Local Amenities and Income Disparities Within Cities

Local Amenities and Income Disparities Within Cities ^*

[PRELIMINARY AND INCOMPLETE, PLEASE DO NOT CIRCULATE]

Morgan Ubeda

September 13, 2019

Abstract

This paper contributes to the study of the causes of income segregation within cities. More precisely, it disentangles the relative importance of two factors in producing segregation: (a) heterogeneity in local productive amenities, mainly driven by agglomeration effects; (b) the role of local residential amenities, that make some places more attractive to residents than others, such as parks, access to shops caf´es and restaurants or public services. It does so by building, calibrating and simulating a quantitative spatial economics model of the inner workings of a city.

Keywords cities, amenities, income sorting, stratification

1 Introduction

A large literature on neighborhood effects and social interactions points to the concentration of poverty in urban areas as detrimental to the achievements of inhabitants of these enclaves (Durlauf, 2004, Gobillon et al., 2011, Ioannides, 2013, Topa and Zenou, 2015). These results show that residential income segregation, i.e.

spatial income disparities, have important consequences on individual trajectories. On the other hand, gentrification, i.e. a strong influx of upper middle-class people into a previously poor area and the resulting renewal, can often lead to the displacement of historical residents and be a source of social conflicts.

The goal of this article is to identify the causes of income segregation within cities. More precisely, it aims at disentangling the relative importance of two factors

*I want to thank my PhD advisors, Pierre-Philippe Combes and Sonia Paty, for their support and guidance. I also thank Sylvie Charlot for helping me with accessing the restricted data, and the people at CASD who made the data available. I am grateful to GATE and the Universit´e Lyon 2 for their finantial support.

†Universit´e de Lyon, Universit´e Lumi`ere Lyon 2, GATE Lyon Saint- ´Etienne, UMR 5824, 93 Chemin des Mouilles 69131 ´Ecully FRANCE.ubeda@gate.cnrs.fr

in producing segregation: (a) heterogeneity in local productive amenities, mainly driven by agglomeration effects, and (b) local residential amenities, that make some places more attractive to residents than others such as parks, access to shops caf´es and restaurants or public services. To do so, I extend the quantitative spatial economics model of the internals of a city of Ahlfeldt et al. (2015) by including workers productivity heterogeneity and Stone-Geary preferences. By allowing for non-homothetic preferences, the model is able to cope with direct sorting of rich households into expensive, high amenities municipalities. Workers productivity heterogeneity is the main driver of incomes heterogeneity (cf. Combes et al., 2008, who report that almost half of the variations in French wages can be explained by individual characteristics.), so that if the model is to capture income heterogeneity in French cities, skill sorting must be taken into account.

This paper is related to urban economics, which usually takes wages and employ- ment locations as given and explains spatial income disparities through differences of location choices for different incomes, depending on the local features of the urban landscape, that constitute residential amenities.

There is a burgeoning empirical literature (Lee and Lin, 2018, Koster and Rouwen- dal, 2017, Glaeser et al., 2018, Garcia-L ´opez et al., 2018) that shows the impact of amenities on gentrification, suburbanization and other transformations of the socio- economic profile of cities. Looking at US metropolitan areas between 1880 and 2010, Lee and Lin (2018) shows that natural amenities tend to anchor neighbourhoods to high income, and that high variability in natural amenities lead to higher persistence in the spatial distribution of wealth. In a similar vein, Garcia-L ´opez et al. (2018) shows that European cities with higher concentrations of historical amenities experienced weaker suburbanization when their transport network improved than their low- heritage counterparts, while Pampill ´on et al. (2017) finds that recent place based policies in Barcelona have been largely unsuccessful in attracting rich households in targeted neighborhoods, except for neighborhoods located in the central, historic districts of the city. Finally, in the US, Couture and Handbury (2017) finds that

”changing preferences of young college graduates for non-tradable service amenities like restaurants, bars, gyms, and personal services account for more than 50 percent of their growth near city centers” from 2000 to 2010. This indicates that these amenities, concentrated in the city centres, act as pull forces keeping the wealthy in the centre and overcoming the classical urban economics result (Fujita, 1989) about suburbanization of high income households due to commuting costs and housing preferences.

Theoretically, these facts would be consistent with a world in which households

have preferences that induce income sorting on the basis of local amenities. In the US,

Lee (2010) reports evidence that income elasticity of demand for local amenities (in

the form of access to local varieties) is greater than income elasticity of demand for

floor space, and shows that this is consistent with a theoretical model where the access

to local amenities are a luxury good. This result therefore offers evidence in favor of

sorting generated by differences in Engel curves for local goods and housing. Gaigne et al. (2017) study this idea in depth by building an income sorting model where locations have two attributes, distance to the CBD and proximity to local amenities.

