IMES DISCUSSION PAPER SERIES

Climate Change Mitigation:
How Effective is Green Quantitative Easing?
Raphael Abiry, Marien Ferdinandusse, Alexander Ludwig,
and Carolin Nerlich
Discussion Paper No. 2022-E-11

INSTITUTE FOR MONETARY AND ECONOMIC STUDIESBANK OF JAPAN
2-1-1 NIHONBASHI-HONGOKUCHOCHUO-KU, TOKYO 103-8660
JAPAN

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IMES Discussion Paper Series 2022-E-11
July 2022

Climate Change Mitigation: How Effectiveis Green Quantitative Easing?
Raphael Abiry*, Marien Ferdinandusse**, Alexander Ludwig***,and Carolin Nerlich****Abstract
We develop a two sector incomplete markets integrated assessment model to analyze
the effectiveness of green quantitative easing (QE) in complementing fiscal policies
for climate change mitigation. We model green QE through an outstanding stock ofprivate assets held by a monetary authority and its portfolio allocation between aclean and a dirty sector of production. Green QE leads to a partial crowding out ofprivate capital in the green sector and to a modest reduction of the global temperatureby 0.04 degrees of Celsius until 2100. A moderate global carbon tax of 50 USD is 4
times more effective.
Keywords: Climate Change; Integrated Assessment Model; 2-Sector Model; Green
Quantitative Easing; Carbon Taxation
JEL classification: E51, E62, Q54* Goethe University Frankfurt (E-mail: abiry.econ@gmail.com)** European Central Bank (E-mail: marine.ferdinandusse@ecb.europa.eu)*** Goethe University Frankfurt, ICIR & CEPR (E-mail: mail@alexander-ludwig.com)**** European Central Bank (E-mail: Carolin.Nerlich@ecb.europa.eu)
We thank Leo Kaas, Alexander Monge-Naranjo, Conny Olovsson, Loriana Pelizzon, Toàn Phan,
Rick van der Ploeg, an anonymous referee and seminar participants at various seminars andconferences for helpful comments and we gratefully acknowledge funding of this research
project through INSPIRE, Climateworks Foundation. We also thank Melina Papoutsi for sharing
the data on the aggregate sectoral shares of the ECB's portfolio holdings. The views expressedin this paper are those of the authors and do not necessarily reflect those of the ECB.

1Introduction

Climate change evoked by mankind will be one of the greatest global challenges in the next
decades. Since pre-industrial times, global temperature has increased by approximately 1.1 degrees
of Celsius as a result of carbon and other greenhouse gas emissions IPCC (2021). If this trend
were to continue, extreme weather events would not only become more frequent—causing large
macroeconomic costs—but the world would also observe irreversible global environmental damages.
To effectively reverse this trend, ambitious policy measures need to be adopted as the window ofopportunity to act is closing rapidly. While there is broad consensus on the effectiveness andusefulness of carbon pricing as a policy tool to combat climate change, recently a vivid debate
emerged on whether and how central banks should play a role in addressing climate change.
This paper examines whether central banks—which we throughout the paper refer to as amonetary authority—can contribute to mitigate climate change through green quantitative easing(QE). Our key research question is whether a portfolio shift of the monetary authority towards thegreen sector of the economy can effectively reduce climate change and how this compares withfiscal mitigation policies, such as a global carbon tax. In our setup, green QE refers to portfolio
reallocation of a given outstanding stock of bonds held by the monetary authority, which is initiallymarket neutral, i.e. proportionally split across the two sectors, towards bonds issued by the green
sector of production.
To address our research question, we develop a two-sector quantitative integrated assessment
model where aggregate output is produced employing clean and dirty intermediate goods. Markets
are incomplete for two reasons. First, households face risky asset returns in the two intermediate
goods sectors and can self insure against this risk by saving in a risk-free bond. Second, there exists
a climate change externality leading to a damage to aggregate output, as frequently employed in
the climate change literature. The model is calibrated to the world economy with one monetaryand one fiscal authority, which comes along with the implicit assumption that both these authorities
coordinate on the introduction of a global carbon tax and green QE.
Intermediate goods in the economy are produced using capital and labour, and either clean or
dirty energy as inputs. Energy production itself takes place using a simple technology employing
some exogenously growing technology level and labour as the only input. Dirty energy production
leads to an accumulation of carbon in the atmosphere, which causes an increase of the global
temperature leading to a damage to aggregate output.
Households live until infinity and maximize their expected discounted life-time utility over consumption streams. Every household runs two intermediate goods firms in the two sectors by
employing its own household capital and by hiring labour and energy on the respective labour andenergy market. Since the return processes on capital in the two firms is stochastic, households are1

heterogeneous, with their heterogeneity resulting from different (histories of) return realizations.
This return risk is idiosyncratic, thus there is no aggregate risk in the economy. The shocks on the
returns of the two capital stocks are imperfectly correlated across sectors. This is an importantfeature of the model as it implies that in response to a reallocation of capital by the monetary
authority, private households will only partially offset its effect by reallocating their savings andthus their will be only a partial crowding out.
Households not only hire labour on the market for production, but also exogenously supply
their own labour on the market and from this labour supply they earn a deterministic wage income.
Given these income processes, households solve a consumption savings problem and choose toallocate their savings between the two capital stocks as well as a risk-free bond that is assumed to
be in zero net supply across households.
We calibrate the model to standard targets in the climate change literature and to a zerocorrelation of asset returns across the two sectors.1 We next simulate the transition of the economy
over the next decades—from 2020 to 2100—and compute the resulting temperature increase. As abaseline scenario, we assume a carbon tax of zero and a constant ratio of assets held by the central
bank of 4 percent of the value of the economy’s capital stock which is split proportionally across thetwo intermediate goods sectors. While we do not model the rationale for such a long-run QE policy,
our assumption can be interpreted as approximating a real world economy in which QE policies takeplace with a certain regularity. Our ad-hoc approach is based on the insight that demographic andclimate change processes will likely lead to a persistently low interest rate environment—which oursimulations also show—and it is therefore reasonable that such unconventional monetary policies
will be implemented again in future recessions. In this baseline scenario, the global temperature
increases until 2100 to about 3.5 degrees of Celsius above pre-industrial levels. This is in line with
the IPCC scenarios of climate change and well above the Paris agreement target of 1.5 degrees ofCelsius.
Next, we consider three policy experiments. First, we model the effects of carbon pricing by afiscal authority, which increases the price of dirty energy.2 We introduce the carbon tax in the year
2020 at an initially low level of 50 USD per ton of carbon emissions, which corresponds to a tax
of 13.6 USD per ton of carbon dioxide (CO2) and to an ad valorem carbon tax of 6.6 percent. We
hold this tax rate constant along the transition so that the absolute tax level increases to 70 USD
per ton of carbon in 2100. With this tax rate in place the global temperature increase would be
reduced by 0.17 degrees of Celsius, compared to the baseline.
1

The assumption of the zero calibration in our baseline calibration is based on the empirical findings in Broadstock
and Cheng (2019).
2
Since there is no aggregate uncertainty, setting the carbon price through an emission trading scheme would beequivalent to a carbon tax.

2

Second, we consider a stylised green QE policy set up to investigate its maximum possible effect.
The share of private assets held by the monetary authority (10 percent of GDP) is calibrated basedon the privately-issued securities holdings of central banks of advanced economies, i.e. including
asset-backed securities as well as commercial bonds. We assume a complete and immediate switch
to green bonds and no uncertainty about the classification of green and dirty bonds. We also
calibrate the model with a very high elasticity of substitution between clean and dirty intermediate
goods. The reallocation of the monetary authority’s portfolio towards the clean sector increases thecapital stock employed for production in that sector relative to the capital stock in the dirty sector.
This triggers a relative increase of labour demand in the clean sector and a relative expansionof output. The monetary authority can thus influence the relative production across the two
sectors in the economy. Despite our calibration tailored to achieve the maximum possible effectof green QE, its impact is rather modest compared to the global carbon tax. We find that the
global temperature reduction achieved through the global carbon tax of initially 50 USD per tonof carbon is 4.3 times larger than what would be achieved through green QE. Put differently, toachieve the same reduction of the global temperature as through green QE, this would require acarbon tax of about 11 USD per ton of carbon.
Third, we consider the two policies, a global carbon tax and green QE, in combination andexamine whether they are substitutes or complements. We find that green QE complements fiscal
policy, i.e., green QE on top of a carbon tax will help to reduce the increase of global temperature
further. However, the whole is less than the sum of its parts: the effect of the two policies
in combination is lower than in isolation. The reason is that the shifts of input factors in theintermediate goods sector induced by a carbon tax, away from dirty energy, are partly diminishedby the impact of the increase in dirty capital costs through green QE, raising the demand for dirty
energy.
We consider a number of alternative calibrations of the model for sensitivity analyses. Twoassumptions turn out to be crucial. First, assuming that the level instead of the share of private
assets held by the monetary authority will be kept constant along the transition implies thatthe share of assets held relative to the global capital stock will converge to zero. In this case,green QE is about 15 times less effective than the assumed initial carbon tax of 50 USD per
ton of carbon. Second, a crucial parameter turns out to be the elasticity of the ratio of energy
inputs with respect to the energy price ratio (short, energy elasticity), which in our baseline wecalibrate to a value of 2 implying a final output substitution elasticity of intermediate goods (short,intermediate goods elasticity) of 26. In our sensitivity analysis we calibrate the model to an energy
elasticity of 1 implying a much lower intermediate goods elasticity of 2.25. With this calibration,
the effectiveness of the global carbon tax is of similar magnitudes but the effectiveness of green QE
is strongly reduced so that the global carbon tax is about 31 times more effective. An in-between
3value of the intermediate goods elasticity of 4 (consistent with an energy elasticity of 1.12) in turnleads to a relative effectiveness of the global carbon tax by a factor of 14.
In conclusion, we find that a global carbon tax is considerably more effective in reducing theincrease of the global temperature than green QE. However, green QE can usefully complement a
carbon tax, in particular if governments only insufficiently coordinate on implementing green fiscal
policies.
Relation to Existing Literature
Our infinitely lived agents integrated assessment model follows the tradition since William Nordhaus(cf. Nordhaus and Boyer (2000) for a detailed description) and borrows elements from Golosov,
Hassler, Krusell, and Tsyvinski (2014) and van der Ploeg and Rezai (2021), in particular withrespect to the calibration of the climate module. We add two central features to this existing
literature. First, we extend this work by exogenously modeling green QE through the monetary
authority. Second, output in the two sectors of the economy is plausibly stochastic and the returns
to capital are imperfectly correlated.
The portfolio choice of private households is a crucial mechanism so that the green QE policyby the monetary authority is not perfectly neutralized on private markets. In this regard, our paper
connects to the literature on the effectiveness of quantitative easing, in particular the so-called
portfolio re-balancing channel of QE. Central bank asset purchases will not influence their price if
in response private investors completely offset the impact by re-balancing their portfolios (Wallace1981). Portfolio re-balancing can affect security prices when private investors are not indifferent
with respect to the composition of their portfolios, for example when they have a preference forcertain maturities (Vayanos and Vila 2021). Different other channels for the effectiveness of QE
that are suggested in the literature, among which signalling the central banks’ intentions or its
impact on the balance sheet constraints of financial intermediaries (see, e.g., Krishnamurthy andVissing-Jorgensen (2011) and Gertler and Karadi (2011)), are not considered in this paper.
We maintain the long-run focus of prototypical integrated assessment models and thus analyze
a stylized long-run green QE policy. This perspective is shared in recent work by Ferrari and Landi(2022) who, as we, develop a two sector integrated assessment model with a monetary authority
to study the effectiveness of green QE in the long-run. Similar to our main finding they reportthat the effect of green QE on the stock of pollution is relatively small. The main differences toour approach are their focus on the EU whereas our model is calibrated to the world economy andthe motive for holding green assets by the private sector, which they model through a preference
for green investments. Our model with idiosyncratic return risk in the two sectors features astandard endogenous portfolio choice instead of an explicit preference based explanation. Throughthis structure we do not directly calibrate a hard-wired motive for holding green bonds but rather

