A.9)
Let a= 1 if σ= 1 and a= 0 otherwise. By (53) and (A.7), we have for all t0
X
L
ωu(C
t)X
L
ωu(µℓ,opt(ω)¯
C
t) = am(ω) + u(¯
C
t)+ (1 σ)m(ω)u(¯
C
t) (A.10)
and
X
L
ωu(C
t) = X
L
ωu(µℓ,opt(ω)¯
Ceff
t) = am(ω)+u(¯
Ceff
t)+(1σ)m(ω)u(¯
Ceff
t).(A.11)
39
Equations (A.10) and (A.11) being true for all t0 and (A.9) then give
V((Ct)t0;ω) =
X
t=0
βtX
L
ωu(C
t)
a m(ω)
1β+
X
t=0
βtu(¯
C
t)!+ (1 σ)m(ω)
X
t=0
βtu(¯
C
t)
a m(ω)
1β+
X
t=0
βtu(¯
Ceff
t)!+ (1 σ)m(ω)
X
t=0
βtu(¯
Ceff
t)
=
X
t=0
βtX
L
ωu(C
t) = V((Ct)t0;ω)
This proves (A.8) and the claim.
A.7 Computing the efficient solution (52)
We adopt a standard Langrangian approach also used in GHKT to characterize the
solution (52). For brevity, use (43) setting E
t:= (E
i,t)iI,D
t:= D(S1,t +S2,t) and adopt
the notational convention from footnote 16 of treating X
i,t as a dummy argument of Fi
if i /Ix. Define Lagrangen multipliers λt:= (λ0,t,((λ
i,t)iI0, λ
N,t)L, λK,t, λS1,t , λS2,t ) for
each t0 and µ= (µi)iIxand the Lagrangian function
L¯
Ct, Kt+1,Kt,Nt,Et,Xt,St,t0,(λt)t0, µ:=
X
t=0
βtu(¯
Ct) (A.12a)
+
X
t=0
λ0,tX
L1D(St)Q
0,tF0K
0,t, N
0,t,E
tCtKt+1 X
LX
iIx
ciX
i,t(A.12b)
+
X
t=0 X
LX
iI
λ
i,tQ
i,tFi(K
i,t, N
i,t, X
i,t)E
i,t(A.12c)
+
X
t=0 X
L
λ
N,tNℓ,s
tX
iI0
N
i,t+
X
t=0
λK,tKtX
LX
iI0
K
i,t(A.12d)
+
X
t=0
λS1,tS1,t S1,t1φLX
LX
iIx
ζiX
i,t(A.12e)
+
X
t=0
λS2,tS2,t (1 φ)S2,t1(1 φL)φ0X
LX
iIx
ζiX
i,t(A.12f )
+X
iIx
µi Ri,0
X
t=0 X
L
X
i,t!.(A.12g)
Standard arguments imply that any solution A= ( ¯
Ct, Kt+1,Kt,Nt,Et,Xt,St)t0to (51)
has to satisfy the first order and complementary slackness conditions. After eliminating
40
as many Lagrangian variables as possible, these conditions can be summarized for all
L,iI, and t0 (suppressing quantifiers when convenient) as:
λ0,t =βtu(¯
Ct) = λK,t+1 (A.13a)
λK,t =λ0,t(1 D
t)Q
0,tKF0K
0,t, N
0,t,E
t=λ
i,tQ
i,tKFi(K
i,t, N
i,t, X
i,t) (A.13b)
λ
N,t =λ0,t (1 D
t)Q
0,tNF0K
0,t, N
0,t,E
t=λ
i,tQ
i,tNFi(K
i,t, N
i,t, X
i,t) (A.13c)
λ
i,t =λ0,t(1 D
t)Q
0,tEiF0K
0,t, N
0,t,E
t(A.13d)
µi=λ
i,tQ
i,tXFi(K
i,t, N
i,t, X
i,t)λ0,t ciζi(φLλS1,t + (1 φL)φ0λS2,t ) (A.13e)
λS1,t =λ0,t X
L
D(St)Q
0,tF0(K
0,t, N
0,t,E
t) + λS1,t+1 (A.13f)
λS2,t =λ0,t X
L
D(St)Q
0,tF0(K
0,t, N
0,t,E
t) + (1 φ)λS2,t+1.(A.13g)
Note that λ0,t can be interpreted as a shadow price of time tconsumption. By the same
token, the time tshadow price of energy of type iIproduced in region Lmeasured
in time tconsumption goods can be defined as
ˆp
i,t := λ
i,t
λ0,t
= (1 D
t)Q
0,tEiF0K
0,t, N
0,t,E
t.(A.14)
Combining (A.13a) and (A.13b) gives the familiar Euler equation
u(¯
Ct1) = βu(¯
Ct)(1 D(St))Q
0,tKF0K
0,t, N
0,t,(E
i,t)iI.(A.15)
As the l.h.s in (A.13e) is independent of and t, we obtain using (A.14)
ˆp
i,tQ
i,tXFi(K
i,t, N
i,t, X
i,t)ζiΛt= ˆp
i,tQ
i,tXFi(K
i,t, N
i,t, X
i,t)ζiΛt=: ˆvi,t (A.16)
for all ℓ, ℓLand t0 and
ˆvi,t ci=βu(¯
Ct+1)
u(¯
Ct)ˆvi,t+1 ci(A.17)
for all t0. Essentially, (A.16) ensures intratemporally efficient allocation and (A.17)
intertemporally effcient extraction of resources.