This allows them to consider the classical urban economics result and this single- crossing induced sorting in an integrated framework. Based on this framework, an empirical exercise using historical amenities shows that distance to employment alone is a poor predictor of residents income, and that amenities distribution is indeed paramount in explaining the location of rich households.

From this result that distance to employment is a very weak predictor of the income of residents when compared to local amenities, one could hastily conclude that at the relatively small spatial scale of cities, the distribution of productivity and firms is of little importance, so that it is enough to study the location choice of household conditional on employment locations and focus on amenities to understand urban segregation and gentrification. Yet, cities with a high employment density tend to be more segregated than cities with more diffuse employment, and strongly polycentric cities that concentrate economic activity in peripheral sub-centers are less segregated than their monocentric counterparts (Garcia-L ´opez and Moreno-Monroy, 2018). This seems to indicate that employment location and the distribution of productive activi- ties are in fact important when explaining spatial income distributions. This apparent contradiction between individual decisions not depending that much on distance to employment and overall segregation depending strongly on urban spatial structure points to the importance of the interaction between the mechanisms underlying the location of economic activities and the ones underlying the location of households for understanding income segregation.

Economic geography on the other hand is mainly focused on firms productivity, and how it is determined by agglomeration economies, skill sorting or firms selection (e.g. Behrens et al., 2014, Combes et al., 2008, 2012). While most of this literature studies regional phenomena and between cities variations, Ahlfeldt et al. (2015) and Tsivanidis (2018) are the only authors who consider a quantitative spatial economics model of the inner workings of a city (Redding and Rossi-Hansberg, 2017). They use their respective models to study the impact of agglomeration forces and transportation infrastructures on urban economic activities.

Ahlfeldt et al. (2015) find very strong and localized agglomeration effects, both in production and in the formation of residential amenities, pointing to centrality as a key determinant of the intensity of economic activities. In particular, these endogenous agglomeration forces are necessary for the model to replicate key facts of the data.

Although they also report variations in production and residential fundamentals, the

model’s homotheticity implies that correlations between local fundamentals and the

income of local residents only comes from agglomeration effects and commuting costs,

whereby more attractive locations have higher populations, thus a bigger workforce,

which implies higher productivity and wages.

Tsivanidis (2018) on the other hand includes non-homothetic preferences, so that his model exhibits sorting on the basis of local amenities. Regarding agglomeration forces, the author specifies productive and residential externalities without spillovers, so that they only depend on own census tract employment and residential population, while a New Economic Geography-style production sector generate market-access driven agglomeration forces between census tracts. Still, he finds strong externalities in the production of local residential amenities and local Total Factor Productivity.

To sum up, the spatial structure of economic activities has a significant impact on urban income segregation, while distance to employment is a weak predictor of residential income when studied conditional on the structure of economic activities.

In order to quantify the respective role of productive infrastructures and residential amenities in producing residential segregation, it therefore seems necessary to take into account the mechanisms through which they interact and not sufficient to study the location of rich and poor households conditional on city structure.

Methodologically, this implies to merge the general structure of economic geography models augmented with commuting, and sorting-inducing preferences as used by the urban economics literature, adapting the models pioneered by Ahlfeldt et al. (2015) and Tsivanidis (2018).

The present paper incorporates Stone-Geary preferences, that are known to generate sorting through sub-unity income elasticities of housing demand (cf. Gaigne et al., 2017), into a quantitative spatial economics model `a la Ahlfeldt et al. (2015) that allows households to choose both their dwelling place and their workplace, and incorporates agglomeration forces through agglomeration economies on production and endogenous amenities on the residential side of the model. It therefore provides a framework where both links between income disparities and amenities have a fair chance of expressing themselves.

The remainder of the paper is organized as follows. Section 2 presents the model

and its properties. Section 3 describes the available data and discusses the estimation

of the models parameters and local amenities. Section

presents the results of

regressing variations of local fundamentals on local governments spending and fiscal

variables. Section

presents some simulation exercises where the effects of equalizing

productive and residential amenities on income disparities are computed. Finally,

section

concludes.

2 Theoretical model

This section outlines the model and discusses workers sorting. The general structure of the model is similar to Ahlfeldt et al. (2015), with the addition of workers heterogeneity and Stone and Geary preferences.

2.1 Workers behaviour

A city or urban area is composed of S municipalities, denoted by i or j, each endowed with some land L

. There are H workers in the city. Each worker has to choose in which municipality to live and in which municipality to work. Workers are perfectly mobile and receive their income from supplying labour to firms in their workplace.

Firms use labour and floor space to produce a final good costlessly traded with the rest of the world.