4

achieve it through a calibration of the risk return structure of the asset classes, which we base on
respective data moments.
Our long-run focus distinguishes us from work on green QE for the short-run. E.g., Ferrari
and Landi (2020) and Benmir and Roman (2020) study climate policies along the business cycleby combining a climate model with a New Keynesian DSGE model with the financial accelerator
framework of Gertler and Karadi (2011). As we do, they understand green QE as a tilting of theportfolio held by the central bank towards the green sector. Ferrari and Landi (2020) avoid aperfect crowding out by introducing costly portfolio rebalancing for private agents. They find thatan aggressive expansion of green QE (i.e., selling dirty and buying clean assets) during expansions
is welfare improving.3 Related, Diluiso, Annicchiarico, Kalkuhl, and Minx (2021) study the interactions between climate change and monetary policies arguing that inflationary pressures causedby climate change policies may demand very strong monetary policy reactions.
Furthermore, our approach of modeling green QE is comparable to policies aiming at a preferential treatment of green corporate bonds in central banks’ collateral frameworks. By increasing
the number of corporate bonds associated with fewer green house gas emissions, central banks
could steer the demand towards greener corporate bonds. Pelizzon, Riedel, Simon, and Subrahmanyam (2020) find that pledgeability as collateral affects the financing conditions and investment
decisions of firms. Analyzing elegibility events in the Eurosystem Collateral Framework they reportthat upon receiving the eligibility status of their corporate bonds, firms increase leverage and expand their balance sheet. Giovanardi, Kaldorf, Radke, and Wicknig (2021) study different degrees
of preferential treatment of green corporate bonds within a DSGE setup, thus focusing on businesscycle frequencies. They find a very limited climate change mitigating effect of such preferential
treatments, which also come at the cost of an increase in entrepreneurial risk-taking. The optimal
green collateral policy is thus characterized by a very modest preferential treatment, with very low
beneficial effects for the climate. Both, green QE and green collateral policies function through anincrease in the demand for green corporate bonds. A major difference between the two instruments
is that with green QE a central bank directly decides about the quantity and composition of green
bond purchases, whereas in the case of a green collateral policy, the central bank incentivizes
private banks to make these choices. These monetary policy instruments differ from a new type of
facility, recently introduced by the central banks of Japan and China, by which low interest funds
are provided to financial institutions to finance firms’ green investment projects and their efforts
to cut carbon emissions.4 In our model setting such a facility would be comparable to providing
subsidies to capital in the clean intermediate sector in order to support green investment.
3

Related, Benmir, Jaccard, and Vermandel (2020) find that optimal carbon taxes should be pro-cyclical.
See
https://greencentralbanking.com/2021/11/10/pboc-launches-targeted-green-lending/and https://www.boj.or.jp/en/announcements/release_2021/rel210716b.pdf, respectively.
45

By modeling idiosyncratic return risk our work also relates to the standard incomplete markets literature in quantitative macroeconomics pioneered in so-called Aiyagari-Bewley-HuggettImrohoglu models (Bewley 1986; Huggett 1993; Aiyagari 1994; Imrohoroglu 1989). More specifically, our model adopts the setup of Angeletos (2007) to a two sector economy with a climate
module. Specifically, the (idiosyncratic) return risk in combination with a risk-free labor income
gives rise to closed form solutions of the household decision functions5 , which is a convenient
property of the model as it allows us to compute the solution over very long horizons in our rather
complex model in limited time.
Finally, we relate to the literature on asset pricing and climate change, e.g., by Hambel, Kraft,and van der Ploeg (2020) who emphasize a trade-off between asset diversification and climatechange mitigation. They further show that green assets feature higher risk premia than brown
assets. The recent empirical literature indeed partially finds lower risk premia for green assets.
Bolton and Kacperczyk (2021) and Bolton and Kacperczyk (2020) analyze the US, respectively theworldwide, stock markets and find a positive carbon premium that has been rising over the recent
years. Kapraun, Latino, Scheins, and Schlag (2021) investigate a large dataset of government andcorporate bonds. In the primary market, they find that green bonds have lower yields than nongreen bonds. However, in the secondary market this reverses and they find green bonds featuring
higher yields. Degryse, Goncharenko, Theunisz, and Vadazs (2020) investigate an international
sample of syndicated loans and find that green firms borrow at significantly lower spreads. For
sake of parsimony, we sidestep these aspects and are agnostic about any mechanisms that may
lead to differential asset returns by calibrating our model to equal mean returns and equal return
variances in both sectors. However, a crucial parameter in our model is the correlation of asset
returns, which in our baseline scenario we calibrate according to the evidence in Broadstock andCheng (2019) suggesting a zero correlation.
The remainder of the paper is organized as follows. Section 2 presents the model and Section 3
discusses the calibration. Section 4 presents our results including our extensive sensitivity analyses
and Section 5 concludes the paper. Detailed derivations are contained in the appendix.

2

A Two-Sector Integrated Assessment Model with Risky Returns

We develop a two sector world economy integrated assessment model with a monetary and a
fiscal authority. Figure 1 provides an overview of the various sectors and entities in the economy,
and Table 1 collects the main indices used throughout. The final consumption good is produced
by a dirty and a clean intermediate goods sector, which itself uses capital, labour and energy as
input. Labour is supplied by households and capital is supplied by households and a monetary5

An insight relating back to Merton (1969) and Samuelson (1969).

6

authority. Energy is supplied by a dirty and a clean energy production sector, using labour suppliedby households as input. We take the total capital stock of the monetary authority supplied tofirms as given and thus the monetary authority solely faces a portfolio choice allocation problem
and can thereby influence the production of clean and dirty intermediate inputs. Profits generated
by the monetary authority flow to the fiscal authority which additionally raises revenue from dirty
energy production by energy (carbon) taxes. Dirty energy production leads via its emissions to an
accumulation of a carbon stock in the atmosphere which creates a temperature increase and with
it causes a damage through a reduction of aggregate output. We now describe the main elementsof the model in more detail.
Figure 1: Overview of the 2 Sector Integrated Assessment ModelD

ClimateFinal Good Sector

Module

Yc

Y di

l

Y

Dirty Intermediate
SectorEdi

Clean Intermediate
Sector

L

di ,

K
l,

K

cl

Lc

di

Households

m

Kclm

K di

Edi

c

Ecl

τ
Dirty
Energy

e

L di

πm

Monetary
Authority

FiscalAuthority

Production

Le
cl

Clean
EnergyProductioneτdiTable 1: Indices
Index
t
i
s
c

Value
t ∈ {0, 1, . . . , ∞}i ∈ {1, 2, . . . , ∞}s ∈ {cl, di}c ∈ {ra, sl}Interpretation
Time
Type
Sector (clean, dirty)
Carbon Stocks (rapidly, slowly depreciating stock)

Notes: List of indices used in the integrated assessment model.

2.1

Time, Risk and Population StructureTime in the model is discrete and runs from t = 0, . . . , ∞. At time t = 0 a continuous distribution of infinitely lived representative agents are born with total initial size N0 = 1, which
7grows exogenously at time varying rate nt . Each period a (heterogeneous) household i earns adeterministic labour income, stochastic returns on physical capital holdings from owning firms andrisk-free returns from owning bonds.
2.2
2.2.1

Production
Final Good Production

The final output good Yt is composed of two intermediate goods produced in a clean and adirty sector Yts , s ∈ {cl, di}, and augmented according to a CES aggregator with substitution
elasticity , henceforth referred to as intermediate goods elasticity. At this outer layer of the
production side we further assume an exogenous technology level Υt , which grows at the exogenous
rate g. Additionally, there is a negative aggregate production externality Dt from air pollution which
proportionally reduces aggregate output and thus
 11

Yt = (1 − Dt ) · Υt · 

X

1− 1

κs Yts

1− 

,



(1)

s∈{cl,di}P
where κs are the sectoral output shares with s∈{cl,di} κs = 1. The representative firm takes asgiven the final goods price pt and the intermediate goods input prices pts and maximizes profits
under perfect competition giving the intermediate goods demand

Yts =

κs
pts /pt((1 − Dt ) · Υt )−1 Yt , for s ∈ {cl, di},(2)

and the price index for the final good as
1
 1−


pt =

1

(1 − Dt )ΥtXκs p1−ts,s∈{cl,di}cf. Appendix A.1.
2.2.2

Intermediate Goods Production

Every household runs the two intermediate goods firms s by employing its own household capital ktis
and hiring labour `tis and energy etis on the respective labour and energy markets. Production ofintermediate goods takes place according to a two-nests Cobb-Douglas technology with inner nest
capital share parameter α and outer nest non-energy share parameter γ (so that the energy shareis 1 − γ). The value of the capital employed in production is subject to an idiosyncratic sector8specific shock ζtis so that gross output isγytis = ψs (ktis )α (`tis )1−α · e1−γ
tis + ζtis ktis ,(3)where ψs is a technology level parameter.
Let ζti = (ζticl ,ζtidi )0 be a vectorcontaining the shocks in both sectors. We assume that ζti isi.i.d. with CDF Ψ (µζcl , µζdi )0 , Σ, . . . , where
 
Σ=

σclζ

2

ζ
ρζcl,di σclζ σdiζρζcl,di σclζ σdi 2  .
ζσdiΣ pins down the variances of the shocks and explicitly allows for them to be correlated acrosssectors when ρζcl,di 6= 0. The details on the shock distribution are described in Appendix A.2.
Households take as given the intermediate goods prices pts , wages, respectively the return on
labour, rtl , energy prices pets and an exogenous depreciation rate on capital δs so that profits are
πts = pts · ytis − rtl `tis − pets etis − δs ktis .

