Carl Gaigné

Hans R A Koster

Fabien Moizeau

Jacques-François Thisse

N Baum- Snow

J Brinkman

M.S P.-P Combes

R J G M Delgado

Florax

M Fujita

J V Henderson

K Mat- Suyama

F Mayneries

M Miyake

Y Murata

H G Overman

P Picard

G Ponzetto

S Riou

F Robert- Nicoud

S L Ross

K Schmidheiny

H Takatsuka

P Ushchev

J N Van Ommeren

Who Lives Where in the City? Amenities, Commuting and Income Sorting

Keywords: cities, social strati…cation, income, amenities, commuting JEL classi…cation: R14, R23, R53, Z13

We develop a new model of a "featureful" city in which locations are di¤erentiated by two attributes, that is, the distance to employment centers and the accessibility to given amenities. The residential equilibrium involves the spatial separation of households sharing similar incomes. Under Stone-Geary preferences, amenities and commuting are subsumed into a location-quality index. Hence, the assignment of households to locations becomes one-dimensional. Since residential choices are driven by the location-quality index, the income mapping may be fully characterized. Using a rich micro-dataset on the Netherlands, we show that household income sorting is indeed driven by amenities and commuting times.

Introduction

Residential segregation by income is a prominent and growing feature of many metropolitan areas. While residential segregation generates negative and persistent e¤ects on economic development and social cohesion, it both re ‡ects and reinforces interpersonal and social inequalities.

For these reasons, we …nd it important to study the spatial sorting of heterogeneous consumers.

We also believe that such an e¤ort is warranted if policy actions against spatial segregation are taken to favor more social cohesion.

The canonical monocentric city model leads to a fairly extreme prediction: households are sorted by increasing income order as the distance to the central business district rises. This result contradicts empirical evidence [START_REF] Glaeser | Why do the poor live in cities?[END_REF][START_REF] Rosenthal | Change and persistence in the economic status of neighborhoods and cities[END_REF]. One missing key explanation is the existence of amenity endowments, such as historic buildings and architecture, scenic landscapes, river and sea proximity. That such amenities matter in residential choices has been well documented [START_REF] Brueckner | Why is Central Paris rich and Downtown Detroit poor? An amenity-based theory[END_REF][START_REF] Glaeser | Consumer city[END_REF] Koster and Rouwendal, 2017; Lee and Lin, 2018; [START_REF] Cuberes | Household location in English cities[END_REF]. Other facilities that appeal to people, such as restaurants and shops, are often located together with households [START_REF] Glaeser | Consumer city[END_REF][START_REF] Couture | Valuing the consumption bene…ts of urban density[END_REF][START_REF] Hoelzlein | Two-sided sorting and spatial inequality in cities[END_REF]. Such endogenous consumption amenities are ignored in this paper because they are likely to be the consequence and not the cause of residential choices, and thus the result of spatial income sorting. This paper proposes a new approach that combines housing consumption and heterogeneity in incomes and locations. To be precise, we propose a new approach in which cities are "featureful" in that locations are distinguished by two vertically di¤erentiated attributes, that is, commuting costs and the accessibility of given and dispersed amenities. While the demand for amenities has a tendency to rise with income, high-income commuters bear higher costs than low-income commuters. Our aim is then to study how local amenity endowments and commuting costs interact to determine the spatial sorting of income-heterogeneous households.

To check the relevance of our approach, we estimate the so-obtained income mapping by using rich datasets from the Netherlands.

The study of income sorting when locations are di¤erentiated by more than one attribute brings about new and di¢ cult issues. The determination of a residential equilibrium has the nature of a matching problem between landlords and households in which land at speci…c locations is di¤erentiated by two characteristics and households by one characteristic. Multidimensional matching problems typically assume quasi-linear preferences and indivisible choices [START_REF] Chiappori | Matching with transfers. The economics of love and marriage[END_REF]. By contrast, each household chooses a single location but her housing consumption is variable and income-dependent. The bid-rent approach o¤ers a way to square the circle, as this function gives the maximum price a household is willing to pay for one unit of housing at each location at a given utility level when housing consumption is variable. This function thus represents the Pareto frontier between a household and the owner of land at each location. The key point is then to …nd conditions for the existence of a unique positive assortative matching between households and landowners.

What are our main contributions? First, we model space as a topological network. Attempts made to move away from the standard one-dimensional framework of urban economics typically use the Euclidean plane (see, e.g. Lucas and [START_REF] Rossi-Hansberg | On the internal structure of cities[END_REF]. Such an approach neglects one the most fundamental features of a space-economy, that is, the shape and density of its transportation networks. Residences and jobs are thus distributed over the topological network.

Second, recent surveys, such as [START_REF] Duranton | Urban land use[END_REF] and Behrens and Robert-Nicoud (2015), express some skepticism about the ability of the bid-rent approach to deal with a continuum of heterogeneous households to be mapped on a continuum of locations. We …rst demonstrate that the bid-rent approach is still applicable to such settings. We then show that the interaction between amenities and commuting gives rise to turning points in the spatial income distribution. In other words, provided that amenities are distributed unevenly across space, a greater geographical distance between households does not necessarily implies a wider income gap.

Third, when commuting costs vary with income, characterizing the equilibrium income mapping requires non-homothetic preferences. Indeed, empirical evidence suggests that housing expenditure shares decline with income [START_REF] Albouy | Housing demand, costs-of-living inequality, and the a¤ordability crisis[END_REF]. Furthermore, we show that in our setting homothetic preferences such as the Cobb-Douglas or the CES generate a continuum of equilibria. In this case, the standard approach is to select an equilibrium via stability.

However, doing this is especially hard here because we work with a continuum of heterogeneous households and a continuum of locations (de Palma et al., 2019). We therefore prefer to use non-homothetic preferences. More speci…cally, we consider Stone-Geary preferences -but we could equally assume a generalized CES -and establish the following two main results. In the …rst place, there exists a unique spatial equilibrium. Households with di¤erent incomes always choose locations with di¤erent characteristics, but households sharing the same income may live in separated neighborhoods. Second, we …nd a location-quality index (LQI) whose behavior re ‡ects the properties of the equilibrium income mapping. This index is built from the primitives of the model, as its value at any particular location is pinned down by the amenity endowments and households'commuting behavior. It is worth stressing that these results are not speci…c to Stone-Geary preferences. They hold true for other non-homothetic preferences; what changes is the functional form of the LQI.

The upshot is that households'bliss point is the global maximizer of the LQI over space, thus implying that the a-uent ‡ock to this location because they propose the highest bid. As one moves away from this location along all admissible directions, households are sorted by decreasing incomes until a local minimizer of the LQI is reached where poorest households are located. Around this minimizer, household income starts rising. Hence, households are more exposed to other social groups when the number of turning points of the LQI rises.

Fourth, our model is ‡exible enough to determine analytically the equilibrium outcome when incomes and the location-quality index are Fréchet-distributed. The so-obtained mapping between the location-quality index and income can be straightforwardly estimated using real-life data.

We then aim to empirically investigate whether the two main forces in our theoretical model -amenities and commuting costs -also drive income sorting in practice. We estimate a reduced-form version of the income mapping for the Netherlands. The choice of the Netherlands is motivated by (i) the availability of large disaggregated data sets and (ii) the fact that the public services that underpin social cohesion (e.g., education and health) are centrally …nanced and/or administered [START_REF] Ritzen | School …nance and school choice in the Netherlands[END_REF]. First, we have access to rich microdata for more than 10 million households covering the years 2010 to 2015 on incomes, residential and job locations at the household level, commuting time, as well as land values and amenities at each location. Second, the central provision of education implies that the quality of schools, which is a major determinant of residential choices in many U.S. metropolitan areas, is much less of an issue in the Netherlands.

Dutch cities, which were established long ago, are known to o¤er a high quality of life, which is at least partly due to the presence of amenity endowments. Despite being a small country, the Netherlands hosts no less than 10 UNESCO world heritage sites, which is more than London and Paris together, while it hosts 61; 908 listed buildings, which is more than three times the number of listed buildings in Greater London. With a population density of 407:4 pop/sq km, the Netherlands is almost as dense as the San Francisco Bay area whose area is similar to that of the Netherlands. By focusing on the whole country rather than a subset of metropolitan areas, we are able to capture relationships that are deployed within the entire Dutch urban system, from the large cities to the small villages.

It is well known that measuring the quality of amenities is a hard task. In this paper, we use a proxy suggested by [START_REF] Ahlfeldt | Urbanity[END_REF] and [START_REF] Saiz | Crowdsourcing architectural beauty: Online photo frequency predicts building aesthetic ratings[END_REF]: the number of outside geocoded pictures taken by residents at a certain location. One key advantage of this LQI is that it lets consumers choose the aesthetic quality of buildings and locations they like best by "voting with their clicks" [START_REF] Carlino | Beautiful city: Leisure amenities and urban growth[END_REF]. This allows us to move beyond the approach of de…ning amenities implicitly, as in [START_REF] Ahlfeldt | The economics of density: Evidence from the Berlin Wall[END_REF] and [START_REF] Albouy | What are cities worth? Land rents, local productivity, and the total value of amenities[END_REF]. We show the robustness of our results by using a completely di¤erent proxy for amenities based on a hedonic price approach developed by [START_REF] Lee | Natural amenities, neighborhood dynamics, and persistence in the spatial distribution of income[END_REF], who use variation in housing prices.

Since there is no proxy that perfectly captures the full amenity potential at a certain location, amenities are measured with error. Commuting time is also likely to be endogenous due to correlation with unobservable household characteristics and agglomeration economies -the latter being more prevalent in dense areas where commutes are shorter. We address the endogenous nature of amenities and commuting time in our econometric analysis in several ways, e.g., by obtaining Oster's (2019) bias-adjusted estimates and by constructing historic instruments.

Since the strategy of using instruments based on historic data raises several issues, we devote considerable attention to the validity of such an identi…cation strategy.

In a …rst step, we show that picture density is strongly correlated to the presence of amenity endowments, such as the density of historic listed buildings and the presence of water bodies.

We then show that both amenities and commuting costs are important in determining the spatial income mapping. More speci…cally, we …nd that doubling the amenity level attracts households whose incomes are 2:6% higher, while doubling commuting time leads to households whose incomes are 15% lower. Hence, the impact of commuting time seems to be somewhat stronger than the impact of amenities. We also …nd a strong impact on land prices: doubling amenities leads to an increase in land prices of 13% , which is sizable. We further estimate the key parameters of the model, which enables us to predict the LQI and income mapping at each location. Because of historic amenities that are prevalent in Dutch cities, we show that higher incomes are clustered in city centers. Thus, our results unambiguously suggest that both amenities and commuting costs are important in determining the spatial income distribution. Nevertheless, commuting costs seem to be a more important driver of income sorting than amenities. Hence, our results support the emphasis put on commuting costs in standard models of urban economics. However, our results also show that the featureless model of urban economics is far too restrictive to explain the city's social structure. Yet, focusing on amenities only is unwarranted because commuting costs are too important to be disregarded. A relevant theory of the space-economy must account explicitly for both amenities and commuting costs.

