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pdftitle={Geographic Constraints and the Housing Supply Elasticity in Germany},
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\title{Geographic Constraints and the Housing Supply Elasticity in
Germany}
\author{Eyayaw Beze\footnote{Eyayaw Beze, University of Duisburg-Essen.
-- I am grateful to my advisor, Tobias Seidel, for his insightful
comments and guidance. I also thank Thomas Bauer, Nadine Riedel,
Andreas Mense, and members of the Research Training Group (RTG)
Regional Disparities \& Economic Policy, the Ruhr Graduate School in
Economics (RGS Econ), and the Mercator School of Management,
University of Duisburg-Essen, for helpful comments and discussions.
Finally, I am highly grateful for the financial support from the
Stiftung Mercator via the RGS Econ. -- All correspondence to: Eyayaw
Beze, eyayaw.beze@gmail.com, Essen, Germany}}
\date{April 21, 2024}
\begin{document}
\maketitle
% \begin{bibunit}
\section*{Abstract}\label{abstract}
\addcontentsline{toc}{section}{Abstract}
The study estimates the housing supply elasticity and the impact of
geographic constraints in Germany from 2008 to 2019 using the Bartik
instrument. The results indicate that the housing supply is, on average,
inelastic, with a floorspace elasticity of 0.22 and a units elasticity
of 0.25. The study also reveals that geographical constraints partially
affect the housing supply elasticity across districts. High development
intensity decreases the elasticity, while the unavailability of land for
development due to restrictive geography has no significant impact. The
housing supply elasticity estimates may prove useful for calibrating
quantitative spatial models in Germany.
\textbf{JEL codes}: R310\\
\textbf{Keywords}: House prices, housing supply, housing supply
elasticity
\blfootnote{A similar version of this chapter has been published as \textit{Beze, E. (2023). Geographic Constraints and the Housing Supply Elasticity in Germany. Ruhr Economic Papers \#1003.} http://dx.doi.org/10.4419/96973169.}
\pagenumbering{gobble} % Remove page number
\newpage
\section{Introduction}\label{introduction}
\pagenumbering{arabic}
The price elasticity of housing supply is a key parameter in urban
economics because it drives the congestion externalities and governs
urban growth dynamics. An inelastic supply of land or housing means that
any positive change in housing demand or location due to a positive
productivity or amenity shock translates into higher prices---a major
source of urban disutility---rather than higher quantities. On the
contrary, if housing is supplied more elastically, we might expect
smaller price changes and larger adjustments in city sizes. Moreover,
inelastic supply aggravates house price differences across regions,
affecting inter-regional labor mobility, and may constrain housing
affordability in cities and regions, particularly in the big and growing
ones. Thus, the housing supply elasticity is central to understanding
the long-term development of cities and regions
\citep{combes_etal_2019, glaeser_etal_2006, saks_2008, lerbs_2014}.
Over the past two decades, house prices have risen in cities worldwide,
often due to a combination of strong and growing demand and a limited
supply of new housing. Significant variations in the level and growth of
house prices across cities and locations within cities and regions have
been recorded \citep{glaeser_2020, hilber_mense_2021}. An unresponsive
housing supply can undoubtedly lead to higher house prices. However,
examining what limits the housing supply and why may take more work. For
example, in the US, house price gaps in rural and urban areas are much
larger than the gaps in construction costs which is understood as a
reflection of the difficulty of building new houses, especially in dense
urban cores \citep{glaeser_2020}. While building new houses may help
reduce house prices and alleviate affordability issues, answering why
some cities and regions can build more houses more flexibly than others
in response to the growing demand for housing is essential. More
precisely, why do changes in demand for housing, triggered by
productivity or amenity shocks, increase house prices rather than
inducing more construction activities and city growth in some cities and
not others?
This study empirically examines these questions by analyzing the growth
of housing supply and house prices between 2008-2019 across the 401
German districts.\footnote{There are 401 districts (``Kreise'' in
German) according to the 2019 end-of-the-year
\href{https://www.destatis.de/DE/Themen/Laender-Regionen/Regionales/Gemeindeverzeichnis/Administrativ/Archiv/Verwaltungsgliederung/31122019_Jahr.html}{(31.12.2019)}
administrative structure breakdown.} More precisely, I estimate the
price elasticity of housing supply from a productivity or amenity shock
induced response in housing supply. The main source of variation
exploited for identifying the housing supply elasticity parameter is a
\citet{bartik_1991} labor demand shock, which has been widely used in
the literature (for example, \citet{saiz_2010} and
\citet{baum-snow_han_2019}) as a proxy for demand. As such, average
housing supply elasticity estimates reflect variations in housing demand
shocks due to changes in labor demand over time across districts.
However, construction costs or productivity, existing land development,
and land use regulations may also influence housing supply differences.
These factors can also mediate the responsiveness of the housing supply
to changes in housing demand. Housing supply variations across districts
are high due to differences in existing development intensity and land
availability constraints. Therefore, parameterizing the housing supply
elasticity with these observed housing supply heterogeneities allows us
to get elasticity estimates at the district level and captures the
importance of these factors in mediating the relationship between
housing quantity and price.
