# Reducing residential emissions: carbon pricing vs. subsidizing retrofits

Alkis Blanz  University of Leipzig, Faculty of Economics and Management
Sciences, Grimmaische Str. 12, 04109, Leipzig, Germany Mercator Research
Institute on Global Commons and Climate Change (MCC), Torgauer Str. 12-15,
10829, Berlin, Germany  Beatriz Gaitan  Potsdam Institute for Climate Impact
Research, Telegrafenberg, 14473, Potsdam, Germany

(August 31, 2024)

In this paper, we compare different mitigation policies when housing
investments are irreversible. We use a general equilibrium model with non-
homothetic preferences and an elaborate setup of the residential housing and
energy production sector. In the first-best transition, the energy demand
plays only a secondary role. However, this changes when optimal carbon taxes
are not available. While providing subsidies for retrofits results in the
lowest direct costs for households, it ultimately leads to the highest
aggregate costs and proves to be an ineffective way to decarbonize the
economy. In the second-best context, a phased-in carbon price outperforms the
subsidy-based transition.

We are very grateful to Matthias Kalkuhl, Kai Lessmann, Ottmar Edenhofer for
useful discussion and comments. This project has received funding from the
European Union’s Horizon Europe research and innovation programme under grant
agreement No 101056891 - CAPABLE - ClimAte Policy AcceptaBiLity Economic
framework. Views and opinions expressed are however those of the author(s)
only and do not necessarily reflect those of the European Union. Neither the
European Union nor the granting authority can be held responsible for them.

##  1 Introduction

Reducing the carbon emissions in the housing sector requires large upfront
investments. Investment needs include retrofitting the existing housing stock
and constructing new, more energy-efficient buildings to meet housing demand
(IEA, 2021). However, vast evidence indicates that current investment levels
are insufficient, increasing the possibility of a carbon lock-in (Cabeza et
al., 2022). In this situation, investments are inconsistent with climate
targets, leaving only expensive reductions through stranded assets as an
option for emission reductions. Preventing a carbon lock-in in residential
housing calls for ambitious mitigation policy that can stimulate and
coordinate various types of investment.

In the context of housing investments, the optimal instrument choice for
mitigation is unclear. Housing-related investments are different. Investments
in housing are irreversible (Miles, 2009) and therefore cannot be converted
after construction. Furthermore, climate policy in housing is challenging from
a political economy perspective, as shelter is a basic need for humans.
Implementing ambitious climate policy can be difficult to implement. Imposing
high carbon taxes on households increase housing costs and utility bills,
which may increase poverty. Furthermore, if the direct costs for households
are too high, households may not be able to invest sufficiently in energy
efficiency and the expansion of renewable energy.

This paper addresses this gap and compares different mitigation policies when
housing investments are irreversible. We build a multisector general
equilibrium model with a special focus on the different investment decisions
affecting the decarbonization of the residential housing sector. The model
combines an elaborate setup of the residential housing market alongside energy
markets. We separate investments to reduce the energy intensity of the
existing housing stock from investments in the construction of new, more
energy-efficient buildings. Furthermore, we allow for investments in fossil
and non-polluting capital used in energy production. We compare different
carbon pricing scenarios with transitions that rely on housing-related
investment subsidies and calibrate the model to the case of Germany. Thereby,
we isolate the impact of policy constraints for the green transition.

When policymakers face no constraints on the availability of mitigation
policies, the conventional wisdom holds and a uniform carbon price is the
cost-effective instrument for the green transition (Pigou, 1932; Goulder and
Parry, 2008; Fischer and Newell, 2008). In the first-best scenario, energy
demand plays only a secondary role. However, this changes when optimal carbon
taxes are not available. When it is not possible to directly increase fossil
resource prices, climate policy has to rely on reductions in energy demand to
reduce fossil fuel consumption in housing. Due to the irreversibility
constraint, the only option is to excessively subsidize investments in energy
efficiency, making the subsidy-based transition very costly from a social
perspective.

This paper builds on several strands of the literature. First, we add to the
literature that analyzes optimal timing and allocation of abatement
investments across sectors. Vogt-Schilb et al. (2018) find that it is optimal
to start with significant short-term abatement investment because we cannot
switch overnight to carbon-free technologies. It matters in which sectors the
short-term effort abatement investment happens (Vogt-Schilb and Hallegatte,
2014; Lecocq et al., 1998; Vogt-Schilb and Hallegatte, 2014). We extend this
literature by explicitly considering housing investments related to the energy
demand of the housing stock. We are able to show that housing investments
differ from investments in industrial capital. In contrast to industrial
capital, investing in housing capital directly increases utility at the cost
of having a higher energy demand. The role of energy efficiency investments is
to solely reduce these direct costs of housing investments. One contribution
is to show how climate policy optimally affects the different investment
incentives over time.

Furthermore, we relate to the large literature that focuses on the energy
efficiency gap (Jaffe and Stavins, 1994; Allcott and Greenstone, 2012;
Gerarden et al., 2017), the difference between optimal and actual energy use.
While the existing literature considers both behavioral failures and market
failures as potential causes for inefficient energy use, it focuses largely on
static frameworks. We complement this literature by analyzing the intersection
within a dynamic, general-equilibrium setting. Thereby, we are able to
consider the interaction of energy demand in housing with climate policy and
macroeconomic activity. We contribute to this literature by analyzing the
energy efficiency gap in a dynamic setting within the context of reducing
residential carbon emissions.

The closest paper to ours is Rozenberg et al. (2020). The authors compare the
impact of different mitigation policies in a multisector general equilibrium
model with irreversible investment and a climate constraint. We consider our
paper to be complementary to their analysis. Rozenberg et al. (2020) discuss a
potential trade-off between the political feasibility and cost-effectiveness
of mitigation policies based on the premature retirement of fossil capital.
Instead, we consider housing costs, including energy bills, as the main source
of transition costs for households. Thereby, we expand their insights for
energy production by focusing on the case of residential housing.

The paper is structured as follows. Section 2 describes the model environment.
Section 3 summarizes the calibration, while section 4 presents the main
analysis. Finally, section 5 concludes.

##  2 Model Environment

This section presents the dynamic multi-sector general equilibrium model used
for analyzing the transition towards a low carbon economy. The model combines
an elaborate setup of the housing and energy sectors, as well as an optimizing
government as in Kalkuhl et al. (2012). The dynamic setting allows for
studying the interaction between sectors and actors. The decisions of
households determine how energy-intensive their housing demand is. Energy
producers decide on how carbon-intensive energy production is, and the
government sets policy instruments optimally to achieve a climate target.

###  2.1 Households

The economy is populated by $n$ households who have preferences of type:

$\max\sum_{t=0}^{T}\frac{1}{\left(1+\rho\right)^{t}}\frac{\left(c_{t}^{\phi}\left(h_{t}-\bar{h}\right)^{1-\phi}\right)^{1-\eta}}{1-\eta},$
(1)

where $\rho$ is the pure rate of time preference, $\eta^{-1}>0$ is the
elasticity of intertemporal substitution and $\phi$ is a share
parameter.111Depending on the application, we either use the pure time
preference rate $\rho$ or the discount factor $\beta$. Recall that they are
directly related: $\rho=(1-\beta)/\beta$. Households derive utility from final
consumption $c_{t}$ and from housing services $h_{t}$. Housing services are
subject to a subsistence level $\bar{h}$ that captures the minimal need for
shelter for humans. Thus, preferences are non-homothetic, as in (Geary, 1950;
Stone, 1954).

Households have access to a wide portfolio of assets. These assets include
industry capital used in the production of the final good and electricity, as
well as different assets related to the production of housing services. As the
focus lies on aggregate dynamics, we assume that all households are property
owners and produce their own housing services. Households produce housing
services according to:

$h_{t}=land_{t}^{a_{h}}(\bar{k}+k_{t}^{H})^{1-a_{h}}$ (2)

The production of housing services requires $land_{t}$ and housing capital
$\bar{k}+k_{t}^{H}$. We distinguish between old housing capital ($\bar{k}$)
with a high energy intensity and capital $k^{H}_{t}$ with a low energy
efficiency. The total amount of land is assumed to be fixed. Thus, by
expanding the housing stock, households can consume more housing services and
thereby increase their utility. However, the construction of new buildings
affects the energy demand of housing $ene_{t}$. The energy demand of the house
covers both the appliances of the house and heating. Housing-related energy
demand is described by:

$ene_{t}=\kappa_{o}(k^{E})\bar{k}+\kappa_{N}k_{t}^{H}$ (3)

Depending on the energy efficiency of new buildings $\kappa_{N}>0$, expanding
the housing stock will increase the total energy demand of the housing stock.
Additionally, the total energy demand depends on the energy demand of the old
housing stock $\bar{k}$. Especially in urban areas, old buildings are
maintained so that they remain in use. The parameter $\kappa_{N}>0$ and the
function $\kappa_{o}$, explained below, describe the energy efficiency of each
housing stock. We assume that households pay maintenance costs on the existing
housing stock to overcome its depreciation so that the old stock remains
constant. The existence of maintenance costs is a common feature of housing
models (Piazzesi and Schneider, 2016; Henderson and Ioannides, 1983; Li and
Yao, 2007). Essentially, this simplification captures the fact that part of
the housing stock is of such high value to society that it will not be
demolished even if the energy intensity of these buildings is high. Instead,
we allow for investments in energy efficiency capital $k^{E}$. The stock of
efficiency capital captures all kinds of retrofits, such as improvements to
the insulation of walls, windows, and the roof. Thus, investments in
efficiency capital reduce the energy intensity $\kappa_{o}$ of the existing
stock, which is:

$\kappa_{o}(k^{E})=\frac{\bar{\kappa}}{k_{t}^{E}}+\kappa_{N}$ (4)

where $\bar{\kappa}$ is a positive constant. Thus, there are limits to the
improvements in energy efficiency of the existing housing stock. In detail,
the function form implies that in the limit, the energy demand of old
buildings cannot exceed the energy efficiency of new buildings $\kappa_{N}$.
Investments in housing capital and efficiency capital determine the total
energy demand of the housing stock. Increasing the production of housing
services by expanding the housing stock decreases the overall energy
intensity, as new buildings are considered to be more energy efficient but
increases the overall energy demand. In contrast, efficiency capital does not
increase the production of housing services but decreases the energy demand of
the housing stock.

The energy demand for housing determines utility bills and, thus, housing
costs. Depending on their investment choices into housing and efficiency
capital, utility bills will be higher or lower. The energy demand related to
housing can be satisfied by using electricity and a fossil resource. We assume
that electricity $e_{t}$ and the fossil resource $res_{t}$ are imperfect
substitutes for households.

$ene_{t}=\left(a_{ene}e_{t}^{\frac{\sigma_{ene}-1}{\sigma_{ene}}}+(1-a_{ene})res_{t}^{\frac{\sigma_{ene}-1}{\sigma_{ene}}}\right)^{\frac{\sigma_{ene}}{\sigma_{ene}-1}}$
(5)

The elasticity of substitution between electricity and the fossil resource is
denoted $\sigma_{ene}>0$, and $a_{ene}$ and $(1-a_{ene})$ are their respective
share parameters. Households pay for electricity and for the use of the fossil
resource. The use of the fossil resource may be subject to a carbon price
$p_{HC,t}$. Therefore, the expenditure side of the households consists of
consumption expenditures, investment $inv^{i}_{t}$ in the portfolio of
industry capital, land, and the two stocks of housing-related capital, and the
energy costs attached to them.