Related literature. Suggesting the complexity of the issue, only a handful of papers in urban economics have studied the social strati…cation of cities. [START_REF] Beckmann | On the distribution of urban rent and residential density[END_REF] was the …rst attempt to take into account a continuum of heterogeneous households in the monocentric city. Unfortunately, the assignment approach used by Beckmann was ‡awed [START_REF] Montesano | A restatement of Beckmann's model of distribution of urban rent and residential density[END_REF]. [START_REF] Hartwick | Comparative statics of a residential economy with several classes[END_REF] and [START_REF] Fujita | Urban Economic Theory[END_REF] proposed a rigorous analysis of the residential pattern for a …nite number of income classes of otherwise identical households in a featureless monocentric city.

Our work is related to two di¤erent strands of literature. First, there is a huge number of contributions on the sorting of heterogeneous households across jurisdictions that supply local public goods (see, e.g., [START_REF] Epple | Fiscal decentralization[END_REF], for a survey). This literature relies on a particular geography in which jurisdictions are small enough for commuting within jurisdictions to be costless while jurisdictions are far enough for households to consume only public goods provided in their own jurisdictions. The attempt made by [START_REF] Epple | Tiebout: Integrating location-speci…c amenities into multi-community equilibrium models[END_REF] shows how di¢ cult it is to introduce commuting in those models. Interestingly, Gravel and Oddou (2014) have obtained a necessary condition for perfect income sorting across jurisdictions: the Marshallian demand for the public good must be a monotone function of the price index of housing and a tradable consumption good. To obtain this necessary condition, Gravel and Oddou assume that households's preferences are separable between public goods and private goods, very much like us, and homothetic between consumption goods and housing, unlike us. We do not use their preferences because they are akin to homothetic preferences and would generate a continuum of equilibria in our set-up. What is more, unlike these authors, we focus on the trade-o¤ between amenities, commuting and land in a setting that involves a continuum of locations. Even though perfect sorting may emerge in our set-up, there is no contradiction between results because the two analytical frameworks are too di¤erent for the results to be directly comparable.

Second, quantitative spatial models are also related to us because they involve amenities and commuting [START_REF] Ahlfeldt | The economics of density: Evidence from the Berlin Wall[END_REF][START_REF] Diamond | The determinants and welfare implications of US workers diverging location choices by skill: 1980-2000[END_REF][START_REF] Monte | Commuting, migration, and local employment elasticities[END_REF][START_REF] Tsivanidis | The aggregate and distributional e¤ects of urban transit infrastructure: Evidence from Bogotá's TransMilenio[END_REF][START_REF] Couture | Income growth and the distributional e¤ects of urban spatial sorting[END_REF]. Their main purpose is to calibrate the impact of various shocks or policies, such as new transportation infrastructure, on the spatial equilibrium. Moreover, these models do not provide properties of the spatial equilibrium because doing so is often hard under a …nite location set. We di¤er from this literature in that our primary aim is to characterize the residential equilibrium in a setting with a continuum of locations distributed along a transportation network. Stated di¤erently, our model is in the tradition of urban economics à la Alonso-Fujita.

The remainder of the paper is organized as follows. We provide a detailed description of our model in Section 2. In Section 3, we show how the bid-rent function may be used to determine the social strati…cation of the city. In Section 4, we study the properties of the residential pattern for preferences that generate a LQI. Since the equilibrium is undetermined under homothetic preferences, we illustrate our results for Stone-Geary preferences. Section 5 presents data, the identi…cation strategy and reduced-form empirical results on the impact of amenities and commuting time on income sorting. We further provide a simple strategy to obtain structural parameters of the income mapping in order to be able to undertake counterfactual analyses. In Section 6, we summarize our main results and discuss the role of endogenous amenities.

The model 2.1 The spatial economy

The economy involves a unit mass of income-heterogeneous households. A household's gross income is given by ! units of the numéraire, with ! 2 [!; !] and 0 < ! !. The income c.d.f. F (!) and the density f (!) are continuous over [!; !]. We are agnostic about the reasons that explain inequality in earnings. 1 The economy involves two normal consumption goods: (i) land h, which is a proxy for housing, and (ii) a homogeneous …nal consumption good q. Shipping the …nal good within the economy is costless, so that its price is the same across locations. This good is used as the numéraire.

Our spatial setting di¤ers from the standard linear (sometimes planar) space used in urban economics. The map formed by streets, roads, highways, and railway junctions (in a city, region or country) is modeled by means of a topological network. A topological arc, denoted a , is the image in R 2 of the interval [0; 1] by a continuous one-to-one mapping [START_REF] Kuratowski | Topology[END_REF]).

The endpoints of arcs are called vertices. Clearly, an arc contains a continuum of locations.

A topological network N = [ T =1 a is de…ned as the union of a …nite number of topological arcs that have a …nite length. Here, N is the location set. Households can travel on arcs in both directions. One unit of land is available at each location x 2 N and the network is supposed to be large enough for the total supply of land to accommodate the whole population of households.

The set N is connected, that is, for any two points x 1 and x 2 belonging to N there is at least one concatenation of arcs and subarcs that links these two points. The distance d(x 1 ; x 2 ) between x 1 and x 2 is given by the length of the shortest path that connects these locations.

Clearly, d( ) is a metric de…ned on N . We assume that vertices are not colinear, so that (N; d) is not a one-dimensional metric space. An example of transportation networks similar to ours can be found in [START_REF] Allen | Trade and the topography of the spatial economy[END_REF]. In what follows, we assume that functions are almost everywhere (a.e.) di¤erentiable along each arc of the network N ; at the vertices of N , di¤erentiability is de…ned by the directional derivatives along the corresponding arcs.

Consumption

Household preferences are separable between amenities b, on the one hand, and the two private goods (q; h), on the other:

U (q; h; b) = b u(q; h); ( 1
)
where q is the numéraire, h is land consumption, u is a well-behaved utility function. The function b(x) > 0 captures the common attributes, such as those discussed in the introduction, that make location x a more or less attractive place to live;2 this function is continuously di¤erentiable function over each arc of N and may display ‡at portions (b x = 0). 3 Preferences

(1) are used in a wide range of models where goods are vertically di¤erentiated because they capture in a simple way the fundamental idea that high-income households are willing to pay more than low-income households to enjoy a given amenity. In other words, the single-crossing condition between incomes and amenities holds.

There is a …nite number of employment locations i 2 N with i = 1; :::; n. Each household works in a single employment location and earns the same gross income ! for all i. However, her income net of commuting costs changes with her residential location and workplace. Since there is ample evidence that the opportunity cost of time varies signi…cantly when households are heterogeneous in incomes (Small, 2012; Koster and Koster, 2015), we assume that commuting costs are income-dependent. More speci…cally, we model commuting as an income loss, not as a utility loss. [START_REF] Ahlfeldt | The economics of density: Evidence from the Berlin Wall[END_REF] and others show that the two approaches are equivalent when preferences are homothetic, but this ceases to hold under non-homothetic preferences.

We also assume that commuting costs lead to a household's net income given by !t i (x) where the function t i (x) decreases with the distance d(x; i) between the residential location x and the employment location i. Since the function t i is speci…c to the employment location, it may capture production fundamentals of this location. Hence, the commuting cost increases with income and distance as it amounts to ! (1 t i (x)). We now show that the above speci…cation displays enough ‡exibility to encompass the two types of income-dependent commuting costs used in urban economics. Let i (x) be the commuting time between locations x and i. We have (i) an iceberg cost when the e¤ective labor supply of a worker living at x decreases with the length of the commute, so that t i (x) = [ i (x)] where > 0 and (ii) an additive cost when the labor supply is perfectly inelastic while the net income is given by t i (x) = 1 i (x). The household's budget constraint is as follows:

!t(x) = q + R(x)h; (2)
where R(x) is the land rent at x. In line with the literature, we assume that the land rent is paid to absentee landowners [START_REF] Fujita | Urban Economic Theory[END_REF].

The function t(x) used in (2) may be obtained from t 1 (x); :::; t n (x) according to di¤erent speci…cations. First, when a household is free to choose where to work, she chooses the em-ployment centers that maximize her net income, that is, t(x) max i t i (x). 4 In this case, all households located at x work in the same employment center. Second, a household knows her employment location j before choosing her residential location x, so that t(x) = t j (x) holds for all x 2 N . In this event, households sharing the same residential location x may work in di¤erent working places but the model does not explain how households choose their workplace.

In the spirit of quantitative spatial models, one may want households to have idiosyncratic reasons for residing and working in di¤erent location pairs (x; i). In what follows, we are agnostic about how t(x) is obtained. We assume only that t(x) is a.e.

di¤erentiable over N and such that t is never constant on a subarc of N because commuting costs increase with the distance d(x; i).

Maximizing the utility U of a !-household residing at x and working at e i with respect to q and h subject to (2) yields the numéraire demand

q (x; !) q(R(x); !t(x)) = !t(x) R(x)h (R(x); !t(x))
and the housing demand h (x; !) h(R(x); !t(x)), which is the unique solution to the equation:

u h [!t(x) R(x)h ; h ] R(x)u q [!t(x) R(x)h ; h ] = 0: (3)

The residential equilibrium

Given an amenity function b(x), a given mass of heterogeneous households choose where to live, how much land and how much of the composite good to consume. For the moment, we assume that households who reside at x have the same income !. In other words, there is an income mapping !(x) from N to [!; !] that speci…es which !-households are located at x. In other words, households endowed with di¤erent incomes reside in di¤erent locations. We will show in Section 4 that this property holds true for di¤erent non-homothetic preferences.

By contrast, the !-households may be distributed over several locations, so that !(x) need not be monotone. Let s(x; !) 2 [0; 1] is the share of the !-households who reside at x. The land market clearing condition holds if the mapping !(x) satis…es the following condition (Montesano, 1972):

s(x; !(x))f (!(x))h (x; !(x)) jd!j = jdxj ; (4)
where the left-hand and right-hand derivatives of the income mapping are not the same over a …nite set of locations. This condition states that the amount of land available between any

x and x + dx > x along a topological arc while the subarc occupied by the households whose income varies from ! to ! + d! are the same. Since d! need not be positive, the land market clearing condition is expressed in absolute value.

