Non-homothetic housing demand and geographic worker sorting.
Date: 2025-02-06
Institutions: European Central Bank
Subjects: Germany, housing need, property market, quantitative analysis, skilled worker

Working Paper Series 

Nina Furbach 

Non-homothetic housing demand and 
geographic worker sorting 

No 3018 

Disclaimer: This paper should not be reported as representing the views of the European Central Bank 
(ECB). The views expressed are those of the authors and do not necessarily reflect those of the ECB. 

Abstract

Housing expenditure shares decline with income. A household’s income determines its sen-

sitivity to housing costs and drives its location decision. Has spatial skill sorting increased

because low income individuals are avoiding increasingly expensive regions? I augment a

standard quantitative spatial model with flexible non-homothetic preferences to estimate

the effect of the national increase in the relative supply of high skilled workers that has put

upward pressure on housing costs in skill-intensive cities. My model explains 10% of the

increase in average house prices in Germany from 2007 to 2017 and 11% of the regional

differences in house price increases. One third of the effects is due to an increase in spatial

skill sorting driven by differences in housing expenditure shares. The observed degree of skill

sorting was not significantly different from the optimal allocation in 2007 while skill sorting

was larger than optimal in 2017.

JEL Classification: H21, H23, R12, R21

Keywords: Housing demand, non-homotheticiy, geographic worker sorting, Germany, quan-

titative spatial models

ECB Working Paper Series No 3018 1Non-technical summary

Western economies are experiencing a housing crisis unlike anything we have seen for decades.

About half of Americans (49%) say the availability of affordable housing in their local community

is a major problem (Schaeffer, 2022). At the same time, real house prices in the US have risen

by 35% from 2010 to 2020. House price increases are sizable also in other Western economies: In

Germany, house prices in 2020 were 47% higher than in 2010 (OECD, 2022). These developments

are more and more seen as a driver of increasing economic inequality. The growing cost burden

is not equally distributed across households since poorer households spend a significantly larger

share of their income on housing. However, we know very little about how hard the lack of

affordable housing hits individuals across the income distribution, nor do we know much about

how it affects location decisions.

Individuals with lower incomes tend to allocate a larger portion of their budgets to housing,

which might lead them to settle in lower-cost cities. Higher income individuals are less sensitive

to housing costs and often choose to live in higher-cost cities. The higher the share of individuals

with large incomes, the higher will be housing demand and house prices in large cities, and the

more pronounced geographic sorting becomes. This paper studies whether the increase in the

national supply of high skilled individuals can explain the trends in location choices observed

over past decades: Skilled individuals have clustered in high-cost cities while unskilled workers

avoid increasingly expensive cities. I ask whether individuals with lower skills (and therefore

lower incomes) tend to avoid big cities to a larger extent than workers with higher skills since

they suffer more from rising housing costs. A large degree of sorting, in turn, amplifies house

price differences between large cities and rural areas. I further analyse how place-based policies

optimally respond to the increase in spatial skill sorting.

I start by documenting that low income households spend a significantly larger share of their

expenditures on housing. To do so, I run household-level regressions using large-scale German

survey data from 2010 to 2014. Controlling for regional house prices, year fixed effects and a

large set of household characteristics, I find that a 100% increase in total household expenditure

causes a 30% decrease in the housing expenditure share.

To analyze how differences in housing expenditure shares affect location choices, I use an

exogenous shock to housing costs: The national rise in the relative supply of workers with a

university degree in Germany has increased the demand for housing. It thereby put upward

ECB Working Paper Series No 3018 2pressure on house prices that was more pronounced in skill-intensive regions.

With the help of a spatial general equilibrium model, I analyze whether high school graduates

avoided increasingly expensive regions to a larger extent than college graduates. The model is

calibrated to be consistent with the empirically documented reduced-form estimates on the

expenditure elasticity of the housing expenditure share. I estimate the model on the level of 141

labour market regions in Germany. My findings suggest that the increase in the national supply

of high skilled workers from 2007 to 2017 led to an increase in spatial skill sorting. It accounts

for roughly one tenth of the national increase in house prices and one tenth of the regional

dispersion in house price increases, while one third of the effects is due to the heterogeneity in

housing expenditure shares.

The results of this paper, which demonstrate that poorer individuals spend a significantly

larger share of their income on housing, suggest that policies aimed at increasing the supply

of affordable housing could substantially improve the well-being of low income individuals. My

model provides evidence that low income individuals are avoiding increasingly expensive cities

because they are hit harder by increases in housing costs. I find that place-based policies that

aim at lowering the degree of geographic sorting have small welfare effects.

ECB Working Paper Series No 3018 31 Introduction

Western economies are experiencing a housing crisis unlike anything we have seen for decades.

About half of Americans (49%) say the availability of affordable housing in their local community

is a major problem (Schaeffer, 2022). At the same time, real house prices in the US have risen

by 35% from 2010 to 2020. House price increases are sizable also in other Western economies: In

Germany, house prices in 2020 were 47% higher than in 2010 (OECD, 2022). These developments

are more and more seen as a driver of increasing economic inequality. The growing cost burden

is not equally distributed across households since poorer households spend a significantly larger

share of their income on housing. However, we know very little about how hard the lack of

affordable housing hits individuals across the income distribution, nor do we know much about

how it affects location decisions.

Individuals with lower incomes tend to allocate a larger portion of their budgets to housing,

which might lead them to settle in lower-cost cities. Higher income individuals are less sensitive

to housing costs and often choose to live in higher-cost cities. The higher the share of individuals

with large incomes, the higher will be housing demand and house prices in large cities, and the

more pronounced geographic sorting becomes. This paper studies whether the increase in the

national supply of high skilled individuals can explain the trends in location choices observed

over past decades: Skilled individuals have clustered in high-cost cities while unskilled workers

avoid increasingly expensive cities. I ask whether individuals with lower skills (and therefore

lower incomes) tend to avoid big cities to a larger extent than workers with higher skills since

they suffer more from rising housing costs. A large degree of sorting, in turn, amplifies house

price differences between large cities and rural areas. I further analyse how place-based policies

optimally respond to the increase in spatial skill sorting.

I start by estimating the degree of non-homotheticity using large-scale survey data of German

households. My results establish that non-homotheticity in housing demand is both econometri-

cally and economically significant. I set up a spatial general equilibrium model of non-homothetic

housing demand to estimate the effect of the national rise in the skill share. Calibrating the

model with the estimated preference parameters, I find that the rising skill share explains 3% of

the increase in spatial sorting by skill, 10% of the national increase in house prices and 11% of

the regional differences in house price increases from 2007 to 2017 in Germany. With homothetic

preferences, the national shock does not change skill sorting which implies that it can explain

ECB Working Paper Series No 3018 4only 6% of the national house price increase and 7% of the regional dispersion in house price

increases. The national increase in the supply of high skilled workers has decreased welfare of

low skilled workers by 0.9% and welfare of high skilled workers by 0.8%.

I next ask how a social planner should optimally respond to the observed changes in skill

sorting. I analyze the optimal allocation in 2007 and 2017 using a utilitarian welfare function.

I choose to study the allocation that equally benefits all worker types. The social planner

maximizes welfare taking into account redistribution and efficiency considerations. To do so,

she chooses regional type-specific taxes and transfers that set incentives for individuals to move

across space. I find that the observed degree of skill sorting was not significantly different from

the optimal allocation in 2007, while skill sorting was larger than optimal in 2017. From a social

planner perspective, it would be optimal to set incentives for all workers to move toward rural

areas, but to a larger extent for high skilled workers. Moving from the observed to the optimal

allocation implies welfare gains of 0.4% in 2007 and 0.5% in 2017.

I arrive at these conclusions by studying sorting in a spatial general equilibrium model

with heterogeneous workers that have non-homothetic preferences. I include two worker types:

workers with and without a university degree.

In my model, heterogeneous individuals with

Generalized Elasticity of Substitution (GES) preferences trade off wages, housing costs and

regional amenities when making their location decision. Locations differ exogenously in terms

of housing supply, group-specific productivity and group-specific amenities. I further include

heterogeneous preference shocks for locations that act as a form of migration costs. Identical

firms combine labor from different worker groups to produce a final good that is traded between

regions at zero cost.

The quantification follows the basic steps known from literature on quantitative spatial mod-

els (see Redding and Rossi-Hansberg, 2017 for an overview). First, I use observed data and the

structure of the model to calibrate the key structural parameters. I estimate non-homothetic

preferences over housing and non-housing consumption utilizing large-scale consumption micro-

data from the German Socio-Economic Panel (GSOEP). I start by linearizing the relationship

between the housing expenditure share and total expenditure derived from the model. From the

reduced-form estimation of this first-order approximation, I obtain parameters that are directly

interpretable as elasticities and therefore comparable to estimates from the literature. Guided

by the structure of the model, I control for local house prices since households’ sorting decisions

introduce a positive correlation between prices and incomes at the regional level (see Albouy et

ECB Working Paper Series No 3018 5al., 2016 and Finlay and Williams, forthcoming). I find that a 100% increase in total expendi-

ture causes a 30% decrease in the housing expenditure share. My estimates are well in line with

those found in comparable studies (see Finlay and Williams, forthcoming for an overview). For

the calibration of the model parameters, I estimate the non-linear relation between the housing

expenditure share, total expenditure and house prices derived from the model directly by Gen-

eralized Method of Moments (GMM). I reject two alternative preferences used in the literature:

Cobb-Douglas and a unit housing requirement.

In the second step of the quantification, I use observed data, the structure of the model,

and the structural parameters to invert the structural productivity, housing and amenity funda-

mentals. For the model inversion, I leverage on matched employer-employee data from German

social security records. In particular, for every year, I observe the local labor market in which

individuals work (Kosfeld and Werner, 2012), the nominal wage and a range of individual-level

characteristics. I use this information to construct a regional wage measure for every skill type

that is purged from differences in observable worker characteristics between regions. Aggrega-

tion of the micro data yields total employment and average wage by region and worker group

for 2007 and 2017. To these data, I merge a regional property price index, which is generated

from property micro data from the largest German listing website (Ahlfeldt et al., 2022).

The model is quantified to match the observed data on house prices, skill-specific wages

and skill-specific employment on the regional level.