Land, industry capital (final good capital ($k_{t}^{Y}$) and fossil and
renewable energy capital ($k_{t}^{F}$ and $k_{t}^{N}$)), housing, and
efficiency capital evolve according to

$\displaystyle land_{t+1}=$ $\displaystyle i_{t}^{Land}+land_{t}$ (6a)
$\displaystyle k_{t+1}^{Y}=$ $\displaystyle
i_{t}^{Y}+\left(1-\delta_{Y}\right)k_{t}^{Y}$ (6b) $\displaystyle
k_{t+1}^{F}=$ $\displaystyle i_{t}^{F}+\left(1-\delta_{F}\right)k_{t}^{F}$
(6c) $\displaystyle k_{t+1}^{N}=$ $\displaystyle
i_{t}^{N}+\left(1-\delta_{H}\right)k_{t}^{N}$ (6d) $\displaystyle
k_{t+1}^{H}=$ $\displaystyle i_{t}^{H}+\left(1-\delta_{F}\right)k_{t}^{H}$
(6e) $\displaystyle k_{t+1}^{E}=$ $\displaystyle
i_{t}^{E}+\left(1-\delta_{N}\right)k_{t}^{E}$ (6f)

where $i^{j}_{t}$ denotes the respective investment, and $\delta_{j}$ is the
rate of capital depreciation. In a dynamic general equilibrium model, agents
must be indifferent between investing in industry capital, housing capital,
and energy efficiency capital. Thus, households will consider not only energy
and land prices but also the interest rate when making investment decisions.
This is reflected in the non-arbitrage conditions, which can be found in the
appendix.

Housing-related investments differ from investments in industry capital.
Investments in housing capital and energy efficiency capital are irreversible
as in Arrow and Kurz (1970). Once households invest in housing, capital cannot
be transformed into consumption or industry capital. The irreversibility is
summarized by:

$inv_{t}^{H}\geq 0$ (7) $inv_{t}^{E}\geq 0$ (8)

###  2.2 Production

On the production side of the economy, we assume the production of a final
good and energy. The final good is used for consumption and for investments.
The production technology of the final good producer is described by:

$Y_{t}=\left(aZ_{t}^{\frac{\sigma-1}{\sigma}}+\left(1-a\right)E_{Y,t}^{\frac{\sigma-1}{\sigma}}\right)^{\frac{\sigma}{\sigma-1}}$
(9)

For production, the final good producer relies on two intermediate inputs. The
final good producer combines final electricity $E_{Y,t}$ from the electricity
producer and a capital-labor composite. The capital-labor composite equals:

$Z_{t}=K_{Y,t}^{a_{Z}}\left(A_{Y,t}L_{t}\right)^{1-a_{Z}}$ (10)

where $K_{Y,t}$ and $L_{t}$, respectively, denote the capital and labor
employed in the production of $Y_{t}$, and $A_{Y,t}$ is exogenous labor
augmenting technological change. We follow Barrage and Nordhaus (2023) so that
$A_{Y,t}$ evolves according to

$A_{Y,t}=A_{Y,t-1}/\left(1-a_{Y}e^{-b_{Y}\left(t-1\right)}\right)$ (11)

where $a_{Y}$ and $b_{Y}$ are positive constants. The production side of the
economy allows for an elaborate description of the electricity sector that
resembles in many aspects the approach employed in Kalkuhl et al. (2012). The
final electricity producer demands electricity from fossil ($E_{F,t}^{d}$) and
non-polluting renewable sources ($E_{N,t}^{d}$). The production function of
the final energy firm is:

$E_{t}=\left(a_{E}\left(E_{F,t}^{d}\right)^{\frac{\sigma_{E}-1}{\sigma_{E}}}+\left(1-a_{E}\right)\left(E_{N,t}^{d}\right)^{\frac{\sigma_{E}-1}{\sigma_{E}}}\right)^{\frac{\sigma_{E}}{\sigma_{E}-1}}$
(12)

where $a_{E}$ is a share parameter and $\sigma_{E}$ is the elasticity of
substitution between $E_{F,t}^{d}$ and $E_{N,t}^{d}$. The role of the final
energy firm in our setting is that of an electricity provider. The firm does
not produce the electricity itself by operating wind farms and power plants,
but buys the electricity from fossil and renewable electricity producers.
Thus, the profits of the final electricity firm are described by:

$\pi_{E,t}=p_{E,t}E_{t}-p_{F,t}E_{F,t}^{d}-p_{N,t}E_{N,t}^{d}$ (13)

where $p_{E,t}$ denotes the final price of energy and $p_{F,t}$ and $p_{N,t}$,
respectively, denote the prices of fossil and renewable energy.

The fossil energy producer uses capital and fossil resources to produce
electricity. The production function of the polluting electricity firm is:

$E_{F,t}=\left(a_{F}K_{F,t}^{\frac{\sigma_{F}-1}{\sigma_{F}}}+\left(1-a_{F}\right)Res_{F,t}^{\frac{\sigma_{F}-1}{\sigma_{F}}}\right)^{\frac{\sigma_{F}}{\sigma_{F}-1}}$
(14)

where $a_{F}$ is a are share parameters, and $\sigma_{F}$ is the elasticity of
substitution between$K_{F,t}$ and $Res_{F,t}$. Based on the setup in Kalkuhl
et al. (2012), we assume that the use of a fossil resource is the source of
carbon emissions and not the use of a dirty capital stock. We abstract from
fossil resource production within the economy. Instead, we assume a small-open
economy that imports the fossil resource at a price $p_{R,t}$. Climate policy
can affect the full price of the fossil resource by imposing a carbon price
$p_{C,t}$. Thus, the profits of the polluting fossil electricity firm are
described by:

$\pi_{F,t}=p_{F,t}E_{F,t}-R^{F}_{t}K_{F,t}-(p_{R,t}+p_{C,t})Res_{F,t}$ (15)

In contrast to the polluting firm, the renewable energy producer does not rely
on fossil resources to produce electricity. Producing non-polluting
electricity is a function of capital. The production technology and profits of
the non-polluting electricity firm are described by:

$E_{N,t}=A_{N,t}K_{N,t}$ (16)

$\pi_{N,t}=p_{N,t}E_{N,t}-R^{N}_{t}K_{N,t}$ (17)

$A_{N,t}$ is exogenous technological change that evolves according to

$A_{N,t+1}=A_{N,t}/\left(1-g_{N}\right).$ (18)

Given a rental rate of capital $r_{t}$, the cost of producing renewable energy
declines at the exogenous rate $g_{N}$. This specification resembles the
decrease in cost of the backstop technology employed in Barrage and Nordhaus
(2023).

###  2.3 Government

The primary focus of the government in our setup is climate policy. As in
Kalkuhl et al. (2012), the government is an agent that, acting as a
Stackelberg leader of its own economy, chooses policy instruments optimally
under a given set of constraints. We assume that the government has a fixed
emission target in the form of a carbon budget and a set of policy tools to
achieve the target. For expositional simplicity, we assume that carbon
emissions are directly associated with the use of the fossil resource both in
housing and energy production. By assuming a common emission target, we fix
the level of ambition across scenarios and allow for a comparison of the
effectiveness of different policy instruments. The climate target is given by:

$\overline{M}=\sum_{t=0}^{T}(Res_{F,t}+nres_{t})$ (19)

The main climate policy tool is a carbon price that taxes the use of the
fossil resource. However, we allow for differentiated carbon prices in housing
and industry. Carbon pricing is the sole revenue source for the government in
our setting. The budget constraint of the government in the baseline analysis
is:

$\Gamma_{t}+n\tau_{INVE,t}inv_{t}^{E}=p_{C,t}Res_{F,t}+np_{HC,t}res_{t}$ (20)

where $\tau_{INVE,t}$ is a subsidy to energy efficiency capital investment.
Depending on the set of policies considered, the revenues from carbon pricing
can be used either to pay transfers to households $\Gamma_{t}$ or to subsidize
investments into efficiency capital.222Theoretically, it could also be an
option to subsidize investments into housing capital, but as it is never
optimal in our setting, we omit it from the budget constraint for expositional
simplicity.

###  2.4 Market clearing

This section briefly presents the market clearing conditions in our setup.
Capital market clearing equals

$K_{Y,t}=nk_{t}^{Y}\text{, }K_{F,t}=nk_{t}^{F}\text{,
}K_{N,t}=nk_{t}^{N}\text{;}$ (21)

Market clearing in electricity equals

$E_{Y,t}+ne_{t}=E_{t};$ (22)

The market clearing in fossil energy is:

$E_{F,t}^{d}=E_{F,t};$ (23)

Similarly, the market clearing in renewable energy equals

$E_{N,t}^{d}=E_{N,t};$ (24)

Finally, the balance of payments is balanced if

$\displaystyle Y_{t}=$ $\displaystyle
n\left(c_{t}+i_{t}^{Y}+i_{t}^{F}+i_{t}^{N}+i_{t}^{H}+i_{t}^{E}+\delta_{H}\overline{k}\right)$
(25) $\displaystyle+p_{R,t}\left(n\times res_{t}+Res_{F,t}\right)$

Table 1: Calibration Parameter | description | value |
---|---|---|---
$a$ | share of capital-labor composite $Z$ | $0.95$ |
$\sigma$ | elasticity of substitution $Z-E_{Y}$ | $0.40$ |
$a_{Z}$ | share of capital $K_{Y}$ in $Z$ | $0.35$ |
$A_{Y,0}$ | labor productivity at $t=0$ | $1.00$ |
$a_{E}$ | share of fossil energy $E_{F}$ | $0.57$ |
$\sigma_{E}$ | elasticity of substitution $E_{F}-E_{N}$ | $4.70$ |
$a_{F}$ | share of capital $K_{F}$ | $0.80$ |
$\sigma_{F}$ | elasticity of substitution $K_{F}-Res_{F}$ | $0.20$ |
$A_{N,0}$ | capital $K_{N}$ productivity at $t=0$ | $0.69$ |
$\phi$ | share of final good consumption | $0.80$ |
$\bar{h}$ | housing services minimum consumption | $1.62$ |
$a_{H}$ | share of land in housing services | $0.35$ |
$a_{ene}$ | share of electricity $e$ in housing energy | $0.14$ |
$\sigma_{ene}$ | elasticity of substitution $e-res$ | $0.89$ |
$\kappa_{N}$ | new housing capital energy intensity | $0.02$ |
$\bar{\kappa}$ | parameter in $\kappa_{o}\left(k_{t}^{E}\right)$ function | $0.005$ |
$\eta^{-1}$ | elasticity of intertemporal substitution | $1.00$ |
$\rho$ | consumers rate of time preference | $0.02$ |
$\delta_{Y}=\delta_{F}=\delta_{N}$ | industry depreciation rate of capital | $0.10$ |
$\delta_{H}$ | depreciation rate of housing capital | $0.02$ |
$\delta_{E}$ | depreciation rate of energy efficiency capital | $0.03$ |
$Land=n\ast land$ | Aggregate land endowment | $501.23$ |
$k_{0}^{E}/\overline{k}$ | Ratio $k_{0}^{E}$ to $\overline{k}$ | $0.08$ |
$a_{Y}$ | constant in labor augmenting technological change | $0.0859$ |
$b_{Y}$ | growth parameter in labor augmenting technological change | $0.0072$ |
$g_{N}$ | growth parameter of technological change | $0.01$ |

##  3 Calibration

We calibrate the model to the German economy and indicate the parameter values
used throughout the simulations in Table 1.

###  3.1 Consumers

For the share $\phi$ of final good consumption, we use German input-output
data for the year 2014 from the World Input-Output Database (WIOD, refer to
Timmer et al. (2015)). Let $x_{c}$ denote aggregate household expenditures
except for real estate activities and energy-related expenditures (mining and
quarrying, manufacture of coke and refined petroleum products, and
electricity, gas, steam, and air conditioning supply) from the WIOD. Let
$x_{h}$ denote the expenditure on real estate activities, and $x_{ene}$ denote
energy-related expenditures from the WIOD. Aggregate expenditure, thus, equals
$x\equiv{x_{c}+x_{h}+x_{ene}}.$ The share $x_{c}/\left(x_{c}+x_{h}\right)$
equals $0.81$, we round this number and set $\phi=0.80$. The housing services
composite is similar to Combes et al. (2021) who find a share parameter for
capital equal to $0.65$ using French data. For lack of a German estimate we
set $\left(1-a_{h}\right)$ equal to $0.65$ so that the share of land $a_{h}$
equals $0.35$.