Last, the network is supposed to be large enough to accommodate the whole population of households:

Z N s(x; !(x))f (!(x))h (x; !(x))dx < Z N dx; (5)
where the second integral stands for the total supply of land.

A residential equilibrium is such that no household has an incentive to move, all households sharing the same income have the same maximum utility level, and the land market clears.

Formally, a residential equilibrium is de…ned by the following vector:

(! (x); s (x; ! (x)); R (x); h (x; s (x)); q (x; s (x)))

with x 2 N , such that b(x) u[q (x; ! (x)); h (x; ! (x))] b(y) u[q (y; ! (x)); h (y; ! (x))] for all x; y 2 N holds under the budget constraint (2) and the land market clearing condition (4).

If the above inequality is strict for all y 6 = x, then all ! (x)-households are located at

x (i.e. s (x; ! (x)) = 1). Otherwise, there exist at least two locations x 1 and x 2 such that the ! (x)-households are indi¤erent between the locations x 1 and x 2 . In this case, we have 0 < s ( ; s (x)) < 1 at x 1 and x 2 , while the sum of the shares is equal to 1. Hence, there is spatial splitting of identical households.

The residential equilibrium with amenities

Since u is strictly increasing in q, the equation u(q; h) = U=b(x) has a single solution Q(h; U=b(x)), which describes the consumption of the numéraire when the utility level is U=b(x) and the land consumption h. The bid rent (x; !; U ) of a !-household is the highest amount it is willing to pay for one unit of land at x when its utility level is equal to U . Formally, the bid-rent function is de…ned as follows:

(x; !; U ) max q;h !t(x) q h s.t. b(x) u(q; h) = U = max h !t(x) Q(h; U=b(x)) h : (6)
Since each household treats the utility level parametrically, di¤erentiating [START_REF] Altonji | An evaluation of instrumental variables strategies for estimating e¤ects of Catholic schooling[END_REF] with respect to h yields the equation:

Q(h; U=b(x)) hQ h (h; U=b(x)) !t(x) = 0 (7)
whose solution, denoted H(!t(x); U=b(x)), is the quantity of land consumed at x when the household pays its bid rent ( ); it is called the bid-max lot size [START_REF] Fujita | Urban Economic Theory[END_REF]. Hence, the budget constraint implies that the bid-rent function may be rewritten as follows:

(x; !; U ) !t(x) Q(!t(x); U=b(x)) H(!t(x); U=b(x)) : (8)
This shows that a household's bid rent at x depends on both b(x) and t(x) while its housing consumption H also varies with these two attributes of x.

Since we consider a mapping that assigns a speci…c income to a location, the bid-rent function has a unique maximizer ! (x) at each location x. Consequently, ! (x) solves the …rst-order condition: @ (x; !; U (!))) @! = 0;

while the second-order condition @ 2 =@! 2 < 0 holds at ! (x).

Totally di¤erentiating [START_REF] Behrens | Working from home: Too much of a good thing?[END_REF] with respect to x yields:

d! dx = @ 2 (x; !; U (!)) @! 2 1 @ 2 (x; !; U (!))
@!@x a.e., [START_REF] Boppart | Structural change and the Kaldor facts in a growth model with relative price e¤ects and non-Gorman preferences[END_REF] which implies that x! (x; !; U (!)) and d! (x)=dx have a.e. the same sign.

Set

B(x) b x (x) b(x) ; T (x) t x (x) t(x) ;
and let E U;! (!=U ) U ! ; the income elasticity of indirect utility,

E H;! (!=H) (H ! + H U U ! )
the income elasticity of housing demand/bid-max lot size while E uq;! (!=u q ) (@u q =@!) is the income elasticity of the marginal utility of the numéraire consumption. 5We may now characterize the slope of the equilibrium income mapping.

Proposition 1. The equilibrium mapping ! (x) is a.e. strictly increasing (decreasing) at

x if x! (x; !; U (!)) = t H (x; !) with (x; !) 1 E H;! + E uq;! E U;! B(x) (1 E H;! )T (x) (11)
is strictly positive (negative) at this location.

Proof. The proof is given in Appendix A.1.

The expression [START_REF] Brueckner | Gentri…cation and neighbourhood cycles: Will America's future downtowns be rich?[END_REF] shows that for any given utility function u the interaction between the amenity and commuting cost functions determines the social strati…cation of the city through the behavior of the function . It also shows that the sign of , whence the slope of the spatial income distribution, changes at any solution ! (x) to the equation

(x; !) = 0 if x (x; !) 6 = 0.
Hence, [START_REF] Brueckner | Gentri…cation and neighbourhood cycles: Will America's future downtowns be rich?[END_REF] emphasizes that preferences play a key role in explaining the pattern of social strati…cation.

To illustrate, consider the benchmark case of a monocentric, linear and featureless city [0; L] where b(x) = 1 and t x (x) < 0 for all x since there is a single employment center whose location is at x = 0. We know from [START_REF] Fujita | Urban Economic Theory[END_REF] that household locations are determined by ranking the bid-rent slopes with respect to income. It then follows from [START_REF] Brueckner | Gentri…cation and neighbourhood cycles: Will America's future downtowns be rich?[END_REF] that the sign of (x; !) depends on whether the income elasticity E H;! of the bid-max lot size is positive because land is a normal good; it is not constant and may be smaller or larger than 1 (Wheaton, 1977).

Since the empirical evidence shows that the expenditure share allocated to housing declines as income rises, the income elasticity of housing is smaller than 1 [START_REF] Glaeser | Why do the poor live in cities?[END_REF], Albouy et al., 2016). Therefore, when income increases, the slope of the bid-rent function gets steeper.

A longer commute shifts the utility of a high-income household downward more than that of a low-income household because the former has a higher opportunity cost of time than the latter. However, this e¤ect is not o¤set by the higher housing consumption because the income elasticity of housing is smaller than 1. By implication, at the residential equilibrium, households are sorted by decreasing order of income as the distance to the CBD increases.

Consider now the case of a featureful monocentric and linear city (B(x) 6 = 0). Owing to the existence of amenities, even when the bid-rent functions are downward sloping, the equation (x; !) = 0 may have several solutions. In this case, there is imperfect sorting, that is, greater income di¤erences are not mapped into more spatial separation. The following three cases may arise. The …rst two cases are straightforward as we assume that (x; !) keeps the same sign for all x, even though the elasticities E H;! , E uq;! and E U;! , as well as B (x) and T (x), could change sign.

(i) Assume that (x; !) > 0 for all x, which implies that ! (x) increases. As ! rises, the bid-rent curve becomes ‡atter. Since the bid rent of a high-income household is always ‡atter than that of a low-income household, individuals are sorted out by increasing income. In other words, the richer is the household, the closer to the city limit is its location. Consumers are willing to pay more to reside at a distant location because the corresponding hike in amenity consumption is su¢ cient to compensate them for their longer commute [START_REF] Fujita | Urban Economic Theory[END_REF].

(ii) If (x; !) < 0 for all x, which implies that ! (x) decreases. Indeed, as the income ! rises, the bid-rent curve becomes steeper. Therefore, the bid-rent curves associated with any two di¤erent incomes intersect once and, for each !, there exists a unique x(!) such that s(x(!); !) = 1. In this case, x = 0 is the most-preferred city location. To put it di¤erently, the utility loss incurred by an increase in distance to the workplace is exacerbated by a drop in the consumption of central amenities [START_REF] Brueckner | Why is Central Paris rich and Downtown Detroit poor? An amenity-based theory[END_REF].

(iii) The most interesting case arises when (x; !) changes its sign over [0; L], so that the bid-rent functions may intersect several times. In this case, there is imperfect sorting: household income rises over some range of sites and falls over others. We develop this argument in more detail in the next section.

The expression [START_REF] Brueckner | Gentri…cation and neighbourhood cycles: Will America's future downtowns be rich?[END_REF] highlights the fact that the impact of the amenity and commuting cost functions on the sign of (x; !) depends on the elasticities E H;! , E uq;! and E U;! . When the utility u(q; h) is speci…ed, the condition ( 11) may be used to determine how households are distributed according to the behavior of B(x) and T (x) by calculating those elasticities. Even when these elasticities are constant, B(x) and T (x) are monotone, imperfect income sorting may arise because the signs of the coe¢ cients of B(x) and T (x) in ( 11) are undetermined. 6 4 The social structure of cities

Non-homothetic preferences

In this section, we characterize the equilibrium mapping ! (x) and the equilibrium land rent R (x) for preferences u(q; h) that reduce the dimensionality of the matching problem.

It is common place to work with homothetic preferences, as they include the CES, Cobb-Douglas and translog. Here, there are at least two reasons to rule out such preferences. First, using homothetic preferences implies that the elasticity of the housing demand is constant in price and income, which contradicts its variability across heterogeneous consumers [START_REF] Albouy | What are cities worth? Land rents, local productivity, and the total value of amenities[END_REF]. Second, when the utility u is homogeneous linear, we show in Appendix A.2 that 6 We recognize that our modelling of commuting cost has an impact on the resulting sorting patterns. To illustrate, consider an income-independent commuting cost c(x) 0. When B(x) = 0; the expression [START_REF] Brueckner | Gentri…cation and neighbourhood cycles: Will America's future downtowns be rich?[END_REF] must be replaced by

x! = [t x (1 E H;! ) + (c x =!) E H;! ] =H (see Appendix A.1
). Thus, income-dependent and income-independent commuting costs have opposite impacts on sorting when E H;! < 1. Our empirical study supports the evidence that income is decreasing in distance from the nearest employment center, thus suggesting that the e¤ect of c (x) is o¤set by the impact of t (x).

E U;! = E H;! = 1 and E uq;! = 0, so that (11) yields (x; !) = 0 for all x. In other words, there exists a continuum of residential equilibria under homothetic preferences.

For these reasons, we consider Stone-Geary preferences which obviate the multiplicity of equilibria. 7 They also account for the plausible assumption of a minimum lot size h > 0 8 :

u(q; h) = q 1 (h h) ; (12)
where 0 < < 1. Maximizing ( 12) with respect to q and h subject to (2) leads to the following demands for the numéraire:

q (x; !) = (1 )[!t(x) R(x)h] (13)
and for housing:

h (x; !) = (1 )h + !t(x) R(x) : (14)
Thus, the demand for housing at any location x increases less than proportionally with income [START_REF] Albouy | Housing demand, costs-of-living inequality, and the a¤ordability crisis[END_REF]. Note that the housing demand is perfectly inelastic when = 0, which corresponds to the assumption of a …xed lot size.

We show in Appendix A.3 that (x; !) is given by

(x; !) = (x) B(x) (1 )T (x); (15)
which is independent of !. Set

(x) b(x)[t(x)] 1 : (16)
Di¤erentiating [START_REF] Couture | Valuing the consumption bene…ts of urban density[END_REF] shows that x (x) and (x) have the same sign. Hence, (x) changes sign at any extrema of (x).