(4)Assuming free entry and exit, profit maximization yields the demand for energy and labour as l − 1−α e − 1−γ(1−α)
α
αγ
1−γ
rt
pts
etis = Γ(ψs , α, γ) ···· ktis(1 − α)γpts
pts l − α1  e − 1−γ
αγ
rt
pts
`tis = Γ(ψs , α, γ) ··· ktis ,
pts
pts

(5a)
(5b)

where the constant Γ(ψs , α, γ) is
1−γ

1Γ(ψs , α, γ) = [ψs (1 − γ)] αγ · [ψs (1 − α)γ] α .

(6)Using (5) in (3) we can rewrite output as1
ytis = Γ(ψs , α, γ) ··(1 − α)γ



rtl
pts

 e − 1−γ
− 1−αααγ
pts·· ktis + ζtispts

which is linearly increasing in ktis and, using this in (4) gives profits as"απtis = Γ(ψs , α, γ) ·· pts
(1 − α)



rtl
pts

#
− 1−α e − 1−γ
α
αγ
pts
·
− δs + pts ζtis ktis ,
pts

9

(7)

which are also proportional to ktis . Defining the idiosyncratic return on capital asαrtis = Γ(ψs , α, γ) ·· pts
(1 − α)



rtl
pts

− 1−α e − 1−γααγpts− δs + pts ζtis
·
pts

(8)

we can thus rewrite profits as
πtis (ktis ) = rtis · ktis .
2.2.3

(9)

Energy Production

Energy employed for production in the two intermediate goods sectors s is produced in two perfectly
separated (across the two sectors) energy producing firms that employ labour Lets and a technology
stock Υets , which grows exogenously and deterministically at the sector specific rates gs . Theenergy production technology is linear and accordingly
Ets = Υets Lets .
eDirty energy production is subject to proportional carbon taxes τts=di≥ 0, whereas energy in theeclean sector is untaxed (or may be subsidized), τts=cl≤ 0, and thus profits in the two energy
producing firms areeπts= pets (1 − τtse ) Υets Lets − rtl Lets .

Assuming free entry and exit drives profits in the energy sector to zero and thus energy prices are
given bypets =
2.3

rtl.
(1 − τtse ) Υets

(10)

Carbon Stock Accumulation, Temperature and the Damage Function

As in Golosov, Hassler, Krusell, and Tsyvinski (2014) and Kotlikoff, Kubler, Polbin, Sachs, andScheidegger (2021), the total carbon stock St in the atmosphere is composed of two stocks,
a rapidly and a slowly depreciating stock, Stc for c ∈ {ra, sl}, thus
St =

X

Stc

c∈{ra,sl}

10which accumulate through dirty energy emissions and feature persistence parameters ρc , where 1 >ρc=sl > ρc=ra > 0, thus
Stc = ρc St−1c + φc ξEts=di

(11)

P

c∈{ra,sl} φc = 1. Each unit of St leads to an increase of the global

where ξ > 0 and φc > 0 andtemparature according to
Tt = λlog(St /Spre ),log(2)(12)

where Spre is the pre-industrial area carbon stock in the atmosphere, and λ > 0. The temperature
increase in turn leads to the negative externality on aggregate output through the damage function
Dt = 1 −1.
1 + νTt2(13)

for ν > 0.
2.4

Fiscal and Monetary Authorities

The model features a fiscal and a monetary authority. The fiscal authority levies Carbon taxes ate≥ 0 and receives profits from the monetary authority πtm . These sources of income arerates τts=didistributed to households in the form of subsidies on consumption, τtc ≤ 0 and thus each period
the fiscal authority features a balanced budget ofeτts=di
Ets=di + πtm + τtc Ct = 0.

(14)

The monetary authority in turn holds an exogenous amount of capital Ktm in the economy
which is growing at exogenous time varying rate gtm ≥ 0. This capital is exogenously split acrossthe two capital stocks in the intermediate goods production sectors, thus
Ktm =

X

m
Kts
.

s∈{cl,di}The monetary authority earns the average marginal products in the two sectors and its profits are
thus
πtm =

X

m
.
E[rts ]Ktss∈{cl,di}

11

2.5
2.5.1

Households
Preferences

Each household i at time t has Epstein-Zin-Weil (Epstein and Zin 1989; Epstein and Zin 1991;
Weil 1989) recursive preferences uti over consumption cti and continuation utility ut+1i which is
discounted at factor β ∈ (0, 1) and features risk aversion parameterized by θ and resistance tointertemporal substitution υ. Thus, preferences are given by
uti =

h

c1−υ
+β·
ti

E[u1−θ
t+1i ]

1
i 1−υ
 1−υ
1−θ

,

(15)

where E is an expectations operator with expectations taken with respect to the idiosyncratic
shocks to the return on physical capital.
2.5.2

Endowments

Households operate the two intermediate goods firms. Accordingly, household i enters into model
period t with capital stocks ktis in the two firms and earns in the current period stochastic profits
generated from production in those firms πtis . Households also earn a deterministic labour income rtl `t where rtl denotes the wage rate on the exogenous labour endowment `t , which is thesame for all households. Furthermore, households enter the period with bond holdings bti , which
are in zero net supply across all households and earn a risk-free return rtf . The household spends
its income from these sources on consumption of the final good cti —which has price pt and istaxed, respectively subsidized, at rate τtc —, on savings in the two capital goods kt+1is as well as
on risk free bond purchases bt+1i . Thus the dynamic budget constraint of household i isX

kt+1is + bt+1i + (1 + τtc )pt cti =s∈{c,d}X

ktis (1 + rtis ) + (1 + rtf )bti + rtl `ts∈{c,d}tis
is the stochastic return on capital in sector s.
where rtjs = πktis

2.5.3

Analysis of the Household Problem

Conditional on the aggregate law of motion of the economy, i.e., for given prices, wages, interest
rates and taxes, the household model permits a closed form solution. To derive it, first rewrite thebudget constraint in terms of cash-on-hand
xti =

X

ktis (1 + rtis ) + (1 + rtf )bti + rtl `ts∈{c,d}12

to getX

kt+1is + bt+1i = xti − (1 + τtc )pt cti .

s∈{c,d}Next, define the portfolio shares as shares invested in the respective asset as a function of totalsavings xti − (1 + τtc )cti asϑtis =

kt+1is,xti − (1 + τtc )pt cti

X1−

ϑtis =s∈{cl,di}bt+1isxti − (1 + τtc )pt cti

to note thatX 

xt+1i =

ffl(xti − (1 + τtc )pt cti ) + rt+1
`t+1 .
1 + rt+1+ ϑtis rt+1is − rt+1

(16)

s∈{c,d}Next, denote by ht the human capital wealth of a household at date t, which is the discounted
sum of future labour income
ht =

∞
X

l
rt+1+j`t+1+j

j=0

j 
Y

f
1 + rt+k+1

−1

k=0

which thus obeys the human capital wealth accumulation equationflht+1 = ht (1 + rt+1
) − rt+1`t+1 .

(17)

Finally, define total wealth of the household as the sum of cash-on-hand and human capital wealth,wti = xti + ht ,and take the sum of (16) and (17) to getp{ϑ̂tis }s∈{cl,di} ,wt+1i = (wti − (1 + τtc )pt cti ) Rt+1i

(18)

where
p
Rt+1i



{ϑ̂tis }s∈{cl,di}f= 1 + rt+1
+

X

ϑ̂tis

s∈{cl,di}13

f
rt+1is − rt+1



is a portfolio return on total savings wti − (1 − τtc )cti and whereϑ̂tis =

kt+1is,wti − (1 + τtc )pt cti

1−Xϑ̂tis =s∈{cl,di}bt+1iswti − (1 + τtc )pt cti

are the portfolio investments in the respective asset in relation to total savings.
Maximization of (15) subject to the resource constraint (18) gives rise to optimal decisionsin terms of consumption policy functions and portfolio allocation decisions as stated in the next
proposition, which we formally prove in Appendix A.3:
 Consumption policy functions are linear functions of total wealth

Proposition 1.

cti = mt wti
where the marginal propensities to consume are




p
c
, Rt+1{ϑ̂ts }s∈{cl,di} , β, υ, θ, Ψ mt+1i
Θ pt , pt+1 , τtc , τt+1




,
mt =
p
c
1 + (1 + τtc )Θ pt , pt+1 , τtc , τt+1
, Rt+1{ϑ̂ts }s∈{cl,di} , β, υ, θ, Ψ mt+1i

(19)

where




p
c
Θ pt , pt+1 , τtc , τt+1
, Rt+1{ϑ̂ts }s∈{cl,di} , β, υ, θ, Ψ =pt (1 + τtc )βcpt+1 (1 + τt+1
)







p
Et Rt+1{ϑ̂∗ts }s∈{cl,di}1−θ  1−υ
1−θ

!− υ1

 The optimal portfolio shares are given byϑ̂∗tclϑ̂∗tdi

!