For the housing energy parameters $a_{ene}$ and $\sigma_{ene}$, we focus on
heating space and hot water, which together account for about $80\%$ of total
residential energy use in Germany (refer to IEA (2020)). For the housing
fossil energy share parameter $\left(1-a_{ene}\right)$, we employ the
estimates of direct residential fossil energy (coal, gas, and oil) from the
Institute for Housing and Environment (IHE, refer to Loga et al. (2012)). To
account for the large share of fossil energy in district heating, we add
$70\%$ of residential district heating energy (as reported by the IHE) to the
direct fossil energy use and generate a total fossil energy use.333More than
$70\%$ of district heating is produced using fossil resources according to the
Federal Ministry for Economic Affairs and Climate Action of Germany (2021).
The ratio of total fossil energy use to total energy use equals 0.854. Since
more than $70\%$ of district heating is produced using fossil resources, we
round up $0.854$ and set $\left(1-a_{ene}\right)$ equal to $0.86$, so that the
share of electricity $\left(a_{ene}\right)$ equals $0.14$.

Let $s_{e}$ denote the household expenditure share of electricity out of total
household energy expenditure. Let $\varepsilon_{res}$ denote the households’
fossil resource partial own price elasticity of demand. With a CES function
$\varepsilon_{res}=-s_{e}\sigma_{ene}$ (cf. Allen (1938) p. 373) where
$\sigma_{ene}$ is the elasticity of substitution between electricity and
fossil resources by households. Since natural gas is the largest source of
energy for space and water heating in Germany, we use Nilsen et al. (2012)
estimate of the short-run price elasticity of demand for natural gas of German
households. Their estimate equals $-0.131$. We set
$s_{e}=\left(1-0.854\right)$, setting $\varepsilon_{res}=-0.131$ and using
$\varepsilon_{res}=-0.131=-s_{e}\sigma_{ene}=-\left(1-0.854\right)\sigma_{ene}$
implies $\sigma_{ene}=0.89,$ which is the value we use for the elasticity of
substitution between electricity and fossil resources used by households.444An
elasticity smaller than 1 indicates that fossil fuels and electricity are
complements rather than substitutes in housing. While historically this seems
intuitive, investments in district heating or heat pumps may affect this
elasticity. Due to the focus on energy efficiency, we leave this aspect for
future research.

We set the rate of time preference $\rho$ equal to $0.02$, a value often used
elsewhere (see Barro and Sala-i Martin (2003)). In view of Guvenen (2006)
findings, we set the inverse of the elasticity of intertemporal substitution
$\eta$ equal to 1.

###  3.2 Depreciation and production

Davis and Van Nieuwerburgh (2015) survey the macroeconomic housing literature
and find that the estimates of the depreciation rate of housing structures are
in the range of 0.01 to 0.03. Based on this, we set the depreciation rate of
housing capital $\left(\delta_{H}\right)$ equal to $0.02$. For the
depreciation rate of energy efficiency capital, $\left(\delta_{E}\right)$, we
use the depreciation rate of electricity, oil, and wood heaters of $0.03$ used
by Nesbakken (2001) who analyzes the energy consumption of the heating of
space for the case of Norway. For the rate of industry capital depreciation
($\delta_{K}$), we employ the value of 0.10 used by Kiyotaki et al. (2011), a
value that is also consistent with the long-run rate of capital depreciation
in manufacturing found in Albonico et al. (2014). The separation of housing
capital leads to larger estimates for the depreciation of manufacturing
capital than the values typically employed for aggregate capital.

Regarding the elasticity of substitution between the capital-labor composite
and electricity of the final good ($\sigma$), we employ estimates of Koesler
and Schymura (2015). Their estimate of all industries is equal to 0.38. We
round this number and set $\sigma=0.4$. The share of the capital-labor
composite of the final good is proxied by making a sectoral aggregation of the
input-output matrix of the WIOD. We aggregate the input-output matrix into
four sectors, namely i) fossil resources, ii) electricity, gas, steam, and air
conditioning supply, iii) real estate activities, and iv) all the remaining
sectors which we think of as the final good. We set the share of the capital-
labor composite in the production of the final good $\left(a\right)$ equal to
the share of value-added in energy expenditures (i and ii) plus value added.
This leads to $a=0.95$.

Regarding the capital share of the capital-labor composite
$\left(a_{Z}\right)$, we use Valentinyi and Herrendorf (2008) estimate. They
find that the omission of intermediate inputs leads to biased values if
capital and labor shares are directly estimated from input-output tables. They
solve this by calculating the amount of capital and labor embodied in
intermediate inputs and impute this to sectoral capital and labor shares. No
estimates for Germany are available, but the US and German economies are
sufficiently similar such that we use their estimate. Valentinyi and
Herrendorf estimate capital shares of agriculture, manufactured consumption,
services, equipment investment, and construction; and consider various
aggregations of these five sectors. Their aggregation of agriculture,
manufactured consumption, and services leads to a capital share of $0.35$. We
set the capital share equal to this value $\left(a_{Z}=0.35\right)$

For the elasticity of substitution between fossil and clean energy,
$\left(\sigma_{E}\right)$ and the respective share parameters of the energy
composite, we use the German estimates of Stoeckl and Zerrahn (2020). We use
the average of their current and future elasticity of substitution estimates
(the latter takes into account capacity constraints), the average equals
$4.7$, we thus set $\sigma_{E}=4.7$. We take a similar average from Stoeckl
and Zerrahn in the case of the share of clean energy $\left(1-a_{E}\right)$
leading to $0.43$, and thus set the share of fossil energy
$\left(a_{E}\right)$ equal to $0.57$.

In the case of the elasticity of substitution between capital and the fossil
resource $\left(\sigma_{F}\right)$ and the capital share parameters
$\left(a_{F}\right)$ in the production of fossil energy, we use the values
used by Kalkuhl et al. (2012) and thus set $\sigma_{F}=0.15$, and $a_{F}=0.8$.

The parameters $a_{Y}$ and $b_{Y}$ in Table 1 reproduce the values employed by
Barrage and Nordhaus (2023), and $g_{N}$ decreases the cost of clean energy by
1

The rest of the parameters $\bar{h},$ $\bar{\kappa},$ $\kappa_{N}$, $A_{N,0},$
and the $Land,$ and the ratio $k^{E}/\bar{k}$ of respective capital endowments
are set so that a steady-state solution, absent of policy and technological
progress, satisfies the following properties.

According to the German Federal Statistical Office (Statistisches Bundesamt
(2022)), housing costs (including energy) were on average $23.3\%$ of
disposable income of households. We solve the steady state of the model
without policy so that the expenditure on housing services and energy on
disposable income equals $0.233$.

In Germany during the year 2019, a $70m^{2}$ well-insulated apartment building
with gas heating had an average heating cost of $485$ euros, instead, the
average heating costs of a $70m^{2}$ apartment in a badly insulated building
amounted to $1,030$ euros (see CLEW (2020)). We use this information and solve
the steady state of the model without policy so that the ratio of the energy
efficiency of old housing capital to that of new housing capital, that is
$\left(\bar{\kappa}/k_{E}+\kappa_{N}\right)/\kappa_{N}$ equals $1,030/485$.

According to Federal Ministry for Economic Affairs and Climate Action of
Germany (2022), the share of renewables in the German electricity sector was
$41\%$ in the year 2021. We use this value and solve the steady state of the
model without policy so that the ratio of clean energy to clean and fossil
energy $E_{N}/(E_{N}+E_{F})$ equals 0.41.

The emissions generated in the German electricity and building sectors
(residential, commercial, and military), respectively, amounted to $247$ and
$115$ million tonnes of CO2 equivalents in 2021 (refer to Statista (2022a)).
Of the building emissions, $76\%$ were residential emissions; in other words,
residential emissions amounted to about $115*0.76=87.4$ million tonnes of
$CO_{2}$ equivalents (see Statista (2022b)). We solve the steady state of the
model without policy so that the ratio of residential fossil consumption to
residential plus electricity fossil consumption $n*res/(n*res+Res_{F})$ equals
87.4/(87.4+247) =0.26

The Institute for Housing and Environment (see Loga et al. (2012)) provides
the area of single and multifamily housing units in Germany. They divide
housing into different groups depending on when they were built. We aggregate
the single and multifamily houses that were constructed until 1978, the year
in which the Ordinance on Thermal Insulation went into effect. We consider
those houses constructed until 1978 to be the old housing capital stock with a
higher energy intensity and those constructed after to be of low energy
intensity. The share structures constructed before 1978 equals $66\%$. We
solve the steady state without policy so that the ratio
$\bar{k}/\left(\bar{k}+k^{H}\right)=0.66.$

In the steady-state solution without policy, we set the price of land equal to
one so that the aggregate land endowment can be determined. The values of
$\bar{h}$, $\bar{\kappa}$, $\kappa_{N}$, $A_{N,0}$, and the $Land$, and the
ratio $k^{E}/\bar{k}$ are indicated in Table 1.

##  4 Results

The transition to a carbon-free economy depends on ambitious climate policy.
In the absence of climate policy, the economy continuously relies on the use
of fossil resources for both housing and energy production. The aim of this
section is to study various transitions towards a carbon-free economy. All
transitions are optimal given the respective constraints, as all agents,
including the government, behave optimally. However, in all scenarios, the
government has access to distinct sets of policy instruments. Specifically, we
analyze the first-best transition without limitations on the availability of
different policy instruments and compare it to several second-best settings.
The main restriction we consider is on carbon pricing in the housing sector.
In detail, we compare the first-best transitions to transitions where carbon
pricing in housing is phased-in or is entirely unavailable.

When analyzing the first-best transition, we rely additionally on analytical
derivations from the model setup. Thereby, we are able to isolate how climate
policy can incentivize the investment shifts necessary for decarbonizing the
economy. Additionally, we can confirm that the conventional wisdom holds in
our setup. The optimal instrument is a uniform carbon price in all sectors of
the economy. Interestingly, the irreversibility constraint does not affect
optimal instrument choice. Instead, the constraint matters for the housing-
related energy demand across the transition path. To analyze the optimal
instrument choice and the implications of the irreversibility constraint, we
rely on numerical simulations of the model.

###  4.1 Optimal Transition

The analysis begins by comparing the optimal transition to the no-policy
benchmark. In all scenarios, the economy grows due to exogenous technological
progress. The increase in economic output and household prosperity lead to a
higher demand for energy and housing. Thereby, the demand for the fossil
resource increases as well. The fossil resource enters both the supply, as
well as the demand side of the economy. Fossil energy producers use the
resource alongside capital to produce energy. Additionally, households use the
fossil resource alongside electricity to satisfy the energy demand of their
housing demand. In the laissez faire, these decisions are not affected by
climate policy.

The transition requires changing the price of using the fossil resources, as
the externality linked to the use of the resource is not accounted for.
However, apart from the resource use, internalizing these external costs may
also influence investment behavior of households. In our setup, households
invest in both industry and housing capital. Industry capital covers capital
that relates to energy and final good production. Housing and efficiency
capital are relevant for producing housing services. These two types of
investment exhibit fundamental differences and react differently to climate
policy. Through analytical derivations of the model, we can explore this nexus
in more detail.

At the core of our general equilibrium model is a rich investment portfolio.
As these investments are interdependent, households choose investments such
that they are indifferent between the different investment options.
Conceptually, investing in capital for energy production, whether it be fossil
or non-polluting, provides an alternative to investing in physical capital.
All three are forms of industrial capital. Households choose consumption and
investment in the laissez-faire equilibrium such that the values of physical
capital, as well as clean and fossil capital, are equalized:

$\psi_{t}=\psi_{t}^{C}=\psi_{t}^{F}=\lambda_{t}$ (26)

where $\psi_{t}$ is the value of investments in physical capital, $i\in(F,N)$
are the shadow values of investments in fossil and non-polluting capital, and
$\lambda_{t}$ is the shadow value of income. In the end, fossil and clean
capital are simply another way to invest in physical capital. As perfect
capital mobility exists, the return of these assets is determined essentially
by the Euler equation:

$\displaystyle R_{t+1}=$
$\displaystyle\frac{1}{\lambda_{t+1}}\left[\frac{\psi_{t}}{\beta}-\left(1-\delta_{Y}\right)\psi_{t+1}\right]=\frac{u_{c_{t}}}{\beta
u_{c_{t+1}}}-\left(1-\delta_{Y}\right)$ (27)

Without climate policy, the marginal productivity of fossil and clean capital,
as well as their returns, are equal:555 The only possible difference in
returns are differences in depreciation rates.