The function (x) subsumes the amount of time devoted to work and the amenity level at

x into a single scalar, which has the nature of a location-quality index (LQI), which is de…ned over the interval [ ; ] where ( ) is the minimum (maximum) value of (x) over N . Note that the LQI depends on location x but not on income ! (x). The higher , the stronger the preference for housing. In other words, as the intensity of preference for housing increases, commuting matters less than the accessibility of amenities. Since the commuting cost function is never constant over a subarc of the network N , (x) is never ‡at on a positive measure interval. 7 Schmidheiny (2006) also uses Stone-Geary preferences to show income segregation in a set-up where a …nite number of jurisdictions/locations are di¤erentiated by their sole provision of local public goods to households heterogeneous in taste and income. 8 A minimum housing threshold will be especially relevant if working from home becomes widespread (Behrens

et al., 2021).

The spatial income distribution

Our objective is now to determine the equilibrium income mapping ! (x). Since housing consumption is chosen optimally at each x, what makes a site attractive to households is both its amenity level and the corresponding working time. The next proposition shows that incomes are distributed over the location set N according to the values of the LQI.

For a given value > 0, we de…ne the function g( ) by the number of locations x 2 N such that (x) = . Since (x) is a.e. continuously di¤erentiable, g(x) is a step function. Let G( ) be the c.d.f. whose density is equal to g( ) except at the values of where g( ) is not continuous. Hence, G( ) is piecewise linear and strictly increasing over [ ; ].

We are now equipped to characterize the equilibrium income mapping.

Proposition 2. Assume that preferences are given by [START_REF] Brueckner | Why is Central Paris rich and Downtown Detroit poor? An amenity-based theory[END_REF]. Then, (i) each location hosts at most one household type; (ii) there exists a unique residential equilibrium; (iii) the equilibrium income mapping ! (x) and the LQI (x) vary together with x; and (iv) the equilibrium income mapping is given by

! (x) = F 1 [G( (x))]; (17)
Proof. The proof is given in Appendix A.4.

Note that (17) implies:

d! dx = g( (x)) f (! (x)) d dx a.e. ( 18
)
Since the bid-rent function allows describing the Pareto frontier between households and locations, it is su¢ cient to show that the equilibrium utility level U (!) satis…es the Spence-Mirrlees condition with respect to ! and , and hence the bid-rent function is supermodular.

This in turn implies the existence of a unique positive assortative matching between incomes and the values of the LQI [START_REF] Chiappori | Matching with transfers. The economics of love and marriage[END_REF]. Furthermore, the expression [START_REF] Couture | Income growth and the distributional e¤ects of urban spatial sorting[END_REF] tells us that the equilibrium income mapping is determined by the income distribution F ( ) and the distribution function G( ) associated with the LQI function ( ).

Equally important, Proposition 2 shows that it is su¢ cient to study how (x) varies with

x to determine how households are distributed over N . In other words, the initial two-to-one matching is reduced to a one-to-one matching ! ( ), which is unique and strictly increasing in . Consequently, the highest income households locate where the LQI reaches its global maximum. As (x) starts decreasing with x, the income level of the corresponding residents also decreases. The lowest income households reside at a global minimizer of the LQI. Around this location, the income level rises together with . As a result, a wider income gap is no longer matched with a greater distance between two households because the function (x) is in general not monotonic in x. In other words, we have:

@ @x dU d! ? 0:
To illustrate, consider a linear city where (x) is v-shaped over [0; L] with two maximizers at x = 0 and x 2 < L and two minimizers at x 1 < x 2 and x = L. Proposition 2 implies that households are sorted by decreasing income over [0; x 1 ). As x increases, (x) increases and households are sorted by increasing income until x 2 , while households'income decreases over (x 2 ; L]. Since (y) = (z) for some y 2 (0; x 1 ) and z 2 (x 1 ; x 2 ), households who reside at y and z have the same income ! . In other words, households sharing the same income do not live in the same place. Although these consumers enjoy the same equilibrium utility level and consume the same lot size (see Appendix 4(i)), they do not have the same net income, do not pay the same land rent and do not consume the same amount of the numéraire because amenities and commuting costs are not the same at y and z.

It remains to determine the equilibrium values of the shares s(x; ! (x)). If x 1 6 = x 2 ::: 6 = x n exist such that (x 1 ) = (x j ) for j = 2; :::; n, it follows from Proposition 2 that ! (x 1 ) = ! (x j ) for j = 2; :::; n. Under Stone-Geary preferences, the bid-max lot size H( ) depends on b(x) and t(x) only through the LQI, (x). As H[!t(x); U=b(x)] H(!; (x); U ), it must be that

H(!; (x 1 ); U ) = H(!; (x j ); U ) for j = 2; :::; n. Since f (! (x 1 )) = f (! (x j
)) for j = 2; :::; n, using ( 4) and ( 18) leads to

s(x 1 ; ! (x 1 )) d (x 1 ) dx 1 = s(x j ; ! (x j )) d (x j )
dx j ; j = 2; :::; n:

Since n X i=1 s(x i ; ! (x i )) = 1;
the unique solution to these n linear equations is given by

s(x i ; ! (x i )) = 1 n P k=1 1 jd (x k )=dx k j 1 jd (x i ) =dx i j ; i = 1; :::; n: (19)
In words, households who have the same income ! (x 1 ) are split across locations whose LQI is equal to (x 1 ) according to location-speci…c shares s(x i ; ! (x i )) because the LQI gradients need not be the same across locations x 1 ; :::; x n (see Appendix A.4.(iv) for a detailed proof).

Note also that in a linear city where (x) is monotone increasing or decreasing over [0; L] would lead to s = 1.

It is worth stressing that Proposition 2 holds true for non-homothetic preferences other than Stone-Geary. In other words, our results hold for a substantial variety of preferences. First, consider the generalized CES given by u(q; h) = q 1 + h 2 with 0 < i < 1 and 1 6 = 2 [START_REF] Mukerji | A generalized S.M.A.C. function with constant ratios of elasticity of substitution[END_REF]. The elasticity of substitution between land and the numéraire is variable and equal to 1=(1

1 1 2 2
) where i is the expenditure share on good i = 1; 2. When 1 > 2 , i.e., the composite good matters more than land, it can be shown that the above preferences generate the index (x)

[b(x)] 1= 1 t(x). This index plays the same role as [START_REF] Couture | Valuing the consumption bene…ts of urban density[END_REF] and may be substituted in [START_REF] Couture | Income growth and the distributional e¤ects of urban spatial sorting[END_REF] to obtain the corresponding income mapping. Second, under

Price-Independent-Generalized-Linear (PIGL) preferences [START_REF] Boppart | Structural change and the Kaldor facts in a growth model with relative price e¤ects and non-Gorman preferences[END_REF], the indirect utility function has the following form:

V (!; x) = b(x) f[!t(x)] = [R(x)]
= g where 0 < 1. When = 0; we have homothetic preferences while V becomes an indirect (logtransformed) Cobb-Douglas utility function when = = 0. For any given location x, we have

V ! = (x)! 1 , so that (x) b(x)t(x)
is the new LQI. Last, if preferences are quasi-linear preferences u(q; h) = v(h)+q where v is strictly increasing and concave, we have E H;! = E uq;! = 0, so that the bracketed term of ( 11) yields (x) = b(x)t(x).

Note also that our setting displays enough ‡exibility to account for a special, but empirically relevant, type of heterogeneity across households. To be precise, assume that a household is characterized by her type (!; ) where may represent, e.g., households are heterogeneous in their attitudes toward leisure. In this case, a household's income is given by ! where is an inverse measure of her demand for leisure. This is accomplished by considering the following indirect utility:

V (x; !; ) / (x) ! R(x) t(x) h R(x) t(x) :
In this case, it can be shown that (15) becomes

(x; ! ) = h h(x; ! ) [A(x) (1 )T (x)] :
Repeating the argument of Proposition 2 implies that the exists a unique PAM between ! and . In other words, the sorting of households is based on the trade-o¤ between income and leisure, so that households who live at x may enjoy di¤erent incomes and di¤erent demands for leisure that satisfy the condition ! = ! 0 0 . This approach can be extended to deal with di¤erences in family structure.

Land rent

The land market clearing condition (4) may be rewritten as follows:

h (x; !(x)) = 1 s(x; !(x))f (!(x)) 1 jd!=dxj ; a.e.
From [START_REF] Cuberes | Household location in English cities[END_REF], using d =dx (x) (x) ; we get:

h (x; ! (x)) = 1 s(x; ! (x))g( (x)) 1 j (x)j (x)
; a.e.

Hence, a higher LQI gradient at x is associated with a higher mass of households located at x, which in turn leads to a lower housing consumption.

When land at x is occupied by households, [START_REF] Chiappori | Matching with transfers. The economics of love and marriage[END_REF] then implies that the equilibrium land rent at x must solve the following equation:

(1 )h + ! (x)t(x) R(x) = (x) with (x) 1 s(x; ! (x))g( (x)) 1 j (x)j (x)
; a.e. so that

R (x) = 1 (x) (1 )h ! (x)t(x); a.e. ( 20
)
which increases with the minimum lot size h.

To illustrate, consider the case where x 1 6 = x 2 ::: 6 = x n exist such that (x 1 ) = (x j ) for j = 2; :::; n. According to Proposition 2, households with income ! (x 1 ) are indi¤erent between locations x 1 6 = x 2 ::: 6 = x n . However, the land rents at these locations di¤er. Indeed, it follows from (20

) that R (x 1 ) t (x 1 ) = R (x 2 ) t (x 2 ) = ::: = R (x n ) t (x n ) ;
which implies that variations of the rent R (x) are exactly balanced by variations of commuting t(x) within the location set fx 1 ; x 2 ; :::; x n g. Stated di¤erently, the land rent at locations hosting households with a higher disposable income t! is also higher.

We show in Appendix A.5 that the land gradient is given by

R x (x) = ! (x)t(x) (x) 1 (x) (1 )h B(x) T (x) ; a.e. ( 21
)
Since the coe¢ cient of B(x) in ( 21) is positive and smaller than 1, the above expression implies that the land rent gradient is negative when B(x) T (x) < 0 for all x. As x rises, the decreasing land rent compensates the households ! (x) located at x for bearing higher commuting costs and being farther away from places endowed with more amenities. For example, in the standard monocentric city model in which B(x) = 0 and T (x) > 0 the land rent gradient is always negative. When B(x) T (x) > 0 over some interval [x 1 ; x 2 ], the land rent gradient can be positive or negative according to the value of the coe¢ cient of B(x) in [START_REF] Diamond | The determinants and welfare implications of US workers diverging location choices by skill: 1980-2000[END_REF]. Since household income increases over [x 1 ; x 2 ], commuting costs also increase over this interval. Therefore, the land rent is a priori neither monotonic nor the mirror image of the spatial income distribution.