It is further calibrated to be consistent

with the empirically documented estimates on the non-homotheticity of preferences. I use the

model to quantify the importance of accounting for non-homothetic preferences when analyzing

geographic worker sorting. To do so, I estimate the effect of the rise in housing congestion

resulting from the national increase in the relative supply of high skilled workers. The size of

the shock amounts to an increase in the national share of high skilled workers from 16% in 2007

to 22% in 2017. In the model with non-homothetic preferences, this shock leads to intensified

geographic worker sorting since low skilled workers are hit harder by increases in housing costs

that are more pronounced in skill-intensive regions. The increase in sorting, in turn, amplifies

house price increases in large cities as compared to rural areas. The model allows for exogenous

changes in productivity, housing and amenity fundamentals. Feeding the national shock of an

increase in the skill share into the model, the model accounts for 10% of the national increase

in house prices and 11% of the regional dispersion in house price increases. Roughly one third

of the effects is due to the non-homotheticity of preferences. Non-homothetic preferences can

ECB Working Paper Series No 3018 6explain 3% of the observed change in skill sorting. I find that the shock has decreased welfare

of low skilled workers by 0.9% and welfare of high skilled workers by 0.8%.

The experiment explores the importance of allowing for non-homothetic preferences when

analyzing sorting in spatial equilibrium models.

I next ask: What is the optimal degree of

skill sorting? There are two reasons why the social planner allocation differs from the observed

equilibrium. First, because wages are on average lower, marginal utilities of tradable good con-

sumption are larger in rural areas. The social planner thus increases welfare by redistributing

from urban to rural regions which sets incentives for workers to move across space. Secondly,

since congestion forces are not taken into account by workers when choosing their place of res-

idence, there is space for welfare improvement by setting incentives for individuals to move

toward rural areas. Non-homothetic preferences affect the optimal allocation in several ways.

Since lower skilled workers have a larger housing expenditure share than with homothetic pref-

erences, they demand more housing. This implies that they generate stronger congestion forces

than with homothetic preferences, while at the same time being more sensitive to housing con-

gestion. Workers with higher skills, on the other hand, generate lower congestion forces than

with homothetic preferences, while being less sensitive to housing congestion. Furthermore,

by changing the marginal utililities of tradable good consumption, non-homothetic preferences

affect the optimal degree of redistribution between regions.

I use the model to compute the optimal allocation which provides insights into the welfare

implications of spatial skill sorting. I solve the problem of a social planner who maximizes a util-

itarian welfare function taking as given workers’ location choices as well as resource constraints

on housing and tradables. To do so, she chooses transfers between locations and worker types

which I characterize. Welfare weights are calibrated such that both worker types experience the

same welfare gain.

While the observed degree of skill sorting was not significantly different from the optimal

allocation in 2007, skill sorting was larger than optimal in 2017. Since high skilled workers,

by consuming more housing than low skilled workers, generate larger congestion forces, it is

optimal to reallocate them to a larger extent toward regions with less congested housing markets.

Furthermore, since the urban wage premium is larger for high skilled workers, spatial differences

in marginal utilities of tradable good consumption are larger for high skilled than for low skilled

workers. It is therefore optimal to redistribute consumption goods between regions to a larger

extent for high skilled than for low skilled individuals. The smaller degree of agglomeration

ECB Working Paper Series No 3018 7and skill sorting in the optimal allocation imply that house prices are less dispersed as reflected

in a larger increase in house prices in rural as compared to urban areas. My findings indicate

that moving from the observed to the optimal allocation implies welfare gains of 0.4% in 2007

and 0.5% in 2017. I further find that the modeling of preferences matters: the social planner

implements significantly larger transfers in 2017 when assuming GES preferences as compared

to Cobb Douglas preferences.

This paper is related to several strands of literature. One strand aims at explaining the

diverging location choices between skilled and unskilled households. While some studies have

stressed the role of endogenous amenities (Diamond, 2016) or the role of technology in gener-

ating skill-biased wage growth in certain locations (Giannone, forthcoming; Eckert et al., 2020;

Rubinton, forthcoming), few studies show that non-homothetic housing demand significantly

affects spatial sorting.

Ganong and Shoag (2017) connect changes in regional housing supply regulations to slowing

regional income convergence. Finlay and Williams (forthcoming) find that skill-biased techno-

logical change has intensified skill sorting since it made lower skilled workers relatively more

sensitive to housing costs.1 My work is closely related to Gyourko et al. (2013) who analyse

the effects of changes in the relative size of different skill groups. Based on empirical tests of a

number of equilibrium relationships, they argue that the increase in spatial skill sorting can be

explained by an increasing number of high skilled households nationally combined with an in-

elastic supply of land in superstar cities. I set up and estimate a quantitative spatial model that

allows to quantify the effects of the national increase in the number of high skilled individuals.

I find that it led to an increase in spatial sorting even when assuming uniform housing supply

elasticities. My model further enables me to quantify the substantial contribution of the growing

skill share to the increase in house prices, and assess its implications for welfare inequality.

In the second part of my analysis, I take one step further and ask how non-homothetic

preferences shape the optimal allocation and the taxes and transfers that could implement it. I

thereby complement a large literature on the extent of spatial misallocation and the role that

transfer and taxation policies play (see Albouy, 2009, Ossa, forthcoming, Fajgelbaum et al.,

2018 and Colas and Hutchinson, 2021). Rather than evaluating exogenous policies, I derive

the optimal allocation in a quantitative spatial model with local congestion forces. Since my

1The result of Finlay and Williams (forthcoming) are in line with evidence from Couture et al. (forthcoming)

on within-city sorting.

ECB Working Paper Series No 3018 8model is flexible enough to capture any degree of non-homotheticity, my results generalize those

of Fajgelbaum and Gaubert (2020) who assume Cobb-Douglas preferences. Focusing on the

effects of skill-specific productivity and amenity spillovers, Fajgelbaum and Gaubert (2020)

find that the US economy would benefit from a larger skill sorting. According to the results

of Rossi-Hansberg et al. (2021) on the other hand, it would be optimal to take advantage of

scarce cognitive non-routine workers by clustering them in small cognitive hubs to maximize

positive production externalities. My results are in line with Fajgelbaum and Gaubert (2020)

and propose that spatial skill sorting is larger than optimal.

This paper is further related to literature estimating non-homotheticities in housing demand.

At the level of cities, a common assumption is that preferences are Cobb-Douglas and therefore

homothetic (see for example Eeckhout et al., 2014; Diamond, 2016; Fajgelbaum and Gaubert,

2020; Rossi-Hansberg et al., 2021). This assumption is often justified by the fact that housing

expenditure shares vary little across cities with very different income levels (Davis and Ortalo-

Magn´e, 2011). My results are in line with Finlay and Williams (forthcoming) and Albouy et al.

(2016) who offer an alternative explanation for the similarity of housing expenditure shares across

cities: offsetting price and income effects. While Albouy et al. (2016) rely on city-level variation

in incomes, prices, and rental expenditure, I follow Finlay and Williams (forthcoming) and use

consumption microdata. I find demand elasticities in Germany comparable to those estimated

for other regions (see Finlay and Williams (forthcoming) for an overview), and quantify the role

of these elasticities for the optimal degree of sorting and the response of house prices and welfare

to changes in the supply of high-skilled workers.

The remainder of the paper is structured as follows. Section 2 presents stylized evidence

that informs my modeling choices. Section 3 introduces a model with heterogenous workers that

have non-homothetic preferences and Section 4 calibrates a quantitative version of this model.

Section 5 uses the calibrated model to quantify the role of non-homothetic preferences when the

national supply of different worker types changes. It further analyzes optimal region-specific

taxes and transfers. Section 6 concludes.

2 Stylized Facts

To motivate the relevance of non-homothetic housing demand in the context of spatial skill

sorting, I present some stylized facts on the spatial economy using data I describe in Section 4.1.

I start by plotting the housing expenditure share for each percentile of the income distribution

ECB Working Paper Series No 3018 9in Figure 1.

It can be seen that housing expenditure shares are far from constant: Moving

from the 10th percentile of the income distribution to the 90th percentile implies a decrease

in the housing expenditure share from 38% to 21%.2 The result is robust to controlling for

household size.

In Section 4.2, I causally estimate the degree of non-homotheticity and find

that indeed housing expenditure shares decline substantially with income. When plotting the

estimated preferences, they appear to fit the data better than preferences often assumed in the

literature, which is confirmed by formal tests that reject the null hypothesis of constant housing

expenditure shares.

Figure 1: Housing expenditure shares and income

Source: GSOEP, own calculations. Note: Housing share is defined as housing expenditure (including heating and

electricity) divided by total net income. The plot is based on household data from 2017. Number of observations:

6648

Non-homothetic preferences could potentially drive the patterns observed in the data and

illustrated in Figure 2. The upper plots show that from 2007 to 2017, house prices have increased

significantly more in large labor markets. I find that the elasticity of house prices with respect

to employment has increased by almost 43%. At the same time, skill sorting has intensified since

high skilled workers are increasingly attracted by dense regions, even to a larger extent than low

skilled workers. The semi-elasticity of the share of high skilled workers with respect to city size

2A comparison to data 30 years ago reveals that the decrease in housing affordability is mainly a problem for
low income households: The housing expenditure share has increased significantly more for low income than for
high income households (see Figure A1).

ECB Working Paper Series No 3018 10has increased by 44%.

Figure 2: Geographic sorting and house prices

Source: SIAB, Ahlfeldt et al. (2022), own calculations. Note: Unit of observation is 141 labor market areas as

defined by Kosfeld and Werner (2012).

In the following, I study a mechanism that links the three stylized facts. It can explain the

simultaneous increase in regional house price differences and spatial skill sorting with the help of

non-homothetic preferences: An exogenous national increase in the relative supply of high skilled

workers has increased the demand for housing. The shock has put upward pressure on house

prices that was more pronounced in skill-intensive regions. Due to lower housing expenditure

shares, workers with higher incomes suffered less from increasing housing costs in these regions.

91011121314log employment -1-0.500.511.52log house price200791011121314log employment -1-0.500.511.52log house price2017Slope(SE): 0.21(0.03)Slope(SE): 0.3(0.04)91011121314log employment 010203040share high skilled in %200791011121314log employment 010203040share high skilled in %2017Slope(SE): 2.97(0.34)Slope(SE): 4.29(0.38)ECB Working Paper Series No 3018 11Non-homothetic preferences in combination with the national increase in high skilled workers can

therefore explain why skilled households have clustered in dense labor markets while unskilled

workers increasingly avoided these regions. The increase in sorting, in turn, amplified house

price increases in large cities as compared to rural areas.