Workers differ in their skills and abilities, denoted l ∈

. Following the canonical literature on the estimation of agglomeration economies (e.g. Combes et al., 2008) workers heterogeneity is modeled in terms of efficient labour supply differences. More precisely, a worker with ability l is assumed to supply l units of efficient labour.

Therefore, given wages per efficient labour unit w

in municipality j, a worker with ability l simply receives an income of lw

. The distribution of skills in the city is fix and denoted F .

Conditional on her place of residence i = 1, . . . , S and her workplace j = 1, . . . , S, agent n with ability l receives a wage lw

that she spends on a quantity x

of the num´eraire good and a quantity f

of floor space. The num´eraire is not subject to transport costs, and is therefore distributed at a constant price (normalized to unity) everywhere in the city. Finally, commuting entails a monetary cost c per unit of distance, d

. The budget constraint of n is thus

lw

= Q

f

+ x

+ cd

, (1) where Q

is the residential floor space rent in municipality i.

Regarding workers preferences, I focus on the sorting of workers on the basis of local amenities, which precludes the use of homothetic preferences.

Following Gaigne et al. (2017) and Tsivanidis (2018) and departing from the Cobb-Douglas specification

1Empirical models of sorting in the vein ofBayer and Timmins(2007) typically assume that different categories of household are endowed with different preference parameters, thus avoiding the need to generate theoretical dependences between individual characteristics and consumption decisions. This atheoretical formulation is satisfactory and may be appropriate in a partial equilibrium or when agents differ by some exogenoustrait, such as race or religion (as in e.g.Ferreyra, 2007). But as pointed out by Fajgelbaum et al.(2011), general equilibrium approaches in which income levels are endogenously determined require explicit non-homotheticities.

in Ahlfeldt et al. (2015), I assume that workers have Stone and Geary preferences U

≡ z

B

x

1 −

!1−β

f

− f

!β

(2) where B

= B

T

exp( −

) are the local amenities perceived when living in i and working in j. They include B

the proper residential amenities in i, the niceness of the workplace j besides its offered wage, T

, and the utility cost of commuting between i and j, of the iceberg form exp( −

with d

the distance and

a parameter. The random variable z

captures idiosyncratic preferences of n for the commute ij, and

∈ (0, 1) and f ≥ 0 are parameters that govern workers preferences for housing.

Stone and Geary preferences have many interesting properties for the present model. First, whenever f > 0, the (indirect) marginal rate of substitution between floor space costs Q

and local amenities B

is increasing with income. This induces a relatively higher willingness to pay for high amenity levels for rich households than for poor households. It provides a parsimonious and theoretically sound foundation for income sorting on the basis of amenities. When f = 0 preferences are simple Cobb-Douglas.

Second, Stone and Gary preferences imply linear expenditures for housing, which causes the share of total income spent on housing to decrease with income. This decrease is consistent with data on the housing consumption of french households.

Indeed, our analysis of Expenditure Survey data in section 3.2 reports downward Engel curves ranging from 60% to 18% and shows that Stone and Geary preferences fit these curves well (cf Figure 1, section 3.2). This is in line with previous evidence using french data from Combes et al. (2018, p. 32, Table 6) who estimate that the share of housing in French households expenses is significantly decreasing in income.

Finally, the parameter f has a natural interpretation as an incompressible floor space consumption, below which workers cannot obtain a positive utility.

By maximizing (2) subject to the budget constraint (1), the individual demand for the private good (3), the individual demand for floor space (4), and the indirect utility of n when she chooses the commute ij (5) are respectively:

x

(l) = (1 −

− cd

− Q

f ), (3) f

(l) =

lw

− cd

Q

+ (1 −

f , (4)

V

(l) = z

B

(lw

− Q

f − cd

)Q

. (5) As in Ahlfeldt et al. (2015), I assume that households are characterized by their idiosyncratic preference shocks, which are are independent draws from a Fr´echet distribution, with scale parameters E

, shape parameter

> 1 and cumulative distribution function

F

(z

) ≡ exp( − E

z

). (6)

These preference shocks are observed by the workers, but not by the econometrician.

Integrating over these idiosyncratic shocks gives the probability to choose a commute.

From F the distribution of the preference shocks, we get G

the distribution of utilities for residents of i who work in j:

G

(v) = Pr[V

≤ v] = F





vQ

B

(lw

− Q

f − cd

)





. (7)

Then, the probability that a worker will choose commute ij is the probability to realize a higher utility in ij than in any other commute km. Formally, this can be written

π_ij

(l) ≡ Pr[V

(l) ≥ V

(l) ∀ km]. (8) Using G

and after integration

, one gets that

π_ij

(l) =

h

B ˜

(lw

− Q

f − cd

)Q

∑^Si=1∑^Sj=1

h

B ˜

(lw

− Q

f − cd

)Q

≡

^(l)

.