In short, the interaction between amenities, commuting and income sorting may give rise to a wealth of land rent pro…les, which generally di¤er from that of the LQI.

From theory to data

To derive testable predictions about the e¤ects of amenities and commuting costs on the spatial income distribution, we have to determine the explicit form of the income mapping

! (x) = F 1 [G( (x))].
For this, we must specify the distributions F and G. Earning distributions are skewed to the right so that the Fréchet distribution is a good candidate to capture this. More speci…cally, we assume that incomes are drawn from a Fréchet distribution with the shape parameter > 0 and the scale parameter > 0: , where > 0 are the shape parameter and > 0 the scale parameter. By playing with , it is possible to describe a fairly wide array of amenity and commuting cost distributions. Furthermore, this speci…cation allows us to determine an analytical expression for ! (x) that is amenable to the data. In Section 5.3 we provide empirical evidence that (x) is well approximated by a Fréchet distribution.

F (!) = exp [ (!= ) ] over [0; 1)
Using [START_REF] Couture | Income growth and the distributional e¤ects of urban spatial sorting[END_REF], the mapping ! (x) can be retrieved from the condition:

Z 1 ! f (y)dy = 1 exp( (! = ) ) = Z 1 g( )d = 1 exp( ( = ) );
which is the counterpart in the -space of the land market clearing condition (4). Solving the above equation yields the equilibrium income mapping:

! (x) = (x) = : (22)

Empirical analysis

In this section, we …rst describe our data in order to provide reduced-form evidence in support of the model's qualitative predictions that amenities and commuting time a¤ect the income distribution in and between cities; and hence that exogenous amenities are an important determinant of sorting of households. The analysis is complemented by a wide range of controls that provide evidence against alternative possible explanations. We close this section by using the estimated parameters to calculate the LQI and income mapping ! (x) over space.

Datasets

We have gained access to various nationwide non-public microdata from Statistics Netherlands between 2010 and 2015. Unlike the United States or the United Kingdom, the Netherlands does not undertake censuses to register their population, but the register is constantly updated when people move or when there are changes in the household composition. The …rst dataset we use is the Sociaal Statistisch Bestand (SSB), which provides basic information on demographic characteristics, such as age, country of birth, marital status, and gender. We only keep people that could be part of the working population, that is, those between 18 and 65 years and aggregate these data to the household level. Importantly, the SSB data enable us to determine where households reside, up to the postcode level. Hence, space is discrete in the plane.

The data on yearly income of households are obtained from the Integraal Huishoudens Inkomen panel dataset. These data are based on the tax register, which provides information on taxable income, tax paid, as well as payments to or bene…ts from property rents or dividends.

The income data also provide information on whether households are homeowners or renters.

More than 65% of the rental sector applies to public housing. Public housing is rent controlled and there are often long waiting lists for public housing. So, households are not entirely free to choose their utility-maximizing location. Therefore, we will focus on owner-occupied housing, which means that we keep about 70% of the data. 9To estimate the commuting time for each household, we use the tax register information, which provides information on individual jobs and the number of hours worked in each …rm for each year. Using data on location information on each establishment from ABR Regio and network travel time from SpinLab, we calculate for each household the average commuting time.

More information on how we calculate the commuting time between locations is provided in

Appendix B.1.
A location is given by a neighborhood (which is a postcode 4-digit location) as de…ned by Statistics Netherlands. There are 4; 033 neighborhoods; the median size of a neighborhood is only 528ha while the average population is about 4; 000. Information on land values and lot sizes is not directly available. As is common practice, we infer them from data on housing transactions, provided by the Dutch Association of Real Estate Agents (NVM ). The methodology used to calculate land values and lot sizes is described in Appendix B.2. The NVM data contain information on the large majority (about 75%) of owner-occupied house transactions between 2000 and 2015. We know the transaction price, the lot size, inside ‡oor space size (both in m 2 ), the exact address, and a wide range of housing attributes such as house type, number of rooms, construction year, garden, state of maintenance, and whether a house is equipped with central heating. 10 We also know whether the house is a listed building.

We are interested in the impact of amenities on income sorting and land prices. We proxy the amenity level by the picture density in a neighborhood. More speci…cally, we gather data from Eric Fisher's Geotagger's World Atlas, which contain all geocoded pictures on the website

Flickr. The idea is that locations with an abundant supply of aesthetic amenities will have a high picture density. We show in Appendix B.3 that there is a strong positive correlation between picture density and historic amenities or geographical variables, such as access to open water or open space. There are, however, several issues with using geocoded pictures as a proxy for amenities. 11First, to avoid the possibility of inaccurate geocoding, we keep only one geocoded picture per location de…ned by its geographical coordinates. 12 This reduces the number of pictures by about 50%. Second, one may argue that the patterns of pictures taken by tourists and residents may be very di¤erent. Since we have information on users'identi…ers, we can distinguish between residents'and tourists'pictures by keeping users who take pictures for at least 6 consecutive months between 2004 and 2015 in the Randstad. It seems unlikely that tourists stay for 6 consecutive months in the area. Note that the correlation between residents' and tourists' pictures is 0:653, which is rather low.

Third, many recorded pictures may not be related to amenities but to ordinary events in daily life occurring inside the house. Hence, we only keep pictures that are taken outside buildings, using information on all the buildings in the Netherlands from the GKN dataset, which comprises information on the universe of buildings. Furthermore, if pictures are not related to amenities, one would expect almost a one-to-one relationship with population density.

However, if we calculate the population density in the same way as we calculate the amenity level, the correlation is only 0:223. Last, we recognize that people who take pictures may belong to a speci…c socio-demographic group (e.g., young people with a smartphone) by including demographic controls and using instrumental variables.

Though imperfect, we believe that the picture density is probably the best proxy available for the relative importance of urban amenities at a certain location because it captures both the heterogeneity in aesthetic quality of buildings and residents'perceived quality of a certain location. Nevertheless, we test the robustness of our results using a completely di¤erent, hedonic, amenity index in the spirit of Lee and Lin (2018) (see Appendix B.3 for more details).

The hedonic LQI aggregates the average impact of several proxies of amenities, such as the locations of historic buildings, proximity to open space and water bodies, by testing their joint impact on house prices.

For our identi…cation strategy, to be discussed later on, we also construct historic instruments. Those ones were much smaller than current ones and about 4 times the size of the current neighborhoods.

For each occupation we obtain the required skill level. This enables us to calculate the share of households who are medium and high-skilled. We gather additional data on the railway network in 1900 and the stations which by then existed (see Appendix B.4 for more information), enabling us to calculate employment accessibility in 1909.

Figure 1a provides a map of the Netherlands, the studied area, where we indicate the most important cities. The conurbation formed by the four largest cities, i.e., Amsterdam, Rotterdam, The Hague, and Utrecht is known as the Randstad, which has a population of about 7:1 million. Figure 1b displays the commuting pattern across neighborhoods and shows that the Dutch urban structure is really polycentric as many commuting ‡ows occur between di¤erent cities. This underlines the need for a model that allows for location choices in the whole country. Figure 1c is a map of the most important roads and railways that form the transportation network in the Netherlands.

[Figure 1 about here]

Further descriptive statistics are shown in Appendix B.5

Econometric framework and identi…cation

Assume that labor time is given by t i (x) w i (x)[ i (x)] , where i (x) is the commuting time between x and i, > 0 is the elasticity of labor time with respect to commuting time, and

w i (x)
is the given number of working hours per year of a household working in i. 13In this case, the LQI becomes:

(x) = b(x) [w i (x)] 1 [ i (x)] (1 ) : (23)
It then follows from [START_REF] Duranton | Urban land use[END_REF] that the income mapping is given by the following expression:

! (x) = " b(x) [w i (x)] 1 [ i (x)] (1 ) # = :
We observe income conditional on labor supply in the data. For example, if someone has a longer commute and therefore supplies less labor, we observe a lower income net of commuting costs. Hence, let !(x) be the gross income, net of commuting costs, of a household k living at

x and working in i observed in the data:

!ki (x) = ! (x)t i (x) ki (x); (24)
where the ki (x) are hourly labor income shocks that are independently and identically distributed according to some given distribution de…ned on [0; 1). Taking the log of (24), we obtain:

log !ki (x) = 0 + 1 log b(x) + 2 log i (x) + 3 log w i (x) + ~ ki (x); (25)
where We consider the income mapping, which plays a key role in our model, and …rst of all provide reduced-form evidence that the picture density is related to observed proxies for amenities and, in turn, we show that sorting by incomes is indeed related to our proxy for amenities and commuting time -the variables that constitute the LQI (see ( 23)). In line with [START_REF] Fujita | Urban Economic Theory[END_REF], set

0 log( = = ), 1 = , 2 (1 +
log !ki (x) = 1 log b(x) + 2 log i (x) + 4 C k + A (x) + i + ki (x); (26)
where !ki (x) is the observed income of household k living at x and working in i, b(x) is the density of geocoded pictures, i (x) is the observed commuting time to workplace i, C k are household characteristics, A (x) are travel-to-work-area A …xed e¤ects, i are workplace …xed e¤ects, and ki (x) is an error term. The parameters 1 , 2 , 3 , A (x) and i are estimated. 14 In an ideal setting we compare otherwise identical households, but with di¤erences in incomes. We then test to what extent di¤erences in amenities and commuting time can explain the spatial di¤erences in income. There are several issues when using [START_REF] Gaigné | Income sorting across space: The role of amenities and commuting costs[END_REF] to identify the causal impact of b(x) and i (x) on income sorting. First, regarding commuting time i (x), unconditional correlations between incomes and commuting times are generally positive rather than negative (see Susilo and Maat, 2007 for the Netherlands). There are several reasons for that.

Higher income and educated people are more specialized and, therefore, operate in 'thinner' labor markets. Given that there is a strong idiosyncratic component to residential location choices (people are strongly attached to a location and usually dislike moving), this will imply that people with higher incomes live further away from their workplace (see, e.g., [START_REF] Manning | The real thin theory: Monopsony in modern labour markets[END_REF]. Another reason for a bias is that labor markets may not be fully competitive as households may bargain over to get an income compensation for living further away. Hence, observed incomes !ki (x) may be higher when people live further away. Note that about 15% of the costs of a longer commute is paid by the employer [START_REF] Mulalic | Wages and commuting: Quasi-natural experiments'evidence from …rms that relocate[END_REF].