3 Model

3.1 Competitive allocation

In this section, I develop a spatial general equilibrium model with heterogeneous workers that
have non-homothetic preferences. I consider an economy that is populated by L = (cid:80)

k Lk work-
ers who I categorize into groups indexed by k. Heterogeneous workers choose a region i taking

as given the decision of all other individuals. Local labor markets vary exogenously in their

productivity, amenities, and housing supply. I further include worker-specific preference shocks

for locations that act as a form of migration costs. Conditional on their labor market, individ-

uals maximize utility over consumption of housing and tradable goods. I incorporate regional

congestion forces by assuming an inelastic supply of housing. Homogeneous firms employ differ-

ent worker types to produce goods that are traded at zero cost. I discuss potential mechanisms

within the model framework that could rationalize the stylized facts presented in Section 2.

3.1.1 Workers

Preferences of a worker n belonging to group k and working in region i are defined over freely-

tradable homogeneous goods cik, housing hik, regional amenities Eik and the idiosyncratic

amenity shock ϵin. I assume GES preferences that take the following form

(cid:16)

uin =

γ(cik)1− 1

ρ + (1 − γ)(hik)1− η

ρ

(cid:17) ρ

ρ−1 Eikϵin

(1)

with γ > 0, 0 < ρ < 1 and η > ρ. ρ is the elasticity of substitution and η captures the degree

of non-homotheticity. GES preferences nest the specifications commonly used in the spatial

literature. If ρ → 1 and η = 1, I obtain Cobb-Douglas preferences where γ is the expenditure

share on tradables.3 In the case of η → ∞, I get a unit housing requirement which is a very

extreme form of non-homotheticity often assumed in the literature. It implies that individuals

always purchase one unit of housing, so demand is perfectly price and income inelastic. GES

3Note that if η = 1 only, I get constant elasticity of substitution (CES) preferences.

ECB Working Paper Series No 3018 12preferences, in contrast, can accommodate any degree of non-homotheticity. In the empirically

relevant case, η > 1 and expenditure shares on housing decrease with income.

Conditional on working in region i, a type-k worker solves the following problem:

γ(cik)1− 1

ρ + (1 − γ)(hik)1− η

ρ

(cid:17) ρ

ρ−1 Eik

vik = max
cik,hik

(cid:16)

s.t.

cik + pihik = wik + Πik + tik

(2)

where pi is the price of housing, wik is the wage and tik is the net government transfer to a

type-k worker in region i. The tradable good is chosen to be the num´eraire. Πik is the return

on a regional portfolio of housing that equals individual housing expenditure:

Πik = pihik.

(3)

I assume that ϵnk is drawn from a type-1 extreme value distribution with shape parameter

ψ that reflects the extent of preference heterogeneity across regions. If the variation in regional

amenity draws is large, workers show little sensitivity to differences in wages, house prices and

amenities, which implies a low geographic mobility. The distributional assumption on region-

specific amenity draws implies closed-form expressions for the number of workers in each region

Lik =

1
ψ
ik

v

(cid:80)

i v

1
ψ
ik

Lk

(4)

where Lk = (cid:80)

i Lik is the total number of type-k workers.

3.1.2 Firms

Identical firms combine labor from different worker groups to produce the freely-traded final

good.

I assume a linear production function with group- and location-specific productivity

shifters Aik. Firm-level production functions translate directly to city-level production since

firms face constant returns to scale and share an identical production technology. The regional-

level production function is given by

Yi =

(cid:88)

k

AikLik.

(5)

ECB Working Paper Series No 3018 13Firms pay wages that are equal to the marginal product of labor

wik = Aik.

(6)

The regional supply of housing Hi is determined by an exogenous part Ti that captures the

availability of land and an endogenous part that depends on total city population Li = (cid:80)

k Lik:

Hi = TiLγH

i

where γH is the housing supply elasticity.

3.1.3 Equilibrium

In equilibrium, the market for tradable goods clears:

(cid:88)

i

Yi =

(cid:88)

(cid:88)

Likcik

i

k

(7)

(8)

which follows from the household budget constraint in equation (2) combined with a balanced
government budget (cid:80)
i

k Liktik = 0. Housing market clearing requires

(cid:80)

Hi =

(cid:88)

k

Likhik

(9)

where housing demand is given by the combined first-order conditions of the household maxi-

mization problem

hik = (

1
pi

1 − γ
γ

ρ − η
ρ − 1

)

ρ

η (cik)

1
η .

(10)

Labor markets clear when equation (4) and equation (6) hold.

Thus, for given parameters γ, ρ, η, ψ and γH , location-specific fundamentals Aik, Ti, Eik and

taxes tik, an equilibrium is a vector of Lik, wik, cik, hik, Πik, pi, Hi and Yi satisfying equations

(2) to (10).4

4When estimating the model, I impose tik = 0 ∀ i, k both in the observed and counterfactual scenario. In this

case, equation (8) follows from the household budget constraint in equation (2) and is therefore redundant.

ECB Working Paper Series No 3018 143.1.4 Potential mechanisms

In a quantitative spatial model as laid out in this section, a number of different shocks could

rationalize the patterns observed in the data and presented in Section 2. With homothetic

preferences, a simultaneous increase in both spatial skill sorting and regional house price differ-

ences could result from shocks to Aik or Eik, i.e. from region-specific shocks to productivity or

amenity fundamentals that differ across skill types. One example would be skill-biased techno-

logical change with differential effects across regions depending on their industry composition.

If high skilled workers become more productive mainly in dense regions, house prices increase in

these regions since more high skilled workers move there. An alternative explanation would be

regional shocks that are symmetric across skill types in combination with asymmetric spillovers.

Consider the case in which all workers become more productive in denser regions. Such produc-

tivity shock would increase the spatial concentration of population. If high skilled workers benefit

more from knowledge spillovers in dense cities, we would observe an increase in both house prices

and geographic skill sorting. A large literature combines both asymmetric spillovers with asym-

metric region-specific shocks to fundamentals such that spillovers amplify the effects of shocks

to regional fundamentals (see for example Diamond, 2016 and Giannone, forthcoming). Another

explanation is that agglomeration spillovers themselves have changed over time. Baum-Snow

and Pavan (2013) argue that agglomeration forces of high skilled workers have become stronger

relative to those of low skilled workers which led to an increase in spatial skill sorting.

With non-homothetic preferences, a number of additional shocks could rationalize the in-

crease in both regional house price differences and spatial skill sorting. Even in the absence

of spillovers, productivity and amenity shocks that are common across regions might lead to

a change in spatial skill sorting. Finlay and Williams (forthcoming) model skill-biased techno-

logical change as a national shock to productivity fundamentals of high skilled relative to low

skilled workers. Assuming non-homothetic preferences, they find that skill-biased technological

change can explain 23% of the increase in skill sorting in the US since 1980. Gyourko et al.

(2013) argue that the simultaneous increase in geographic skill sorting and spatial house price

differences can be explained by an inelastic supply of land in superstar cities combined with

an increasing number of high income households nationally. In the following, I show that the

national increase in the relative supply of high skilled individuals leads to an increase in spatial

sorting even with uniform housing supply elasticities γh.

ECB Working Paper Series No 3018 153.2 The planner’s problem

In this section, I characterize the optimal allocation and the taxes and transfers that implement

it. There are two reasons why the social planner allocation differs from the observed equilibrium.

First, the decentralized world is inefficient due to congestion forces on regional housing markets.

The fact that housing supply is inelastic (γh < ∞) implies an externality: Workers do not

generate the same degree of congestion in all regions which is not taken into account when

choosing a place of residence. Thus, there is space for welfare improvement by reallocating

workers across space (see Fajgelbaum and Gaubert, 2020). Second, due to differences in wages

across space, marginal utilities of tradable good consumption are not constant across regions.

The social planner increases welfare by redistributing toward regions with low marginal utilities

of consumption.

My aim is to contrast the decentralized allocation with the solution to the planner’s problem.

I solve the problem of a social planner who takes as given that workers can freely move across

labor markets. Under this assumption, expected utility of a type-k worker is given by

(cid:16)

ρ

ρ−1 log

uexp
k ≡ ψ log(

(cid:88)

e

i

1− 1

γ(cik )

ρ +(1−γ)(hik )

1−

ψ

(cid:17)

η
ρ

+log Eik

).

(11)

Then, if ωk denotes the welfare weight for skill type k, I can postulate the generalized social

welfare function

(cid:16)

ρ

ρ−1 log

W =

(cid:88)

k

ωkψ log(

(cid:88)

e

i

1− 1

γ(cik )

ρ +(1−γ)(hik )

1−

ψ

(cid:17)

η
ρ

+log Eik

).

(12)

The planner maximizes the expression in equation (12) subject to workers’ location choices

(equation (4)), the resource constraint on housing (equation (9)), as well as the resource con-

straint on tradables (equation (8)). I turn next to characterizing the solution to this planning

problem. Details on the calculation of the social planner solution can be found in Appendix B.

Competitive equilibria according to the definition in Section 3.1.3 may not correspond to

a point on the Pareto frontier due to spatial inefficiencies: Workers do not internalize the

impact that their location choice has on other workers in the form of housing congestion. The

social planner takes the social costs of additional workers in different regions into account when

setting incentives for workers to move between labor markets. In the optimal allocation, the

ECB Working Paper Series No 3018 16social marginal cost of an additional type-k worker in region i has to equal its social marginal

value. More formally, I can express optimal expenditures as

µY cik + µh

i hik = wik + (cid:101)Πik + λik

(13)

where µY , µh

i and λik are Lagrange multipliers on the government budget constraint, the re-

source constraint on housing and the mobility constraint. (cid:101)Πik denotes the social marginal value

generated in the housing sector:

(cid:101)Πik = µh
i

∂Hi
∂Lik

.

Thus, the social planner implements transfers according to

tik = Φik + λik

(14)

(15)

where Φik = (cid:101)Πik − Πik is the wedge between the private and the social marginal value of an

extra type-k worker in region i.

The proposition generalizes a key insight in Fajgelbaum and Gaubert (2020) to an economy

with non-homothetic preferences and imperfect worker mobility between regions. As pointed

out by Fajgelbaum and Gaubert (2020), the optimal transfers tik take care of inefficiencies due

to spillovers as well as distributional concerns. In the absence of spillovers, I would still have

tik = λik, so that transfers would redistribute according to differences in the marginal utility of

consumption across individuals, as implied by the second welfare theorem. In Fajgelbaum et al.