1
≈ Σ−1
θ

!
f
ln(1 + E [rt+1cl ]) − ln(1 + rt+1),fln(1 + E [rt+1di ]) − ln(1 + rt+1)

(20)

which in case of a zero correlation of return shocks across sectors, i.e. for ρζcl,di = 0, simplifiestofln(1 + E [rt+1s ]) − ln(1 + rt+1)∗
ϑ̂ts ≈,
θ · V ar(ln(1 + rt+1s ))(21)

s ∈ {cl, di}.
Thus, the marginal propensities to consume out of total wealth and the optimal portfolio sharesin t, s are the same for all i, mti = mt , ϑ̂tis = ϑ̂ts . Linearity of policy functions in total wealth and14

identical marginal propensities to consume in any t, s across all households is a very convenient
property of the model as it simplifies the aggregation to the effect that we only need to keep trackof the mean decisions and not their distribution.
2.6

Definition of Equilibrium

We define the equilibrium in this economy sequentially. By the result in Proposition 1 we do notneed to keep track of the distribution of heterogenous households and thus household specificvariables are not indexed by i and it is understood that the household variables in the formal
equilibrium definition indexed by t, respectively by t and s, refer to average allocations.
Definition 1. Given an initial total wealth level w0 , initial carbon stocks {S0c }c∈{ra,sl} , a sequenceof technology levels and of the population {Υt , {Υets }s∈{cl,di} , Nt }∞t=0 and a sequence of policy∞mecparameters {τt , τts=di , {Kts }s∈{cl,di} }t=0 , a competitive equilibrium is an allocation{{Ets , Kts , Lts , Yts , ϑ̂ts }s∈{cl,di} , xt+1 , ht+1 , wt+1 , St , Tt , Dt }∞t=0 , a sequence of pricesfl ∞e{{pts , pts , rts }s∈{cl,di} , rt , rt }t=0 and a sequence of profits {{πts }s∈{cl,di} , πtm }∞t=0 such that∞cem
1. given prices {{pts , pets , rts }s∈{cl,di} , rtf , rtl }∞t=0 and policies {τt , τts=di , {Kts }s∈{cl,di} }t=0 households behave optimally with resulting optimal policy functions for ct , ϑ̂ts , wt+1 as characterized in Proposition 1.

2. prices satisfy (5),(10) andZ
rts =rtis diwhere rtis is given in (8);3. the government budget constraint (14) holds in all t ≥ 0;
4. the sequence of carbon stocks, global temperature and global damage {{Stc }c∈{ra,sl} , Tt , Dt }∞t=0evolve according to (11)–(13);15

5. markets clear:
L t = Nt ` tZ
Kts = Nt ktsi di, for s ∈ {cl, di}Z
Bt = Nt bti di = 0

(22a)
(22b)
(22c)

− α1  e − 1−γαγpts· Kts , for s ∈ {cl, di}
Lts = Nt `tsi di = Nt · Γ(ψs , α, γ) ·
·
pts
(22d)
1−γ(1−α)1−α l − α  e − αγ
Z
1−γ
rt
pts
· Kts ,
Ets = Nt etsi di = Nt · Γ(ψs , α, γ) ···(1 − α)γ
pts
pts


Z

rtl
pts

for s ∈ {cl, di}Z
Yts = NtX
yt,j,idi = Nt · Γ(ψs , α, γ) ·1·(1 − α)γ

for s ∈ {cl, di}



(22e)
1−α
1−γ



− α
− αγ
rtl
pets
·
· Kts ,
pts
pts
(22f)

where Γ(ψs , α, γ) is given in (6).

3Calibration and Policy Experiments

3.1

Overview of Calibration

We calibrate the model by fixing some parameters exogenously (first stage parameters) and by
calibrating others (second stage parameters) to match selected moments in an initial steady state
year, which we pick to be year 2010. While the latter set of parameters are calibrated jointly, forclarity of identification of the parameter values we relate each parameter with a specific target.
Tables 2 and 3 provide an overview of all first- and second-stage parameters and the subsequent
sections provide the details of the calibration by sector in the economy.
3.2

Population and Labour Supply

The exogenous initial size of the population N0 is normalized to one. The population growthrate nt and the working age population ratio (WAPR) ωt are calibrated from the population growthrate information (and projections) provided by the World Population Prospects of United Nations(UN) (United Nations 2020). The initial year 2010 population growth rate is n0 = 0.0121 andthe population growth rate shrinks gradually to reach zero growth by year 2100 thus nt = 0, forall t > 90. Aggregate labour in the model is Lt = ωt Nt . In our baseline scenario, we abstract fromtime variation in the working age population ratio by letting ωt = ω0 (which we normalize to 1 in
16

Table 2: Calibration: First Stage Parameters

Parameter
Population and labour supply
Initial population size N0
Initial population growth rate n0
Initial working age population ratio ω0
Final good technology
Intermediate Goods Elasticity Intermediate good technology
Non-energy share: γCapital share: α
Corr. of depreciation shocks: ρζcl,diClimate Module
Initial carbon stock: S0
Pre-industrial carbon stock: Spre
Stock 1 share: φs
Emission share in atmosphere: ξ
Carbon stock persistence: ρcTemp. increase with S: λTemperature to damage: νPreferences
Elasticity inter-temp. substit., 1/υ

ValueTarget (Source)

1
0.0121
1

Data moment (United Nations)
Data moment (United Nations)
Constant (baseline)

26

Energy Elasticity η = 2

0.96
0.33
0

Kotlikoff et al. (2019)
Standard value
Zero correlation

802 GtC
581 GtC[0.5,0.5]
0.4
ρc = [0.996, 0.999], c ∈ {ra, sl}
30.0028388
0.5Standard value

Notes: Calibration in the baseline model. First stage parameters calibrated with reference to other studies orwithout using the model. Steady state year is year 2010.

17

Table 3: Calibration: Second Stage Parameters

Parameter
Final good technology
Interm.
good weight κs , s ∈{cl, di}Growth rate final good TFP, g
Intermediate good technology
Iterm. productivity factor: ψs=cl =ψs=diExpected depreciation rate: δs , s ∈{cl, di}Std. of depreciation shock: σsζ , s ∈{cl, di}Energy production technology
Clean productivity factor, Υe0s=cl
Dirty productivity factor, Υe0s=dieGrowth rate clean prod. fact., gs=cle
Growth rate dirty prod. fact., gs=diPreferences
Time discount factor: βRelative risk aversion: θCentral bank portfoliomCapital holdings K0s, s ∈ {cl, di}Value

Moment[0.45, 0.55]

E0s=di /E0s=cl = 4

0.0098/ Y2020 ) 80 − 1 = 1.50%( LY2100
2100 L2020

4811

E0s=di = 30 GtCO2[0.015, 0.087]

E[r0s ] = 6.94%, , s ∈ {cl, di}[0.030, 0.021]

σ r0s = 8.4%, s ∈ {cl, di} (std. of capital returns)

128
192
0.020
0.011

pe0s=cl = 810 USD/tC
pe0s=di = 540 USD/tCe1(pe2100s=cl /pe2020s=cl ) 70 − 1 = −0.50%
1( E2035s=di/ E2020s=di) 15 − 1 = −0.50%
Y2035
Y2020

0.997
63.9

K/Y = 2.5
rf = 2.9%[6244, 8930]

m +K mK0cl
m
m0cl= 10% & K0cl
/K0cl = K0di
/K0di
Y0

1

Notes: Calibration in the baseline model. Second stage parameters calibrated endogenously by matching of moments
in steady state year 2010.

18

the base year) and thus the aggregate of labour grows at the same rate as the population. As asensitivity analysis, we feed into the model a time varying working age population ratio. Figure 2
displays the evolution of the aggregate population size in panel (a) and the working age population
ratio in panel (b), from year 2015 to year 2100. Population features a gradually decreasing growthrate and is thus hump-shaped over the next 80 years. This reflects the increase in the world
population from 7.8 Billion in year 2020 to about 10.9 Billion people in year 2100 according to themedian variant of the UN projections.
Figure 2: Population and Working age population ratio(a) Population size(b) WAPR

Notes: Aggregate population size in panel (a) and working age population ratio (WAPR) in panel (b). Population
size in panel (a) normalized such that is equal to one in the year 2010. Panel (b) shows WAPR in the baseline
setup (held constant at one) and as used in sensitivity analysis WAPR. Population size in baseline and sensitivity
setup and WAPR in case of the sensitivity setup correspond to the median variant of the UN projections.
Source: United Nations (2020).

3.3

Production

Final Good Production We take an indirect approach to the calibration of the parameter
governing the elasticity of final output in the two goods, Yts , s ∈ {cl, di} in equation (1). In ourmodel with the two separate firms for energy production there is no direct parameter that would
govern the energy demand elasticity (short energy elasticity), which is the percent change in thepets=dratio of dirty to clean energy demand EEts=din
response
toapercentchangeofrelative
pricespets=cts=cdenoted as η Ets=d pets=d . According to Papageorgiou, Saam, and Schulte (2017) the energy demand,

Ets=c pets=celasticity is about 2-3, where the lower value refers to the electricity-generating sector and values
close to 3 are in nonenergy industries. We take the lower value as target for the intermediate goods
elasticity . In appendix B.1 we derive that locally—i.e., holding constant the (expected) marginal
19

remuneration of capital Erts , s ∈ {cl, di} and labour rtl —the energy elasticity is given byη Ets=di pets=di =  · (1 − γ) + γ,
Ets=cl

, pe

ts=cl

which we invert to calibrate  for given target η Ets=di pets=di and a given non-energy share paEts=cl

6

, pe

ts=cl

rameter γ. For our calibrated value of γ = 0.96 (see below), this gives a calibrated value ofthe intermediate goods elasticity of  = 26 and thus, as a consequence of the low energy shareof 1 − γ = 0.04, our model requires a high intermediate goods elasticity to match a conservative
energy demand elasticity of 2. It turns out that the intermediate goods elasticity  is a crucial
parameter for our quantitative findings on the relative effectiveness of green QE. In our sensitivity
analysis, motivated by Golosov, Hassler, Krusell, and Tsyvinski (2014) and Hassler, Krusell, andOlovsson (2021), we also calibrate the model with lower values of the intermediate goods elasticity. Specifically, we assume an energy elasticity of η = 1 implying an intermediate goods elasticityof  = 2.25. Since elasticity estimates are downward biased due to adjustment costs (Caballero1994), we also consider an in-between value of  = 4 (implying η = 1.12) as reasonable in light of
the long-run focus of our analysis.
The relative weights on the two goods in (1), κs , s ∈ {cl, di} are calibrated such that (i)we normalize κs=di = 1 − κs=cl and (ii) match the ratio of energy output in the two sectors= 4 giving κs=cl = 0.45 and thus κs=di = 0.55. The final good output growth rate g isof EE0s=di
0s=clcalibrated to generate a total annual output growth of 1.5% in the period from year 2020 to year
2100, giving g = 0.0098.
Intermediate Goods Production We set the capital share, cf. equation (3), exogenously
to α = 0.33, corresponding to standard estimates of capital shares in production. The non-energy
share parameter γ is set to 0.96, following Kotlikoff, Kubler, Polbin, Sachs, and Scheidegger (2021).
The technology levels in both sectors are normalized such that ψcl = ψdi and calibrated to generate
dirty energy production of 30 gigatons of CO2 in the initial steady state equilibrium.
The average depreciation rates δs , s ∈ {cl, di} and the standard deviation of depreciation
shocks σsζ , s ∈ {c, d} are calibrated to yield expected average returns of 6.94% in both sectors
and a standard deviation of expected returns of 8.4%, based on empirical estimates of Piazzesi,
Schneider, and Tuzel (2007). This gives δs = [0.015, 0.087], s ∈ {cl, di}. For our baseline results,
based on Broadstock and Cheng (2019)7 we assume a zero correlation in the returns in both
6