Much of the existing evidence on the housing supply elasticity comes
from the US housing market, but the literature is limited for other
countries due to data availability, particularly for Germany. To my
knowledge, only \citet{lerbs_2014} estimated the housing supply
elasticity for Germany, using a dynamic panel data model from new
construction permits of single-family homes for 2004-2010. By adopting
the recent approaches in the literature
\citep{saiz_2010, hilber_vermeulen_2016, baum-snow_han_2019}, this paper
presents the German housing market's peculiarities regarding the housing
supply elasticity.\footnote{The German housing market is known for its
stability, pro-tenant rental laws, moderate rental income taxes, low
interest and mortgage rates, recent fundamental supply shortages, and
stiff land use regulations
(\href{https://www.dbresearch.com/PROD/RPS_EN-PROD/PROD_0000000000517463/Outlook_for_the_German_residential_property_market.pdf?undefined&realload=OGCzKhorommVYqr3DKzCKCT/4~0T75BVVmGs5XHbQs3QdBI~xyjvWG3i2CAeU6MJ}{German
Property Market Outlook 2021, Deutsche Bank AG, accessed August 10,
2022},
\href{https://www.globalpropertyguide.com/Europe/Germany}{Global
Property Guide, accessed August 12, 2022}). Moreover, it was one of
the few housing markets that experienced less house price volatility
during the 2008-2009 financial crisis. Because of these unique
features (subtle nuances), studying this market, particularly
concerning housing supply and elasticity differences across districts,
would be a great addition to the literature.} For doing so, I leverage
a detailed unique house price dataset by \citet{rwi_redhk_2020} covering
the whole of Germany available for 2007 onward. Furthermore, in line
with the housing production literature, I use total residential
floorspace over housing units as the main measure of housing quantity as
it better captures the true level of housing supply
\citep{baum-snow_han_2019, epple_etal_2010}.\footnote{In the literature,
housing supply has been measured or proxied by several variables,
including housing units (stock), new construction permits,
completions, and starts, and the number of households (e.g.,
\citet{saiz_2010}).}
Using a reduced form approach, I recover the housing supply elasticity
as the impact of housing demand-induced growth in house prices on
residential floorspace growth over 2008-2019. Following the housing
production literature, I derive a housing supply function incorporating
local variations in construction costs or productivity and land
availability. This allows writing the housing supply elasticity as a
function of the same factors. Finally, I estimate the housing supply
elasticity via a two-stage least squares (2SLS) estimation using
predicted employment growth as an instrument for house price growth.
The data show that most districts in Germany have experienced
substantial growth in house prices, an about 32\% change between 2008
and 2019, on average. Urban districts, in particular, have experienced
slightly higher growth than rural districts, but there is little
difference in price growth between the West and East German districts.
In contrast, the housing supply growth has been weak across German
districts, about 7\%, on average. New construction permits and
completions have been continuously declining since 1995, gradually
rising after 2009, yet the 2008-2019 levels remain far below the late
1990s and early 2000s.
Consequently, districts, on average, have been supply-inelastic.
According to the baseline results, on average, a district has about 0.22
elasticity in floorspace, 0.25 in units. That means, over the 2008-2019
period, a 10\% increase in house prices has generated a 2.2\% growth in
residential floorspace, on average, keeping other things constant. These
estimates are similar to what \citet{lerbs_2014} found for 2004-2010 (a
short-run elasticity of 0.25 and a long-run value of 0.4).\footnote{Apart
from the time periods, however, estimates in this paper may be
different from \citet{lerbs_2014} estimates due to differences in the
empirical methods employed. First, permits are used as a housing
quantity measure, as opposed to the residential floorspace used in
this paper. Second, the price data used in the two papers are
different.} \citet{baum-snow_han_2019} also found housing supply
elasticity estimates in a similar range for census tracts in the US for
2000-2010.\footnote{The authors estimate the housing supply elasticity
to be in a range of 0.3-0.5 aggregated to the Metropolitan Statistical
Area (MSA) level (see (3) and (8) columns of Table 6 and Table 11 in
the Appendix of their paper).}
The baseline specification obscures district heterogeneity since it does
not allow the housing supply elasticity to vary across districts.
Instead, the main specifications allow housing supply elasticities to
vary across districts as a function of housing supply constraints. This
is achieved by interacting price growth with fractions of already
developed land and land that cannot be developed because of steep slopes
and water bodies. According to the results, only land development
intensity significantly constrains the housing supply elasticity in
German districts, while land undevelopability due to restrictive
geography has no significant impact. Land development intensity lowers
the housing supply elasticity by about 0.46. Moving in the interquartile
range of existing development intensity (5.4\%, 23.0\%) reduces the
floorspace elasticity by 0.08, from 0.285 to 0.204.
Finally, this paper provides robust housing supply elasticity estimates
for Germany from 2008-2019. These estimates may prove useful for
calibrating quantitative urban or regional models in Germany, which
previously relied heavily on estimates for other markets. In addition,
the study's utilization of supply constraints in Germany, precisely the
measurement of undevelopable land constructed from elevation and land
cover data, can be valuable for other studies examining the housing
supply constraints in Germany.
\section{Literature Review}\label{literature-review}
This study builds on a growing body of literature examining the
determinants of housing supply. The existing literature presents strong
evidence regarding the impact of supply constraints on house prices and
the housing supply elasticity in the US housing market
\citep{glaeser_gyourko_2005, saiz_2010, paciorek_2013, baum-snow_han_2019}.