(2018), workers are perfectly mobile and hold Cobb-Douglas preferences. In this case, λik does

not depend on the region. The burden of dealing with spatial inefficiencies falls on the other

component of the optimal transfer scheme, corresponding to the first term in equation (15).

Since my model does not incorporate positive externalities and incomes are on average lower in

rural areas, which implies higher marginal utilities of tradable goods consumption, the social

planner will transfer resources from urban to rural areas, and thereby incentivise individuals to

move away from expensive large cities.

Both determinants of the optimal transfer scheme are affected by the non-homotheticity

of preferences.

In terms of efficiency considerations, incomes are on average higher in urban

areas, which implies that individuals have lower housing expenditure shares and suffer less from

ECB Working Paper Series No 3018 17the lack of affordable housing. Due to the weaker congestion forces, the social planner will

redistribute less workers towards rural areas. The redistribution of workers is also less efficient

when housing expenditure shares decline with income. Implementing negative transfers in urban

expensive cities leads to a decline in the disposable income of individuals in these regions, which

implies an increase in their housing expenditure share and an increase in their housing demand.

Non-homothetic preferences therefore mitigate the decline in housing congestion achieved from

setting negative transfers in expensive regions that incentivise individuals to move towards less

expensive areas.

Non-homothetic preferences also affect the optimal degree of skill sorting. Since lower skilled

workers have a larger housing expenditure share than with homothetic preferences, they demand

more housing. This implies that they generate stronger congestion forces than with homothetic

preferences, while at the same time being more sensitive to housing congestion. Workers with

higher skills, on the other hand, generate lower congestion forces than with homothetic prefer-

ences, while being less sensitive to housing congestion. Furthermore, by changing the marginal

utililities of tradable good consumption, non-homothetic preferences affect optimal transfers

which determine the optimal degree of redistribution between regions for both skills. With non-

homoethetic preferences affecting the optimal transfer scheme in several different ways and many

of those working into different directions, it is ex-ante not clear how the optimal allocation with

a generalized utility function compares to the results obtained when assuming Cobb Douglas

preferences.

Note that with GES preferences, the housing expenditure share depends not only on indi-

viduals’ income but also on regional house prices. In Section 5.3, I show that accounting for

a positive elasticity of the housing expenditure share with respect to house prices does matter:

the results when assuming CES preferences are significantly different from those obtained when

assuming Cobb Douglas preferences.

4 Quantification

I calibrate the model to German labor market regions in 2007 and 2017. The quantification

of the model consists of two steps that follow the literature on quantitative spatial models (see

Redding and Rossi-Hansberg, 2017 for an overview). First, I obtain values of the structural

parameters. I estimate the preference parameters γ, ρ and η using variables observed in the data

and the structure of the model. The housing supply elasticity γH and the migration elasticity ψ

ECB Working Paper Series No 3018 18are taken from the literature. Second, I use data from 2007 and 2017, the calibrated parameter

values, and the structure of the model to invert the structural fundamentals Aik, Ti and Eik

separately for 2007 and 2017.

4.1 Data

I estimate the model on the level of 141 German labor market regions as defined by Kosfeld and

Werner (2012) based on commuting data. The areas are constructed by combining one or more

districts with the aim of creating self-contained labor markets. The boundaries of local labor

markets are defined such that commuting flows between labor market regions are minimized. I

drop all regions in which the number of observations for any worker group is smaller than 20. I

end up with a sample of 138 labor markets.

I obtain information on regional employment and wages for different worker groups from

the microdata on individual employment histories from the Sample of Integrated Labor Market

Biographies (SIAB) provided by the Institute for Employment Research (Antoni et al., 2019).

The SIAB is a 2% representative sample of administrative data on all workers who are subject

to social security contributions excluding self-employed and civil servants. I restrict the sample

to full-time workers between 20 and 64 and use the consumer price index from Statistisches

Bundesamt (2019) to calculate real wages.

In the SIAB data, I only observe wages up to

the social security contribution ceiling. To impute top-coded wages for the roughly 5% of

observations above the social security contribution ceiling, I use the approach from Dauth and

Eppelsheimer (2021).

I split the sample into 2 groups: Workers with and workers without a university degree.5 I

aggregate wages to the labor market level by running the following regression for every worker

group k and for the years 2007 and 2017 separately:

ln wraw

n = αk + βkXn + dik + ϵn

(16)

where Xn is a set of observable worker characteristics, dik is a group-region dummy, and ϵn is
an error term.6 Given the Mincerian regressions and assuming ϵn is normally ditributed, I can

5Individuals are assigned the highest qualification level that they achieve throughout their working life.
6The controls include sex, a dummy that indicates whether a person is German, detailed level of educational

attainment, duration of past unemployment periods, and duration of past unemployment periods squared.

ECB Working Paper Series No 3018 19rescale average wages according to

wik = exp

(cid:16)

αk +

1
Lk

(cid:88)
(

n∈k

Xn)βk + dik +

(cid:17)

σ2
2

(17)

which represents the average wage of a type-k worker in region i while assuming that workers

have otherwise identical characteristics between regions.

I use a house price index from Ahlfeldt et al. (2022) who utilize data from the FDZ (Forschungs-

datenzentrum) Ruhr on real estate offers published on the largest German listing website Immo-

bilienScout24 with a self-reported market share of about 50% (see Klick and Schaffner, 2019).

By combining a hedonic regression approach with recent extensions that treat spatial units as

the nucleus of a spatial price gradient, Ahlfeldt et al. (2022) generate an index that controls for

property characteristics and distance from the center of the labor market region.

Table 1: Summary Statistics

Wage low skill
Wage high skill
Total employment (in thd)
Share high skill (in %)
House purchase price

2007

2017

mean

sd

mean

sd

83.17
148.17
126.91
12.62
1

9.25
22.42
142.75
4.36
0.34

94.91
157.64
126.13
17.10
1.43

10.09
21.03
146.04
5.57
0.80

Note: The table shows descriptive statistics for cross-sectional data on the level of 138
labor markets. Wages are gross daily wages, house prices are relative to the national
mean in 2007.

To calibrate the preference parameters, I use consumption microdata from the GSOEP

which is a yearly survey with information on income, expenditure and education of individuals

in approximately 11000 private households.7 I aggregate the data set to the household-level and

merge it with the house price index on the district-level from Ahlfeldt et al. (2022).

7I exclude households where the household head is non-employed, doing an apprenticeship, is younger than
18 or older than 64 years, has refugee status or is seeking asylum, as well as all households with owner-occupied
housing.

ECB Working Paper Series No 3018 204.2 Structural parameters

Preference parameters ρ and η

To calibrate the preference parameters of the model, I utilize the household first order condition
pi
xik

as defined in equation (10). Defining total expenditure xik = cik + pihik, multiplying with

and substituting for cik yields

sik =

(cid:16) 1 − γ
γ

ρ − η
ρ − 1

(cid:17) ρ

η p

1− ρ
η
i

(xik)−(1− 1

η )(1 − sik)

1
η

(18)

denotes the housing expenditure share. To estimate this equation, I use the

where sik =

pihik
xik

variation across households h and years t. I interpret α ≡ ( 1−γ
γ

ρ−η
ρ−1 )

ρ
η as an idiosyncratic shock

to a household’s taste for housing, so that equation (18) becomes

sht = αhtp

1− ρ
η
it

(xht)−(1− 1

η )(1 − sht)

1
η .

(19)

I follow Finlay and Williams (forthcoming) and log-linearize equation (19) around the mean

housing expenditure share ¯s to obtain

(cid:98)sht =

η(1 − ¯s)
η(1 − ¯s) + ¯s

(cid:16)

(cid:98)αht + (1 −

ρ
η

)(cid:98)pit − (1 −

(cid:17)

1
η

)(cid:98)xht

(20)

where (cid:98)y denotes the log deviation of a variable y from its mean. Defining βht ≡ η(1−¯s)
θ ≡ η(1−¯s)

η ), equation (20) simplifies to

η ) and ζ ≡ − η(1−¯s)

η(1−¯s)+¯s (1 − ρ

η(1−¯s)+¯s (1 − 1

η(1−¯s)+¯s (cid:98)αht,

(cid:98)sht = βht + θ(cid:98)pit + ζ (cid:98)xht.

(21)

Under the null of homothetic preferences, θ = ζ = 0.

I bring equation (21) to the data by

modeling the demand shifter βht as a function of observables, year fixed effects, and an additive

error. Formally, I get

(cid:98)sht = βt + δXht + θ(cid:98)pit + ζ (cid:98)yht + ϵht

(22)

where Xht is a vector of demographic characteristics which includes household size, the number

of earners in the household as well as the gender and age of the household head.

I observe

total expenditure xht, the housing expenditure share sht, and prices pit. The error term ϵht

represents measurement error in expenditure plus random shocks to housing demand which

ECB Working Paper Series No 3018 21both are assumed to be uncorrelated with expenditure and prices conditional on the controls.

In my preferred specification, I estimate the nonlinear equation (19) directly by GMM.

Since expenditure data is only available from 2010 to 2014, I restrict my sample to these

years. I drop households in the top and bottom 1% of the income and expenditure distribution

each year to guard against serious misreporting errors. I further restrict the sample to renters.

Since homeownership rates are increasing with income and homeowners spend on average less on

housing than renters, I expect my estimates to be a lower bound of non-homotheticity. Finlay

and Williams (forthcoming) use data on housing expenditures of homeowners and find similar

results to those for renters.

Table 2: Preference Estimates
Dependent variable: Log housing expenditure share

(1)
OLS

(2)
OLS IV

(3)
OLS

(4)
OLS IV

(5)
GMM

(6)
GMM IV

Log expenditure

-0.477∗∗∗
(0.013)

-0.212∗∗∗
(0.021)

-0.529∗∗∗
(0.012)

-0.270∗∗∗
(0.021)

Log price

eta

rho

0.311∗∗∗
(0.017)

0.233∗∗∗
(0.017)

2.576∗∗∗
(0.079)

1.520∗∗∗
(0.057)

1.746∗∗∗
(0.025)

1.649∗∗∗
(0.066)

1.072∗∗∗
(0.044)

0.643∗∗∗
(0.034)

Demographic controls

Year FE

R2
adj. R2
First stage F-statistic
N
No. of clusters

✓

✓

.331
.331

8232
4659

✓

✓

.331
.33
122970
8232
4318

✓

✓

.055
.054

8232
4659

✓

✓

.055
.054
118590
8232
4659

∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

✓

✓

1.807∗∗∗
(0.028)

0.598∗∗∗
(0.036)

✓

✓

8232
4318

8232
4318

Note: Standard errors in parentheses, clustered at the household level. Renters only.
Instrument is log
household income. Demographic controls include household size, number of earners as well as gender and
age of the household head.