Recall that 1 − γ is the energy elasticity in intermediate goods production, cf. equation (3).
For the US bond market, Broadstock and Cheng (2019) report a negative correlation before mid-2013, and apositive correlation thereafter so that an on average zero correlation is an appropriate assumption for the long-run.
7

20

sectors, i.e. we set ρζcl,di = 0. We further consider a positive correlation for sensitivity analysis,
based on Reboredo, Ugolini, and Aiube (2020).8
Energy Production Recall from equation (10) that energy prices are inversely proportional to
the technology level in the energy sector. Based on this relationship we calibrate the technology
parameters Υe0s , s ∈ {cl, di} to match the absolute price levels9 in the two sectors per ton of carbon
emission (tCe) of USD 810, respectively 540, which requires Υe0s=cl = 128 and Υe0s=di = 192. We
denote by gse the time constant growth rates in the two sectors, i.e., Υets = Υet−1s (1 + gse ). Weesuch that energy prices
endogenously determine the growth rate in the clean energy sector gs=cl
fall by 0.5% on average over the 80 years between year 2020 and 2100. This calibration is basedon Nordhaus (2017), also see Kotlikoff, Kubler, Polbin, Sachs, and Scheidegger (2021). We
e
endogenously determine the growth rate in the dirty energy sector gs=disuch that CO2 emissions
relative to GDP reduce at a rate of −0.5% annually over the period from year 2020 to 2035, whichcorresponds to the average value of the share of CO2 emissions relative to world GDP over theperiod 1995 to 2018 as measured in PPP units at constant prices, which we compute from WorldBank (2021). Our calibration gives g e = [0.010, 0.011] for the two clean and dirty energy sector
growth rates, respectively.
3.4

Household Preferences

The elasticity of inter-temporal substitution 1/υ = 0.5, corresponding to the standard estimate
in the literature. The remaining household preference parameters are calibrated endogenously tomatch a capital output ratio of 2.5 by choice of the discount factor, which gives β = 0.997, and arisk-free rate of return of 2.9% by choice of the coefficient of risk aversion which requires θ = 63.9.
This high value is not surprising because shocks in our model are assumed to be distributed as
log-normal (thus, there are no extreme events), and there are no additional income shocks (nobackground risk) for households.
3.5

Carbon Stock Accumulation, Temperature and the Damage Function

The calibration of the climate module closely follows Golosov, Hassler, Krusell, and Tsyvinski(2014), where we draw on van der Ploeg and Rezai (2021) to adjust for the yearly periodicity of
our calibration approach.10
Carbon Stock The initial carbon stock in the atmosphere is set to S0 = 802 gigatons of carbon,
where S0c=sl = 684 GtC and S0c=ra = 118 GtC. As to the dynamics of the two carbon stocks in
8

Reboredo, Ugolini, and Aiube (2020) identify network connectedness as a source of a potential positive correlation across asset classes.
9
Recall that final consumption is the numeraire good in the economy so that these absolute price levels are equal
to the relative prices in units of the final consumption good.
10
See Kotlikoff, Kubler, Polbin, and Scheidegger (2021) for an adoption to a quinquennial frequency.

21

equation (11) we assume that 40% of dirty energy output leads to an accumulation of the carbon
stocks and thus ξ = 0.4, which is split up equally across the two stocks, thus φc = 0.5, c ∈ {sl, ra}.
The slow decumulating carbon stock features a persistence of ρc=sl = 0.999, and the rapidly
decumulating of ρc=ra = 0.995.
Temperature and Damage Function We calibrate the temperature function in (12) by setting λ = 3 and Spre = 581, and the damage function in (13) by letting ν = 0.0028388.
3.6

Fiscal and Monetary Authorities

Monetary Authority The monetary authority’s portfolio is calibrated such that in the initialsteady state capital held by the monetary authority (both clean and dirty) equals 10% of GDP,K m +K mi.e. 0clY0 0cl = 0.1. This is a rough estimate of the overall privately-issued security holdings ofadvanced-economy central banks at the beginning of the year 2021.11 Furthermore, capital holdings
in clean and dirty assets by the monetary authority are set such that they are proportional to private
m
K0cl
Km
capital holdings, i.e. K0cl
= K0di. Since the capital-to-output ratio is 2.5, this results in 4% of
0dicapital of both sectors held directly by the central bank. Given the endogenously determined sizesmmof the two sectors in the economy, this requires K0s=cl= 6244 and K0s=di= 8930. In our baselinemKtsexperiment, we hold the shares constant, i.e., Ks = 0.04 for s ∈ {cl, di} for all t > 0, so thatthe wealth holdings of the monetary authority grow with the capital stock of the economy. As
sensitivity analyses we consider two alternative scenarios regarding the stock of capital held by
the central bank. First, we assume a constant absolute size of the outstanding capital held by
the central bank, which implies that the relative size diminishes to zero over time. Second, werecalibrate the model to an initially higher share of assets held by the monetary authority in the
dirty sector of the economy.
Fiscal Authority In the initial steady state equilibrium, the fiscal authority does not levy carbon
taxes on emissions, thus τtse = 0. Since revenues from asset holdings of the monetary authority
are paid back to households in the form of consumption subsidies, the consumption tax rate is
negative, τtc < 0. With −0.54% in the initial steady state this subsidy is small.
3.7

Thought Experiments

Taking as given the exogenous dynamics of population and technology, we compute transitions
under alternative fiscal and monetary policy scenarios over 200 model periods, starting in year
2010 with an initial steady state.12 We treat the first 10 years as a phase-in period and show
results until 2100, that is overall we focus on the evolution of key model outcome variables for the
next 80 years from 2020-2100.
11

Our estimates are based on balance sheet data publicly available by July 2021 of the European Central Bankand the central banks of Canada, Japan, Sweden, United Kingdom and the United States.
12
The model is closed by setting the final period equal to the final steady state.

22

First, we conduct a baseline experiment, where all policy parameters are held constant attheir respective 2010 values, that is the initial carbon tax is zero and the capital allocation of
the monetary authority relative to the total capital stock is held constant and in equal proportion
across the two sectors. Consequently, the claims on private capital held by the monetary authority
grow with the time varying growth rate of the aggregate capital stock. This assumption canbe interpreted as approximating a real world economy in which asset purchases by the monetary
authority take place with a certain regularity. Our economy does not feature aggregate risk andthus there are no recessions, which would endogenously lead to repeated non-standard monetary
policy interventions (QE) if a zero lower bound on interest rates becomes binding. Since two of theworldwide secular economic mega-trends—demographic change and climate change—will likely leadto a persistently low interest rate environment, we regard it as plausible that such unconventionalmonetary policies will be implemented again in future recessions and our assumption of a constantshare is therefore a reasonable approximation.13
Next, we consider a carbon tax policy reform scenario where a carbon tax is introduced, withthe tax rate being held constant. The initial carbon tax in year 2020 is set to 50 USD per toneof carbon. The implied carbon tax rate is τts=di= 0.066 which we hold constant for all t ≥ 10.
Revenues from carbon taxation are redistributed to households through consumption subsidies. 14
In our second policy reform scenario, the green QE scenario, the portfolio composition of themonetary authority changes such that it reshuffles all of its privately-issued capital holdings towards
the green sector. In the green QE policy scenario we assume that, first, the monetary authority’s
private asset portfolio holdings are relatively large, second, that the private asset stock held by themonetary authority is growing in line with the world capital stock, third, all countries in the world
execute QE policies, fourth, clean and dirty asset returns are uncorrelated so that the QE policy
will not lead to a full crowding out of private clean capital investments and fifth, that the elasticity
of substitution between clean and dirty energy production is relatively high so that the alreadylarge changes induced by green QE policies do have relatively mild price effects only, which in turnleads to low crowding out. We thus evaluate the maximum climate change mitigating potency ofsuch policies. We acknowledge that we model green QE as a stylised scenario, in particular by
abstracting from modelling financial intermediaries. We also do not consider the role of frictions
13
Our simulations support this argument. We show in Appendix C that the risk-free interest rate is flat over the
projection period, which is a result of the accumulating climate change damages suppressing aggregate productivity.
This is reinforced in our sensitivity analysis, where we model population aging through a gradual reduction of theworking age population ratio leading to a relative shortage of labor. As is well established from the macroeconomics
of demographic change literature, this leads to a reduction of marginal productivities thus decreasing returns, cf.,
e.g., Krueger and Ludwig (2007) and, for a model with differential asset returns, Geppert, Ludwig, and Abiry (2016).
14
An alternative use of revenues from carbon taxation would be to subsidize clean energy production, which would
support the green transition further. Likewise, profits of the monetary authority could be used for such purpose.

23

in the monetary policy transmission mechanism, which would possibly amplify the effects of green
QE.15
Finally, as a full policy scenario we consider both instruments jointly and thereby investigate
whether both policies are substitutes or complements in mitigating the adverse effects of climatechange.

4

How Effective are Green Quantitative Easing and Carbon Taxation?

Before we address our main research question on the effectiveness of green QE in comparison tocarbon taxation in mitigating climate change damages, we compute a baseline path of the economy
along which we hold constant all policy instruments.
4.1

Climate Change with Constant Policies

Intermediate Goods Production Figure 3 displays prices in the clean and the dirty energy
sector in panel (a). The increasing prices of dirty energy and the falling price of clean energy isa consequence of the calibrated increase of relative productivity in the clean energy sector. With
regard to the ensuing dirty energy production and thus dirty emissions, two mechanisms are at workin the model. On the one hand, demand for goods through population growth and technologicalprogress in the final goods sector will lead to an increase of harmful emissions, Ets=di . On theother hand, the technological progress in the clean sector Υts=cl by increasing the relative price ofdirty intermediate goods leads to a substitution of intermediate goods production towards the clean
sector. Two forces lead to this substitution. The one is a reduction of demand for dirty energy
in the intermediate goods sector. The second is a substitution towards clean intermediate goodsin the production of the final good. Over our projection period, the first mechanism dominates.
Consequently, dirty energy emissions are increasing over the entire period, but at a decreasing rateand clean energy emissions gain relative importance. Since by this gradual substitution the cleanintermediate goods sector expands relative to the dirty sector, the aggregate input factors capitaland labor in the economy are increasingly employed in the production of clean intermediate goods,
cf. panels (c) and (d) of the figure.
Overall, these dynamic adjustments lead to an increase of the relative price of dirty intermediate
by 1% and a reduction of relative output yyts=diby about −27%, cf. Figure 4.
inputs ppts=di
ts=cl
ts=cl
Climate Implications The implications of the above shown gradual substitution towards cleaner
intermediate goods production for the global climate are shown in Figure 5, where Panel (a) shows
the level of the emissions of the dirty sector Ets=di . Panel (b) displays the resulting time paths ofthe carbon stocks that accumulate as a consequence of these emissions according to the calibrated
15

Within the structure of our model there is no role for a triggering mechanism of a form that private investors
may follow the example of the monetary authority. If such a mechanism were at work, then we would underestimate
the role of green QE policies.