$R_{t}=R^{F}_{t}=R^{N}_{t}$ (28)

Climate policy affects investment in energy-related capital stocks indirectly
through its effect on incentives for energy producers. Without a carbon price,
the fossil energy producer chooses fossil resource use by setting its value
marginal productivity equal to the exogenous price $p_{R,t}$. However, the
price of fossil energy is too low from a societal perspective since it is not
high enough to satisfy the emissions budget. With a carbon price, the demand
for non-polluting capital increases. Through a relative price change,
electricity producers rely less on fossil energy, which increases the
incentive to move capital to renewable energy production. The change in
marginal productivity of fossil and non-polluting capital affects the return
on investments and, thus investment behavior of households. Optimal climate
policy shifts investments from fossil capital to non-polluting capital.

Housing investments differ from investments in industrial capital. An increase
in housing capital leads to a higher level of housing services, making housing
investments a direct source of utility rather than increasing income in the
future. The implicit return of investing in housing is utility. Non-homothetic
preferences determine the increase in utility from additional housing
consumption and, thereby, the return from investing in housing capital.
Housing services are subject to a subsistence level. Depending on household
income levels, the increase in housing capital increases utility differently.
For example, an additional unit of housing consumption increases utility more
strongly at low-income levels as households spend a relatively larger share of
their income on housing. Since an increase in housing capital increases
utility more strongly at low-income levels, it becomes more attractive to
invest in housing capital. Thus, investing in housing capital becomes less
attractive over time as households become wealthier.

However, investments in housing capital have direct costs. Increasing the
stock of housing capital also increases the aggregate energy demand of the
housing stock. This energy demand is met by households using electricity and
the fossil resources. Consequently, an increase in housing capital results in
higher energy expenditures for households. This increase in energy
expenditures lowers the return of an increase in housing capital (cf. non-
arbitrage conditions in the appendix). The size of this increase depends on
the energy efficiency of newly constructed buildings. With a higher energy
efficiency of new buildings, the impact of these direct costs decreases.

The rich investment portfolio in our setup allows for an alternative that
lowers the direct costs of housing capital. By retrofitting the existing
housing stock, households can directly lower the housing-related energy
demand. This decrease in energy demand can offset the direct costs of housing
investments. Increasing the energy efficiency capital stock lowers the direct
costs of housing capital by lowering the overall energy expenditures of
households. Because the housing stock becomes less energy-intensive,
households need less electricity and fossil resources to meet the energy
demand. Expanding the housing stock with new, energy-efficient buildings
affects less the energy expenditure of households.

Another difference from industrial capital, in our setting, is that housing
investments are irreversible. Irreversibility refers to the fact that once
capital has been invested in a certain activity, it cannot be transformed into
consumption goods (Arrow and Kurz, 1970). Housing is reversible only at very
high costs, which is equivalent to irreversible investment (Miles, 2009).
Therefore, households need to anticipate that investing in housing-related
capital means that it cannot be converted back into consumption or industry
capital. The irreversibility constraints have a direct effect on the shadow
value of housing and efficiency capital. The shadow values of housing capital
and efficiency capital can be expressed as:

$\psi_{t}^{H}=\lambda_{t}+\phi_{t}^{H},$ (29)

and

$\psi_{t}^{E}=\lambda_{t}+\phi_{t}^{H},$ (30)

where $\phi_{t}^{i}$, $i\in\left\\{H,E\right\\}$ is the multiplier of the
respective irreversibility constraint of housing and efficiency capital (for
more details, refer to the appendix).

If the irreversibility constraint is binding, then $\phi_{t}^{i}>0$ which
creates a wedge between the shadow value of housing-related capital and
industry capital. However, when the irreversibility constraints are not
binding, the shadow values of all types of capital are equal. To determine
whether these constraints are binding, it is useful to analyze how climate
policy affects the incentives to invest in both types of housing capital.

In contrast to industry capital, climate policy directly affects the
incentives for investing in housing-related capital. In the absence of climate
policy, the price of fossil resources for heating does not account for
external costs associated with the burning of fossil fuels. This implies that
the private and social value of energy demand in housing differ (In appendix
A, we provide expressions for the shadow value of energy demand
$\nu_{t}^{ene}$ in both cases. Without climate policy, the shadow value of
energy demand is lower. Thereby, a marginal increase in housing capital is
less costly in terms of increased energy demand:

$\displaystyle\frac{\partial h^{ene}}{\partial k_{t+1}^{H}}=$
$\displaystyle\frac{1}{\nu_{t+1}^{ene}}\left[\chi_{t+1}\frac{\partial
h^{k}}{\partial
k_{t+1}^{H}}+\left(1-\delta_{H}\right)\psi_{t+1}^{H}-\frac{\psi_{t}^{H}}{\beta}\right],$
(31)

where $\chi_{t}$ equals the marginal utility of housing services and
$\delta_{H}$ is the depreciation rate of housing capital. The direct costs of
investing in housing capital are too low from a social perspective. In this
situation, households reasonably invest more in housing capital, as this
increases utility. The rise increase in energy costs due to the higher energy
demand is less significant due to the low price of both electricity and fossil
fuels. Therefore, compared to the scenario with optimal climate policy,
households over-invest in housing capital.

From a societal perspective, the low direct costs of housing investments make
it less attractive to invest in reducing the energy demand. Investing in
efficiency capital solely lowers energy expenditures by lowering the energy
demand of the existing housing stock. Without climate policy, the marginal
benefit of decreasing the energy demand is smaller:

$\displaystyle\frac{\partial h^{ene}}{\partial k_{t+1}^{E}}=$
$\displaystyle\frac{1}{\nu_{t+1}^{ene}}\left[\left(1-\delta_{E}\right)\psi_{t+1}^{E}-\frac{\psi_{t}^{E}}{\beta}\right],$
(32)

where $\delta_{E}$ is the depreciation rate of efficiency capital. In the
laissez faire, the return to investing in retrofitting the existing stock is
too low. Therefore, the stock of efficiency capital is lower than in the
optimal policy case. Climate policy affects housing investments by increasing
the shadow value of energy demand. From this, we can derive first insights for
the irreversibility constraints. First, as the stock of efficiency capital is
always lower in the no-policy scenario, the irreversibility constraint on
efficiency capital will not be a barrier to the transition to a carbon-free
economy. It will always be optimal to expand the stock of efficiency capital.
Therefore, in the subsequent analysis, we only focus on the irreversibility
constraint on housing capital investments.

The irreversibility constraint on housing investments may be binding, as the
stock of housing capital is higher without climate policy. In principle, it
may be optimal to have a lower energy demand once the carbon budget is
introduced, as energy is more expensive. Decreasing the housing capital stock
partially mitigates this problem by reducing the energy demand and, thus,
energy expenditures. However, housing capital differs from industrial capital.
Lowering the housing capital stock decreases housing services and,
consequently, the level of utility. The extent of this loss of utility depends
on non-homothetic preferences. Furthermore, there is an alternative available
in the situation. By Investing in efficiency capital, households effectively
use a backstop technology. When the irreversibility constraint binds,
households can invest in efficiency capital and directly decrease the pressure
of the constraint. In our setting, only households only want to decrease
sharply housing capital, if the energy demand is too high.

Essentially, the question is what households do when they realize that the
housing stock is too large. The first option is to decrease the energy demand
by decreasing the housing capital at the expense of utility. If this is the
cheapest option, then the irreversibility constraint is a barrier. The second
option is to forego present consumption in favor of investing in efficiency
capital. Investing in retrofitting the existing stock may be a suitable
alternative to running down the housing capital stock. Theoretically,
investing in industrial capital through the expansion of capital in non-
polluting electricity lowers the costs of electricity. In our general
equilibrium model, these options interact. It is challenging to determine
analytically if the constraint is binding. Thus, we explore whether the
constraint is binding through the numerical simulation of the model.

The last part of the analytical derivations focuses on optimal climate policy.
Given the differentiated impact on investments, it may be optimal to use
differentiated carbon prices for households and the industry. Furthermore,
decarbonizing electricity production may facilitate the decarbonization of the
housing sector if electricity becomes cheaper. This leads to the question of
whether it is optimal to implement different carbon prices for housing and
energy production.

We find that the conventional wisdom, namely the least-cost theorem (Baumol
and Oates, 1971) holds: the optimal carbon price is equal for households and
industry. The intuition is as follows. In our setting, emissions are
associated with the use of the fossil resource. During the transition, the
substitution possibilities vary on the production and the household side.
While in the industry, the expansion of renewable energy is the main option,
for housing, electricity is the primary alternative. At the same time,
reducing the energy demand in final goods production and housing may be
necessary. In order to incentivize these changes, a shift in relative prices
is necessary. A uniform carbon price optimally achieves this shift. The
optimal carbon tax rate is given by:

$\tau_{t}=\tau_{t}^{Y}=\frac{\mu_{t}^{R}}{u_{c_{t}}p_{R,t}}=\tau_{t}^{h}$ (33)

In turn, the optimal carbon price equals $p_{C,t}=\tau_{t}p_{R,t}$. The level
of the carbon tax is determined by the shadow value of the carbon budget
$\mu_{t}^{R}$ divided by the marginal utility of consumption $u_{c_{t}}$. This
is standard in the literature (i.e., Barrage (2018); Kalsbach and Rausch
(2021)). The only difference is that in our setting, it additionally depends
on the (exogenous) price of the resource $p_{R,t}$. The transition to a
carbon-free economy features a gradual decline in the use of the fossil
resource both in energy production and in housing. A gradual increase in
carbon prices ensures that the decline in fossil resources is optimal.666In
the appendix, we show that this increase is constant and equal to the pure
rate of time preference. The transition relies on a shift in investment
behavior, which we analyze in more depth in the numerical simulation.

Figure 1: Housing capital stock relative to the no-policy benchmark Figure 2:
Housing capital investments in the optimal policy scenario and the no-policy
baseline. Investments are expressed as a ratio over industry capital
investments.

Within the numerical simulation, the climate target is to reduce carbon
emissions by 50 percent within three decades. In our setting, this is
equivalent to reducing fossil resource use by the same amount. Figure 4
describes capital accumulation dynamics throughout the transition for various
forms of industry capital and for efficiency capital. The appendix includes
graphs with further variables such as resource use, carbon prices, and
renewable energy. The variables are presented as deviations from the no-policy
path. Since there is exogenous technological progress, we compare the first-
best transition to the outcome, where there is no an emission budget
constraint but only technological progress during the same period.

During the first-best transition, energy production steadily decreases the
fossil energy share and relies increasingly on renewable energy. By imposing a
carbon tax on the fossil resource, climate policy increases the costs of the
fossil energy producers and subsequently the price of fossil energy. As a
result, the final energy producer substitutes fossil energy with renewable
energy. This is reflected by a strong increase in non-polluting capital,
accompanied by a decline in fossil capital. After a decade, the fossil capital
stock is already half of what it is in the no-policy scenario. At the same
time, the stock of non-polluting capital is 60 percent greater than the
respective stock in the laissez-faire path.

The higher energy costs also influence the production of the final good. As
the green transition requires taxing a previously cheaper good, energy, it
results in a loss in output. Higher energy prices alter energy demand in final
good production, which affects output. While there is a decline in output
during the transition, the decline is not driven by overall lower investments
into industry capital. Interestingly, industry capital investments do not
change significantly. In fact, during the optimal transition, industry capital
investments closely resemble those from the no-policy path. Next, we turn to
the investment dynamics in the housing sector. Retrofitting becomes more
attractive through climate policy, resulting in an increase in the energy
efficiency capital stock ($k^{E}_{t}$). After two decades, the efficiency
capital stock is 50 percent higher than in the no-policy case. The increase in
efficiency capital occurs in two phases. Initially, it is optimal to rapidly
increase the stock of efficiency capital, while the carbon price remains low.
Afterwards, the stock of efficiency capital increases only gradually until it
reaches a plateau. Overall, the energy demand of the housing stock decreases
by 30 percent over this period. However, energy demand does not only depend on
changes in the stock of efficiency capital. Additionally, investments in
housing capital affect the aggregate energy demand of the housing stock.
Figure 1 shows the evolution of the housing capital stock relative to the no
policy benchmark. In the optimal climate policy scenario, the housing stock is
20 percent lower in the long run. To analyze the role of the irreversibility
constraint in more detail, it is useful to look at investment rather than the
stock of housing capital.