The estimation results are shown in Table 2. To deal with measurement error in expenditure,

I follow Finlay and Williams (forthcoming) and use income as an instrument for expenditure. I

ECB Working Paper Series No 3018 22find an expenditure elasticity of ζ = −0.27 which is in line with estimates found in the literature

that range from −0.88 to −0.01 (see the literature review in Finlay and Williams, forthcoming).

Controlling for house prices on the district level increases the absolute size of the expenditure

elasticity. Since higher income households sort into expensive regions, the estimated elasticity

will be biased toward zero when not controlling for house prices. Offsetting price and income

effects are in line with the findings in Albouy et al. (2016) and Finlay and Williams (forthcoming).

Finally, column (6) shows my preferred specification where I estimate the nonlinear equation

(19) directly by GMM and instrument for expenditure. I estimate ρ = 0.60 and η = 1.81. Finlay

and Williams (forthcoming) find estimates of the price elasticity θ = 0.39 and the expenditure

elasticity ζ = −0.25. Assuming a mean housing expenditure share of ¯s = 0.29 as observed in

the data, these estimates imply ρ = 0.78 and η = 1.44. In Appendix A.2, I show that my results

are close to Finlay and Williams (forthcoming) when estimating preference parameters from

NHCES preferences using GMM.

I follow Finlay and Williams (forthcoming) and compare the preferences estimated in Ta-

ble 2 to two benchmarks from the literature: Cobb-Douglas preferences and a unit housing

requirement. The GES preferences estimated above nest both of these special cases. The null

hypothesis of Cobb-Douglas preferences, corresponding to ρ → 1 and η = 1, can be rejected at

the 1% level. A unit housing requirement corresponds to η → ∞. Column (6) allows me to

reject the null hypothesis that η = 1.87 or any number above at the 1% level.

Although my GES specification is more general than these special cases, it still imposes

a particular functional form on the relationship between total expenditure and the housing

expenditure share. I assess the validity of this assumption by constructing a binned scatterplot

of expenditure against housing shares as shown in Figure 3. My estimated preferences appear

to fit the data well. For comparison, I also include housing expenditure shares as implied by

Cobb-Douglas and a unit housing requirement. Neither alternative comes close to matching the

data.

Housing congestion γH

To calibrate the elasticity of housing supply to population, I combine equation (9) and (10) and

solve for pi

pi =

1 − γ
γ

ρ − η
ρ − 1

(cid:16) (cid:80)

1
η
ik

k Likc
Hi

(cid:17) η
ρ .

(23)

ECB Working Paper Series No 3018 23Figure 3: Preference calibration

Note: ‘Estimated Preferences’ plots equation (20) at the parameter values obtained in Table 2, column (4). The
‘Cobb-Douglas’ plots the preferences with ρ → 1 and η = 1
shaded area represents a 95% confidence interval.
and ‘Unit Housing Requirement’ with η → ∞. The scale parameter γ is chosen to match the expenditure share
as in the data. ‘Data’ plots the average housing share in 100 evenly sized bins defined by total expenditure.

The elasticity of house prices with respect to population is

Ek ≡

∂pi
∂Lik

Lik
pi

=

η
ρ

1
η
ik

(cid:16) Likc
(cid:80)

k Likc

1
η
ik

− γH

(cid:17)

.

Lik
Li

Summing over k yields

Ek =

η
ρ

(cid:88)

k

(1 − γH ).

Assuming equal elasticities for all worker types

E =

1
K

η
ρ

(1 − γH ).

(24)

(25)

(26)

where K denotes the number of worker groups. I take the parameter E = 0.208 from Combes
et al. (2019). This implies γH = 1 − K ρ

η E = 0.861.

Scale parameter γ

The scale parameter γ is not identified separately from the scale of prices and consumption, so

ECB Working Paper Series No 3018 24I normalize it to match the aggregate housing share. Plugging xik = cik
1−sik

into equation (18), I

get

sik
1 − sik

=

(cid:16) 1 − γ
γ

ρ − η
ρ − 1

(cid:17) ρ

1− ρ
η
i

η p
(cik)1− 1

η

(27)

which I numerically solve for sik and γ using the additional constraint that the mean housing
expenditure share matches the observed data ( 1
L

k Liksik = 0.29).

(cid:80)
i

(cid:80)

Table 3: Calibrated Parameters

Parameter

Value

Source

Preferences

ρ

η

ψ

Congestion forces

γH

0.60

1.81

0.5

Estimated

Estimated

Gaubert et al. (forthcoming)

0.86

Combes et al. (2019), own calculation

4.3 Structural fundamentals

I obtain the location-specific productivity, housing supply and amenity shifters Aik, Ti and Eik

by inverting the model so that it exactly matches the observed data on pi, wik, tik and Lik for

all regions i and skill types k. Abstracting from income taxes, social security contributions and

transfers, I set tik = 0.8 From equation (6), I calculate productivity fundamentals

Aik = wik.

(28)

Plugging housing supply (equation (10)) and housing demand (equation (7)) in the housing

market clearing condition (equation (9)), I get an expression for the housing supply shifter that

8There are no location-specific income taxes in Germany. Note that linear taxes do not change the results in the
case of Cobb-Douglas preferences. I abstract from non-linearities due to GES preferences and from non-linearities
in income taxes, social security contributions and transfers.

ECB Working Paper Series No 3018 25depends solely on variables that I observe in the data

Ti = L−γH

i

(cid:16) 1
pi

ρ − η
ρ − 1

1 − γ
γ

(cid:17) ρ

η (cid:88)

k

Lik(wik + tik)

1
η .

(29)

Finally, I combine the mobility constraint (equation (4)) with the budget constraint (equation

2) to get an equation that I can numerically solve for amenities Eik

Lik =

(cid:16)

(cid:80)
i

(γ(wik + tik)1− 1
(cid:16)

(γ(wik + tik)1− 1

ρ + (1 − γ)(hik)1− η
ρ )

ρ

ρ−1 Eik

(cid:17) 1
ψ

ρ + (1 − γ)(hik)1− η
ρ )

ρ

ρ−1 Eik

Lk

(cid:17) 1
ψ

with housing demand from equation (10):

hik = (

1
pi

1 − γ
γ

ρ − η
ρ − 1

)

ρ

η (wik + tik)

1
η .

(30)

5 Results

5.1 Changes in house prices and geographic sorting

I use the calibrated model to quantify the importance of accounting for non-homothetic pref-

erences when analyzing geographic worker sorting. To do so, I estimate the effect of the rise

in housing congestion resulting from the national growth in the relative supply of high skilled

workers. The shock amounts to an increase in the share of high skilled workers from 16% in

2007 to 22% in 2017. I plot a decomposition of the increase in the regional dispersion of house

prices and the increase in skill sorting from 2007 to 2017. I compare the decomposition results

obtained from a model with non-homothetic preferences with those from a model with constant

housing expenditure shares. When assuming Cobb-Douglas preferences, I calibrate the scale

paraemter γ to match the mean housing expenditure share observed in the data. However, by

calibrating ρ → 1 and η = 1, I do not match any other moments obtained from the GSOEP

consumption microdata which I match in the case of GES preferences (see Section 4.2).

The left panels of Figure 4 plot changes as observed in the data, while the right panels isolate

changes resulting from the national increase in the relative supply of high skilled workers. I plot

data in a counterfactual scenario in which the national skill share increased as observed in the

data while fundamentals remain at their 2007 level. The upper panels in Figure 4 illustrate that

the increase in housing congestion is more pronounced in large regions. I find that the national

ECB Working Paper Series No 3018 26change in the relative supply of high skilled workers can explain 11% of the regional dispersion

in house price increases. With homothetic Cobb-Douglas preferences, a change in the national

skill share can explain only 7% of the regional dispersion in house price increases.

Figure 4: Decomposition of changes in sorting and house prices

Note: Unit of observation is 141 labor market areas as defined by Kosfeld and Werner (2012). The left panels show
changes in house prices and sorting as observed in the data from 2007 to 2017. The right panels show changes in
house prices and sorting in a counterfactual scenario in which the national skill share increases as observed in the
data while fundamentals remain at their 2007 level.

Why do house price differences increase more with GES preferences? With non-homothetic

preferences, geographic worker sorting intensifies since low skilled workers are hit harder by

increases in housing costs that are more pronounced in skill-intensive regions. The increase in

skill sorting, in turn, amplifies differences in house price increases. The slope parameters in the

9101112131405010015091011121314024681012Slope(SE): CD 0.93(0.11); GES 1.39(0.17)Slope(SE): 12.8(2.67)91011121314-200204060801001209101112131447.247.447.647.84848.248.448.6CDGESSlope(SE): CD 0(0); GES 0.14(0.02)Slope(SE): 4.53(1.88)ECB Working Paper Series No 3018 27lower panels quantify the change in skill sorting. The parameter in the data is 4.53, while in the

counterfactual scenario, I find a slope parameter of 0.14. These values indicate that the national

change in the relative supply of high skilled workers can explain 3% of the observed change in skill

sorting. With homothetic preferences, there is no change in worker sorting. When calculating

the average national increase in house prices in the data and in the counterfactual, I find that

roughly 10% of the national increase in housing costs can be explained by the model, while

roughly one third of this increase comes from the assumption of non-homothetic preferences.

In Section 5.3, I show that the results are robust to the calibration of the non-homotheticity

parameter η.