24Figure 3: Baseline: Intermediate Production Inputs(a) Energy price(b) Energy input(c) Capital input(d) Labor input

Notes: Energy prices pets in panel (a), energy inputs Ets in panel (b), capital stocks Kts in panel (c), and labor
input Lts in panel (d), for s ∈ {cl, di}. Capital is normalized such that clean and dirty capital sum to one in theyear 2010. Labor is normalized to one in the year 2010.

25

Figure 4: Baseline: Final Production Inputs(a) Relative price dirty to clean



pt,di
pt,cl



(b) Relative good input dirty to clean



Yt,di
Yt,cl



Notes: Intermediate production: relative price of dirty to clean goods, ppts=di
in panel (a), and relative intermediate
ts=cl
Yts=digoods input, Yts=cl in panel (b).

process described in (11). By year 2100 the total carbon stock will have increased by about 63%
relative to its year 2020 level. This leads to an increase of global temperature as shown in Panel (c).
According to our model, the year 2020 temperature level is about 1.5 degrees Celsius above thepre-industrial level. Observe that the initial level exceeds the current range of estimates by the IPCC
(2021) of 0.9 − 1.3 degrees slightly.16 According to our model, without policy intervention the
global temperature will increase to about 3.5 degrees, an increase over 80 years by 2 degrees,or 0.025 degrees per year. The resulting damage in terms of a percent output loss, shown inPanel (d) of the figure, increases from 0.6% in 2020 to 3.5% in 2100, a factor of 5.
4.2

Climate Change with Policy Intervention

We now analyze the two policy reform scenarios, the introduction of a carbon tax and a portfolio
shift of the capital holdings by the monetary authority. We first study both policies in isolation
before turning to a joint analysis.
Policies in Isolation As our first policy scenario, we consider the introduction of a global carbon
tax in year 2020 at a level of 50 USD per ton of carbon and hold constant the allocation of capitalby the monetary authority. As the year 2020 price of dirty energy in our model is at 750 USD thiscorresponds to a tax rate of 6.6%. We hold this tax rate constant along the transition, τts=di =16This upward bias is a consequence of the climate module we adopt from Golosov, Hassler, Krusell, and Tsyvinski(2014) (GHKT). The annual variant of the GHKT model calibrated in van der Ploeg and Rezai (2021) also features
a 1.5 degree increase in their baseline year 2010.

26

Figure 5: Baseline: Climate Variables(a) CO2 emissions(b) Carbon stock(c) Temperature(d) Damage

Notes: Climate variables: CO2 emissions Ets=di in panel (a), carbon stocks St , {Stc }c∈{ra,sl} in panel (b), world
temperature Tt in degree Celsius compared to pre-industrial times in panel (c), and aggregate damage Dt (inpercent) in panel (d).

27

0.066, which implies that the absolute amount of carbon taxation increases at the growth rate ofthe dirty energy price pets=di . By 2100 the absolute carbon tax reaches almost 70 USD per tonof carbon. Panel (a) of figure 6 shows the time path of the absolute amount of the carbon tax
expressed in USD per ton of carbon.
As our second policy scenario, we assume that in year 2020 there is a full shift towards capital
holdings in the clean intermediate goods sector but we abstract from an introduction of a carbon
tax. Panel (b) of the same figure shows the according capital holdings of the monetary authority
in the two sectors. While this complete portfolio allocation is, of course, an extreme assumption, it
enables us to investigate the effects of QE on climate change assuming a (hypothetical) situation
where QE is at its maximum potency.
Figure 6: Reforms: Carbon Taxation and Portfolio Reallocation(a) Carbon tax(b) Capital holdings by the monetary authority

Notes: Policy reforms: carbon tax (in US dollars) in panel (a) and portfolio allocation of monetary authority (inUS dollars) in panel (b). Capital held by the monetary authority is normalized such that clean and dirty capital
holdings sum to one in the year 2010.

Figure 7 shows the key outcome variables of our experiments, in terms of changes relative tothe baseline path. Turning to the reduction of global temperature we observe from panel (c) thatthe global temperature reduction in the carbon tax experiment is about ∆T -τ = −0.167 degrees
of Celsius. In the green QE scenario, it is only about ∆T -QE = −0.039. Thus, in terms of∆T -τrelative effectiveness ∆T, the effect the carbon tax is about 4.3 times larger. Carbon taxes-QE
through changing the relative price of dirty energy lead to a reduction of dirty energy production
and thus a reduction of the increase in the global temperature through two mechanisms. First,
the price increase leads to a substitution of dirty energy through clean energy in the production ofintermediate goods, thus more clean intermediate goods are produced (supply side mechanism).
28

Second, the price increase of dirty energy pets=di increases the price of the dirty intermediate
good pts=di which leads to a substitution in the production of the final output away from dirty
intermediate towards clean intermediate goods (demand side mechanism).
The portfolio reallocation of the monetary authority, in contrast, has a theoretically ambiguous
effect on dirty energy demand. First, on impact, i.e., holding factor prices constant, a reduction ofcapital employed for production in the dirty intermediate goods sector and a simultaneous increaseof capital in the clean intermediate goods sector increases the marginal return on capital in the dirty
and decreases it in the clean energy sector. This leads to an adjustment of private capital, which isreallocated from clean to dirty intermediate goods production and thus the portfolio reallocation by
the monetary authority leads to a partial crowding out of private capital in the clean intermediate
goods production. Also, the increased rate of return on capital in the dirty intermediate goods
production increases capital costs for the intermediate goods firms leading to a substitution from
capital towards energy and labour employed in production. Thus, while output is reduced by the
portfolio reallocation, which also reduces energy demand in the dirty intermediate goods sector,
this reduction in energy demand is partially muted by a substitution towards energy in production.
Additionally, the increased capital costs of the firm leads to an increase of the intermediate goods
price pts=di which induces a substitution in the production of the final good towards the cleanintermediate input and through this channel reduces the demand for energy. Quantitatively, itturns out that the energy demand reducing mechanisms dominate.
One key feature of the calibration of our two sector two physical assets model is the assumed
zero correlation of the idiosyncratic returns across the two sectors. It implies that financial investors
will hold a diversified portfolio of wealth across the two sectors. This explains why QE in our model
is not neutral: a portfolio reallocation by the monetary authority towards the clean sector does not
induce a perfect crowding out of private capital in the clean sector, but leads to a partial crowdingout only. To illustrate the extent of this partial crowding out in our model, Figure 8 shows the
allocation of capital in both sectors, by the monetary authority as dashed lines and by the private
sector as solid lines. As a consequence of the portfolio reallocation, the monetary authority shifts
its capital holding towards the clean sector. In response to this, private investors hold less capitalin the clean sector, but this crowding out effect is much smaller than the additional capital held by
the monetary authority in the clean sector. Likewise, the substitution of private investors into dirty
capital holdings is smaller than the reduction of dirty capital holdings by the monetary authority.
Thus the net effect on capital holdings is positive in the clean sector and negative in the dirty
sector.
Equivalent Carbon Tax From the above analysis we observe that QE has a much milder effecton key climate variables than carbon taxation. We can thus conclude that relative to carbon
taxation green QE is a less efficient instrument to mitigate the adverse societal effects of climate
29

Figure 7: Reforms: Climate Variables(a) Emission reduction(b) Carbon stock reduction(c) Temperature reduction(d) Damage reduction

Notes: Policy reforms: emission reduction relative to baseline in gigatons of carbon in panel (a), carbon stock
reduction relative to baseline in gigatons of carbon in panel (b), temperature reduction in degrees of Celsius inpanel (c), and damage reduction in percentage points in panel (d).

30Figure 8: Quantitative Easing: Why no crowding out?

Notes: Policy reforms: Difference in dirty and clean sector capital holdings by private households and the monetary
authority. HH denotes holdings of the households and MA of the monetary authority. The holdings are reported as
the percentage share of total outstanding capital in the baseline scenario.

change. To look at this finding from a different perspective we next compute the carbon tax it
would take to achieve the same effect as for the green QE policy. The corresponding carbon tax
schedule is introduced in 2021 at some time constant carbon tax rate. The resulting equivalentcarbon tax required to achieve in 2100 the same global temperature reduction as for the green QE
policy in levels is only 11.06 USD per ton of carbon (22% of the tax in our baseline scenario of 50
USD per ton of carbon), corresponding to about 3 USD per ton of CO2.
Joint Policies A closely related question is whether green QE can be used complementary tocarbon taxes. We therefore next consider both instruments jointly with results displayed in Figure 9.
This shows that green QE has an additional climate change mitigating effect when it comes on topof a carbon tax policy, see panel (a) of figure 9, and thus green QE complements the green fiscal
policy. Yet, the model results do not support a positive interaction of both policy instruments, seepanel (b) of Figure 9. Thus, the climate change mitigating impact of carbon taxes would not bemagnified when green QE is simultaneously at work. On the one hand, green QE alone leads to a
reduction of the global temperature by 0.036 degrees Celsius. On the other hand, the joint effect ofgreen QE and a carbon tax relative to a scenario where carbon taxation is used in isolation implies
a global temperature reduction of 0.035 degrees. The reason for the negative interaction effect isthe substitution of input factors due to changes in the costs structure in production. Specifically,31

on the one hand, the carbon tax increases the cost of dirty energy, leading dirty firms to partiallysubstitute energy with labour and capital. On the other hand, green QE by increasing the cost of
dirty capital leads to a partial substitution of capital with labour and energy. In combination theseeffects partially offset each other.
Figure 9: Temperature Reduction with Both Policy Instruments(a) Temperature(b) Temperature reduction

Notes: Policy reforms: Temperature compared to pre-industrial in degrees of Celsius in panel (a) for baseline,
carbon tax, QE and carbon tax plus QE; panel (b) shows the temperature reduction from baseline to QE and fromthe carbon tax to the carbon tax plus QE.