(9)

with ˜ B

= E

B

measuring adjusted amenities that take into account systematic deviations in preference shocks. When lw

< Q

f however, utility is null and so is the numerator of the choice probability

. Since

> 1, these choice probabilities are still smooth and differentiable for any w

∈

_{and any} Q

∈

, as long as there is at least one commute in the city in which households can realize a positive utility.

The total probability to reside in i for a worker with skills l,

(l) (respectively working in j,

(l)) is the sum over workplaces j (respectively dwelling places i) of the bilateral probabilities:

π_Ri

(l) =

(l)

φ(l)

, (10)

π_Mj

(l) =

(l)

φ(l)

. (11)

2SeeAhlfeldt et al.(2015) for details.

3Thereafter, I will implicitly assumelw_j >Q_if when writing down choice probabilities. If a worker gets too poor relative to floor space prices in the city, so that they cannot reach their incompressible flor space demand in any municipality, then it is simply assumed that they opt out from the city and leave.

Thus, residential choice probabilities conditional on employment location are

(l) =

(l)

∑^Si=1φ_ij

(l)

=

h

B ˜

(lw

− Q

f − cd

)Q

∑^Si=1

h

B ˜

(lw

− Q

f − cd

)Q

(12)

which correspond to traditional logit residential choice models in the tradition of e.g. McFadden (1978), Nechyba and Strauss (1998), Schmidheiny (2006) where the explanatory variables are taken in logs.

Similarly, workplace probabilities conditional on residential location are

π_ij|i

(l) =

(l)

∑jφ_ij

(l)

=

B ˜

(lw

− Q

f − cd

)

∑j

B ˜

(lw

− Q

f − cd

)

.

(13)

Now that I have derived the choice probabilities that describe the spatial distri- bution of workers conditional on wages, rents and amenities, I can discuss workers’

sorting.

2.2 The sorting of workers

When f > 0, workers exhibit direct sorting, in the sense that high ability workkers are willing to pay more in rents that poorer workers for an increase in residential amenities, and are willing to forego more wage per unit of labor for an increase in workplace niceness.

2.2.1 Residential location

To simplify the exposition, let ˜ B

= B

T

, so that the amenity of the commute can be decomposed in the product of a residential amenity and a “workplace niceness” that measures the attractiveness of j as a workplace, besside wages. For now, we also omit distances to focus on sorting based on amenities. Then the probability for a worker to choose to reside in i conditional on her working in j defined at equation (12) becomes

π_ij|j

(l) =

B

(lw

− Q

f )Q

∑^S_i=1h

B

(lw

− Q

f )Q

. (14)

4The model ofAhlfeldt et al.(2015) and the one in this paper, generalize discrete choice models of residential location `a laBayer et al.(2004),Bayer and Timmins(2007). While the strategy is similar, we no longer work conditional on workplace locations and incomes but allow for workplaces and wages to be endogenously determined.

From this residential location choice probability, we can define a rate of substitution between rents and amenities as the variation in rents in i necessary to keep the share of j workers living in i stable when the amenities in municipality i increase/decrease.

Formally, setting

constant gives dQ

dB

dπ_ij|j(l)=0

(l) = −

^(l)

∂_Qiπ_ij|j

(l) , (15) where

(l) is the partial derivative of residential choice probabilities with respect to rents,

∂_Qiπ_ij|j

(l) = −

"

β

Q

+ f w

l − Q

f

#

π_ij|i

(l)(1 −

(l)), (16) while

(l) is the partial derivative of residential choice probabilities with respect to local amenities,

∂_Biπ_ij|j

(l) =

B

(l)(1 −

(l)). (17) Therefore, the (indirect) rate of substitution between rents and amenities is

B

Q

dQ

dB

dπ_ij|j(l)=0

(l) = w

l − Q

f

βw_j

l + (1 −

f . (18) When f = 0, i.e. when preferences are Cobb-Douglas, this elasticity boils down to 1 /

choose a municipality constant when her rent increases by 1 /

increase in amenities. In fact in this case careful examination of

(l) shows that the conditional residential choice probabilities are independent of income and skills: all else equal, skilled and unskilled households make the same residential choices.

Whenever f > 0 however, this elasticity is strictly increasing in l. This means that when amenities in i increase, more productive and thus richer workers can accept a stronger increase in rents while keeping their probability to live in i constant. This is the basic direct sorting effect that is induced by non-homotheticities in housing demand, and that drives differences in residential location choices between rich and poor workers in the model.