Next, I calculate welfare changes implied by the national increase in the relative supply of

high skilled workers. I measure the percentage change of pre-shock tradable good consumption

that would make the worker as bad off as after the shock. I equalize expected utility as defined

in equation (11) before and after the shock

uexp
k

(cid:0)(1 + ∆k)c 2007

ik

, h 2007
ik

, E 2007
ik

(cid:1) = uexp

k

(cid:16)

c sh
ik , h sh

ik , E sh
ik

(cid:17)

(31)

are consumption of tradables, housing and amenities in the observed

where c 2007

, E 2007
, h 2007
ik
ik
ik
ik , E sh
ik , h sh
equilibrium. c sh

ik are values in the counterfactual scenario in which the national share
of high skilled workers changes as in the data, while fundamentals remain as in 2007. Solving

numerically for ∆k, I find that the increase in the national share of high skilled workers has

decreased expected utility of high skilled workers by 0.8% and expected utility of low skilled

workers by 0.9%. Since low skilled workers spend a larger share of their income on housing, they

were hit harder by increases in housing congestion.

5.2 The size of inefficiencies

After having explored the importance of allowing for non-homothetic preferences when analyzing

sorting in spatial equilibrium models, I next estimate the optimal degree of sorting. To do

so, I solve the problem of a social planner who uses transfers between locations and worker

types which change the spatial distribution of economic activity. By changing the location

incentives of workers, they affect spatial sorting and the spatial concentration of population.

These reallocations in turn impact house prices, which feed back to location choices.

In the

following, I describe the spatial equilibrium resulting from this process.

ECB Working Paper Series No 3018 28I start by calibrating the welfare weights ωk such that both worker types experience the same

welfare gain as compared to the observed allocation. I measure welfare gains as the percentage

change in tradable good consumption that would make individuals as well off as in the optimal

allocation. Similarly to the measurement in equation (31), I obtain the welfare change ∆k from

numerically solving

(cid:16)

uexp
k

(1 + ∆k)c obs

ik , h obs

ik , E obs

ik

(cid:17)

(cid:16)

= uexp
k

c opti
ik

, h opti
ik

, E opti
ik

(cid:17)

(32)

ik , h obs

ik

and E obs

ik

are consumption and amenities in the observed allocation, while

where c obs
, h opti
c opti
ik
ik

and E opti

ik

are values in the optimal allocation.9

Figure 5 illustrates the transfer scheme that implements the optimal allocation. Both in

2007 and 2017, it is optimal to set incentives for high and low skilled workers to move toward

less populated areas. As formally shown in Section 3.2, the social planner takes into account re-

distribution and efficiency considerations when choosing a regional skill-specific transfer scheme.

In terms of efficiency, it is optimal to set larger transfers in rural areas because I include only

negative externalities in the form of housing congestion.

In terms of redistribution, optimal

transfers are larger in rural areas since wages are on average lower than in urban areas which

implies larger marginal utilities of tradable good consumption.

It is therefore intuitive that

the spatial concentration of population is smaller in the optimal allocation as compared to the

observed allocation.

As illustrated in the upper panels of Figure 6, it is further optimal to decrease skill sorting

by moving a larger share of high skilled workers toward less populated regions. Since high skilled

workers, by consuming more housing than low skilled workers, generate larger congestion forces,

it is optimal to reallocate them to a larger extent toward regions with less congested housing

markets. Furthermore, since the urban wage premium is larger for high skilled workers, spatial

differences in marginal utilities of tradable good consumption are larger for high skilled than

for low skilled workers.10 The smaller degree of agglomeration and skill sorting in the optimal

allocation imply that house prices are substantially less dispersed as reflected in a larger increase

9Note that calibrating welfare weights such that welfare gains are equal for both worker types implies welfare
weights to be differentially calibrated for 2007 and 2017. However, when estimating the social planner solution,
I find negligible changes in welfare weights: Calibrating the welfare weights to sum up to 1, I find the social
planner to choose a weight for high skilled workers of 0.627 in 2007 and of 0.639 in 2017. The results are robust
to calibrating the welfare weights in 2017 to the weights found for the calibration in 2007.

10The urban wage premium is plotted in Figure A.3.

ECB Working Paper Series No 3018 29Figure 5: Optimal transfers

Note: Unit of observation is 141 labor market areas as defined by Kosfeld and Werner (2012). The plots show
optimal transfers for low and high skilled workers relative to their wage. The planner’s weights are chosen such
that both types of workers experience the same welfare gains.

in house prices in rural as compared to urban areas (see the lower panels of Figure 6). Doubling

region size is associated with a roughly seven percentage point lower change in house prices.

How should transfers adjust to changes from 2007 to 2017? Figure 5 illustrates that in 2017,

optimal policies imply a greater degree of redistribution between regions compared to 2007. A

larger dispersion in transfers, in turn, implies larger incentives for individuals to move across

space. The upper panels in Figure 6 show that it is optimal to decrease sorting by 70% more

than in 2007: The slope parameter decreases from -0.41 in 2007 to -0.7 in 2017. In Figure A.3,

91011121314-10-505101520200791011121314-10-5051015202007Slope(SE): -2.67(0.4)Slope(SE): -2.78(0.56)91011121314-10-505101520201791011121314-10-5051015202017Slope(SE): -2.69(0.39)Slope(SE): -2.97(0.49)ECB Working Paper Series No 3018 30Figure 6: Optimal sorting and house prices

Note: Unit of observation is 141 labor market areas as defined by Kosfeld and Werner (2012). The plots show
the change in skill sorting and house prices between the observed equilibrium and the optimal allocation. The
planner’s weights are chosen such that both types of workers experience the same welfare gains.

it can be seen that the urban wage premium has decreased for both worker types, which implies

that the stronger decrease in sorting in 2017 is driven by efficiency considerations rather than

differences in marginal utilities of consumption. Since sorting has increased from 2007 to 2017,

congestion externalities generated by high skilled individuals have increased to a larger extent.

It is therefore optimal from a social planner perspective to decrease sorting more than in 2007.

As a consequence, moving from the observed to the optimal allocation implies a larger decrease

in house price differences between rural and urban areas than in 2007, as shown in the lower

91011121314-15-10-50510152025200791011121314-15-10-505101520252017Slope(SE): -0.41(0.57)Slope(SE): -0.7(0.45)91011121314-20-1001020304050200791011121314-20-10010203040502017Slope(SE): -6.74(1.26)Slope(SE): -6.82(1.11)ECB Working Paper Series No 3018 31panels of Figure 6.

Next, I allow for differential welfare gains for workers with and without a university degree

moving from the observed to the optimal allocation. Figure 7 plots the utility frontier obtained

from solving for the optimal allocation on a grid of welfare weights ωk. Welfare gains are

measured as given in Equation (32).

I choose the grid of welfare weights such that welfare

gains for both worker types are positive. I find that for any combination of welfare gains, the

benefits from moving from the observed to the optimal allocation are larger in 2017 than in

2007. When both worker types benefit equally, welfare gains amount to 0.43% of tradable good

consumption in 2007 and 0.45% in 2017. Since sorting and the spatial dispersion in house prices

have increased from 2007 to 2017, it is optimal to redistribute more in 2017, which implies larger

welfare gains.

Figure 7: Utility frontier between high and low skilled workers

Note: The plots show the change in welfare between the observed equilibrium and the optimal allocation. Welfare
changes are measured as the percentage change in tradable good consumption in the observed allocation that
would make the individual as well off as moving to the optimal allocation.

Both in 2007 and 2017, the maximum possible welfare gain of low skilled workers is substan-

tially larger than that of high skilled workers. While low skilled workers can gain more than

1.2% from moving from the observed to the optimal allocation in 2017, the welfare gain for high

skilled workers does not exceed 0.7%. Since low skilled workers spend a larger share of their

income on housing, they benefit more from reduced housing congestion.

00.20.40.600.20.40.60.811.22017(0.45,0.45)00.20.40.600.20.40.60.811.22007(0.43,0.43)ECB Working Paper Series No 3018 325.3 Robustness to the calibration of the preference parameters

In the baseline specification, I have focused on renting households which likely provide a lower

bound of non-homotheticity. To test the robustness of the results to the calibration of the non-

homotheticity preference parameter, I solve the model on a grid of η from one, which corresponds

to CES preferences, to 2.8, which is the average that existing studies find for homeowners.11

Figure 8 shows for different calibrations of the key preference parameter the slope coefficients

of the fitted lines in Figure 4, which capture how changes in house prices and sorting vary across

regions of different size. The results are robust even when calibrating the model with a high

non-homotheticity parameter corresponding to the value for homeowners. With η = 2.8, the

national increase in the supply of high skilled workers explains roughly 13% of the regional

dispersion in house price increases, while it still explains roughly 10% of the national increase

in house prices. In this specification, it accounts for 4% of the increase in spatial skill sorting.

Note that the plotted calibrations are not necessarily empirically relevant, since I keep the price

elasticity ρ constant, which might not be independent of the degree of non-homotheticity.

Figure 8: The effect of region size on model-implied changes in sorting and house prices

Note: Slope parameters from a linear regression of model-implied changes in house prices and geographic sorting
on region size with different calibrations of the preference parameter η. The dashed lines show 90% confidence
intervals. The model-implied changes refer to a counterfactual scenario in which the national skill share increases
as observed in the data while fundamentals remain at their 2007 level. The slope parameters in the data are 12.8
for house prices and 4.5 for skill sorting.

11I take the average of the studies by Finlay and Williams, forthcoming, Ioannides and Zabel, 2008, Larsen,

2014 and Zabel, 2004).

11.522.530.60.811.21.41.61.82BaselineCES11.522.5300.050.10.150.2BaselineCESGESCDECB Working Paper Series No 3018 33Next, I analyse the optimal allocation with different calibrations of the preference param-

eters. Figure 9 and Figure 10 show the results presented in Section 5.2 on a grid of the non-

homotheticity parameter η, in particular the slope parameters of the fitted lines in Figure 5 and

Figure 6. Optimal transfers as well as changes in geographic sorting and house prices are robust

to higher calibrations of η. The results that I obtain with Cobb Douglas preferences are broadly

in line with the findings of Fajgelbaum and Gaubert (2020) for the US. I further find that the

modeling of preferences matters: the social planner implements significantly larger transfers in

2017 when assuming GES preferences as compared to Cobb Douglas preferences.

Figure 9: Optimal transfers with different calibrations of the preference parameters

Note: Slope parameters from a linear regression of optimal transfers relative to wages (in %) on region size
with different calibrations of the preference parameter η. The dashed lines show 90% confidence intervals. The
planner’s weights are chosen such that both types of workers experience the same welfare gains.