In other markets, the literature is somewhat limited, except for a few
notable studies, such as \citet{hilber_vermeulen_2016} and
\citet{buchler_etal_2021}, which looked at the UK and Swiss markets,
respectively. Much of the existing work looks at differences between
cities concerning spatial scale. Few studies take on within cities and
regions, such as neighborhoods, at a granular spatial scale, for
example, \citet{baum-snow_han_2019}.
\citet{glaeser_gyourko_2005}, \citet{saiz_2010}, \citet{paciorek_2013},
\citet{hilber_vermeulen_2016}, and \citet{baum-snow_han_2019} explore
the relationship between housing prices and various factors such as
regulatory approval, geography, and land availability.
\citet{glaeser_gyourko_2005} argues that changes in house prices in the
US appear to result from a changing regulatory regime that has made
large-scale development increasingly difficult in expensive regions of
the country. \citet{saiz_2010} uses a variation of the Alonso-Mills-Muth
model to show that land-constrained metro areas should have more
expensive housing and higher amenities or productivity. Most areas
widely regarded as supply inelastic were found to be severely
land-constrained by their geography, and highly regulated areas were
found to be geographically constrained. \citet{paciorek_2013} focuses on
the relationship between supply constraints and house price volatility
and indicates that permit delays and marginal costs of new investment
explain much of the observable differences in elasticity across markets.
\citet{hilber_vermeulen_2016} finds regulatory constraints to be the
causal impact of various long-run supply constraints on house prices in
England. Finally, \citet{baum-snow_han_2019} provides a comprehensive
characterization of housing supply elasticities for residential
neighborhoods in 306 US metro areas and finds that distance from urban
centers, initial development density, topography, and zoning regimes are
among the most important determinants of local housing supply.
In summary, the literature on the impact of supply constraints on house
prices and supply is extensive in the US but limited for other countries
\citep{hilber_vermeulen_2016}. There is strong evidence emphasizing that
both geographical constraints \citep[see][]{saiz_2010, paciorek_2013}
and regulatory constraints
\citep[see][]{glaeser_gyourko_2005, paciorek_2013, gyourko_etal_2021, glaeser_2020}
explain much of the rapid house price growth across cities and markets
in the US, as well in the UK \citep{hilber_vermeulen_2016}. In most of
the US studies, the Wharton Residential Land Use Regulatory Index
(WRLURI 2006 or 2018) has been exclusively used as a measure of
regulatory restrictiveness. In contrast, \citet{hilber_vermeulen_2016}
used a dataset that includes direct information on actual planning
decisions to measure regulatory restrictiveness. Regarding geographic
constraints, mainly the degree of land development and unavailability,
measures constructed from digital elevation models and land cover
classes have been used. In the literature, housing supply is measured by
several variables, including housing units (stock), construction
permits, completions or starts, and household size. The housing
production literature \citep[see][]{epple_etal_2010, combes_etal_2021}
suggests using housing services, which can be proxied by floorspace, as
a better measure, as used in \citet{baum-snow_han_2019}.
\citet{bartik_1991} is widely used as a main source of variation for
identifying the housing supply elasticity parameter.
This study extends the analysis period of \citet{lerbs_2014} to
2008-2019, providing a more recent assessment of the price elasticity of
housing supply in Germany. Unlike the previous study, which focused on
single-family homes using construction permits for 2004-2010, this study
uses residential floorspace, including for single-family homes,
construction permits, and completion, providing a more comprehensive
measure of housing supply. Additionally, by examining the impact of
geographic constraints on housing supply elasticity at the district
level, this study captures the heterogeneity in housing supply across
districts, which has not been explored in the literature.
\section{Method}\label{method}
I begin with a simple theoretical derivation of the supply of housing on
a fixed plot of land and a simple aggregation to the district level.
Then, the empirical implementation subsection discusses how the housing
supply elasticity is estimated using the \citet{bartik_1991} shocks as
an instrument for housing demand.
\subsection{Model}\label{model}
This subsection demonstrates the derivation of the local housing supply
function, following the housing production function literature
\citep{epple_etal_2010, combes_etal_2021}. Importantly, the model
demonstrates how the local housing supply and housing supply elasticity
can be written as a function of supply determinants, in particular
geographical or physical constraints.
\subsubsection{Housing supply}\label{housing-supply}
\subsubsection*{A simple model of district housing
supply}\label{a-simple-model-of-district-housing-supply}
\addcontentsline{toc}{subsubsection}{A simple model of district housing
supply}
A competitive developer combines a fixed amount of land \(\overline{T}\)
and non-land inputs \(K\), which I simply call capital, to produce
housing \(H\) via a Cobb-Douglas technology:
\begin{equation}\phantomsection\label{eq-cobb}{
H = H(A, \overline{T}, K) = A \overline{T}^{\alpha} K^{1-\alpha},
}\end{equation} where \(\alpha \in (0, 1)\). \(A\) captures supply
heterogeneity across parcels due to local labor costs, productivity,
geography, or ease of construction differences.\footnote{\(K\) captures
a composite of all inputs for housing production other than land,
which can broadly be labor and materials.}
A representative builder maximizes profit by choosing capital \(K\) over
a fixed parcel of land \(\overline{T}\) \[
\Pi = P(x) \cdot H - R - P^K \cdot K,
\] where \(R\) denotes the endogenous price of land of size
\(\overline{T}\), and \(P^K\) is the price of capital which is assumed
to be invariant across parcels and locations and normalized to unity.