Second, a more general concern about 1 and 2 as measures of the impacts of amenities and commuting time on the spatial income distribution is that there is an omitted variable bias due to heterogeneity in preferences for housing quality, agglomeration economies at the workplace, and unobserved spatial features. More speci…cally, households may not only sort on the basis of income, but also on the basis of other household characteristics. Households with children, for example, may aim to locate in neighborhoods with a large amount of green space. If households with children have higher incomes, we may wrongly conclude that amenities attract the rich, while they in fact attract households with children. The variables b(x) and i (x) could also be correlated with unobserved housing attributes because households with di¤erent incomes may have di¤erent preferences for housing quality, such as the age of the housing stock [START_REF] Brueckner | Gentri…cation and neighbourhood cycles: Will America's future downtowns be rich?[END_REF]. For example, a large share of the housing stock in the city center of Amsterdam takes the form of apartments. This may imply that the a-uent are not willing to locate there because they eschew apartment living [START_REF] Glaeser | Why do the poor live in cities?[END_REF].

Third, there may be reverse causality between !ki (x) and b(x) and between !ki (x) and i (x).

For example, the provision of amenities may be a direct result of the presence of high-income 14 One may argue that we should not control for house size, as the income mapping is a result of the trade-o¤ between amenities, commuting time, and land. We can show that our results are virtually the same if we do not control for house size. The results are available upon request.

households. Indeed, anecdotal evidence suggests that cultural and leisure services are often abundantly available in upscale neighborhoods [START_REF] Glaeser | Consumer city[END_REF]. Similarly, high income neighborhoods may attract employers that are in need of specialized and highly educated labor.

Last, since we do not observe the 'exact'amenity level, there may be a measurement error in b(x), which may lead to a downward bias of 1 when the error is random. 15 The …rst step to mitigate the biases associated with these concerns is …rst to 'purge'income from household, job and housing characteristics, C k . For example, C k captures the members of the households who work full-time or part-time, the size of the household and the age of the adults, while housing attributes are, for example, housing type and construction year. This approach reduces the likelihood that we measure sorting on the basis of household characteristics other than incomes. We also control for travel-to-work-area …xed e¤ects (at the home place)

to address the concern of aggregate sorting e¤ects between urban areas, as well as di¤erences in spatial policies that are often spatially autocorrelated. More speci…cally, travel-to-workarea …xed e¤ects imply that we measure sorting of rich and poor within commuting zones.

Furthermore, but importantly, by including workplace …xed e¤ects i (at the neighborhood level) we control for wage di¤erences between workplaces (e.g., due to agglomeration economies and sectoral composition of an area).

Working with an endless string of controls will not fully address the endogeneity concerns raised above. Unfortunately, our data do not allow us to exploit quasi-experimental or temporal variation in b(x) and i (x). Therefore, to investigate the importance of the omitted variable bias we analyze coe¢ cient movements after including controls. [START_REF] Oster | Unobservable selection and coe¢ cient stability: Theory and evidence[END_REF] shows that coe¢ cient movements together with changes in the R 2 can be used to estimate biased-corrected coe¢ cients.

We outline this procedure and discuss the results in detail in Appendix C.2.

Omitted variable bias is not the only endogeneity issue. Our proxies may also su¤er from measurement error and reverse causality. We will, therefore, rely on instrumental variables.

We exploit the fact that b(x) and i (x) are autocorrelated. First, land use in 1900 is used as an instrument. Clearly, land use in 1900 cannot be a¤ected by today's income distribution, but we expect aesthetic amenities to be positively correlated to the share of built-up area in 1900. For example, the historic city center of Amsterdam has many buildings that have been built before 1900, which are now listed buildings. Furthermore, we also expect water bodies available in 1900 to be correlated to current water bodies, which are often considered as an amenity. As an 15 As suggested by the literature on local public goods, there might be reverse causality, meaning that the location of local public goods is determined by the spatial income distribution. To a large extent, this is because the institutional context that prevails in the U.S. implies that the quality of schools and other neighborhood characteristics are often determined by the average income in the neighborhood (see, e.g., [START_REF] Bayer | A uni…ed framework for measuring preferences for schools and neighborhoods[END_REF]. This is to be contrasted with what we observe in many other countries where local public goods such as schools are provided by centralized bodies.

instrument for commuting time, we count the total number of workers a 1909 (x)in 1909 within commuting distance using the railway network in 1900:

a 1909 (x) = n X i=1 F ( i (x))E i;1909 ; (27)
where i (x) is the commuting time between x and employment location i = 1; :::; n, E i;1909 are the number of workers at i in 1909, while F ( i (x)) is the share of people who commute at most minutes in the sample (see Appendix B.1). Hence, F ( i (x)) represents the aggregate cumulative distribution of commuting times. Because of temporal autocorrelation, we expect that a better employment accessibility in 1909 also implies a better employment accessibility today and, therefore, a lower commuting time.

Historic instruments can be criticized because of the (strong) identifying assumption that past unobserved locational features are uncorrelated to current unobserved locational endowments. However, these instruments are more likely to be valid in the context of income sorting because the patterns of income sorting within each city have considerably changed throughout the last century. Around 1900, open water and densely built-up areas were not necessarily considered as amenities. For example, the canals in Amsterdam were essentially open sewers [START_REF] Geels | The hygienic transition from cesspools to sewer systems (1840-1930): The dynamics of regime transformation[END_REF]. Therefore, locations near a canal often repelled high-income households who located in lush areas just outside the city. It was also before cars became the dominant mode of transport. People around 1900 often walked to their working place so that commuting distances were short. However, the rich could a¤ord to live outside the city and take the train to their workplace. The cities in 1900 were not yet in ‡uenced by (endogenous) planning regulations, as the …rst comprehensive city plans date from the 1930s.

Still, one may be concerned that the measure of amenities is itself determined by the wealth of individuals who locate there. The reason is that unobservables that determine the concentration of wealthy individuals in the past also determine the locations of landmarks today, and thus determine where pictures are taken. Moreover, one may argue that historic employment accessibility, which is correlated to current commuting times, makes it easier to …nd jobs for all household members, and thus increases household income due to better matching, rather than shorter commutes. We address these concerns in several ways.

(i) We estimate speci…cations where we control for the current share of built-up areas and population density. Locations that were attractive in the past attracted people and consequently have a high share of built-up area in 1900. The share of built-up areas in 1900 is likely to be correlated to the current population density and to shares of built-up areas nowadays. By controlling for the current share of built-up areas and population density we mitigate the issue that our proxy for amenity just captures contemporary population density, rather than a higher amenity level because of the historic buildings.

(ii) We gather data from the 1909 census on occupations and skills in each municipality.

We then control in various ways for the average skill level of households in 1909 as a proxy for the income in the past. Controlling for the skill level should also address the issue that employment density in 1909 may be correlated to better matching opportunities. Since this proxy may be imperfect, we also use the share of Protestants in 1899 at the municipality level as another proxy for income/skill. Indeed, at that time Protestants had a higher education level and were wealthier.

(iii) We also consider another instrument for commuting time. From the 1899 census, we gather data on the share of locally born people (i.e., within the same municipality). If the (lack of) mobility of households is correlated over time, the share of locally born people should be correlated positively to current commuting times because immobile households have to commute on average longer to their jobs.

(iv) Finally, we estimate speci…cations where we exclusively focus on areas of reclaimed land since 1900. These are areas that are reclaimed from the sea (about 5% of the land) just before and after World War II. As these reclaimed locations are otherwise identical, and as no one was living in those locations at that time, we address reverse causality, and strongly mitigate any remaining omitted variable bias.

Results

Baseline results. We …rst check that picture density is a good proxy for amenity endowments. The results in Appendix B.3 show the expected signs: there is a higher picture density areas where there are many historic buildings, in built-up areas, and in areas with more water bodies (e.g., the Amsterdam canal district). The same holds if we control for sorting on other household characteristics (e.g., sorting based on household composition) and commuting time.

Table 1 reports the baseline reduced-form results for the income mapping. Column (1) shows a simple regression of log income on log amenities and log commuting times, while we only control for demographic characteristics and year …xed e¤ects. This shows that more amenities are associated with higher incomes. Doubling amenities implies an increase in neighborhood income of (log 2 log 1) 0:0387 = 2:7%. However, doubling of commuting time seems to attract households whose incomes are 4:6% higher -contrary to the expectations. In column

(2), we add a wide array of controls related to housing quality and add travel-to-work-area …xed e¤ects to address spatial heterogeneity. Although the R 2 increases by about 30%, the coe¢ cients related to amenities and commuting time are hardly a¤ected. This suggests that amenities and commuting time are not so much correlated to building quality and aggregate characteristics of the urban area.

[Table 1 about here]

In column [START_REF] Albouy | What are cities worth? Land rents, local productivity, and the total value of amenities[END_REF] we further aim to address potential measurement error in the picture density and address reverse causality issues by instrumenting amenities and commuting time with historic variables. The instruments are the number of households within commuting distance in 1909 using the railway network in 1900; the shares of water bodies and of built-up area in 1900 within a neighborhood, as well as within 500m and between 500 and 1000m of a neighborhood's border. 16 In Appendix C.1, we report the corresponding …rst-stage results. The share of built-up area, the share of water bodies in 1900 are strongly and positively correlated to the current amenity level, while employment accessibility in 1909 is negatively related to current commuting time. Overall, the Kleibergen-Paap F -statistic is above the rule-of-thumb value of 10 in all speci…cations, suggesting that the instruments are su¢ ciently strong.

Going back to Table 1, the coe¢ cient of amenities is now somewhat higher: doubling amenities attracts households whose incomes are 2:6% higher; doubling commuting time implies an e¤ect of 15%. Column ( 4) is the most comprehensive speci…cation where we include job characteristics and workplace …xed e¤ects. We consider this as the preferred speci…cation.

Doubling amenities attracts households whose incomes are 1:6% higher. Doubling commuting time leads to households whose incomes are 14% lower. Hence, the impact of commuting time seems to be somewhat stronger than the impact of amenities.

Despite the inclusion of controls, travel-to-work-area and workplace …xed e¤ects, one may argue that we do not convincingly address the omitted variable bias. We deal with this issue by estimating bias-corrected regressions following Oster (2019) in Appendix C.2. We show that when we choose the appropriate maximum attainable R 2 (as only part of the variation in incomes can be explained by variables varying at the neighborhood level), the estimates are very close to the preferred IV estimates (so a positive amenity elasticity of around 0:025 and a commuting time elasticity between 0:05 and 0:45). This strongly suggests that the historic instruments address omitted variable bias.