Of course, in equilibrium households will choose simultaneously where to work

and where to live, and agglomeration economies will generate an indirect relationship

between amenities and income. Indeed, higher populations in high amenity munic-

ipalities and commuting costs will imply a higher workforce for the surrounding

workplaces. In turn, the bigger workforce will induce productivity gains through

agglomeration effects. In the end, this will result in higher wages for residents of the

high amenities, high population municipalities. Still, this agglomeration effect only

provides and indirect link between amenities and income, while the urban economics

literature has documented the importance of skills sorting as discussed above. The key

contribution of the paper is to be able to disentangle between these direct and indirect links between local features and local incomes.

2.2.2 Workplace

Looking at workplace location choice, conditional on residential location, choice probabilities in (13) become

π_ij|i

(l) =

T

(lw

− Q

f )

∑^Si=1

h

T

(lw

− Q

f )

, (19) so that one can define, in a similar fashion, the rate of substitution that captures the willingness to forego wages to benefit from higher amenities at the workplace:

dw

dT

dπ_ij|i(l)=0

(l) = −

^(l)

∂_wjπ_ij|i

(l) , (20) where

(l) is the partial derivative of workplace choice probability with respect to wages,

∂_wjπ_ij|i

(l) =

w

l − Q

f

(l)(1 −

(l)), (21) while

(l) is the partial derivative of workplace choice probabilities with respect to amenities at the workplace,

∂_Tiπ_ij|i

(l) =

T

(l)(1 −

(l)). (22) This gives a willingness to pay for workplace amenities equal to

T

w

dw

dT

dπ_ij|i(l)=0

(l) = Q

f

lw

− 1. (23)

Whenever the commute has a positive probability to be selected, this quantity is strictly between zero and one, and monotonically decreasing with skills. All workers are willing to forego some income for an increase in their workplace quality, but for poorer workers the percentage increase needed to compensate a reduction in wages tends to infinity. This elasticity is also decreasing in Q

, so that workers living in more expensive municipalities will be less willing to forego wages for workplace niceness.

2.3 Aggregation

From individual choice probabilities, aggregate quantities at the municipal level can

be computed as follows:

• Total residential population in i is given by summing residential probabilities over skill levels

H

= H

0 π_Ri

(l)d F (l), (24)

• Total income of residents in i is given by summing wages over workplaces and skill levels

W

= H ∑

j

w

0

lπ

(l)d F (l). (25)

• Total supply of effective labour in j is given by summing the supply from all skills l

H

= H

0

lπ

(l)d F (l), (26)

2.4 Production

Each municipality j produces a final good using floor space F

and its workforce, H

. Because goods are costlessly traded with the rest of the economy, their price is taken as exogenous and normalized.

The production function is taken to be Constant Returns To Scale (CRS) Cobb- Dougglas:

y

= A

( H

)

(F

)

, (27) where productive amenities are measured by A

, a total factor productivity (TFP) term that varies between municipalities. Firms pay a rent Q

per unit of floor space, pay a wage w

per unit of effective labour. Under these assumptions, the profit of firms in j is thus

A

(H

)

(F

)

− Q

F

− w

H

. (28) The first order conditions of profit maximization yield the following conditional demands:

H

=

y

w

F

= (1 −

y

Q

.

(29)

Rearranging these conditions gives the demand for commercial floor space, given workforce:

F

= 1 −

w

H

Q

. (30)

Moreover, plugging these two equations into the firms production function gives the zero profits condition that has to hold if profit maximizing firms operate in municipality j:

A

= Q

1 −

!1−α

w

α

. (31)

Armed with these relationships, we can now move on to the market for floor space and the exposition of the equilibrium conditions for the model.

2.5 The market for floor space

We assume that floor space is produced by a competitive development sector under CRS technology, using elastically supplied capital and land that is completely inelastically supplied. This implies an elastic supply of floor space, with a price elasticity inversely proportional to the share of land in the construction technology of the construction sector.

Formally, F

the total floor space in i, available for both commercial and residential use, is supplied by a competitive development sector. Following Combes et al. (2017) and Epple et al. (2010), developers use land L

with rental price R

and capital K

with rental price P (common to all municipalities) as inputs to a CRS Cobb-Douglas technology:

F

= C

K

L

. (32) Developers are assumed to treat land available for construction as given and fixed, L

= ¯ L

,

and maximize their profit by choosing how much capital to invest for land development in i. The first order conditions of this program give

(33) K

=

(1 −

P Q

_µ¹

L ¯

, which yields the following supply function:

(34) F

= ˜ L

Q

,

where ˜ L

≡ L ^¯

C

(

)

is a measure of land in i corrected by the constructibility in i and ˜

≡

µ

is the rent elasticity of floor space supply.

On the demand side, the demand of floor space from firms is given, as a function of workforce, by equation (30). For residents, total demand can be computed by aggregating the individual demand in (4) over skills and commute probabilities:

(35) F

=

W

− cD

Q

+ (1 −

,

where W

, D

and H

are total income of residents, total distance travelled by residents and residential populations respectively, as per equations (25) and (24). Therefore, the market clearing condition is given by equating floor space supply (34) to both these demands:

˜ (36)

L

Q

= 1 −

w

H

Q

+ (1 −

f H

+

W

− cD

Q

.