The price of a unit of housing developed on a parcel of size
\(\overline{T}\) is given by \(P\) and depends on \(x\), a vector of
observed or unobserved parcel or location characteristics that reflect
the housing demand on the parcel.\footnote{The full detail of this
section is delegated to the Appendix; see
Section~\ref{sec-hs-appendix}.}
Builder's profit maximization delivers the factor demand for capital
\(K\).\footnote{Since land is fixed, the developer chooses capital to
maximize profit.} \[
K^* = \Bigl((1-\alpha)AP(x)\Bigr)^{\frac{1}{\alpha}}\overline{T}\equiv K^*(P(x), \overline{T}, A)
\]
By substituting the factor demand equation for capital back into the
housing production function Equation~\ref{eq-cobb}, the per parcel
supply function can be written as: \[
H(A, \overline{T}, K^*(P(x), \overline{T}, A))=\mu A^{\frac{1}{\alpha}}{P(x)}^{\frac{1-\alpha}{\alpha}}\overline{T} \equiv H^S(P(x), A)\,,\quad \text{with } \mu = (1-\alpha)^{\frac{1-\alpha}{\alpha}}.
\] In log-linear form, \begin{equation}\phantomsection\label{eq-hs}{
\ln H^S(P(x), A, \overline{T}) = \ln \mu + \frac{1}{\alpha}\ln A + \underbrace{\Bigl(\frac{1-\alpha}{\alpha}\Bigr)}_{\varepsilon}\ln P(x) + \ln \overline{T}.
}\end{equation} This shows that the supply of housing developed on a
parcel depends on local supply heterogeneity \(A\), local housing demand
conditions captured by \(P(x)\), and on the size of the parcel
\(\overline{T}\).
Following \citet{baum-snow_han_2019}, aggregation of the housing supply
on a parcel in Equation~\ref{eq-hs} over all developed parcels in the
district delivers the total supply of housing in the district. Let
\(\mathcal{L}_{i}\) denote the total (developable) land endowment of
district \(i\) and \(\Lambda_i (P_i)\) the fraction of partitioned
parcels that are developed in \(i\), then the stock of developed land in
\(i\) is defined as
\(\mathcal{T}(P_i) = \Lambda_i(P_i) \cdot \mathcal{L}_{i}\). The
implicit district-level aggregate housing supply function
\(S_i(P_i(x))\) can then be defined as the product of the (average)
housing supply per parcel and the stock of developed land:
\begin{equation}\phantomsection\label{eq-agg-hs}{
\begin{aligned}
S_i(P_i) = H_i^S(P_i, A_i)\cdot \mathcal{T}_{i}(P_i)\\
\ln S_i(P_i) = \Bigl[\ln \mu_i + \frac{1}{\alpha}\ln A_i + \varepsilon\ln P_i \Bigr]+ \Bigl[\ln\Lambda_i(P_i) + \ln\mathcal{L}_i\Bigr]
\end{aligned}
}\end{equation}
Differentiating Equation~\ref{eq-agg-hs} with respect to \(\ln P\)
delivers the housing supply elasticity,
\begin{equation}\phantomsection\label{eq-elasticity}{
\varepsilon^S_{i} \equiv \varepsilon + \frac{\partial \ln\Lambda_i(P_i)}{\partial \ln P_i},
}\end{equation} where the first term captures the intensive margin of
development (floorspace per parcel) and the second reflects the
extensive margin (parcel development).
Districts with more developable (more flat or less rugged) land, low
initial level of development density, and unrestrictive regulation may
respond more along the extensive margin, increasing the level of land
development. In contrast, districts that have a high level of existing
development (high built-up) or are restricted by their geography or by
restrictive land regulation may respond along the intensive margin,
increasing floorspace per parcel.
\subsubsection{Housing demand}\label{housing-demand}
The canonical housing demand is derived from a utility maximization
problem of a representative household living in district \(i\) that
consumes final goods \(C\) priced at 1 and a unit of housing \(H_i\)
priced at \(P_i(x)\). \[
\begin{aligned}
\underset{C,H}{\max}\, U(C, H) = \theta H_i^\beta C_i^{1-\beta}\\
\text{s.t.}\quad w_i = C_i+P_i(x)\cdot H_i\,,
\end{aligned}
\]
where \(w_i\) denotes average wage or productivity in \(i\).\footnote{For
simplicity, I assume that market imperfections are minimal such that
workers earn wages equal or proportional to their productivity.} The
first order condition for utility maximization with respect to \(H\)
delivers the housing demand function \[
H^{*}_i(P_i(x), w_i) = \beta\frac{w_i}{P_i(x)} \equiv H^d_i(P_i(x), w_i).
\]
Assuming that locations within district \(i\) are perfect demand
substitutes for given values of parcel characteristics, then \(P_{i}\)
representing the average price of housing in the district, summing up
over the housing consumption of all the residents \(N\) of \(i\), the
log aggregate housing demand function for district \(i\) can be written
as \begin{equation}\phantomsection\label{eq-agg-hd}{
\ln P_{i}=\ln \beta + \ln w_{i}-\ln H_{i}^{d} + ln N_{i}.