To sum up, the results unequivocally indicate that the impact of amenities on income sorting is positive and highly signi…cant. As for the commuting time, its e¤ect on income sorting is negative and strong. To compare the e¤ects of commuting and amenities, it is informative to look at a standard deviation of a log change in commuting time or amenities. In the preferred speci…cation, a standard deviation increase in log amenities attracts households whose incomes 16 Since we have more instruments than endogenous variables, one might object that two-stage least squares estimates are biased (Angrist and Pischke, 2009). Hence, we also have experimented with other estimators that are (approximately) median unbiased, such as LIML or GMM estimators. The results are virtually identical.

For this reason, we do not report them in the paper.

are about 1:656 0:023 = 3:7% higher. On the other hand, a standard deviation increase in log commuting time attracts households whose incomes are 0:645 0:207 = 13:3% lower.

Hence, commuting time seems to be a more important driver of income sorting than amenities.

However, the impact of amenities is far from negligible. For example, if we compare an area with the lowest amenity level (some rural area in the north) with the area with the highest amenity level (the center of Amsterdam), the predicted income di¤erence is 28:6%, which is clearly non-negligible.

Alternative proxies for amenities. One may worry that our results hinge on the particular choice of the amenity index. We therefore consider an alternative proxy for amenity endowments. Following Lee and Lin (2018), we construct an aggregate hedonic amenity index that describes the amenity provision at every location using house prices. The procedure is described in Appendix B.3. We report the results in Table 2, which replicates the speci…cations in Table 1 but replaces picture density by the log of the hedonic amenity index.

[Table 2 about here]

The e¤ect of commuting time is very much comparable to the results reported in Table 1.

Regarding the hedonic amenity index, once we control for housing, location, job characteristics, as well as workplace …xed e¤ects in column (2), we …nd an elasticity of 0:8, implying that a 100% increase in the hedonic amenity index leads to an increase in income of 55%. We …nd a somewhat lower amenity elasticity in column (4) when we include control variables, workplace and travel-to-work-area …xed e¤ects, and instrument for amenities and commuting time with historic variables. Doubling the amenity level is then associated with an increase in incomes of 36%.

To make the results comparable, we rescale the hedonic amenity index in such a way that the standard deviation of the log of the hedonic amenity index is the same as that of the log of the picture density. The estimated elasticity of the rescaled hedonic amenity index then varies between 1:6% and 2:4%, which is very much comparable to the impact of picture density.

E¤ects on land prices. We also investigate the reduced-form impacts of amenities and commuting times on land prices, so we estimate a reduced-form version of [START_REF]Mobiliteitsbalans 2010[END_REF]. In our setup the signs of the e¤ects of amenities and commuting time on land prices and incomes are the same (although magnitudes may di¤er). Therefore, we now estimate the e¤ects of amenities and commuting time on land prices. The results are reported in Table 3.

[Table 3 about here]

We start in column (1), Table 3, with a simple OLS speci…cation including amenities and commuting time, while controlling for demographic characteristics. This implies a strong positive e¤ect of amenities on land prices: doubling amenities leads to an increase in land price of (log 2 log 1) 0:1874 = 13%, while doubling commuting time leads to land prices that are 2:5% lower. Column (2) adds job controls and workplace …xed e¤ects, leading to a coe¢ cient that is 50% lower.

When we instrument for commuting time with accessibility of employment in 1909, the preferred speci…cation in column ( 4) with job controls and workplace characteristics indicates that doubling amenities leads to an increase in land prices of 13% , which is sizable. The impact of commuting is somewhat imprecisely estimated, but the coe¢ cient is negative. The point estimate indicates that a 100% increase in commuting time is associated with a decrease in land prices of 27%. Hence, the reduced-form e¤ects on land prices do indeed have the same signs as the e¤ects on income, but are stronger in magnitude. For example, if we compare the land price di¤erential between the location with the lowest and the highest amenity level, it is 238%, which is considerable.

Other sensitivity checks. We consider various additional robustness analyses in Table 4.

First, we show that our results are similar once we focus solely on urban areas. In column [START_REF] Ahlfeldt | Urbanity[END_REF] we only include observations in the Randstad. This reduces the total number of observations by more than 50%. However, our results are very similar.

[Table 4 about here]

In column [START_REF] Ahlfeldt | The economics of density: Evidence from the Berlin Wall[END_REF], Table 4, we include municipality …xed e¤ects. This implies that we identify commuting and amenity e¤ects within municipalities. Municipality …xed e¤ects absorb any e¤ects related to municipal policies and tax di¤erentials. We …nd very similar coe¢ cients.

Column (3) controls for the current share of built-up areas and population density to make sure that our amenity proxy is not just capturing population density or built-up land. We …nd very similar e¤ects for amenities, but the e¤ect of commuting time is now stronger. This may be because we have weak instruments, with a Kleibergen-Paap F -statistic that is barely 10.

Another concern is that clusters of high-income households are autocorrelated so that our instruments are correlated to the concentrations of high-income households in 1909. To investigate whether this is an issue, we calculate the share of medium and high-skilled households in 1909. Municipalities then were much smaller so this is a rather …ne-grained measure of skill sorting across space. We also gather data on the share of Protestants in each municipality in 1899 and control for population accessibility in 1900. Including those measures in column (4) does not materially impact our coe¢ cients. Note that the share of high-skilled and medium-skilled households in 1909 is negatively correlated to current incomes, which suggests that the determinants of residential choices in the two periods are fairly di¤erent. In addition, conditional on commuting time, population accessibility in 1900 does not in ‡uence the current income distribution.

In column [START_REF] Allen | Trade and the topography of the spatial economy[END_REF] in Table 4 we further study the sensitivity of our results by choosing another instrument for commuting time. We use the share of the population in 1909 born in the same municipality. If mobility of households is correlated over time, the share of locally born people should be positively correlated to current commuting time, as the areas that host a high number of jobs (so have a low commuting time) are expected to attract workers from other places and in turn have a lower share of locally born people. Indeed, we …nd that the share of locally born people in 1909 is positively associated with current commuting time. The Kleibergen-Paap Fstatistic again indicates that these are strong instruments. We …nd a similar coe¢ cient related to commuting time.

To the extent one is still concerned that household income sorting is autocorrelated, in column ( 6) we only include neighborhoods on reclaimed land. The Netherlands is well known for its large-scale projects that reclaim land from the sea. We consider the three main projects (Wieringermeer, Noordoostpolder, Oostelijk, and Zuidelijk Flevoland) that occurred between 1930 and 1968, but permission by the government to reclaim those areas was already given in 1930. Most of the land was intended for agriculture, but a few small settlements were planned on the newly reclaimed land. Moreover, Lelystad was planned to be the largest city in the area, but nowadays Almere is by far the largest one. Hence, the plans di¤er considerably from the current spatial distribution of activities. Since only a small share of the population lives in those areas, we only keep about 2:5% of the observations. The latter approach should address any remaining concerns related to reverse causality as no one was living in those locations at that time, and thus income was zero.

We then instrument for amenities and commuting time with the share of planned built-up and green areas and planned accessibility in column [START_REF] Altonji | An evaluation of instrumental variables strategies for estimating e¤ects of Catholic schooling[END_REF]. The instruments are unfortunately very weak, leading to imprecise coe¢ cients. Still, the point estimates are again similar to the baseline estimates. Instrumenting either amenities or commuting time leads to similar outcomes, but with somewhat lower standard errors.

Appendix C.3 further shows that our results still hold for a wide range of alternative robustness checks and sample selections. We aim to minimize measurement error regarding commuting time and workplace productivity, by running speci…cations where we only keep households (i)

with a single job, (ii) with a single job in a single-plant …rm, and (iii) with a company car who are more likely to use the car for commuting. We further test whether our results change when using the share of highly educated adults in the household, which is a measure that is very much correlated to income, but arguably is measured with little error. We …nd very similar e¤ects, both in terms of sign and magnitude, which con…rms that looking at income or skill levels is more or less equivalent. We also use commuting time by rail instead of commuting time over the road and focus on areas close to city centers of large cities. Overall, the impact of amenities and commuting time on income sorting choice is robust.

Structural parameters

Estimating structural parameters We think it is instructive if the reduced-form estimates are interpreted through the lens of the structural parameters. Recall that 1 = and

2 (1 + (1 ) = )
, where and are the shape parameters of the Fréchet distributions regarding the LQI and incomes respectively, is the preference for land, and is the commuting time elasticity. To identify all the necessary parameters we take three steps.

First, to determine , we use information on land prices R(x) and lot sizes h(x) for a subset of the sample. Rewriting ( 14) yields:

R(x) = !(x) h(x) (1 )h : (28)
We use nonlinear least squares to obtain an estimate for , given a discount rate (to calculate yearly payments to land) and a minimum lot size h. We set h = 25m 2 , which corresponds to the minimum lot size in the sample, and we use a discount rate of 3:5% to go from land prices to land rents (see Koster and Pinchbeck, 2021). 17Second, we estimate the reduced-form income mapping [START_REF] Gaigné | Income sorting across space: The role of amenities and commuting costs[END_REF] to obtain ^ 1 and ^ 2 . Then, we have:

^ = ^ 2 (^ 1)^ 1 1 : (29)
Third, we calculate the LQI, given by equation [START_REF] Epple | Tiebout: Integrating location-speci…c amenities into multi-community equilibrium models[END_REF], and then …t a Fréchet distribution through the estimated ^ (x) using Maximum Likelihood to obtain an estimate for ^ . Hence, = ^ =^ 1 . Table 5 reports the results.

[Table 5 about here]

We …rst …nd ^ = 0:0955, which con…rms that richer households spend less of their income on land [START_REF] Albouy | Housing demand, costs-of-living inequality, and the a¤ordability crisis[END_REF]. One may be concerned that ^ may seem low compared to the existing literature, but we only include payments to land, not to housing itself. We further show that the commuting time elasticity ^ = 0:2024. This elasticity seems reasonable: doubling the commuting time decreases gross income by 6%. 18 Last, we …nd that there is considerably more heterogeneity in the LQI as compared to incomes, as ^ ^ . To put it di¤erently, locations are much more di¤erentiated (in terms of amenities and commuting time) than households (in terms of incomes).

Determining the LQI and the income mapping. Using the estimated structural parameters and ( 23), we can calculate the LQI for each household living at di¤erent locations. We show a histogram of the LQI in Figure 2.

[Figure 2 about here]

This shows that the LQI is well approximated by a Fréchet distribution, which is ‡exible enough to capture the positively skewed distribution of the LQI.

We can further use the estimated parameters to calculate the income mapping ! (x) throughout the Netherlands. Note that this mapping is identi…ed up to a multiplicative constant. We display the results in Figure 3 where we focus on the most urbanized part of the Netherlands:

the Randstad. We observe that the highest income households locate in city centers of Amsterdam, Utrecht, Rotterdam and the Hague, where amenity levels are high and commuting times relatively short.