5Assuming that the supply of land is fixed does not seem to be a strong assumption in an urban context, where alternative uses of land such as agriculture are not a concern.

This market clearing condition, which relates commercial and residential uses of floor space in each municipalities, allows us to define the equilibrium conditions of the model.

2.6 Equlibrium

In this subsection I define an equilibrium of the model and discuss its existence.

Definition 2.1 Equilibrium.

An equilibrium of the model, conditional on parameter values {

f ,

} , exogenous amenities (b

), exogenous total factor produc- tivity shifters (a

), land use regulations (ξ

), developable land areas ( ˜ L

) and total city population H, is a set { (Q

), (w

), (A

), (B

) } of vectors of rents, wages, TFPs and endogenous amenities such that for all municipality i:

1. the profit maximization condition for firms (31) holds A

=

ξ_i

Q

1 −

1−α

w

α α

; 2. the market for floor space clears according to equation (36)

L ˜

Q

=

(1 −

Q

¹

α

H

+ (1 −

f H

+

W

Q

; 3. amenities are given by equation (??);

B

= b





∑

exp( −

)W





η

;

4. TFPs are given by equation (??).

A

= a

"

∑

exp( −

) H

L

.

Proposition 2.1 Equilibrium existence.

An equilibrium exists for this economy.

Proof. Repeating the equations that define the equilibrium, we have for all i L ˜

Q

=

(1 −

Q

¹

α

H

+ (1 −

f H

+

W

Q

, A

=

ξ_i

Q

1 −

1−α

w

α α

,

A

= a

"

∑

exp( −

) H

L

,

B

= b





∑

exp( −

)W





η

,

where W , H

and H

are defined in the text in Section 2.3. Rearranging gives the following set of equations:

Q

= F

(Q, w, A, B) =













(1 −

Q

¹_α

H

+ (1 −

f H

+

W

Q

L ˜













1˜ µ

,

A

= F

(Q, w, A, B) = a

"

∑

exp( −

) H

L

,

B

= F

(Q, w, A, B) = b

(

W

)





∑

exp( −

)W





η

,

so that an equlibrium of the model is a fixed point of F = (F

, F

, F

) where the w

s are treated as auxiliary variables which values are given by the zero-profit condition (31). Note that the amenities B

have been rescaled by

W

the total income generated in the urban area. This is of co consequence since the amenities only enter the model through the choice probabilities up to a multiplying constant.

For all i the wage w

is homogenous of degree one in (Q, A, B). Thus, commute choice probabilities

(l) are all homogenous of degree zero in (Q, A, B), and so are H

and H

. Consequently, F

, F

and F

are all homogeneous of degree zero in (Q, A, B).

Our goal now will be to define a rescaled continuous function that takes value in some compact convex subset of the 3S-dimentional Euclidian space, so that we can apply Brouwer’s fixed point theorem on that new function. Define x ≡ (Q, A, B) and F ¯ :

→

as

F(x) ¯ ≡ ¹

∑^Sk=1

F

(x) + F

(x) + F

(x) F(x).

Since the choice probabilities are continuous in x, and all the components of ¯ F are continuous in x and choice probabilities, ¯ F is continuous in x. Thus, ¯ F is a continuous function from a convex, closed and bounded (and thus compact) subset of

(namely the unit simplex) to itself: from Brouwer’s fixed point theorem ¯ F a fixed point.

Now we need to recover the equilibria of our economy from these fixed-points.

Denote x

a fixed point of ¯ F. Then by homogeneity of degree zero, x

= ¯ F(λx

)

for all positive scalar

=

F(x

) ≡

F

(x

) + F

(x

) + F

(x

), and writing x

≡

, we get that x

/

= ¯ F(x

). Thus, x

= F(x

) ×

F(x

) /

F(x

).

By homogeneity of F, x

is a fixed point of F and an equilibrium of the model.

Finally, from the equilibrium rents Q

and TFPs A

the zero profits condition (31)

gives equilibrium wages w

.

3 Data and calibration

The following section discusses the various sources of data and empirical specifications used to estimate the model parameters.

Table 1: Summary of the estimations and calibrations.