}\end{equation}
The housing demand function in Equation~\ref{eq-agg-hd} shows how
exogenous productivity (\(w_i\)) changes can be used as demand shifters
for identifying the housing supply elasticity.\footnote{Higher levels of
productivity are associated with higher levels of economic development
and income, which can lead to increased demand for housing by
attracting more people to an area
\citep{glaeser_gottlieb_2009, glaeser_2008}. This increased demand can
drive up local house prices, as people are willing to pay more for
housing in areas with strong job market opportunities and higher
wages. Home builders will react to the increased demand and higher
prices by building new houses, other things held constant.}
\subsection{Empirical implementation}\label{empirical-implementation}
Fundamental to recovering the housing supply elasticity \(\varepsilon\)
is finding an exogenous shifter that comes from the housing demand
function Equation~\ref{eq-agg-hd}. This shifter can be used as an
instrument for house prices provided it is uncorrelated with supply
shifters (such as construction costs or productivity).
The main estimation equation is the aggregate housing supply function
Equation~\ref{eq-agg-hs} in discrete changes
\begin{equation}\phantomsection\label{eq-main}{
\Delta\ln H_i^S = a^S_i + \varepsilon^S_i\Delta\ln P_i + \boldsymbol{\beta}\mathbf{X}^S_i + u^S_i,
}\end{equation} where \(i\) indexes districts across Germany and changes
are computed from long (log) differences between 2008-2019, and
\(\mathbf{X}\) includes a set of district controls.\footnote{The control
variables include construction and labor costs, housing supply
constraints, and dummy variables for whether the district is urban and
in West Germany. Depending on the variable type, controls are either
in levels or logarithms.} The parameter vector of interest is
\(\varepsilon^S_i=\mathbf{Y}_i\varepsilon\). Following \citet{saiz_2010}
and \citet{baum-snow_han_2019}, such a setup allows us to compute an
elasticity estimate for each district \(i\). For doing so,
\(\varepsilon^S_i\) can be defined as a function of observed district
supply characteristics \(\mathbf{Y}_i\) such as undevelopable fraction
of land, level of existing land development (developed fraction), and
regulation. First, as a baseline specification, and following
\citet{saiz_2010}, I start with a common price elasticity of housing
supply across all districts, i.e.,
\(\varepsilon^S_i=\varepsilon^S \;\forall i\), which corresponds to
\(\varepsilon\) in the aggregate housing supply function in
Equation~\ref{eq-agg-hs}. Then, I consider estimating the housing supply
elasticity levels that vary across districts as a function of these
observed district supply conditions.
\subsubsection{The housing supply elasticity as a function of supply
constraints}\label{sec-decompose}
In this section, I demonstrate how land unavailability (due to
geographical constraints), the level of existing land development, and
restrictive land use regulations impact the housing supply elasticity. I
follow the reasoning by \citet{saiz_2010} and \citet{baum-snow_han_2019}
as to why these constraints mediate the impact of growth in house prices
on housing supply.
As districts are inherently different, they will respond differently if
they receive the same amount of housing demand shock. More precisely,
the same level of demand shock will produce different results in housing
supply growth because of differences in factors that affect housing
supply. For instance, districts that are more flat, growing, and less
regulated are expected to react more to a given change in housing
demand, other things held constant. Therefore, the extent to which
changes in housing demand translate into more construction rather than
higher prices depends on physical and regulatory factors. More land
availability (for instance, through rezoning) shifts the supply curve
outward. The question is whether land and regulatory constraints also
impact the housing supply elasticity, i.e., whether \(\varepsilon^S_i\)
is a function of land availability, regulation, and development
intensity.
The first hypothesis I test is whether districts that have a high level
of existing development, measured by the fraction of land that is
already developed (\emph{Developed}), have more inelastic housing supply
than newer or physically growing districts, i.e., \(\varepsilon^S_i\) is
a function of development intensity:
\begin{equation}\phantomsection\label{eq-case-2}{
\Delta\ln H_i^s = \varepsilon^S\Delta\ln P_i + \beta^{Developed} \Delta \ln P_i\times\text{Developed}_i + \boldsymbol{\beta}\boldsymbol{X}_i^S + u^S_i\,,
}\end{equation} where \(\beta^{\text{Developed}} \le 0\). Then, we have
district-specific housing supply elasticities that incorporate
development intensity:
\(\varepsilon^S_i = \varepsilon^S + \beta^{\text{Developed}}\times\text{Developed}_i\).
Second, I test whether districts that are land-constrained due to
geography (measured by \emph{Unavail}) have more inelastic housing
supply than relatively unconstrained districts, i.e.,
\(\varepsilon^S_i\) is a function of land unavailability:
\begin{equation}\phantomsection\label{eq-case-3}{
\Delta\ln H_i^s = \varepsilon^S\Delta\ln P_i + \beta^{\text{Unavail}} \Delta \ln P_i\times\text{Unavail}_i + \boldsymbol{\beta}\boldsymbol{X}_i^S + u^S_i\,,
}\end{equation} where \(\beta^{\text{Unavail}} \le 0\). Then, the
elasticity is given by
\(\varepsilon^S_i = \varepsilon^S + \beta^{\text{Unavail}}\text{Unavail}_i\).