11.522.53-5-4-3-2-102007BaselineCES11.522.53-5-4-3-2-102007BaselineCES11.522.53-5-4-3-2-102017BaselineCES11.522.53-5-4-3-2-102017BaselineCESGESCDECB Working Paper Series No 3018 34Figure 10: Optimal spatial equilibrium with different calibrations of the preference parameters

Note: Slope parameters from a linear regression of optimal changes in house prices and the share of high to low
skilled workers (both in %) on region size with different calibrations of the preference parameter η. The dashed
lines show 90% confidence intervals. The planner’s weights are chosen such that both types of workers experience
the same welfare gains.

Figure 9 shows that the social planner implements substantially larger transfers with CES

preferences than with Cobb Douglas preferences. The reason is that relocating workers is more

efficient with CES preferences. When the social planner sets incentives for workers to move

towards rural areas, housing costs decline in urban regions. With CES preferences, a reduction

in housing costs leads to a decline in the housing expenditure share, which in turn helps to

bring down the high costs of housing in dense expensive regions. This mechanism is reflected

in a signifiantly larger decline in the regional dispersion of house prices with CES preferences

11.522.53-2-1.5-1-0.500.512007BaselineCES11.522.53-2-1.5-1-0.500.512017BaselineCES11.522.53-10-9-8-7-6-5-4-3-22007BaselineCES11.522.53-10-9-8-7-6-5-4-3-22017BaselineCESGESCDECB Working Paper Series No 3018 35than with Cobb Douglas preferences (see Figure 10). The slope paramter with CES and GES

preferences is more than twice as high as the coefficient obtained when calibrating the model

with Cobb Douglas preferences.

The non-homotheticity of preferences, in contrast, makes the redistribution of workers less

efficient.

Implementing negative transfers in urban expensive cities leads to a decline in the

disposable income of individuals in these regions, which implies an increase in their housing ex-

penditure share and an increase in their housing demand. Non-homothetic preferences therefore

mitigate the decline in housing congestion achieved from setting negative transfers in expensive

regions that incentivise individuals to move towards less expensive areas. Note, however, that

optimal transfers take care of inefficiencies due to spillovers as well as distributional concerns,

as laid out in Section 3.2. Both determinants of the optimal transfer scheme are affected by the

assumptions on the utility function.

6 Conclusion

With non-homothetic preferences, an increase in the national supply of high skilled individuals

intensifies spatial sorting. The reason is that low skilled workers are hit harder by increases in

housing costs that are more pronounced in skill-intensive regions. The increase in sorting, in

turn, amplifies house price increases in large cities as compared to rural areas. I find that 10%

of the increase in average house prices in Germany from 2007 to 2017 can be explained by the

growth in the national share of high skilled workers. My model further explains 11% of the

observed regional differences in house price increases. Roughly one third of these effects is due

to the increase in spatial sorting.

The results suggest that low income individuals are hit harder by increases in housing costs

which is why they are avoiding increasingly expensive cities. I find that place-based policies that

aim at lowering the degree of geographic sorting have small welfare effects. However, geographic

sorting might have implications for welfare inequality beyond of the effects estimated in this

paper. Workers’ location choices might determine, besides housing market developments, also

their labour market conditions and access to education and health services, amplifying the

increases in inequality observed in many Western economies over the past decades.

ECB Working Paper Series No 3018 36References

Ahlfeldt, Gabriel, Stephan Heblich, and Tobias Seidel, “Micro-geographic property price

and rent indices,” Regional Science and Urban Economics, 2022, p. 103836.

Albouy, David, “The Unequal Geographic Burden of Federal Taxation,” Journal of Political

Economy, 2009, 117 (4), 635–667.

, Gabriel Ehrlich, and Yingyi Liu, “Housing demand, cost-of-living inequality, and the

affordability crisis,” National Bureau of Economic Research No. w22816, 2016.

Antoni, Manfred, Philipp vom Berge, Tobias Graf, Stephan Grießemer, Steffen

Kaimer, Markus K¨ohler, Claudia Lehnert, Martina Oertel, Alexandra Schmucker,

Stefan Seth, and Christan Seysen, “Weakly anonymous Version of the Sample of Inte-

grated Labour Market Biographies (SIAB) – Version 7517 v1,” 2019.

Baum-Snow, Nathaniel and Ronni Pavan, “Inequality and City Size,” The Review of

Economics and Statistics, 2013, 95 (5), 1535–1548.

Colas, Mark and Kevin Hutchinson, “Heterogeneous Workers and Federal Income Taxes in

a Spatial Equilibrium,” American Economic Journal: Economic Policy, 2021, 13 (2), 100–134.

Combes, Pierre-Philippe, Gilles Duranton, and Laurent Gobillon, “The Costs of Ag-

glomeration: House and Land Prices in French Cities,” Review of Economic Studies, 2019, 86

(4), 1556–1589.

Comin, Diego, Danial Lashkari, and Mart´ı Mestieri, “Structural Change With Long-Run

Income and Price Effects,” Econometrica, 2021, 89 (1), 311–374.

Couture, Victor, Cecile Gaubert, Jessie Handbury, and Erik Hurst, “Income growth

and the distributional effects of urban spatial sorting,” Review of Economic Studies, forth-

coming.

Dauth, Wolfgang and Johann Eppelsheimer, “Preparing the sample of integrated labour

market biographies (SIAB) for scientific analysis: a guide,” Journal for Labour Market Re-

search, 2021, 54 (1), 1–14.

ECB Working Paper Series No 3018 37Davis, Morris A. and Fran¸cois Ortalo-Magn´e, “Household expenditures, wages, rents,”

Review of Economic Dynamics, 2011, 14 (2), 248–261.

Diamond, Rebecca, “The Determinants and Welfare Implications of US Workers’ Diverging

Location Choices by Skill: 1980-2000,” American Economic Review, 2016, 106 (3), 479–524.

Eckert, Fabian, Sharat Ganapati, and Conor Walsh, “Skilled Scalable Services: The New

Urban Bias in Economic Growth,” Available at SSRN 3736487, 2020.

Eeckhout, Jan, Roberto Pinheiro, and Kurt Schmidheiny, “Spatial Sorting,” Journal

of Political Economy, 2014, 122 (3), 554–620.

Fajgelbaum, Pablo D. and Cecile Gaubert, “Optimal Spatial Policies, Geography, and

Sorting,” The Quarterly Journal of Economics, 2020, 135 (2), 959–1036.

, Eduardo Morales, Juan Carlos Su´arez Serrato, and Owen Zidar, “State Taxes and

Spatial Misallocation,” The Review of Economic Studies, 2018, 86 (1), 333–376.

Finlay, John and Trevor Williams, “Sorting and the Skill Premium: The Role of Nonho-

mothetic Housing Demand,” Journal of International Economics, forthcoming.

Ganong, Peter and Daniel Shoag, “Why has regional income convergence in the US de-

clined?,” Journal of Urban Economics, 2017, 102, 76–90.

Gaubert, Cecile, Patrick M. Kline, and Danny Yagan, “Place-Based Redistribution,”

American Economic Review, forthcoming.

Giannone, Elisa, “Skilled-biased technical change and regional convergence,” Journal of Po-

litical Economy, forthcoming.

Gyourko, Joseph, Christopher Mayer, and Todd Sinai, “Superstar Cities,” American

Economic Journal: Economic Policy, 2013, 5 (4), 167–99.

Ioannides, Yannis M and Jeffrey E Zabel, “Interactions, neighborhood selection and hous-

ing demand,” Journal of urban economics, 2008, 63 (1), 229–252.

Klick, Larissa and Sandra Schaffner, “FDZ data description: Regional real estate price

indices for Germany (RWI-GEO-REDX),” RWI Projektberichte, 2019.

ECB Working Paper Series No 3018 38Kosfeld, Reinhold and Alexander Werner,

“Deutsche Arbeitsmarktregionen-

Neuabgrenzung nach den Kreisgebietsreformen 2007-2011,” Raumforschung und Rau-

mordnung, 2012, 70 (1), 49–64.

Larsen, Erling Røed, “The Engel curve of owner-occupied housing consumption,” Journal of

Applied Economics, 2014, 17 (2), 325–352.

OECD, “Housing prices (indicator),” Accessed on 08 December 2022, 2022.

Ossa, Ralph, “A quantitative analysis of subsidy competition in the US,” Journal of Public

Economics, forthcoming.

Redding, Stephen J. and Esteban Rossi-Hansberg, “Quantitative Spatial Economics,”

Annual Review of Economics, 2017, 9 (1), 21–58.

Rossi-Hansberg, Esteban, Pierre-Daniel Sarte, and Felipe Schwartzman, “Local in-

dustrial policy and sectoral hubs,” AEA Papers and Proceedings, 2021, 111, 526–531.

Rubinton, Hannah, “The geography of business dynamism and skill biased technical change,”

Review of Economic Studies, forthcoming.

Schaeffer, Katherine, “Key facts about housing affordability in the U.S.,” Pew Research

Center, 2022.

Statistisches Bundesamt, “Preise - Verbraucherpreisindizes f¨ur Deutschland,” 2019.

Zabel, Jeffrey E, “The demand for housing services,” Journal of Housing Economics, 2004,

13 (1), 16–35.

ECB Working Paper Series No 3018 39A Quantification Appendix

A.1 Stylized Facts

To analyze how hard increases in housing costs have hit households across the income distribu-

tion, I plot housing expenditure shares in 1987 compared to housing expenditure shares in 2017.

Figure A1 provides suggestive evidence that the decrease in housing affordability is mainly a

problem for low income households: The housing expenditure share has increased significantly

more for low income than for high income households.

Figure A1: Housing expenditure shares

Source: GSOEP, own calculations. Note: West Germany only. Housing share is defined as housing expenditure

(including heating and electricity) divided by total net income. Number of observations: 2725 in 1987, 5188 in

2017

Next, I investigate whether differences in housing expenditure shares are driven by education

levels rather than income. Figure A2 plots housing expenditure shares for 50 evenly sized bins

defined by total income separately for households with a household head not holding a university

degree and households with a household head holding a university degree. It can be seen that for

given income levels, the two types of households do not spend different shares of their expenditure

on housing.