4.3

Sensitivity Analyses

We examine the sensitivity of our main findings with respect to some key model parameters andassumptions of the policy analyses. First, rather than assuming that the stock of private assets
held by the monetary authority is constant in relative amounts as in our main analysis, we assume
in the sensitivity exercise that it is constant in absolute amounts so that over time the relative sizeof private assets held by the monetary authority convergences to zero. As shown in column 2 oftable 4 (scenario “Flat QE”), the carbon tax is then substantially more effective than green QE∆T -τ(the relative effectiveness ∆T
increases to 15.2, compared to 4.3 in the baseline calibration).
-QE
Second, following the notion of market neutrality we assume in our main analysis that the private
asset holdings of the monetary authority in clean and dirty capital are proportional to the market
shares. However, Papoutsi, Piazzesi, and Schneider (2021) show that the corporate bond portfolioof the European Central Bank has a carbon bias, i.e. has larger weights in sectors associated withhigher emissions. Column 3 of table 4 (scenario “CO2 Bias”) reports results where we replicate
the carbon bias detailed in Papoutsi, Piazzesi, and Schneider (2021) by increasing the monetary32

Table 4: Sensitivity Analyses

T in 2100∆T -τ
∆T -QE∆T -τ /∆T -QE

Baseline
3.505-0.167-0.0394.3

Flat QE
3.503-0.167-0.01115.2

CO2 Bias
3.505-0.167-0.0563

Pos. Corr.
3.408-0.170-0.032
5.3

Low SE
3.539-0.155-0.005
31

WAPR
3.190
-0.149-0.036
4.1

CO2 Re.
2.566-0.107-0.029
3.7

Notes: Different calibrations for sensitivity analyses. “Flat QE”: Size of monetary authority’s balance sheet held
constant over time. “CO2 Bias”: Size of dirty assets on monetary authority’s balance sheet 43% larger, i.e.
mK0s=di= 12770. “Pos. Corr.”: Correlation between clean and dirty returns set to ρζcl,di = 0.4. “Low SE”:
Low energy elasticity η Ets=di pets=di = 1 so that intermediate goods elasticity is ε = 2.25. “WAPR”: time varying
Ets=cl , pe
ts=clworking age population ratio ωt . “CO2 Re.”: strong CO2 reduction in baseline such that share of CO2 in GDP∆T -τdecreases at −1.5% annually. ∆T-QE : relative effectiveness of the carbon tax.

authority’s holdings of dirty capital by 43%.17 The shift in the monetary authority’s portfolio
towards clean capital is thus larger in absolute size, giving green QE more power. The additionaltemperature reduction equals about 43%, and is thus approximately equal to the additional size of
dirty capital held by the central bank relative to our baseline scenario. Consequently, the relative
effectiveness decreases to 3.
The two preceding sensitivity analyses (constant absolute size of the monetary authority’sbalance sheet and carbon bias of its assets holdings) illustrate the importance of the size of thedirty assets held by the monetary authority. The more dirty assets it holds, the more effective isgreen QE.
Third, as discussed in section 4.2 (see also figure 8) the impact of green QE hinges on theimperfect correlation between the returns in the clean and dirty intermediate production sectors. In
our main analysis we assumed a zero correlation. Column 4 of table 4 reports results with a positive
correlation of ρζcl,di = 0.4 between the shocks return (scenario “Pos. Corr.”), which is based on theempirical findings in Reboredo, Ugolini, and Aiube (2020).18 Central bank intervention throughgreen QE thus triggers a stronger reaction by private investors and we observe a larger crowdingout of private capital. This lowers the efficacy of green QE compared to a carbon tax. However,
the effect is not that large; the relative effectiveness factor increases to 5.3.
Fourth, a key parameter in our model is the intermediate goods elasticity . For our main analysis we determine this parameter such that the resulting energy elasticity is η = 2, corresponding
17

We thank Melina Papoutsi for sharing the data on the aggregate sectoral shares of the ECB’s portfolio holdings.
Reboredo, Ugolini, and Aiube (2020) report in table 1 correlations between indices of green (corporate andgovernment) bonds and investment grade corporate bonds of 0.89 and 0.91 in the EU and US markets, respectively.
Correlations between indices of green bonds and (large and mid-cap) equities are at −0.10 and −0.26 in the EU
and US markets, respectively. An equal weighted average of these correlations stands at 0.36.
1833

to empirical estimates ranging from 2 to 3. Column 5 of Table 4 (scenario “Low SE”) reports
results for an energy elasticity of about one, η ≈ 1, which is consistent with an intermediate goods
elasticity of  = 2.25, cf. Appendix B.1. The effects of green QE are then substantially smaller,
so that the relative effectiveness is now at 31. To understand this finding note that green QE canonly lower emissions by increasing the price of the dirty intermediate good. The lower intermediate
goods elasticity  implies that demand for the dirty intermediate good does not react much to its
price changes. Since this is the only lever of green QE, it becomes much less effective. Assuming
instead an in-between calibration with an intermediate goods elasticity of  = 4 (consistent withan energy elasticity of η = 1.12), which we also regard as reasonable given the long-run focus of
our analysis, leads to a relative effectiveness of 14.4.
Fifth, we feed into the model a time varying working age population ratio as described in ourcalibration section 3 (scenario “WAPR”). In the baseline scenario without any adjustment of policy
instruments, a shrinking income per capita (because of decreasing productive labor force relative
to total population) implies lower dirty emissions so that the global temperature increases by lessuntil 2100. Consequently, also in the policy experiments both policy instruments are less potent
in reducing emissions and thereby in reducing the trend increase of the global temperature so that
also the relative effectiveness is not affected much.
Sixth, we follow Nordhaus (2017) and assume a faster rate of reduction of CO2 emissionsof 1.5% annually rather than 0.5% (scenario “CO2 Re.”). This leads to a stronger price increaseof dirty relative to clean energy prices so that the gradual substitution towards clean intermediate
goods is faster in the baseline analysis and the global temperature thus increases by less. Bothinstruments turn out to be less potent, while the effectiveness of the carbon tax decreases morestrongly so that the relative effectiveness is reduced to 3.7.

5Concluding Discussion

We develop and calibrate a two-sector (clean and dirty) integrated assessment model to study
the roles of green quantitative easing (QE) and carbon taxation for mitigating global warming.
Green QE is modelled through an exogenous portfolio reallocation by the monetary authority. Asa key element of our model we assume imperfectly correlated risky returns in the two sectors of
production, which we calibrate in accordance with empirical findings. Consequently, the assumed
exogenous reallocation of capital by the monetary authority does not lead to a perfect crowdingout of private capital employed for production in the green sector and therefore green QE has realeffects.
We consider an ambitious green QE policy by assuming a complete reallocation of capital
towards the clean sector instead of a proportional split between the clean and dirty sector. We
compare the climate change mitigating effects of the green QE policy to those of a global carbon
34

tax, which is introduced at a moderate tax of 50 USD per ton of carbon and grows exogenously
such that the ad valorem tax stays constant. We find that the effects of the moderate global
carbon tax on climate change mitigation are substantially larger—by a relative effectiveness factor
of about 4.3—than what is achieved through green QE. Put differently, it would only require acarbon tax of initially 11 USD per ton of carbon to achieve the same global temperature reduction
with green quantitative easing. We show that our quantitative results crucially depend on two
assumptions, first, a calibration of a high intermediate goods elasticity in final output and second,
a constant share (instead of a constant level) of the capital stock held by the monetary authority.
We also find that pursuing a green QE policy on top of the introduction of the carbon tax
leads to an additional climate change mitigation. Thus, while the effects of green QE are rather
mild, they can make a positive contribution in a world where fiscal policy instruments are in place.
However, we do not find positive interaction effects. In fact, green QE has a larger effect if used
in isolation than in combination with a carbon tax.
We view our model with an explicit portfolio choice between assets as an important first step
to incorporate a finance perspective into the evaluation of long-run climate change mitigating
policies. On the policy side, we treat the amount of assets held by the monetary authority as given
and assume that it grows with the size of the economy. We thus assume that in a persistent low
interest rate environment—which is an endogenous outcome of our model—the monetary authority
will repetitively resort to asset purchases, which we do not explicitly model. Among various otheravenues, we leave for future research an extension of our model towards endogenous quantitativeeasing policies, which requires extending our model by adding aggregate shocks and an explicit rolefor (non-)conventional monetary policy. This would allow us to address the trade-off between on
the one hand undoing QE policies during economic booms and on the other hand pursuing green
QE to combat climate change.
Furthermore, financial frictions in the model reflecting that the clean sector potentially faces
stronger credit frictions than the dirty sector may lead to larger effects of green QE, at least in
the short-run. An additional—complementary—extension of our model would be to add directed
technical change (cf., e.g. Acemoglu, Aghion, Bursztyn, and Hemous (2012) and Acemoglu, Akcigit, Hanley, and Kerr (2016)), which may increase the effectiveness of green QE if through theuse of the instrument the initial setup costs of the green sector are mitigated and technical change
is directed more strongly towards the clean sector of the economy. In such a set-up, it would also
be interesting to study the impact of fiscal subsidies to the clean sector of the economy—financed
from the proceeds of untilted asset purchases or other fiscal revenue sources—as a policy experiment, also given the real world governance problems of implementing green QE, which we sidestep
in our analysis.