Third, I combine the above two cases and test the importance of both
variables (developed and unavailable land fractions) in affecting the
response of supply growth to price growth, i.e., \(\varepsilon^S_i\) is
a function of both developed land and land unavailability:
\begin{equation}\phantomsection\label{eq-case-4}{
\begin{aligned}
\Delta\ln H_i^s = \varepsilon^S\Delta\ln P_i + \beta^{\text{Developed}} \Delta \ln P_i\times\text{Developed}_i + \\
\beta^{\text{Unavail}} \Delta \ln P_i\times\text{Unavail}_i + \boldsymbol{\beta}\boldsymbol{X}_i^S + u^S_i\,,
\end{aligned}
}\end{equation} where
\(\beta^{\text{Developed}} \le 0\,, \beta^{\text{Unvail}}\le0\).Then,
the elasticity is given by
\(\varepsilon^S_i = \varepsilon^S + \beta^{\text{Developed}}\text{Developed}_i + \beta^{\text{Unavail}}\text{Unavail}_i\).
\subsubsection{Constructing the Bartik
instrument}\label{constructing-the-bartik-instrument}
Since, in equilibrium, housing quantity and price are jointly
determined, the classic endogeneity problem needs to be addressed to
correctly identify \(\varepsilon^S_i\) in Equation~\ref{eq-main}.
Solving this problem requires an exogenous shifter, sourced from the
demand equation in Equation~\ref{eq-agg-hd}, that generates exogenous
housing demand changes across locations. Ideally, this shock then causes
a shift in the housing demand curve so the housing supply curve can be
traced out and the parameter \(\varepsilon^S_i\) is identified. The
\citet{bartik_1991} instrument, also known as the shift-share
instrument, has been widely used in the literature for identification
(see, for example, \citet{saiz_2010}, \citet{hilber_vermeulen_2016},
\citet{baum-snow_han_2019}), as a proxy for or source of variation in
housing demand. Below I explain how the Bartik instrument has been
constructed and used for identification in this study.
The Bartik instrument or shock is a labor demand shock that is
constructed from predicted industry employment growth
\citep{goldsmith-pinkham_etal_2020, blanchard_katz_1992, bartik_1991}.
By construction, local employment growth is the weighted mean of local
industry growth rates, where the weights are local employment shares of
the industries, \[
g_{it} = \sum_k z_{ikt}\cdot g_{ikt},
\] where the subscripts \(i\), \(k\), and \(t\) index district,
industry, and time (year), respectively. \(z_{ikt}\) denotes industry
\(k\)'s employment share in district \(i\)'s total employment
\(L_{it}\), and \(g_{ikt}\) denotes industry \(k\)'s employment growth
in \(i\) from \(t-1\) to \(t\).
The local industry growth rate \(g_{ikt}\) can be decomposed into a
national industry growth rate \(g_{kt}\) and an idiosyncratic local
industry growth rate \(\widetilde{g}_{ikt}\) components:
\[g_{ikt} = g_{kt} + \widetilde{g}_{ikt}.\]
The Bartik instrument uses the national industry growth rate \(g_{kt}\)
and local industry composition at some base or initial time period to
predict the local industry employment growth \(g_{ikt}\). Following
\citet{bartik_1991}, ``predicted employment growth'' can be written as
\[
\begin{aligned}
\widehat{g_{it}} &=\sum_{k=1}^K z_{ikb}\cdot g_{kt}\\
&\quad\quad\text{with }\;g_{kt} = \frac{L_{kt} - L_{kt-1}}{L_{kb}}\,,
\end{aligned}
\]
where \(L\) represents the actual level of employment, \(b\) denotes
some initial time period, variables without index \(i\) represent values
at the national level, and variables without index \(k\) are aggregates
over industries. \(g\) denotes the growth rate of employment from
\(t-1\) to \(t\) as a proportion of the base year value. \(L_{kt}\) can
be calculated for each \(i\) from leave-one-out aggregate of
\(L_{ikt}\), denoted \(L_{(i^\prime)kt}\).
From the data, I constructed seven (hence \(K=7\)) broad industry
classes.\footnote{The industry classification follows the
\href{https://www.destatis.de/DE/Methoden/Klassifikationen/Gueter-Wirtschaftsklassifikationen/klassifikation-wz-2008.html}{German
Classification of Economic Activities, Edition 2008 (WZ\_2008)}. I
aggregate the district and national industry employment data that I
used for constructing the Bartik instrument to 7 broad industry
classes: (1) agriculture, forestry, and fishing, (2) mining and
quarrying, energy, and water, sewage, and waste, (3) manufacturing,
(4) construction, (5) trade, transport, hospitality, and information
and communication, (6) finance and insurance, and real estate, (7)
public and other services, education, and health.} Moreover, following
the Bartik instrument convention, I take the first period of the study
as the initial period (i.e., \(b=2008\)).
The Bartik instrument used in the estimation is the predicted change in
the logarithm of employment between the base period (2008) and the last
(2019),
\(\widehat{\Delta\ln L_i} = \sum_k{z_{ik_2008} \left(\ln L_{(i^\prime)k_2019}-\ln L_{(i^\prime)k_2008}\right)}\).
\section{Data}\label{data}
I construct a house price index from a granular and rich set of house
price data. The housing stock and floorspace, population, and employment
data are all obtained from the German Regional Statistical Offices
Database.