Quantity Description Method Source

β,f Preference parameters OLS housing expenditure Expenditure Survey e Preference shocks MLE residential location DADS, TT and DVF τ Commuting disutility MLE residential location DADS, TT and DVF w₁, . . . ,w_J Local wages Fixed-effects Mincer eq. DADS

B₁, . . . ,BJ Residential amenities MLE residential location DADS, TT and DVF T₁, . . . ,T_J Workplace niceness MLE workplace location DADS, TT and DVF L˜₁, . . . , ˜L_J Adjusted land Housing mket clearing DADS and DVF

A₁, . . . ,A_J TFP Housing mket clearing DADS and DVF

α Labor share in prod. Calibrated Aggregate figures

˜

µ Housing supply elasticity Calibrated Combes et al.(2017)

For the estimations and simulations, I consider the Paris and Lyon Urban Areas, which are the two biggest Urban Areas in France. For the delimitations of the Urban Areas, I use the definitions from the National Statistical Institute, based on commuting flows and housing continuity.

Table 1 lists the parameters and fundamentals of the model, and the source and methods used to estimate or calibrate them, where the data sources are as follows.

Workers Microdata (DADS):

The D´eclarations Automatiques de S´ecurit´e Sociale are an administrative, restricted-access dataset on the universe of French workers. Sent by employers to the social security administration on a yearly basis for the computation of social security participation. They contain the salaries, hours worked, occupation, workplace and dwelling place of every French employee. They are exhaustive on the universe of French private payroll employees, and available yearly from 1993 to 2015.

However, it is not a proper panel as individual IDs are changed every two years.

Household Expenditure survey:

The Enquˆete Budget des Familles is a representative survey of French households expenditures conduced by the National Statistical Institute. It contains household composition, housing expenditures, household income and housing surface area. For the estimations, I pool the 2006 and 2011 waves of the survey.

Housing transactions (DVF):

The Demande de Valeurs Fonci`eres is an open dataset,

exhaustive on the universe of housing transactions in France.

Travel Times (TT):

Travel times between municipalities are computed using the OpenRoute services API, computing travel times by road between municipal centroids.

Cf. Appendix for a more complete discussion.

3.1 Construction sector

For the construction sector technology, I calibrate 1 −

= .65, as estimated on French microdata by Combes et al. (2017). This gives a supply elasticity of ˜

= 1.86. This implied elasticity is in line with estimated long-term elasticities of housing supply in the literature. Saiz (2010) reports unweighted mean elasticities across US Metropolitan Areas (MSAs) of 2.5, while Harter-Dreiman (2004) reports ranges of elasticities of [1.8 − 3.2] for constrained housing markets and [2.6 − 4.3] for unconstrained cities, still in the US.

3.2 Housing consumption

I estimate the Stone Geary demand parameters

and f using the Family Expenditure Survey of the French Statistics Institute (Enquˆete Budget des Familles, INSEE). Multiply- ing the demand equation (4) by rents, and adding an error term, we get the linear expenditure function famously associated with Stone and Geary preferences:

E

=

w

+

Q

+

,

where E

denotes the expenditures of o on housing, Q

is the rent per square meters paid by worker o, and

is a residual. The structural parameters are identified as

=

and f =

/ (1 −

). This is our estimating equation.

For simplicity, I focus on private sector renters, for which Q

is readily available.

For each renting household, I have data on total rent paid for the dwelling unit, its surface area, the total income of the household and the number of workers in the household. For each individual member of each household, I also have access to employment status, wages and other incomes.

Since the goal is to calibrate the model for workers, I restrict the analysis to

individual workers. Every employed worker of every renting household is treated

as an observation. For the right hand side, E

is constructed by dividing the total

yearly rent for the dwelling by the number of workers in the household, and w

is the

yeary wage of the worker. I drop workers for which the share of housing expenditures

in wage, E

/ w

, does not belong to (0, 1). Pooling data from the 2006 and 2011 waves

of the survey, the sample contains 2027 workers. Column (1) of Table 2 reports the

results of this regression. The minimum housing consumption f is estimated at

24.23m

/ worker. Compared to the legal minumum housing size of 12m

, this estimate

may seem high. However, remember that I only consider workers but account for

the housing consumption of the whole household, therefore f captures the minimum

consumption of a worker taking into account the fact that, on average, workers also support non-working household members. The estimated

parameter is .0891.

Table 2: Estimates of the housing preference parameters from the Family Expenditure Survey, from OLS estimation of the housing expenditure function.

(1) (2) (3) (4)

fbar 24.21

29.86

24.57

24.01

(1.056) (2.022) (2.333) (1.786) beta 0.0892

0.0940

0.0857

0.124

(0.00689) (0.0102) (0.0182) (0.0176)

r2 0.860 0.878 0.851 0.867

Observations 2025 2301 2301 471

Standard errors (in parenthesis): robust for household level regressions, clustered by household for individual level equations.^c: significant at 10% level,^b: significant at 5% level,^a: significant at 1% level.