Figure 2 describes the dynamics of housing investments for both the no-policy
and optimal-policy paths. Due to the irreversibility constraint, we focus on
investments themselves rather than the housing capital stock. Housing capital
investments are expressed as a ratio over industry capital investments.
Without climate policy, housing investments steadily increase until they reach
their peak. Afterwards, households do not invest further into housing capital.
With advancements in technology, households become richer and eventually reach
a point where additional housing service do not provide sufficient utility to
outweigh increased energy expenditures. The total expansion of housing capital
is significant without climate policy. At their peak, housing investments are
one-tenth of physical capital investments.

In contrast, with optimal carbon pricing in place, the total expansion of the
housing stock is limited. When considering the external costs of fossil fuel
use, the direct costs of expanding the housing stock make it less attractive
to invest in housing capital. Consequently, housing investments peak earlier
and at a lower level. Since the increase in energy demand is more costly,
households reach faster the point where an increase in utility from additional
housing services is not worth it anymore. Thus, it is optimal to stop earlier
investing further in housing capital.

Additionally, housing investments start later than in the no-policy case and
only after a few periods. This indicates that the irreversibility constraint
is binding. It is optimal to reduce sharply residential energy demand in the
first period. Households achieve this in two ways. First, households initially
run-down part of the housing stock. Since the irreversibility constraint
prevents negative investments, the only option is depreciation. Furthermore,
the presence of the irreversibility constraint triggers the large investments
in efficiency capital. Thus, in the first-best transition, it is optimal to
reduce energy demand in the short term by both reducing the housing capital
stock and sharply increasing the amount of retrofits.

Figure 4: Evolution of the different capital stocks in the optimal climate
policy scenario in deviation from the no-policy benchmark.

###  4.2 Second-best Transition

Implementing optimal carbon pricing schemes may not be feasible. Imposing high
carbon prices on households may cause resistance and thus may lack political
support. Households may have misperceptions about the effectiveness or
fairness of carbon taxes (Douenne and Fabre, 2022). In addition, depending on
their political background, policymakers may prefer subsidies or regulatory
policies to the introduction of taxes. Given that high carbon prices are
needed to decarbonize the housing sector, we compare the first-best transition
with transitions, where carbon prices for households can only slowly be
introduced or are not available at all. Thus, we consider a phased-in carbon
price scenario and one scenario, where only subsidies for efficiency capital
are used. Note that these transitions are optimal in the sense that all
actors, including the government, behave optimally. Thus, the government sets
taxes and subsidies optimally given the constraints it faces.

In all scenarios, we do not affect climate policy on the production side. The
aim is to consider different policy scenarios for the housing sector while
leaving energy production unaffected. We achieve this by separating the carbon
budget. The carbon budget for the industry is set to the optimal burden in the
first best. Since we do not impose any restrictions on climate policy in
energy production, the resulting carbon price path for the industry is
identical to the first best. This allows us to isolate the effects of
different restrictions on climate policy on the household side. This approach
is inspired by climate policy on the European level, where separate emission
trading systems are being established for firms and households. In the present
analysis, our focus is on climate policy on the household side. In detail, we
consider the scenario of phased-in carbon prices as in Rozenberg et al.
(2020), as well as investment subsidies for efficiency capital.777In B
appendix, we include a graphical comparison of investment patterns for housing
capital and efficiency capital for all scenarios. The scenarios include the
no-policy benchmark, first-best climate policy, and the described second-best
settings.

Figure 5: Optimal versus phased-in carbon prices. Variables are expressed as a
ratio over fossil resource price.

The phased-in carbon price scenario focuses on short-term restrictions on the
level of carbon prices. Figure 5 compares the phased-in carbon price for
households to the optimal carbon price. The underlying intuition is that a
slower implementation of carbon pricing will lead to lower immediate costs for
households and thus increase political support compared to a higher carbon
price regime. This approach is common for the introduction of carbon price
schemes. Examples include Phased-in carbon prices reduce emissions more slowly
in the short run, because we restrict them in the first decade. During this
period, the carbon price is fixed to a lower level than the first-best carbon
price. In detail, due to the phased-in carbon price, the price of fossil
resource use increase by 25 percent. Afterwards, the government optimally sets
the carbon price.

Additionally, we consider the case where carbon pricing is not available at
all for the household side. In this case, the alternative is investment
subsidies. In principle, the government can subsidize both the construction of
new energy-efficient buildings and the retrofit of the existing stock.
However, subsidizing investments in housing capital is never optimal in our
setting. While new construction increases the energy efficiency of the overall
housing stock, it also increases the energy demand. Since a higher level of
energy demand makes the transition more costly, it is not useful from a
climate policy perspective to subsidize housing capital investments. Thus, we
solely consider subsidies to energy efficiency capital in the case where
carbon pricing for housing is not available.

From the first-best transition, we understand that the irreversibility
constraint on housing capital interacts with the necessity to reduce energy
demand. One solution is to lower the energy demand of the existing stock. To
measure the difference in energy efficiency between old and new buildings, we
use the energy efficiency gap.888The energy efficiency gap is defined as the
energy efficiency ratio of old to new buildings
$EEG=\frac{\kappa_{N}+\frac{\bar{\kappa}}{k_{t}^{E}}}{\kappa_{N}}$. We
normalize it by the initial level of the ratio. When the efficiency gap
reaches zero, the stock of old buildings has the same energy intensity as new
buildings. Thereby, the energy efficiency gap determines the limits to
investments in efficiency capital. Particularly in subsidy-based transitions,
the energy efficiency gap may act as a barrier for climate policy.

Figure 6 describes the evolution of the energy efficiency gap in housing for
both the first-best and second-best transitions, as well as the no-policy
benchmark. Without climate policy, the energy efficiency gap grows over time.
Energy costs decrease over time, reducing the return to investing in energy
efficiency and, thereby, the level of efficiency capital. In contrast, all
scenarios with climate policy result in a significantly smaller energy
efficiency gap. However, energy demand plays a fundamentally different role in
carbon price and subsidy-lead transitions. In transitions with carbon pricing,
energy demand plays a secondary role. While energy demand is relevant for the
irreversibility constraint, the reduction of fossil resource use is managed by
affecting the price directly. If carbon pricing is not available, lowering the
energy demand becomes the primary tool to reduce fossil fuel consumption in
housing. Thus, investments in efficiency capital are essential. Notably, even
in this scenario, it is never optimal to close the efficiency gap completely.
We will analyze next the second-best transitions in greater depth.

We start with the case of phased-in carbon prices. As the carbon budget is
identical to the first best, carbon prices must be higher once the policy
constraint on the level of carbon taxes is no longer binding. Thus, the
phased-in carbon price scenario focuses on a shift in the level of ambition
during the transition. In exchange for a lower level of ambition in earlier
periods, the level of ambition increases more afterwards to meet the climate
target. The lower carbon price in the initial period leads to relatively lower
energy expenditures for households. While the direct burden on households is
lower, the lower carbon price weakens the price signal to increase energy
efficiency investments.

Figure 6: Evolution of the Energy Efficiency Gap in the first-best, second-
best scenarios and the no-policy benchmark

When carbon prices are phased in, the investment pattern for efficiency
capital differs from the first best. While in the first best, it is optimal to
increase investments in energy efficiency sharply in the initial period, the
initial expansion in the phased-in scenario is much smaller. In contrast to
the first best, there is a second wave of investment in energy efficiency in
the last period of the constrained carbon prices. Specifically, households
invest a second time in efficiency capital. Notably, the magnitude of the
increase in investment is almost identical to the initial expansion. The
underlying intuition is that it is optimal for households to split investments
into efficiency capital, as the initial increase addresses the comparatively
lower carbon price. The second increase takes advantage of the fact that the
burden is still low from the initial carbon price scheme, but due to
technological progress, households are richer than in the initial period.

In contrast, when the government uses subsidies for efficiency capital,
investment dynamics differ both qualitatively and quantitatively. First, the
price of the fossil resource cannot be directly influenced. Reducing the use
of fossil resource requires a reduction in energy demand. By reducing the
energy demand of the housing stock, households need less electricity and less
fossil resource to meet their energy demand. Consequently, very large
investments in efficiency capital are needed to decrease the energy demand of
the housing stock sufficiently. It is optimal to invest early and strongly in
efficiency capital. In detail, the investments are five times higher than in
the first-best scenario. After the large investments in the early periods,
energy efficiency investments are zero thereafter.

Next, we turn to investments in housing capital. Compared to the first-best
investment pattern, investments in housing capital are moved forward. While
similar in magnitude to the first best, investment in housing capital occurs
after the investment hike in efficiency capital, but as long as carbon prices
are constrained. Due to the lower carbon price, the irreversibility constraint
on the housing stock is less binding in the early periods. In contrast,
housing capital investment is larger when climate policy relies on efficiency
capital subsidies. Intuitively, when the energy demand of existing stock
decreases due to the large expansion of efficiency capital, the direct cost of
investing in housing capital is smaller. Hence, it is more attractive to
expand the stock of housing capital to receive additional utility.

| Uniform | Phased-in | Subsidy
---|---|---|---
Welfare | -0.0443 | -0.0449 | -0.0565
Energy Costs | 0.597 | 0.616 | -0.243
Housing Costs (net) | 0.022 | 0.023 | 0.01
Housing Costs | 0.0553 | 0.0587 | -0.001
Disp. Income | 0.0057 | 0.006 | -0.018
Transfers | 0.015 | 0.015 | -0.003
Output | -0.03 | -0.03 | -0.034
Energy Prod. | 0.058 | 0.058 | 0.044
Table 2: Comparison of uniform carbon price, phased-in carbon price and the
investment subsidy for efficiency capital. Changes in cumulative welfare,
energy costs, net housing costs, housing costs (incl. energy), disposable
income, output and energy production relative to the laissez-faire scenario.
Household transfers are expressed as percent of income (net of transfers). All
variables are discounted at the internal discount rate

The irreversibility constraint is directly related to investments in housing
capital. In the first best, households want to decrease housing capital stock
during the initial period once they realize that the stock is excessively
large, resulting in inefficiently high energy expenditures. Ideally,
households immediately want to reduce the size of the housing stock, but the
irreversibility constraint is preventing this. In the phased-in carbon price
scenario, the irreversibility constraint is less problematic. Due to the lower
carbon prices in the earlier periods, the need to reduce the energy demand in
the first period is lower. In contrast, the irreversibility constraint is a
key barrier in the transition that relies on subsidies. The underlying reason
is that this transition has to use energy demand reduction as the primary tool
to reduce fossil fuel use. Due to the irreversibility constraint, reducing the
housing stock is no option, making investments in efficiency capital
necessary. However, the large amounts of efficiency-related investments crowd
out other types of investments, such as investments in industry capital.

Apart from the investment dynamics, the different policies affect welfare.
Table 2 summarizes the cumulative welfare levels relative to the no-policy
benchmark. In our setting, all transitions result in a welfare loss compared
to the no-policy baseline. Achieving net zero emissions requires factoring in
previously ignored external costs. This increases the relative price, directly
or indirectly, of using the fossil resource both in housing and in energy
production. The cost-effective way to achieve this is to impose a uniform
carbon price on both housing and energy production. When comparing different
constrained policy scenarios to the first best, the scenarios can be ranked
according to their welfare impact. While the welfare is on lower levels in
both second-best transitions, the phased-in carbon price transition leads to a
higher welfare level than the subsidy scenario.