\subsection{House prices}\label{sec-hedonic-hse}
I use the RWI-GEO-Real Estate Data of the FDZ Ruhr at RWI
\citep{rwi_redhk_2020} to construct quality-adjusted house
prices.\footnote{The original data is provided by
\emph{ImmobilienScout24}, Germany's largest online platform for
listing real estate (for both selling and renting houses and
apartments). The house prices are self-reported offer prices by the
respective home seller or agent and may therefore differ from the
actual transaction prices.} The data are highly detailed (at a scale
of 1km\textsuperscript{2} grid), cover all of Germany, and have been
available since 2007. Moreover, the data come with a rich set of
property characteristics, enabling us to compute a hedonic price index
to quality-adjust house prices.
I construct a mix-adjusted house price index from the following panel
hedonic regression
\begin{equation}\phantomsection\label{eq-hedonic-hse}{
\ln P_{hit} = \delta_{it} + \mathbf{X}_{hit}\boldsymbol{\beta} + e_{hit},
}\end{equation} where \(h\) indexes houses, \(i\) districts and \(t\)
years 2008-2019, \(P\) price of houses in euros per
m\textsuperscript{2}, \(\delta_{it}\) denotes district-year fixed
effects that are of main interest to estimate, and \(\mathbf{X}\)
includes a set of house characteristics.\footnote{House attributes
included in the hedonic regression are: floorspace, plot area, number
of rooms, number of floors, number of bedrooms, number of bathrooms,
type of the house, type of heating, years of construction and
renovation, condition and facilities of the property, whether the
property has a basement, a guest washroom, is or in a protected
building, and is usable as a holiday house. The mix-adjusted house
price index computation is based on the method described by
\citet{ahlfeldt_etal_2020}.} In Equation~\ref{eq-hedonic-hse}, the
estimated intercepts \(\widehat{\delta}_{it}\) represent the
quality-adjusted prices for each district \(i\) in every year \(t\).
After estimating Equation~\ref{eq-hedonic-hse} with fixed effects, the
hedonic price index is given by
\(\widehat{\delta}_{it} = \ln P_{hit} - \mathbf{X}\boldsymbol{\widehat{\beta}}\).
\subsection{Housing quantity}\label{housing-quantity}
Housing quantity is measured by housing units (stock) and housing
services proxied by total residential floorspace. As houses vary in both
observable and unobservable characteristics, they need to be
standardized to account for these differences. In fact, the housing
production literature views houses as only differing in the housing
services they provide, which are homogeneous and perfectly divisible
\citep{epple_etal_2010, combes_etal_2021}.
In light of that, this paper uses total residential floorspace as the
main measure of housing quantity, as a stock count of houses or
buildings may not accurately reflect the true level of housing supply in
a city or region. Moreover, using housing units as a measure of housing
supply does not account for differences in size or other attributes. For
example, newly built or renovated houses are often larger and better
equipped with features than older houses, which units may not capture.
Data on housing quantity variables such as residential units,
floorspace, and construction activities (i.e., permits and completions)
were obtained from the Regional Atlas of Germany.
\subsection{Geographic data}\label{geographic-data}
I use geographical data to construct land development intensity, land
unavailability, and terrain ruggedness index (TRI) measures.\footnote{TRI
is an objective measure of terrain heterogeneity. I computed TRI
according to \citet{riley_etal_1999}, which calculates TRI by
comparing changes in elevation between a central pixel and its eight
neighbors as the square root of the squared sum of these elevation
differences. Grid cell level TRI values were then averaged across grid
cells within districts for TRI value at the district level. TRI is
derived from the same DEM data.}
From the Digital Elevation Model (DEM) of Germany at \(200\times200\)
meter resolution, I calculated slope to extract the share of land
corresponding to steep slopes, which makes up the undevelopable land
(along with land covered by wetland and water bodies)
measure.\footnote{\href{https://gdz.bkg.bund.de/index.php/default/digitale-geodaten/digitale-gelandemodelle/digitales-gelandemodell-gitterweite-200-m-dgm200.html}{Digitales
Geländemodell Gitterweite 200 m (DGM200)}} An area exhibiting steep
slopes (for example, above 15\% in the US context) is considered
unsuitable for construction \citep{saiz_2010}.
From the Corine Land Cover (CLC) Germany, compiled by the
\href{https://www.dlr.de/eoc/en/desktopdefault.aspx/tabid-11882/20871_read-48836}{German
Remote Sensing Data Center (DFD) of DLR} and the
\href{https://www.bkg.bund.de}{Federal Agency for Cartography and
Geodesy (BKG)}, I calculated the exact share of land covered with
wetlands and water bodies to measure land unavailability. More,
development intensity is constructed from the exact share of
``artificial surfaces'', a comprehensive land cover class that includes
continuous and discontinuous urban fabric, defined by CLC.
\subsubsection{Undevelopable and developed land}\label{sec-constraints}
I define undevelopable or unavailable land as land covered by wetlands
and water bodies or as potentially developable land with an average
slope greater than 15\%. Undevelopable land may also include already
developed land (an area covered by ``artificial surfaces'' such as
buildings) if we rule out redevelopment through renovation or demolition
as a development option. In other words, land already developed may not
be regarded as undevelopable as it can be redeveloped. In this paper,
already-developed land is not part of the undevelopable land. I use the
land cover classes defined by
\href{https://land.copernicus.eu/user-corner/technical-library/corine-land-cover-nomenclature-guidelines/html/index.html}{Corine
Land Cover (CLC)} that are relevant to Germany.