Equations (1) at the worker level, (2) (3) and (4) at the household level. Parameters areβ=θ₁and f=θ₂/(1−θ₁), where theθparameters are estimated via OLS and the standard errors ofβandf are obtained via the Delta method. In regressions (2), (3) and (4), housing sampling weights are used to weight observations.

Table 2 also reports some robustness checks to considering only employed workers and wages, neglecting other sources of income. Column (2) reports the regression, at the household level, of total housing expenditure on total household income and rent per squared meter. Households with negative or greater than one housing expenditure shares are dropped, but households with no employed member are kept.

The estimated incompressible housing consumption is 30m

/ household, while the marginal expenditure

is .094. Because the model is written in terms of individuals, and not households, it is necessary to control for household size for the incompressible consumption to make sense. Column (3) reports results from the same regression as column (2), but expenditures and income are divided by the number of adults in the household. The estimated incompressible consumption is now 24, 7m

/ adult, while the marginal expenditure is .086. Finally, column (4) duplicates column (3) but restricts the sample to households with no employed member. In this specification, I estimate an incompressible consumption of 24.1m

/ adult and a marginal expenditure of .124.

Overall, the results are stable across the specifications. In particular, the coefficients for wages per worker (column (1)) and total income per adults (column (3)) are almost identical, showing the stability of consumption patterns. Even when restricting the sample to households with purely non-wage incomes (column (4)), the estimated in- compressible consumption stays stable and only the marginal consumption parameter increases slightly.

Finally, Figure 1 shows actual versus predicted Engel curves for housing for each

of the regressions. First, we note that Engel curves are highly downward slopping,

Figure 1: Observed and predicted Engel curves for housing for each regression sample.

0.2.4.6.8

0 .2 .4 .6 .8 1

Workers

0.2.4.6.8

0 .2 .4 .6 .8 1

Households

0.2.4.6.8

0 .2 .4 .6 .8 1

HouseholdsPC

0.2.4.6.8

0 .2 .4 .6 .8 1

HouseholdsPCInact

Share of housing expenditure

Income percentile

95% CI Data

95% CI Model

Local polynomial regressions of the share of housing expenditure on income percentile. Each panel corresponds to an estimation column in Table2: Workerscorresponds to column (1),Householdsto column (2),Households PCto column (3), andHouseholdsPCInactto column (4). For each panel, they variable is housing expenditure, either observed or predicted, over income. Thexvariable is the position of the observation on the income distribution. See the text on the regressions for the expenditure and income measures corresponding to each panel.

validating the inadequaty of homothetic preferences, in line with Tsivanidis (2018), Combes et al. (2018). Second, the Stone and Geary specification of preferences is able to closely replicate the shape of the curve.

For the rest of the analysis I calibrate f = 24.6 and

= .085, from column (3) of Table 2.

3.3 Commuting costs, preference heterogeneity and residential amenities

The commuting costs parameter

and the preference heterogeneity parameter

are

estimated by maximum likelihood on the residential choice of French workers using

administrative data. The estimations use a 10% random sample of the exhaustive data

drawn at the Urban Area level.

Table 3: Estimates of the location choice parameters.

epsilon 16.76

(1.258)

tau 0.0636

(0.00370)

Log. lik. -1287704.3

Urban Areas 2

Alternatives 530

Workers 256476

MLE estimate of residential location choice conditional on workplace and wages. Robust standard errors in parentheses.

epsilon =eis the Fr´echet scale parameter of workers choices.

tau =eτis the parameter on commute times.

βand f calibrated to .085 and 24.6 respectively.

Monetary cost of commuting calibrated to 55 euros per minute.

Because

c and the B

s are hardly identified simultaneously from the location choice data, I calibrate the monetary costs per minute of travel time using gas prices, average car consumption, average commuting time and average commuting distance for French workers.

reports that average commuting travel time in minutes and distance in kilometers for French workers are equal, with an average distance of 24 Km (resp. 30,1Km) and an average travel time of 25 min (resp 31 min) for urban (resp.

suburban) workers. Therefore, I calibrate the cost per minute to be equal to the cost per kilometer. Assuming a consumption of 7.5 L/Km (3.5 miles per gallon) and a price of 1.5€/L, this gives an annual cost of 55€/min. I thus calibrate c = 55€/min.

Having calibrated c, the remaining parameters

and

can be estimated from conditional choice probabilities in equation (12), that rewrite

π_i|j

(l) = B

exp( −

)(lw

− f Q

− cd

)Q

∑i

B

exp( −

)(lw

− f Q

− cd

)Q

. (37) I estimate

and B

for all i by maximum likelihood. From the above conditional choice probability, the conditional probability of worker n with observed wage w

in her workplace j

to reside in i with rent Q

is

π_i|n

= B

exp( −

)(w

− f Q

− cd

)Q

∑i

B

exp( −

)(w

− f Q

− cd

)Q

. (38)