From a political economy perspective, transition costs are relevant. The
transition creates direct costs for households by increasing both housing and
energy costs, making energy demand more costly. We compare the cumulative
housing and energy costs, during the transition. These costs do not correspond
to the ranking in welfare. Although a uniform carbon price is the most cost-
effective option, it creates substantial direct costs for households. Total
housing expenses rise by 5 percent compared to the no-policy case. For the
phased-in carbon price, the rise in housing costs is of a similar magnitude.
In contrast, if the transition relies on subsidies for efficiency capital, the
direct costs of households are lower than in the no policy case. The
underlying driver is the difference in cumulative energy costs. While in both
carbon pricing scenarios utility bills are 60 percent higher than in the no-
policy benchmark, they are 25 percent lower in the subsidy-based transition.
Hence, the least preferable option from a welfare perspective leads to the
smallest direct costs of all scenarios. The underlying reason is that the
energy demand is drastically reduced, which has a significant impact on
housing costs.

Redistributing carbon pricing revenues is an effective tool to lower the
burden on households. The source of transfers are revenues from carbon
pricing, both industry and housing. As the revenues from carbon pricing from
the industry are identical in all scenarios, we focus only on the revenues
from carbon pricing in housing. Thereby, we can measure the fiscal burden of
the subsidy scenario. Subsidizing efficiency capital is costly and the
government has to use lump-sum taxes to finance the subsidies. In the carbon
price scenarios, households receive transfers that are equal to 1.5 percent of
their cumulative income. At the same time, in the subsidy scenario, households
have to pay taxes equal to 0.3 percent of their income. Consequently,
households have a lower disposable income in the subsidy-based transition.This
indicates how costly it is to decarbonize the economy through investment
subsidies when the irreversibility constraint is binding. The need to invest
in efficiency capital is largest in the first period, where revenues of carbon
pricing in energy production are at their lowest level. Consequently, the
government has to impose high taxes on households to finance the subsidies
that are necessary to incentivize the large increase in efficiency capital due
to the binding irreversibility constraint.

Relying on investment subsidies has important implications for housing demand.
In detail, we find a housing-related rebound effect in the subsidy-based
transition.999A graphical comparison of housing capital stocks for all
scenarios is included in appendix B. While the housing stock declines compared
to the no policy benchmark, it is significantly higher than in the carbon
price scenarios. This indicates that due to higher aggregate energy
efficiency, households increase their housing consumption relative to the
social optimum. Thereby, instrument choice not only matters for climate
policy, but potentially affects also aggregate housing demand.

##  5 Conclusion

Due to a lack of investments, the building sector could become a barrier to
the transition to a carbon-free economy. Investment needs include the
construction of new, energy-efficient buildings, along with the retrofitting
of existing housing stock. In addition to housing-related investments, the
green transition requires investments in renewable energy. Climate policy must
incentivize and coordinate these various investments while recognizing the
direct burden on households. In this nexus, we compared different transitions
based on the availability of specific instruments in residential housing. We
rely on a general equilibrium model with an elaborate setup of housing, energy
production, and an optimizing government that chooses policy instruments
optimally, given a set of constraints.

Climate policy has to not only incentivize these investments, but also
coordinate between the different types of investments. Additionally, climate
policy in housing can be difficult to implement, as it creates direct costs
for households. In this nexus, we compared different transitions based on the
availability of specific instruments in housing. We rely on a general
equilibrium model with an elaborate setup of housing, energy production, and
an optimizing government that chooses policy instruments optimally, given a
set of constraints.

If there are no constraints on mitigation policies, the conventional wisdom
holds. The first-best policy is to impose a uniform carbon price on both
households and the industry. Despite differences in housing investments,
optimal climate policy remains unchanged. Energy demand only plays a secondary
role in the optimal transition, as renewable energy expansion is not
restricted in our setting. When comparing second-best transitions, the
situation changes. While energy demand is not problematic with phased-in
carbon prices, this is no longer the case in the subsidy-led transition. Due
to the binding irreversibility constraint, the only option is to heavily
subsidize investments in energy efficiency to reduce fossil fuel consumption.
While providing subsidies for retrofits results in the lowest direct costs for
households, it ultimately leads to the highest aggregate costs. Therefore,
subsidies to efficiency investments prove to be an ineffective way to
decarbonize the economy.

We leave several important aspects for future research. This paper is on the
role of energy demand, but other investments can additionally affect the
substitution possibilities. For example, expanding infrastructure that enables
district heating, especially in urban areas. Furthermore, the housing sector
is highly heterogeneous and is characterized by many barriers and additional
market failures, including principal-agent problems and behavioral failures.
This could be an promising aspect to consider, since the present analysis
relies on perfect commitment and perfect foresight and thus anticipation of
climate policy. In addition, market failures may limit the expansion of
renewable energy and the amount of retrofits per year. Although we have an
understanding of these obstacles in isolation, studying them in a general
equilibrium context is essential for effective climate policy. A better
understanding of their interactions will enhance climate policy’s ability to
navigate the complex task of decarbonizing the housing sector.

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##  Appendix A Analytical Derivations

###  A.1 No-Arbitrage Conditions

Since the household’s energy composite is given by

$ene_{t}=\left(a_{ene}e_{t}^{\frac{\sigma_{ene}-1}{ene}}\left(1-a_{ene}\right)res_{t}^{\frac{\sigma_{ene}-1}{ene}}\right)^{\frac{\sigma_{ene}-1}{ene}}$
(34)

one can readily verify that the composite price of $ene_{t}$ equals

$\displaystyle p_{ene,t}=$
$\displaystyle\left(a_{ene}^{\sigma_{ene}}p_{E,t}^{1-\sigma_{ene}}+\left(1-a_{ene}\right)^{\sigma_{ene}}\left(p_{R,t}+p_{HC,t}\right)^{1-\sigma_{ene}}\right)^{\frac{1}{1-\sigma_{ene}}}$
(35)

Using the consumer’s first-order conditions lead to the following non-
arbitrage conditions

$\displaystyle R_{t}-\delta_{K}=$
$\displaystyle\left[MRS_{k_{t}^{H},c_{t}}-\delta_{H}-p_{ene,t}\kappa^{N}\right]+\frac{\phi_{t-1}^{H}}{\lambda_{t}}\left(1+\rho\right)-\frac{\phi_{t}^{H}}{\lambda_{t}}\left(1-\delta_{H}\right)$
(36) $\displaystyle R_{t}-\delta_{K}=$
$\displaystyle\frac{\left(1-\tau_{INVE,t}\right)-\left(1-\tau_{INVE,t-1}\right)}{1-\tau_{INVE,t-1}}+$
(37)
$\displaystyle+\frac{\left[\frac{p_{ene,t}\overline{\kappa}}{k_{t}^{E}}\frac{\bar{k}}{k_{t}^{E}}-\left(1-\tau_{INVE,t}\right)\delta_{E}\right]+\frac{\phi_{t-1}^{E}}{\lambda_{t}}\left(1+\rho\right)-\frac{\phi_{t}^{E}}{\lambda_{t}}\left(1-\delta_{E}\right)}{1-\tau_{INVE,t}}$

where $MRS_{k_{t}^{H},c_{t}}$ is the marginal rate of substitution between the
consumer’s housing capital and the final good, $\tau_{INVE,t}$ is the subsidy
to energy efficiency capital investment, $\lambda_{t}$ is the Lagrange
multiplier associated with the consumer’s budget constraint, and
$\phi^{H}_{t}$ and $\phi^{E}_{t}$ are the multiplies associated with the non-
negativity of investments.

###  A.2 Decentralized Setup

The representative household-investor maximizes:

$\max\sum_{t=0}^{T}\beta^{t}u(c_{t},h_{t})$ (38)

subject to:

$\displaystyle
c_{t}+i_{t}^{Y}+i_{t}^{F}+i_{t}^{N}+i_{t}^{H}+i_{t}^{E}+\delta_{H}\overline{k}+p_{E,t}e_{t}+p_{R,t}res_{t}$
$\displaystyle\left(\lambda_{t}\right)$ (39)
$\displaystyle=w_{t}l+R_{t}k_{t}+R_{t}^{F}k_{t}^{F}+R_{t}^{N}k_{t}^{N}$

$\displaystyle ene_{t}=$ $\displaystyle
h^{e}\left(e_{t},res_{t}\right)=\left(a_{ene}e_{t}^{\frac{\sigma_{ene}-1}{ene}}+\left(1-a_{ene}\right)res_{t}^{\frac{\sigma_{ene}-1}{ene}}\right)^{\frac{\sigma_{ene}-1}{ene}}$
$\displaystyle\left(\nu_{t}^{e}\right)$ (40a) $\displaystyle ene_{t}=$
$\displaystyle
h^{ene}\left(k_{t}^{E},k_{t}^{H}\right)=\left(\kappa_{N}+\frac{\overline{\kappa}}{k_{t}^{E}}\right)\overline{k}+\kappa_{N}k_{t}^{H}$
$\displaystyle\left(\nu_{t}^{ene}\right)$ (40b) $\displaystyle h_{t}=$
$\displaystyle
h^{k}\left(land_{t},k_{t}^{H}\right)=land_{t}^{a_{h}}\left(\overline{k}+k_{t}^{H}\right)^{1-a_{h}}$
$\displaystyle\left(\chi_{t}\right)$ (40c) $\displaystyle k_{t+1}^{Y}=$
$\displaystyle i_{t}^{Y}+\left(1-\delta_{Y}\right)k_{t}^{Y}$
$\displaystyle\left(\psi_{t}\right)$ (40d) $\displaystyle k_{t+1}^{F}=$
$\displaystyle i_{t}^{F}+\left(1-\delta_{F}\right)k_{t}^{F}$
$\displaystyle\left(\psi_{t}^{F}\right)$ (40e) $\displaystyle k_{t+1}^{N}=$
$\displaystyle i_{t}^{N}+\left(1-\delta_{H}\right)k_{t}^{N}$
$\displaystyle\left(\psi_{t}^{N}\right)$ (40f) $\displaystyle k_{t+1}^{H}=$
$\displaystyle i_{t}^{H}+\left(1-\delta_{F}\right)k_{t}^{H}$
$\displaystyle\left(\psi_{t}^{H}\right)$ (40g) $\displaystyle k_{t+1}^{E}=$
$\displaystyle i_{t}^{E}+\left(1-\delta_{N}\right)k_{t}^{E}$
$\displaystyle\left(\psi_{t}^{E}\right)$ (40h) $\displaystyle i_{t}^{H}\geq$
$\displaystyle 0$ $\displaystyle\left(\phi_{t}^{H}\right)$ (40i)
$\displaystyle i_{t}^{E}\geq$ $\displaystyle 0$
$\displaystyle\left(\phi_{t}^{E}\right)$ (40j)

The first-order conditions are as follows:

$\displaystyle\frac{\partial\mathcal{L}}{\partial c_{t}}=$ $\displaystyle
u_{c_{t}}-\lambda_{t}=0$ $\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}=u_{c_{t}}$ (41a)
$\displaystyle\frac{\partial\mathcal{L}}{\partial h_{t}}=$ $\displaystyle
u_{h_{t}}-\chi_{t}=0$ $\displaystyle\Rightarrow$ $\displaystyle
u_{h_{t}}=\chi_{t}$ (41b) $\displaystyle\frac{\partial\mathcal{L}}{\partial
ene_{t}}=$ $\displaystyle-\nu_{t}^{e}+\nu_{t}^{ene}=0$
$\displaystyle\Rightarrow$ $\displaystyle\nu_{t}^{e}=\nu_{t}^{ene}$ (41c)
$\displaystyle\frac{\partial\mathcal{L}}{\partial e{}_{t}}=$
$\displaystyle\nu_{t}^{e}\frac{\partial h^{e}}{\partial
e{}_{t}}-\lambda_{t}p_{E,t}=0$ $\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}p_{E,t}=\nu_{t}^{e}\frac{\partial h^{e}}{\partial
e{}_{t}}$ (41d) $\displaystyle\frac{\partial\mathcal{L}}{\partial res{}_{t}}=$
$\displaystyle\nu_{t}^{e}\frac{\partial h^{e}}{\partial
res{}_{t}}-\lambda_{t}p_{R,t}=0$ $\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}p_{R,t}=\nu_{t}^{e}\frac{\partial h^{e}}{\partial
res{}_{t}}$ (41e) $\displaystyle\frac{\partial\mathcal{L}}{\partial
i_{t}^{Y}}=$ $\displaystyle-\lambda_{t}+\psi_{t}=0$ $\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}=\psi_{t}$ (41f)
$\displaystyle\frac{\partial\mathcal{L}}{\partial i_{t}^{F}}=$
$\displaystyle-\lambda_{t}+\psi_{t}^{F}=0$ $\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}=\psi_{t}^{F}$ (41g)
$\displaystyle\frac{\partial\mathcal{L}}{\partial i_{t}^{N}}=$
$\displaystyle-\lambda_{t}+\psi_{t}^{N}=0$ $\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}=\psi_{t}^{N}$ (41h)
$\displaystyle\frac{\partial\mathcal{L}}{\partial i_{t}^{H}}=$
$\displaystyle-\lambda_{t}+\psi_{t}^{H}+\phi_{t}^{H}=0$
$\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}=\psi_{t}^{H}+\phi_{t}^{H}$ (41i)
$\displaystyle\frac{\partial\mathcal{L}}{\partial i_{t}^{E}}=$
$\displaystyle-\lambda_{t}+\psi_{t}^{E}+\phi_{t}^{E}=0$
$\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}=\psi_{t}^{E}+\phi_{t}^{E}$ (41j)

$\displaystyle\frac{\partial\mathcal{L}}{\partial k_{t+1}^{Y}}=$
$\displaystyle-\beta^{t}\psi_{t}+\beta^{t+1}\lambda_{t+1}R_{t+1}+\left(1-\delta_{Y}\right)\beta^{t+1}\psi_{t+1}=0$
(42) $\displaystyle\Rightarrow R_{t+1}=$
$\displaystyle\frac{1}{\lambda_{t+1}}\left[\psi_{t}/\beta-\left(1-\delta_{Y}\right)\psi_{t+1}\right]$
(43) $\displaystyle=$
$\displaystyle\frac{\lambda_{t}}{\beta\lambda_{t+1}}-\left(1-\delta_{Y}\right)$
(44) $\displaystyle=$ $\displaystyle\frac{u_{c_{t}}}{\beta
u_{c_{t+1}}}-\left(1-\delta_{Y}\right)$ (45)

$\displaystyle\frac{\partial\mathcal{L}}{\partial k_{t+1}^{j}}=$
$\displaystyle-\beta^{t}\psi_{t}^{j}+\beta^{t+1}\lambda_{t+1}R_{t+1}^{j}+\left(1-\delta_{j}\right)\beta^{t+1}\psi_{t+1}^{j}=0$
for $j=F,N$ (46) $\displaystyle\Rightarrow
R_{t+1}^{j}=\frac{1}{\lambda_{t+1}}$
$\displaystyle\left[\psi_{t}^{j}/\beta-\left(1-\delta_{j}\right)\psi_{t+1}^{j}\right]=R_{t+1}\text{
\text{for $j=F,N$}}$ (47)

$\displaystyle\frac{\partial\mathcal{L}}{\partial k_{t+1}^{H}}=$
$\displaystyle-\beta^{t}\psi_{t}^{H}+\beta^{t+1}\left[-\nu_{t+1}^{ene}\frac{\partial
h^{ene}}{\partial k_{t+1}^{H}}+\chi_{t+1}\frac{\partial h^{k}}{\partial
k_{t+1}^{H}}+\left(1-\delta_{H}\right)\psi_{t+1}^{H}\right]=0$ (48)
$\displaystyle\Rightarrow\frac{\partial h^{k}}{\partial k_{t+1}^{H}}=$
$\displaystyle\frac{1}{\chi_{t+1}}\left[\psi_{t}^{H}/\beta+\nu_{t+1}^{ene}\frac{\partial
h^{ene}}{\partial k_{t+1}^{H}}-\left(1-\delta_{H}\right)\psi_{t+1}^{H}\right]$
(49) $\displaystyle\Rightarrow\frac{\partial h^{ene}}{\partial k_{t+1}^{H}}=$
$\displaystyle\frac{1}{\nu_{t+1}^{ene}}\left[\chi_{t+1}\frac{\partial
h^{k}}{\partial
k_{t+1}^{H}}+\left(1-\delta_{H}\right)\psi_{t+1}^{H}-\psi_{t}^{H}/\beta\right]$
(50)

$\displaystyle\frac{\partial\mathcal{L}}{\partial k_{t+1}^{E}}=$
$\displaystyle-\beta^{t}\psi_{t}^{E}+\beta^{t+1}\left[-\nu_{t+1}^{ene}\frac{\partial
h^{ene}}{\partial
k_{t+1}^{E}}+\left(1-\delta_{E}\right)\psi_{t+1}^{E}\right]=0$ (51)
$\displaystyle\Rightarrow\frac{\partial h^{ene}}{\partial k_{t+1}^{E}}=$
$\displaystyle\frac{1}{\nu_{t+1}^{ene}}\left[\left(1-\delta_{E}\right)\psi_{t+1}^{E}-\psi_{t}^{E}/\beta\right]$
(52)

First, we can recognize the implicit rental rates of the different types of
capital in spirit of Jorgenson (1967). The implicit rental rate of physical
capital $k_{t}$ is defined as:

$R_{t}=\frac{\psi_{t-1}}{\beta\psi_{t}}-\left(1-\delta_{Y}\right)$ (53)

where $\psi_{t}$ is the shadow price of physical capital. As capital freely
moves between final good production and fossil and clean energy production,
their rental rates are equalized. That is

$R_{t}=R^{F}_{t}=R^{N}_{t}$ (54)

###  A.3 Optimal Carbon Price

The aim of this section is to derive some analytical results. In what follows
we write the emissions budget is a recursive form. Since the economy consists
of $n$ identical households, the social planner problem is as follows:

$\max\sum_{t=0}^{T}\beta^{t}nu(c_{t},h_{t})$ (55)

subject to equations
$\left(\ref{eq:ene}\right)-\left(\ref{eq:iEPositive}\right)$ multiplied by the
number of households $n$ and also subject to

$\displaystyle F^{Y}\left(nk_{t}^{Y},E_{Y,t}\right)=$ $\displaystyle
n\left(c_{t}+i_{t}^{Y}+i_{t}^{F}+i_{t}^{N}+i_{t}^{H}+i_{t}^{E}+\delta_{H}\overline{k}\right)$
$\displaystyle\left(\lambda_{t}\right)$ (56)
$\displaystyle+p_{R,t}\left(n\times res_{t}+Res_{F,t}\right)$ $\displaystyle
E_{t}=F^{E}\left(E_{F,t},E_{N,t}\right)=$
$\displaystyle\left(a_{E}E_{F,t}^{\frac{\sigma_{E}-1}{\sigma_{E}}}+\left(1-a_{E}\right)E_{N,t}^{\frac{\sigma_{E}-1}{\sigma_{E}}}\right)^{\frac{\sigma_{E}}{\sigma_{E}-1}}$
$\displaystyle\left(\chi_{t}^{Y}\right)$ (57) $\displaystyle
E_{F,t}=F^{F}\left(nk_{t}^{F},Res_{F,t}\right)=$
$\displaystyle\left(a_{F}\left(nk_{t}^{F}\right)^{\frac{\sigma_{F}-1}{\sigma_{F}}}+\left(1-a_{F}\right)Res_{F,t}^{\frac{\sigma_{F}-1}{\sigma_{F}}}\right)^{\frac{\sigma_{F}}{\sigma_{F}-1}}$
$\displaystyle\left(\chi_{t}^{F}\right)$ (58) $\displaystyle
E_{N,t}=F^{N}\left(nk_{t}^{N}\right)=$ $\displaystyle A_{N,t}nk_{t}^{N}$
$\displaystyle\left(\chi_{t}^{N}\right)$ (59) $\displaystyle M_{t+1}=$
$\displaystyle M_{t}-\left(n\times res_{t}+Res_{F,t}\right)$
$\displaystyle\left(\mu_{t}^{R}\right)$ (60) $\displaystyle M_{t}\geq$
$\displaystyle 0$ $\displaystyle\left(\mu_{t}\right)$ (61) $\displaystyle
E_{t}=$ $\displaystyle n\times e_{t}+E_{Y,t}$
$\displaystyle\left(\chi_{t}^{E}\right)$ (62)

Note that the variable in parenthesis to the right denotes the Lagrangian
multiplier associated with the equation at hand. The first order conditions of
the social planner problem are as follows:

$\displaystyle\frac{\partial\mathcal{L}}{\partial c_{t}}=$ $\displaystyle
nu_{c_{t}}-n\lambda_{t}=0$ $\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}=u_{c_{t}}$ (63a)
$\displaystyle\frac{\partial\mathcal{L}}{\partial h_{t}}=$ $\displaystyle
nu_{h_{t}}-n\chi_{t}=0$ $\displaystyle\Rightarrow$
$\displaystyle\chi_{t}=u_{h_{t}}$ (63b)
$\displaystyle\frac{\partial\mathcal{L}}{\partial ene_{t}}=$
$\displaystyle-n\nu_{t}^{e}+n\nu_{t}^{ene}=0$ $\displaystyle\Rightarrow$
$\displaystyle\nu_{t}^{e}=\nu_{t}^{ene}$ (63c)
$\displaystyle\frac{\partial\mathcal{L}}{\partial e{}_{t}}=$ $\displaystyle
n\nu_{t}^{e}\frac{\partial h^{e}}{\partial e_{t}}-n\chi_{t}^{E}=0$
$\displaystyle\Rightarrow$
$\displaystyle\chi_{t}^{E}=\nu_{t}^{e}\frac{\partial h^{e}}{\partial e_{t}}$
(63d) $\displaystyle\frac{\partial\mathcal{L}}{\partial res_{t}}=$
$\displaystyle-n\lambda_{t}p_{R,t}+n\nu_{t}^{e}\frac{\partial h^{e}}{\partial
res_{t}}-n\mu_{t}^{R}=0$ $\displaystyle\Rightarrow$
$\displaystyle\mu_{t}^{R}+\lambda_{t}p_{R,t}=\nu_{t}^{e}\frac{\partial
h^{e}}{\partial res_{t}}$ (63e)
$\displaystyle\frac{\partial\mathcal{L}}{\partial i_{t}^{Y}}=$
$\displaystyle-n\lambda_{t}+n\psi_{t}=0$ $\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}=\psi_{t}$ (63f)
$\displaystyle\frac{\partial\mathcal{L}}{\partial i_{t}^{F}}=$
$\displaystyle-n\lambda_{t}+n\psi_{t}^{F}=0$ $\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}=\psi_{t}^{F}$ (63g)
$\displaystyle\frac{\partial\mathcal{L}}{\partial i_{t}^{N}}=$
$\displaystyle-n\lambda_{t}+n\psi_{t}^{N}=0$ $\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}=\psi_{t}^{N}$ (63h)
$\displaystyle\frac{\partial\mathcal{L}}{\partial i_{t}^{H}}=$ $\displaystyle
n\left(-\lambda_{t}+\psi_{t}^{H}+\phi_{t}^{H}\right)=0$
$\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}=\psi_{t}^{H}+\phi_{t}^{H}$ (63i)
$\displaystyle\frac{\partial\mathcal{L}}{\partial i_{t}^{E}}=$ $\displaystyle
n\left(-\lambda_{t}+\psi_{t}^{E}+\phi_{t}^{E}\right)=0$
$\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}=\psi_{t}^{E}+\phi_{t}^{E}$ (63j)