ECB Working Paper Series No 3018 40Figure A2: Housing expenditure shares by skill

Source: GSOEP, own calculations. Note: The plot shows mean housing expenditure shares for 50 evenly sized

bins defined by total income. Housing share is defined as housing expenditure divided by total net income. The

plot is based on household data from 2017. Skill refers to the skill of the household head. Number of observations:

3259 low skill, 1322 high skill

A.2 Calibration

I compare my non-homotheticity estimates to those assuming non-homothetic constant elasticity

of substitution (NHCES) preferences as applied in Finlay and Williams (forthcoming)

(Uik)

σ−1
σ = Ω

1

σ (hik)

σ−1

σ (Uik)

ϵ

σ + (cik)

σ−1
σ

(33)

where 0 < σ < 1, ϵ ≥ σ −1, and Ω > 0 are parameters.12 Cobb-Douglas preferences are obtained

by taking ϵ = 0 and σ → 1. The opposite case, a unit housing requirement, is obtained by taking

ϵ = −1 and σ → 0. Instead of equation (18), I get

sik = Ωp1−σ

i

(xik)ϵ(1 − sik)1+ ϵ

1+σ .

(34)

12Note that the preferences in equation (33) are equivalent to those in Comin et al. (2021)

1 = γ

1
σ

cik(uik)−1(cid:17)1− 1
(cid:16)

σ + (1 − γ)

1
σ

(cid:16)

hik(uik)−(1+ϵ′)(cid:17)1− 1

σ

with uik = Uikγ

1

σ−1 , ϵ′ = ϵ

1−σ and Ω = 1−γ

γ1+ϵ′ .

ECB Working Paper Series No 3018 41I estimate equation (34) by GMM. The results as given in Table A1 are similar to those in Finlay

and Williams (forthcoming) who find ϵ = −0.306 and σ = 0.522.

Table A1: Preference Estimates for NHCES preferences

ϵ

σ

Demographic controls

Year FE

N

No. of clusters

(1)
GMM

-0.299∗∗∗
(0.003)
0.882∗∗∗
(0.002)
✓

✓

8231

4333

(2)
GMM IV

-0.301∗∗∗
(0.003)
0.886∗∗∗
(0.002)
✓

✓

8231

4333

∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001
Note: Standard errors in parentheses, clustered at the household level.
Instrument is log family income. Demographic controls
Renters only.
include household size, number of earners as well as gender and age of the
household head.

ECB Working Paper Series No 3018 42A.3 The urban wage premium

Figure A3: The urban wage premium by skill

Source: SIAB, own calculations. Note: Unit of observation is 141 labor market areas as defined by Kosfeld and
Werner (2012).

910111213144.14.24.34.44.54.64.74.82007910111213144.64.74.84.955.15.25.35.42007Slope(SE): 0.059(0.01)Slope(SE): 0.063(0.014)910111213144.14.24.34.44.54.64.74.82017910111213144.64.74.84.955.15.25.35.42017Slope(SE): 0.052(0.009)Slope(SE): 0.061(0.012)ECB Working Paper Series No 3018 43B Model Appendix - Constrained efficient allocation

The program of the constrained social planner maximizing ex-ante utility with the constrained

that she does not know the realizations of the idiosyncratic shocks and therefore needs to respect

location choices can be written as

(cid:16)

ρ

ρ−1 log

ωkψ log(

(cid:88)

e

1− 1

γ(cik )

ρ +(1−γ)(hik )

1−

ψ

(cid:17)

η
ρ

+log Eik

)

L =

(cid:88)

k

(cid:88)

(cid:88)

(cid:16)

λik

−

k

i

i
(cid:16)

(cid:80)

i

(γ(cik)1− 1
(cid:16)

ρ + (1 − γ)(hik)1− η
ρ )

(cid:17) 1
ψ

ρ

ρ−1 Eik

(γ(cik)1− 1

ρ + (1 − γ)(hik)1− η
ρ )

(cid:17) 1
ψ

ρ

ρ−1 Eik

(cid:17)

Lk − Lik

− µY (cid:88)

(cid:88)

(Likcik − LikAik)

i

k

−

(cid:88)

µh
i

(cid:16) (cid:88)

i

k

Likhik − Hi

(cid:17)

where I omit notation for the non-negativity constraints and solve for interior solutions. Fa-

jgelbaum and Gaubert (2020) show that the social planner problem is concave when congestion

forces are at least as large as agglomeration forces. The fact that my model features conges-

tion forces only and that the generalization of non-homothetic preferences does not act as an

agglomeration force ensure that there is a unique solution to the maximization problem. The

first-order conditions are given by

[hk

i ] ηh

i Likhik =

ρ−η

ρ−1 (1 − γ)(hik)1− η

ρ

γ(cik)1− 1

ρ + (1 − γ)(hik)1− η

ρ

(cid:104)

ωkLik −

λikLik
ψ

(cid:88)

+

λjk

j

(cid:105)

LjkLik
ψ

[cik] ηY Likcik =

ρ

γ(cik)1− 1
ρ + (1 − γ)(hik)1− η

ρ

γ(cik)1− 1

(cid:104)
ωkLik −

λikLik
ψ

(cid:88)

+

λjk

j

(cid:105)

LjkLik
ψ

[Lik] − λik = µY (cik − Aik) − µh
i

(cid:16)

hik −

(cid:17)

∂Hi
∂Lik

ECB Working Paper Series No 3018 44I have 4xIxK+I+1 equations in Lik, cik, hik, λik, µh

i and µY :

(cid:136) Dividing [hik] through [cik] gives housing demand

hik =

(cid:16) µY
µh
i

ρ − η
ρ − 1

1 − γ
γ

(cid:17) ρ

η (cik)

1
η

(35)

(cid:136) Optimal consumption [cik]

µY
γ

(cid:16)

cik

γ + (1 − γ)

(cid:136) Optimal labor allocation [Lik]

ρ

(hik)1− η
(cik)1− 1

ρ

(cid:17)

= ωk −

λik
ψ

(cid:88)

+

λjk

j

Ljk
ψ

≡ xik

(36)

−λik = µY (cid:16)

Aik − cik

(cid:17)

+ µh
i

(cid:16) ∂Hi
∂Lik

(cid:17)

− hik

(37)

(cid:136) Mobility constraint

Lik =

(cid:16)

(cid:80)

i

(cid:136) Housing market clearing

(γ(cik)1− 1
(cid:16)

ρ + (1 − γ)(hik)1− η
ρ )

ρ

ρ−1 Eik

(cid:17) 1
ψ

(γ(cik)1− 1

ρ + (1 − γ)(hik)1− η
ρ )

ρ

ρ−1 Eik

Lk

(cid:17) 1
ψ

(38)

(cid:88)

k

Likhik − Hi = 0

(cid:136) Balanced government budget

(cid:88)

(cid:88)

i

k

(Likcik − LikAik) = 0

(39)

(40)

ECB Working Paper Series No 3018 45Decentralized vs planner

From [Lik], I get

where I define

and

µY cik + µh

i hik = wik + Πik + tik

wik ≡ µY Aik

Πik ≡ µh
i

1
η

(cik)
k Lik(cik)

(cid:80)

Hi

1
η

tik ≡ λik + µh
i

(cid:16) ∂Hi
∂Lik

− Πik

(cid:17)

.

(41)

(42)

(43)

Together with housing demand in equation (35), I have the first order conditions from the

decentralized allocation with µY = 1 and µh

i = pi.

B.1 Computation

To solve the system of equations numerically, I can substitute in the Lagrange multipliers and

housing to express the system of equations in terms of Lik and cik. Combining housing demand

in equation (10) with housing market clearing in equation (9), I get

hik =

1
η

(cik)
k Lik(cik)

(cid:80)

Hi.

1
η

Summing [cik] over i and normalizing population of each type to 1 ((cid:80)

i Lik = 1) yields

ωk =

µY
γ

(cid:88)

(cid:16)

i

γLikcik + (1 − γ)Lik(hik)1− η

ρ (cik)

(cid:17)

1
ρ

which, after summing over k, can be rearranged to

µY =

(cid:16)

(cid:80)
i

(cid:80)
k

γ
γLikcik + (1 − γ)Lik(hik)1− η

ρ (cik)

(cid:17) .

1
ρ

ECB Working Paper Series No 3018 46Summing housing demand over k and rearranging, I obtain

i = µY 1 − γ
µh
γ

ρ − η
ρ − 1

(cid:16) (cid:80)
(cid:80)

1
η
k c
ik
k hik

(cid:17) η
ρ .

The first-order condition with respect to [Lk

i ] gives an expression for λk
i :

−λik = ηY (Aik − cik) + µh
i

(cid:16) ∂Hi
∂Lik

(cid:17)

.

− hik

Thus, after substituting in housing hik and the Lagrange multipliers µY , µh

i and λik, I have a

system of 2xIxK equations in Lik and cik:

µY
γ

(cid:16)

cik

γ + (1 − γ)

ρ

(hik)1− η
(cik)1− 1

ρ

(cid:17)

= ωk −

λik
ψ

(cid:88)

+

λjk

j

Ljk
ψ

Lik =

(cid:16)

(cid:80)
i

γ(cik)1− 1
(cid:16)

ρ + (1 − γ)(hik)1− η

ρ

γ(cik)1− 1

ρ + (1 − γ)(hik)1− η

ρ

(cid:17) ρ

(cid:17) 1
ψ

ρ−1 Eik
(cid:17) ρ

ρ−1 Eik

Lk.

(cid:17) 1
ψ

ECB Working Paper Series No 3018 47Acknowledgements 

I thank Philip Jung, Lukas Buchheim, participants of several conferences and seminars and an anonymous referee for their helpful 
comments and suggestions. This paper uses confidential data from the Research Data Centre of the German Federal Employment 
Agency at the Institute for Employment Research. Details on possibilities of data processing can be found on the homepage of the 
Research Data Centre (fdz.iab.de). 

Nina Furbach 
Central Bank, Frankfurt am Main, Germany, email: nina.furbach@ecb.europa.eu 

© European Central Bank, 2025 

Postal address  60640 Frankfurt am Main, Germany 
Telephone 
Website 

+49 69 1344 0 
www.ecb.europa.eu 

All rights reserved. Any reproduction, publication and reprint in the form of a different publication, whether printed or produced 
electronically, in whole or in part, is permitted only with the explicit written authorisation of the ECB or the authors.  

This paper can be downloaded without charge from www.ecb.europa.eu, from the Social Science Research Network electronic library or 
from RePEc: Research Papers in Economics. Information on all of the papers published in the ECB Working Paper Series can be found 
on the ECB’s website. 

PDF 

ISBN 978-92-899-7113-3 

ISSN 1725-2806 

doi:10.2866/0803901 

QB-01-25-059-EN-N