Assuming that limn→∞ βn+1λS1,t+n= limn→∞ ((1φ)β)n+1λS2,t+n= 0, equations (A.13f)
and (A.13g) can be solved forward to obtain
λS1,t
λ0,t
=
X
n=0
βnu(Ct+n)
u(Ct)X
L
D(St+n)Q
0,t+nF0(K
0,t+n, N
0,t+n,E
t+n) (A.18a)
λS2,t
λ0,t
=
X
n=0
βnu(Ct+n)
u(Ct)(1 φ)nX
L
D(St+n)Q
0,t+nF0(K
0,t+n, N
0,t+n,E
t+n).(A.18b)
Finally, define for t0
Λt:= φL
λS1,t
λ0,t
+ (1 φL)φ0
λS2,t
λ0,t
.(A.19)
Then, using (A.18a) and (A.18b) in (A.19) gives precisely the condition (56).
41
A.8 Proof of Theorem 2
Combining (64) with the conditions derived in Section 2, one can show directly that
the aggregate equilibrium allocation solves the same equations as the efficient solution
which were derived in Section 3.5 resp. A.7. This proves the claim.
A.9 Proof of Theorem 3
The assertion follows directly from Proposition 1 and the transfer policy (67).
A.10 Proof of Theorem 4
Let Lbe arbitrary and ( ¯
Ceff
t)t0and ( ¯
CLF
t)t0be the aggregate consumption se-
quences along the efficient and laissez faire allocation, respectively. By Lemma 3(ii)
and Assumption 2, utility of region along the LF allocation is U((µLF
¯
CLF
t)t0) =
a+b U(( ¯
CLF
t)t0) where aand b > 0 are constants that depend only on µLF. Further,
by construction and Lemma 3(ii), utility of region along the allocation A(τeff , θLF ) is
U((µLF
¯
Ceff
t)t0) = a+b U(( ¯
Ceff
t)t0). Thus, U((µLF
¯
CLF
t)t0)< U((µLF
¯
Ceff
t)t0) if and
only if U(( ¯
CLF
t)t0)< U(( ¯
Ceff
t)t0) which follows directly from the optimality of the
efficient allocation (52).
A.11 Proofs for extensions in Section 6
Establishing the following results requires mostly straightforward modifications of the
Lagrangean approach used in Section A.7. The following sections highlight how the
arguments change depending on the respective modification. Furthermore, introducing
damages in the utility function requires a slight modification of the arguments to prove
Theorem 1 which are also explained.
A.11.1 Climate change affecting utility
First, we show that Theorem 1 still holds under the alternative specification (70). Using
the same notation as in the proof of Theorem 1, denote the efficient solution (52) by
¯
Aeff defining efficient aggregate consumption ( ¯
Ceff
t)t0and climate states (Seff
t)t0. Let
µopt(ω) be the unique solution to (53) which is independent of the additive term v(St)
in (70) and thus the same as before. Let AAbe any other feasible allocation
defining disaggregated consumption (C
t)t0with C
t= (C
t)L, climate states (S
t)t0,
and aggregate consumption ( ¯
C
t)t0where ¯
C
t:= PLC
t. Clearly,
X
L
ωlog(C
t)v(S
t)X
L
ωlog(µℓ,opt(ω)¯
C
t)v(S
t).(A.20)
42
Exploiting the logarithmic structure in (70) and (A.20) gives the desired result
V((Ct, S
t)t0;ω) =
X
t=0
βtX
L
ωlog(C
t)v(S
t)x
X
t=0
βtX
L
ωlog(µℓ,opt(ω)¯
C
t)v(S
t)
=
X
t=0
βt X
L
ωlog(µℓ,opt(ω)) + log( ¯
C
t)v(S
t)!
X
t=0
βt X
L
ωlog(µℓ,opt(ω)) + log( ¯
Ceff
t)v(Seff
t)!
=
X
t=0
βtX
L
ωlog(µℓ,opt(ω)¯
Ceff
t)v(Seff
t)=V((µopt(ω)¯
Ct, Seff
t)t0;ω).
To compute the efficient solution, replace utility in the objective function in (A.12a) by
(70). The first order conditions (A.13f) and (A.13g) now change to
λS1,t =βtv(St) + λ0,t X
L
D(St)Q
0,tF0(K
0,t, N
0,t,E
t) + λS1,t+1
λS2,t =βtv(St) + λ0,t X
L
D(St)Q
0,tF0(K
0,t, N