$\displaystyle\frac{\partial\mathcal{L}}{\partial E_{Y,t}}=$
$\displaystyle\lambda_{t}\frac{\partial F^{Y}}{\partial
E_{Y,t}}-\chi_{t}^{E}=0$ $\displaystyle\Rightarrow$
$\displaystyle\chi_{t}^{E}=\lambda_{t}\frac{\partial F^{Y}}{\partial E_{Y,t}}$
(64) $\displaystyle\frac{\partial\mathcal{L}}{\partial E_{t}}=$
$\displaystyle-\chi_{t}^{Y}+\chi_{t}^{E}=0$ $\displaystyle\Rightarrow$
$\displaystyle\chi_{t}^{Y}=\chi_{t}^{E}$ (65)
$\displaystyle\frac{\partial\mathcal{L}}{\partial E_{F,t}}=$
$\displaystyle\chi_{t}^{Y}\frac{\partial F^{E}}{\partial
E_{F,t}}-\chi_{t}^{F}=0$ $\displaystyle\Rightarrow$
$\displaystyle\chi_{t}^{F}=\chi_{t}^{Y}\frac{\partial F^{E}}{\partial
E_{F,t}}$ (66) $\displaystyle\frac{\partial\mathcal{L}}{\partial E_{N,t}}=$
$\displaystyle\chi_{t}^{Y}\frac{\partial F^{E}}{\partial
E_{N,t}}-\chi_{t}^{N}=0$ $\displaystyle\Rightarrow$
$\displaystyle\chi_{t}^{N}=\chi_{t}^{Y}\frac{\partial F^{E}}{\partial
E_{N,t}}$ (67) $\displaystyle\frac{\partial\mathcal{L}}{\partial Res_{F,t}}=$
$\displaystyle-\lambda_{t}p_{R,t}+\chi_{t}^{F}\frac{\partial F^{F}}{\partial
Res_{F,t}}-\mu_{t}^{R}=0$ $\displaystyle\Rightarrow$
$\displaystyle\lambda_{t}p_{R,t}+\mu_{t}^{R}=\chi_{t}^{F}\frac{\partial
F^{F}}{\partial Res_{F,t}}$ (68)

$\displaystyle\frac{\partial\mathcal{L}}{\partial k_{t+1}^{Y}}=$
$\displaystyle 0$ (69) $\displaystyle\Rightarrow\frac{\partial F^{Y}}{\partial
nk_{t+1}^{Y}}=$
$\displaystyle\frac{1}{\lambda_{t+1}}\left[\psi_{t}/\beta-\left(1-\delta_{Y}\right)\psi_{t+1}\right]$
(70)

$\displaystyle\frac{\partial\mathcal{L}}{\partial k_{t+1}^{F}}=$
$\displaystyle 0$ (71) $\displaystyle\Rightarrow\frac{\partial F^{F}}{\partial
nk_{t+1}^{F}}=$
$\displaystyle\frac{1}{\chi_{t}^{F}}\left[\psi_{t}^{F}/\beta-\left(1-\delta_{F}\right)\psi_{t+1}^{F}\right]$
(72)

$\displaystyle\frac{\partial\mathcal{L}}{\partial k_{t+1}^{N}}=$
$\displaystyle 0$ (73) $\displaystyle\Rightarrow\frac{\partial F^{N}}{\partial
nk_{t+1}^{N}}=$
$\displaystyle\frac{1}{\chi_{t+1}^{N}}\left[\psi_{t}^{N}/\beta-\left(1-\delta_{N}\right)\psi_{t+1}^{N}\right]$
(74)

$\displaystyle\frac{\partial\mathcal{L}}{\partial k_{t+1}^{H}}$
$\displaystyle=0$ (75) $\displaystyle\Rightarrow\frac{\partial h^{k}}{\partial
k_{t+1}^{H}}=$
$\displaystyle\frac{1}{\chi_{t+1}}\left[\psi_{t}^{H}/\beta+\nu_{t+1}^{ene}\frac{\partial
h^{ene}}{\partial k_{t+1}^{H}}-\left(1-\delta_{H}\right)\psi_{t+1}^{H}\right]$
(76) $\displaystyle\frac{\partial h^{ene}}{\partial k_{t+1}^{H}}=$
$\displaystyle\frac{1}{\nu_{t+1}^{ene}}\left[\chi_{t+1}\frac{\partial
h^{k}}{\partial
k_{t+1}^{H}}+\left(1-\delta_{H}\right)\psi_{t+1}^{H}-\psi_{t}^{H}/\beta\right]$
(77)

$\displaystyle\frac{\partial\mathcal{L}}{\partial k_{t+1}^{E}}=$
$\displaystyle 0$ (78) $\displaystyle\Rightarrow\frac{\partial
h^{ene}}{\partial k_{t+1}^{E}}=$
$\displaystyle\frac{1}{\nu_{t+1}^{ene}}\left[\left(1-\delta_{H}\right)\psi_{t+1}^{H}-\psi_{t}^{E}/\beta\right]=0$
(79)

$\displaystyle\frac{\partial\mathcal{L}}{\partial M_{t+1}}=$
$\displaystyle-\beta^{t}\mu_{t}^{R}+\beta^{t+1}\mu_{t+1}^{R}+\beta^{t+1}\mu_{t+1}=0$
(80) $\displaystyle\Rightarrow\mu_{t+1}^{R}+\mu_{t+1}$
$\displaystyle=\frac{\mu_{t}^{R}}{\beta}$ (81)

Since the marginal product of $Res_{t}$ approaches infinity as $Res_{t}$
approaches zero and since $res_{t}$ must be used for housing services then
$M_{t}$ approaches zero only at the end of the horizon $t^{*}$. Implying that
$\mu_{t}=0$ for all $t<t^{*}.$ Therefore

$\frac{\mu_{t+1}^{R}-\mu_{t}^{R}}{\mu_{t}^{R}}=\frac{1-\beta}{\beta}$ (82)

The aim of this subsection is to analyze optimal carbon taxation. The
underlying question is whether fossil resource use in housing and in energy
production should be taxed in the same way. Let us start with the optimal
carbon tax for households. Their optimal resource use is described by:

$\lambda_{t}p_{R,t}(1+\tau^{H}_{t})=\nu_{t}^{e}\frac{\partial
h^{e}(e_{t},res_{t})}{\partial res_{t}}$
$(1+\tau^{H}_{t})=\frac{\nu_{t}^{e}}{\lambda_{t}p_{R,t}}\frac{\partial
h^{e}(e_{t},res_{t})}{\partial
res_{t}}=\frac{\nu_{t}^{e}}{u_{c_{t}}p_{R,t}}\frac{\partial
h^{e}(e_{t},res_{t})}{\partial res_{t}}$

A carbon price can be implemented instead of a carbon tax, by substituting
$p_{R,t}\tau^{H}_{t}$ with $p_{HC,t}$. Recall that from Social planner
optimization, we know:

$\nu_{t}^{e}\frac{\partial h^{e}(e_{t},res_{t})}{\partial
res_{t}}=\mu_{t}^{R}+\lambda_{t}p_{R,t}=\mu_{t}^{R}+u_{c_{t}}p_{R,t}$

If we plug it in, we get:

$(1+\tau_{t}^{H})=\frac{\nu_{t}^{e}}{\lambda_{t}p_{R,t}}\frac{\partial
h^{e}(e_{t},res_{t})}{\partial
res_{t}}=\frac{\mu_{t}^{R}+u_{c_{t}}p_{R,t}}{u_{c_{t}}p_{R,t}}$

Thus, we arrive at the optimal carbon tax of households:

$\tau_{t}^{H}=\frac{\mu_{t}^{R}}{u_{c_{t}}p_{R,t}}$ (83)

Next, we can study the optimal carbon tax in fossil energy production. The
profit maximization problem of the fossil energy producer is:

$\pi_{F,t}=p_{F,t}F^{F}(K_{F,t},Res_{F,t})-R^{F}_{t}K_{F,t}-p_{R,t}(1+\tau_{t}^{Y})Res_{F,t}.$
(84)

The first-order condition with regard to capital equals

$p_{F,t}\frac{\partial F^{F}(K_{F,t},Res_{F,t})}{\partial K_{F,t}}=R^{F}_{t}.$
(85)

The optimal resource use for fossil energy producers is described by:

$p_{F,t}\frac{\partial F^{F}(K_{F,t},Res_{F,t})}{\partial
Res_{F,t}}=p_{R,t}(1+\tau_{t}^{Y})$ (86)

$(1+\tau_{t}^{Y})\frac{p_{R,t}}{p_{F,t}}=F^{F}_{Res}$

From social planner optimization, we use:

$F^{F}_{Res}=\frac{\mu^{R}_{t}}{\chi^{F}_{t}}+\frac{u_{c_{t}}p_{R,t}}{\chi^{F}_{t}}$

If we plug it in the optimal resource use condition, we get:

$(1+\tau_{t}^{Y})\frac{p_{R,t}}{p_{F,t}}=\frac{\mu^{R}_{t}}{\chi^{F}_{t}}+\frac{u_{c_{t}}p_{R,t}}{\chi^{F}_{t}}$

or

$(1+\tau_{t}^{Y})=\frac{\mu_{t}^{R}p_{F,t}}{\chi^{F}_{t}p_{R,t}}+\frac{u_{c_{t}}p_{F,t}}{\chi^{F}_{t}}$

we arrive at:

$\tau_{t}^{Y}=\frac{\mu_{t}^{R}p_{F,t}}{\chi^{F}_{t}p_{R,t}}+\frac{u_{c_{t}}p_{F,t}}{\chi^{F}_{t}}-1$
(87)

The open question is how the shadow price of fossil energy $\chi^{F}_{t}$
relates to the multiplier of the resource constraint $\lambda_{t}$ and thus to
the marginal utility $u_{c_{t}}$. Recall that the shadow price of fossil
energy is:

$\chi^{F}_{t}=\chi^{Y}_{t}F_{E_{F,t}}^{F}=\chi^{E}_{t}F_{E_{F,t}}^{F}=\lambda_{t}\frac{\partial
F^{Y}}{\partial E_{Y,t}}F_{E_{F,t}}^{F}$

We now use expressions from the profit maximization of final energy production
and the final good producer to simply. Recall that the final energy producer
maximizes:

$\pi_{E,t}=p_{E,t}F^{E}(E_{F,t},E_{N,t})-p_{F,t}E_{F,t}-p_{N,t}E_{N,t}$ (88)

with the respective first order conditions:

$p_{F,t}=\frac{\partial F^{E}(E_{F,t},E_{N,t})}{\partial E_{F,t}}p_{E,t}$ (89)

$p_{N,t}=\frac{\partial F^{E}(E_{F,t},E_{N,t})}{\partial E_{N,t}}p_{E,t}$ (90)

From the profit maximization of the final good producer we can get an
expression for the price of final energy $p_{t}^{E}$:

$p_{E,t}=\frac{\partial F^{Y}(K_{Y,t},E_{Y,t})}{\partial E_{Y,t}}$ (91)

Now, we can combine both to get:

$p_{F,t}=\frac{\partial F^{Y}(K_{Y,t},E_{Y,t})}{\partial
E_{Y,t}}\frac{\partial F^{E}(E_{F,t},E_{N,t})}{\partial E_{F,t}}$ (92)

we can substitute the shadow price of aggregate energy:

$\chi_{t}^{F}=\lambda_{t}p_{F,t}$

If we plug it into the formula for the optimal carbon taxes in energy
production, we get:

$\tau_{t}^{Y}=\frac{\mu_{t}^{R}}{u_{c_{t}}p_{R,t}}=\tau_{t}^{H}$ (93)

Thus, the optimal carbon price for fossil energy producers is identical to the
carbon price for households.

##  Appendix B Figures

Figure 7: Evolution of housing-related energy demand, resource use in housing
and industry and the expansion of renewable energy in the first best.
Variables are expressed in deviations from the no-policy benchmark. Figure 8:
Efficiency capital investments in the optimal policy scenario, second-best
scenarios and the no-policy baseline. Investments are expressed as a ratio
over physical capital investments. Figure 9: Housing capital stock in the
optimal policy scenario, second-best scenarios and the no-policy baseline.
Investments are expressed as a ratio over physical capital investments. Figure
10: Housing capital investments in the optimal policy scenario, second-best
scenarios and the no-policy baseline. Investments are expressed as a ratio
over physical capital investments.