The area with a slope greater than 15\% is defined over the district's
total ``developable stock'' of land. I define the developable stock as
the district's total administrative area, excluding the area covered
with wetlands and water bodies. In other words, areas covered by forests
or agriculture make up the developable stock. Note that developable
stock does not exclude areas with a slope greater than 15\%. The
fraction of area with a slope greater than 15\% is defined over this
quantity, i.e., developable stock as a denominator.
More concisely, denoting the district's total administrative area by
\(T\), developed land by \(T^{\text{artificial}}\), area covered by
wetland by \(T^{\text{wetland}}\), water bodies by \(T^{\text{water}}\),
agriculture by \(T^{\text{agri}}\), and forests by
\(T^{\text{forest}}\), then developable stock \(T^{\text{developable}}\)
is given by \[
T^{\text{developable}} = T - T^{\text{wetland}} - T^{\text{water}} = T^{\text{agri}} + T^{\text{forest}}.
\]
Then, the fraction of developable land that is lost to steep slopes is
defined as \(r^{steep} = \frac{T^{steep}}{T^{\text{developable}}}\),
where
\(T^{steep} = T^{\text{developable}} \cdot\mathbb{1}\left[{\text{slope} > 15\%}\right]\),
the developable area with a slope greater than 15\%.
The fraction of undevelopable land is defined as the ratio of the total
undevelopable land to the total administrative land of the district.
Undevelopable land \(T^{\text{undevelopable}}\), is given by \[
T^{\text{undevelopable}} = T^{steep} + T^{\text{wetland}} + T^{\text{water}}.
\] Then, the share of undevelopable land (out of the total land),
\(r^{\text{undevelopable}} = \frac{T^{\text{undevelopable}}}{T}\).
Similarly, the share of ``developed land'' is
\(r^{\text{developed}} = \frac{T^{\text{artificial}}}{T}\).
Undevelopable share is this paper's main measure of geographical
constraint, with TRI and slope as alternative measures. Finally, the
developed share measures the existing level of development intensity.
\subsection{Bartik shocks: Predicted employment
growth}\label{bartik-shocks-predicted-employment-growth}
In the main regression analysis, the fundamental source of variation in
changes in housing demand is predicted employment, also known as Bartik
or local labor demand shock. I use employment data decomposed by seven
industries to construct this shock using the 2008 industry employment
levels in German districts and the national industry-specific employment
growth rates from 2008 to 2019. Labor demand shock is local employment
growth in each district that would have resulted, given the district's
industry composition in the initial period (2008), had employment in
each industry developed over time (2008-2019) in the same way as at the
national (Germany) level.
Finally, other controls, including the price of land, population,
income, and other socioeconomic control variables used in this study,
are all obtained from the \citet{atlasde_2022}.
\section{Results}\label{results}
\subsection{Descriptive analysis}\label{descriptive-analysis}
As shown in Figure~\ref{fig-price-growth}, average prices of houses in
Germany have significantly increased over the 2008-2019 time period
across districts. In level terms, the average price of houses (of all
types) was about €2,563 per m\textsuperscript{2} in 2008, which rose by
about 32.2\% to about €3,388 per m\textsuperscript{2} in 2019.
Single-family homes followed the same trend; the average price of
single-family homes increased by about 28.9\%.\footnote{These house
prices are quality-adjusted, i.e., these are the hedonic values (as
discussed in Section~\ref{sec-hedonic-hse}). Additionally, prices are
adjusted for inflation using Germany's 2015 general Consumer Price
Index.} Variation in average house prices (measured by the standard
deviation) across districts has significantly risen over this period.
For all homes, the standard deviation of prices in €\(/m^2\) has
increased by 84\%, from about 883 in 2008 to about 1,625 in 2019. For
single-family homes, price variation has gone up by 78\%, from 991 in
2008 to 1,760 in 2019.
Across space, as shown in Figure~\ref{fig-spatial-pattern}, there has
been high variation in the level and the growth of house prices,
especially in and around big ``city-districts''. High-price places in
2008, such as Berlin, Munich, and Hamburg, remained expensive also in
2019 and experienced a higher house value appreciation. Urban districts
have experienced relatively higher growth than their rural counterparts.
Price levels have risen by about 34.5\% in urban and by 28.9\% in rural
districts, on average, over the 2008-2019 time period. Prices of
single-family homes have grown by about 30.7\% and 26.2\%, in urban and
rural districts, respectively. In contrast, there has been little
disparity along the West-East divide; house prices have risen by about
32.4\% in the West and 30.7\% in the East districts. However, the growth
of prices of single-family homes has been higher in the East German
districts, by 32.5\%, while in the West districts by 28.3\%.
\begin{figure}
\centering{
\includegraphics{output/figs/fig-price-growth-1.pdf}
}
\caption{\label{fig-price-growth}The development of house prices in
Germany.}
\begin{minipage}{0.975\textwidth}
\small
\emph{Notes:} \(\ln P\) is the log
of average house prices across districts each year. The house type
category ``all'' captures all house types, including ``single-family''
homes. House prices are in 2015 prices. Data obtained from
\citet{rwi_redhk_2020}.
\end{minipage}
\end{figure}%
\begin{figure}
\centering{
\includegraphics{output/figs/fig-spatial-pattern-1.pdf}
}
\caption{\label{fig-spatial-pattern}Spatial trends: house prices across
districts in Germany.}
\begin{minipage}{0.975\textwidth}
\small
\emph{Notes:} This