35

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39

A

Analytical Derivations and Proofs

A.1

Intermediate Goods Demand

The final representative firm operates under perfect competition maximizing



max

{Yts }s∈{cl,di} =

p t Yt −pts Yts{Yts }s∈{cl,di} 






s∈{cl,di}





max

X

 11


pt · (1 − Dt ) · Υt · 

X

1− 1

κs Yts  

1− 

−

X
s∈{cl,di}s∈{cl,di}pts Ytswhich gives the price of intermediate good s aspts
= κs ((1 − Dt ) · Υt )−1
pt



Yt
Yts


, for s ∈ {cl, di}.

and thus the intermediate goods demand
Yts =

κs
pts
pt

!
((1 − Dt ) · Υt )−1 Yt , for s ∈ {cl, di}.

and the price of the final good as
1
 1−


pt =

X
1


κs p1−ts(1 − Dt )Υt.

s∈{cl,di}A.2

The Shock Distribution

The distribution of ζti = (ζticl , ζtidi )0 , Ψ is defined implicitly via the distribution of the gross returns
on capital. The gross return on capital is assumed to follow a multivariate log-normal distribution
with


2 
!
(σclζ )
log(1 + rticl )log(1 + Ertcl ) − 2  ∼ N ,Σ2(σ ζ )
log(1 + rtidi )log(1 + Ertdi ) − di2
whereZ
Erts =

αErtis di = Γ(ψs , α, γ) ·· pts
(1 − α)
40



rtl
pts

− 1−α e − 1−γ
α
αγ
pts
·
− δs
pts

is the average marginal profit from an additionalunit of capital in sector s. Given this distributional2
assumption we get V ar (rtis ) = (1 + Erts )2 · exp σsζ − 1 and Cov (rtis=cl , rtis=di ) = (1 +




ζ
− 1 . So that the correlation is Corr (rtis=cl , rtis=di ) =Erts=cl )(1 + Erts=di ) · exp ρζcl,di σclζ σdiζ ζ
exp(ρζcl,di σcl
σdi )
ζ
r
h
i . Using exp(x) ≈ 1 + x, the correlation is Corr (rtis=cl , rtis=di ) ≈ ρcl,di .
Qζ 2s∈{cl,di} exp(σs ) −1
A.3

Proof of Proposition 1

The proof is by guess and verify using the method of undetermined coefficients. We start byshowing linearity of policy functions in total wealth, which differs across all i through optimal
portfolio shares ϑ̂∗tis . In a second step we show that ϑ̂∗tis = ϑ̂∗ts for all i and thereby that m∗tis = m∗ts
for all i.
Proof.

1. Claims: The consumption policy function in each period t for household i isc(wti ) = mti wti

for some mti and the associated value function is
U (wti ) = %ti wti
for some %ti .
2. Induction step: In any period t we get under the induction claim, writing U (wti ) = %ti wti(
U (wti ) = max
cti ,ϑ̂ti



h

c1−υ
+ β Et (%t+1i wt+1i )
ti

1−θ

1
 1−υ
i 1−υ
1−θ

).

Using the resource constraint we get
Uti (wti )









1−θ  1−υ
1−θ


1
! 1−υ


p
c1−υ
+ β Et %t+1i (wti − (1 + τtc )pt cti ) Rt+1i{ϑ̂∗tis }s∈{cl,di}ti
cti ,ϑ̂ti ,wt+1i


1
! 1−υ
 



1−θ  1−υ
1−θ
1−υ
p
c
∗
= max
c1−υ
+
β
(w
−
(1
+
τ
)p
c
)
E
%
R
{
ϑ̂
}
t ti
t
t+1i t+1i
tit
tis s∈{cl,di}ticti ,ϑ̂ti 
n
1 o
 1−υ
1−υ
1−υ
c
= max cti + β (wti − (1 + τt )pt cti )
Λt+1i

=

max

cti ,ϑ̂ti41

where Λt+1i ≡


Et

p
%t+1i Rt+1{ϑ̂∗tis }s∈{cl,di}1−θ  1−υ
1−θ

.

Take the first-order condition w.r.t cti to obtain−υcc−υ(1 + τtc )pt Λt+1i
ti = β (wti − (1 + τt )pt cti )⇔

cti = (wti − (1 + τtc )pt cti ) Ξt+1i

for1Ξt+1i = (β(1 + τtc )pt Λt+1i )− υ ,and thus
cti = mti wti
where
mti =

Ξt+1i
1 + (1 + τtc )pt Ξt+1i

42

Use this back in the objective to get
U (wti )=(mti wti )1−υ + β Et





p
%t+1i (1 − (1 + τtc )pt mti )wti Rt+1i{ϑ̂∗tis }s∈{cl,di}1−θ  1−υ
1−θ 1= (mti )1−υ + β(1 − (1 + τtc )pt mti )1−υ Λt+1i 1−υ wt
1

1−υ

1−υ ! 1−υ
Ξ−υ
Ξt+1i
1
t+1i
=
+
wt
1 + (1 + τtc )pt Ξt+1i(1 + τtc )pt 1 + (1 + τtc )pt Ξt+1i1
1−υ 
! 1−υ

1
Ξt+1i
1+
wt
=
1 + (1 + τtc )pt Ξt+1i(1 + τtc )pt Ξt+1i
1
! 1−υ

1−υ
Ξt+1i
1 + (1 + τtc )pt Ξt+1i=
wt
1 + (1 + τtc )pt Ξt+1i(1 + τtc )pt Ξt+1i
1
! 1−υ

1−υ
Ξt+1i
1
1 + (1 + τtc )pt Ξt+1i=
wt(1 + τtc )pt 1 + (1 + τtc )pt Ξt+1i
Ξt+1i
1

−υ ! 1−υ
1
Ξt+1i
=
wt
(1 + τtc )pt 1 + (1 + τtc )pt Ξt+1i
1
 1−υ

1
−υ
m
=
wt .
(1 + τtc )pt ti
We therefore get

%ti =

1
m−υ
(1 + τtc )pt ti

1
 1−υ

,

which is non-stochastic, and we can accordingly rewrite Λt+1i as
Λt+1i ≡

1c
(1 + τt+1
)pt+1

m−υ
t+1i




Et

p
Rt+1



{ϑ̂∗tis }s∈{cl,di}1−θ  1−υ
1−θ

and thus
!− υ1
 

1−θ  1−υ
1−θ
(1 + τtc )ptpΞt+1i = βEt Rt+1{ϑ̂∗tis }s∈{cl,di}
mt+1ic(1 + τt+1
)pt+1




p
c
= Θ pt , pt+1 , τtc , τt+1
, Rt+1{ϑ̂∗tis }s∈{cl,di} , β, υ, θ, Ψ mt+1i

43

1
! 1−υ

and thuspc, Rt+1
pt , pt+1 , τtc , τt+1







{ϑ̂tis }s∈{cl,di} , β, υ, θ, Ψ mt+1i



.
p
c
, Rt+1{ϑ̂tis }s∈{cl,di} , β, υ, θ, Ψ mt+1i
1 + (1 + τtc )Θ pt , pt+1 , τtc , τt+1
Θ

mti =





3. Finally, from the FOC w.r.t. ϑ̂tis we get

∂Et

p
Rt+1



{ϑ̂tis }s∈{cl,di}1−θ 
=0∂ ϑ̂tis∗∗and we thus
 get ϑ̂tis = ϑ̂ts for all i, which implies that mtis = mts for all i. Assumingpthat Rt+1{ϑ̂tis }s∈{cl,di} is distributed as log-normal we get as an approximation applying
results in Campbell and Viceira (2002) that under the assumed cross-sectional independenceof the returnsfln(1 + E [rt+1s ]) − ln(1 + rt+1)∗,
ϑ̂ts ≈
θ · V ar(ln(1 + rt+1s ))B

Calibration Appendix

B.1

Intermediate Goods Elasticity  and Energy Elasticity ηStart from equation (8) and integrate out across all i to get using E[ζtis ] = 0 thatαErts = Γ(ψs , α, γ) ·· pts
(1 − α)



rtl
pts

 e − 1−γ
− 1−α
α
αγ
pts
·
− δs
pts

from which we get

pts =

1−α
αΓ(ψs , α, γ)

αγ· (Erts + δs )αγ rtl(1−α)γ e 1−γ
pts

(23)

and thus
pts=cl=pts=di

Erts=cl + δs=cl
Erts=di + δs=diαγ  e1−γ
pts=cl.
pets=di44(24)

From the demand for intermediate goods by the final firm (2) we get the intermediate goods
demand ratioκs=di pts=cl
Yts=di=.
(25)
Yts=clκs=cl pts=diUsing (24) in the above we obtain
Yts=di= Ξ {Erts , δs , κs }s∈{cl,di}
Yts=cl(1−γ) e
pts=clpets=di(26)for some time varying Ξ {Erts , δs , κs }s∈{cl,di} .
Next, on the supply side for intermediate goods, we get from (22e) and (22f)
Yts =1 petsEts
1 − γ pts

and using (23) in the above we obtain

 pe
−γ
Ets=di
Yts=di= Λ α, γ, {Erts , δs , Γ(ψs , α, γ)}s∈{cl,di} , rtl ets=cl
Yts=clpts=di Ets=cl
for some time varying Λ

α, γ, {Erts , δs , Γ(ψs , α, γ)}s∈{cl,di} , rtl



(27)

.

Combining the intermediate goods demand and supply side, i.e., equations (26) and (27), we
thus getΛ
Ets=di=
Ets=cl

α, γ, {Erts , δs , Γ(ψs , α, γ)}s∈{cl,di} , rtl
Ξ {Erts , δs , κs }s∈{cl,di}





 e
(1−γ)+γ
pts=cl.
pets=di(28)

Holding constant the (expected) returns {Erts }s∈{cl,di} , rtl we thus find that the energy demand
elasticity is given byη Ets=di pets=di =  · (1 − γ) + γ.
Ets=cl

, pe

ts=cl

Observe that the energy elasticity is thus bounded from below by γ if the final output production
features perfect complements ( = 0). Also note that it is equal to 1 if we assume Cobb-Douglas
production of final output ( = 1).

45

C

Additional Results

Figure 10 shows the average rates of return in the two intermediate goods production sectors,
rts , s ∈ {cl, di} as well as the risk-free rate rtf in the baseline economy of the main setup with a fixedworking age population ratio in the left panel and that of the baseline economy of the sensitivity
analysis with a time-varying working working age population ratio. Both panels display decreasing
risky asset returns and an eventually decreasing risk-free rate rtf . The risk-free rate is weakly humpshaped for the case of the main setup, while with a time-varying working age population ratio it
decreases from the beginning until 2055, from where on it stays roughly constant. The decreasein rates in the left panel is a consequence of increasing climate change damages. The decrease in
the right panel is stronger as on top of climate damages also the demographic change suppresses
rates.
While the level of our calibrated model risk-free rate exceeds current market interest rates (fornumerical instability reasons), we argue on the basis of this finding that it is appropriate to assume
that the world economy will continue to be in a low interest rate environment over the projection
period. In a world with cyclical fluctuations, this will likely lead to repeated application of nonconventional monetary policy through quantitative easing so that our assumption of a constantshare of total assets held by the central bank is a reasonable approximation.
Figure 10: Baseline: Climate Variables(a) Fixed WAPR - Main setup(b) Time-varying WAPR - Sensitivity setup

Notes: Financial returns in the main setup with a constant working age population ratio in panel (a) and with atime-varying working age popoulation ratio as in sensitivity analysis WAPR in panel (b). Each panel shows the risky
returns rts , s ∈ {cl, di} and the risk-free return rtf .

46