[Figure 3 about here]

Concluding remarks

We have used a new setup in which location are di¤erentiated by two attributes, i.e., amenity endowments and commuting costs. The bid-rent function of urban economics may be used to show that the uneven distribution of amenity endowments is su¢ cient to break down the perfect sorting of households across the city space. Under Stone-Geary, quasi-linear or a generalized CES preferences, there exists a LQI that blends amenities and commuting costs into a single aggregate whose behavior drives households's residential choices. Studying this LQI has allowed us to gain insights about how governments and urban planners could design policies whose aim is to redraw the social map of cities. For example, our analysis suggests that promoting equal access to amenities across individuals is likely to favor residential segregation, whereas a multi-modal provision of amenities across the city fosters income mixing. The relevance of local amenity endowments and commuting costs to explain the residential choices of heterogeneous 18 Using a gravity model of commuting ‡ows, Ahlfeldt et al. (2015) …nd a commuting time semi -elasticity of about 0:1. Although not directly comparable to our estimate (because we estimate an elasticity), they show that utility decreases by about 7% when commuting time increases from 5 to 10 minutes; and by 14% when commuting time increases from 10 to 20 minutes. Hence, our estimate is in the same ballpark.

consumers is con…rmed by the empirical analysis where both e¤ects are found to be economically and statistically signi…cant.

Tables Table 1

-Baseline regression results

(Dependent variable: the log of household gross income) Di¤erentiating (8) with respect to x and using [START_REF] Bayer | A uni…ed framework for measuring preferences for schools and neighborhoods[END_REF], we obtain:

+ Job controls Historic instruments (1) (2) (3) (4)
x (x; !; U (!)) = !t H t x t Q b !t b x (A.1)
Di¤erentiating (A.1) with respect to ! and rearranging terms yields the following expression:

x! (x; !; U (!)) = t H t x t h 1 ! H (H ! + H U U ! ) i + b x t H ! + H U U ! H Q b (Q bH (H ! + H U U ! ) + Q bU U ! ) : (A.2)
Since Q is the solution to the equation u(q; h) = U=b(x), the following expressions must hold:

Q b = U b 2 u q Q bU = 1 b 2 u q + U b 2 u 2 q u qq Q U Q bH = U b 2 u 2 q (u qq Q H + u qh ) :
Assume that the !-households are located at x. Di¤erentiating u = U (!)=b with respect to ! and using the budget constraint Q = !t(x) H and (9), we obtain:

[t (H ! + H U U ! ) ]u q + (H ! + H U U ! )u h = U ! b : Since u q + u h = 0
at the residential equilibrium, we have:

t = U ! bu q : (A.3)
Plugging this expression, Q b ; Q bU and Q bH in (A.2), we get

x! (x; !; U (!)) = t H t x t h 1 ! H (H ! + H U U ! ) i b x U ! bu q H ! + H U U ! H U b 2 u q b x U ! bu q U b 2 u 2 q (u qq Q H + u qh ) (H ! + H U U ! ) U ! b 2 u q + U b 2 u 2 q u qq Q U U ! ;
which is equivalent to

x! (x; !; U (!)) = t H n T (x) h 1 ! H (H ! + H U U ! ) i +B(x) 1 H ! + H U U ! !H !U U ! !U u q !U ! (u qq Q H + u qh ) (H ! + H U U ! ) U u q u qq Q U : Using du q d! = u qq Q H (H ! + H U U ! ) + u qq Q U U ! + u qh (H ! + H U U ! ) ;
we can rewrite x! as follows:

x! (x; !; U (!)) = t H 1 E H;! + E uq;! E U;! B (1 E H;! ) T ; (A.4)
which proves Proposition 1.

Income-independent commuting costs If commuting costs are given by ! (x) + c(x), the bid rent function (x; !; U (!)) is given by

(x; !; U (!)) max h !t(x) c(x) Q(h; U=a(x)) h : Consequently, x (x; !; U (!)) = 1 H !t x c x + b x b u u q : and x! (x; !; U (!)) = t x H (1 E H;! ) + b x bH u !u q E U;! E H;! E uq;! + E H;! H c x ! :
Using ( 11) and (A.1), one can rearrange the above expression:

x! (x; !; U (!)) = t H B(x) 1 E H;! + E uq;! E U;! T (x)(1 E H;! ) + c x !t E H;! :
When t(x) is constant, T (x) = 0. Therefore, when B(x) = 0, we obtain

x! (x; !; U (!)) = c x !t E H;! > 0:
A.

Homothetic preferences

Assume that the utility u(q; h) is homothetic, that is, homogeneous linear. Then, it must be that E h;! = E q;! = 1. The …rst-order condition for utility maximization implies

u h = Ru q :
It follows from Euler's theorem that

hu h + qu q = u , h u h u + q u q u = 1;
that is,

E U;h + E U;q = 1:
Since the income elasticity of utility is given by

E U;! = E U;h E h;! + E U;q E q;! ;
we get

E U;! = 1:
It remains to determine @u q =@!. Using the …rst-order condition u h = Ru q , the budget constraint Rh + q = !t and Euler's theorem, we obtain:

u q = u !t :
Taking the total derivative of this expression with respect to ! yields:

du q d! = 1 t (du=d!)! u ! 2 = u ! 2 t (E U;! 1) = u q ! (E U;! 1)
so that E uq;! = 0.

To sum up, we have E U;! = 1, E H;! = E h;! = 1 and E uq;! = 0.

A.3 Stone-Geary preferences

It is readily veri…ed from (12) that

Q(h; U=b(x)) = 1 (h h) U b 1 1

:

It follows from this expression that

Q U = 1 1 U 1 1 1 1 b(h h) 1 1 = 1 (1 ) Q U ; Q U b = Q U (1 )b ; Q b = U b Q U ; Q h = 1 1 (h h) U b 1 1 Q bH = U b 1 (h h) 1 Q U :
Given [START_REF] Bayer | A uni…ed framework for measuring preferences for schools and neighborhoods[END_REF], the expression (8) boils down to

(x; !; U ) = Q H (H; U=b(x)): (x; !; U ) = 1 (H h) 1 1 U b 1 1

:

Plugging Q b ; Q bH and Q bU into (A.2) and rearranging terms leads to

x! (x; !; U (!)) = t H t x t h 1 ! H (H ! + H U U ! ) i + b x b H ! + H U U ! H U t Q U h (1 )h (1
)(h h)

+ Q U (1 )t U ! : (A.5)
Plugging Q h and Q in [START_REF] Bayer | A uni…ed framework for measuring preferences for schools and neighborhoods[END_REF] and solving the corresponding equation yields

h (1 )h (1 )(h h) = !t b U (h h) 1 1

:

(A.6)

Given the expression of Q U ; it turns out that

U t Q U h (1 )h (1
)

(h h) = ! 1 : (A.7)
Di¤erentiating [START_REF] Beckmann | On the distribution of urban rent and residential density[END_REF] with respect to ! and using [START_REF] Bayer | A uni…ed framework for measuring preferences for schools and neighborhoods[END_REF], we obtain:

! (x; !; U (!)) = t H 1 Q U t U ! (A.8)
which is equal to 0 if

U ! = t Q U : (A.9)
Using (A.7) and (A.9), (A.5) can be rewritten as follows

x! (x; !; U (!)) = t H h 1 ! H (H ! + H U U ! ) i 1 1 (1 ) t x t + b x b : (A.10)
Applying the implicit function theorem to (A.6) yields

H U = (h (1 )h)(h h) U h and H ! = t(1 ) 2 h U 1 1 b 1 1 (h h) 1+ 1 1 :
Given Q U ; (A.9) can be expressed as the following di¤erential equation:

U ! = t (1 ) b (h h) 1 1
(U (!)) 1 :

(A.11)

H ! + H U U ! = t (1 )(h h) b U (h h) 1 1
Therefore, by implication of (A.6), we have:

1 ! H (H ! + H U U ! ) = (1 )h h :
Substituting this expression into (A.10) yields:

x! (x; !; U (!)) = t H h H [B (1 )T ] :
A.4 Proof of Proposition 2

The proof involves four steps.

(i) The bid-max lot size. From the de…nition of the LQI given by ( 16), i.e.

b(x)[t(x)] 1 , (A.6) can be rewritten as follows:

H h + h (1 )(H h) = ! 1 1 (H h) U 1 1

:

(A.12)

The LHS of this expression is decreasing and tends to 1 when H ! h and to 1=(1 ) > 0 when H ! 1. The RHS is increasing in H, tends to 0 when H ! h and to 1 when H ! 1.

Therefore, (A.12), or equivalently [START_REF] Bayer | A uni…ed framework for measuring preferences for schools and neighborhoods[END_REF], has a single solution H(!t(x); U=b(x)). In other words, the bid-max lot size is uniquely determined. It also follows from (A.12) that the bid-max lot size H( ) depends on b(x) and t(x) only through the LQI (x), that is, H(!t(x); U=b(x))

H( (x); !; U ).

Applying the implicit function theorem to (A.12) yields

@H @ = U 1 1 (H h) 1 1 1 H 1 1 ! 1 < 0: (A.13)
(ii) Equilibrium utility level. Using the de…nition of the LQI, (A.11) implies that the equilibrium utility level is a solution to the following di¤erential equation in U :

U ! = 1 1 (1 )(H h) 1 (U (!)) 1 ; (A.14)
so that U (!) depends on only.

(iii) Supermodularity of the equilibrium utility level. Di¤erentiating (A.14) w.r.t. , we obtain:

@ @ dU d! = 1 (H h) 1 (U (!)) 1 1 + (H h) 1 @H @ :
Plugging (A.13) into this expression yields:

@ @ dU d! = 1 (H h) 1 (U (!)) 1 " 1 (H h) 1 1 (1 )! 1 1 (U (!)) 1 1 H # :
It follows from (A.12) that the bracketed term of this expression is equivalent to

1 (H h) 1 1 (1 )! 1 1 (U (!)) 1 1 H = (1 ) h h > 0:
Therefore, @ @ dU d! > 0:

The Spence-Mirrlees condition thus holds, which implies the existence of a unique positive assortative matching between incomes and the values of the LQI. In other words, there is a unique one-to-one and increasing relationship between ! and [START_REF] Chiappori | Matching with transfers. The economics of love and marriage[END_REF]. Since a single value of is associated with x, a unique value of ! must be associated with x. Hence, in equilibrium the households who reside at x must have the same income, so that ! (x) is a mapping.

Furthermore, there exists a unique residential equilibrium because H( (x); ! (x); U (! )) is univocally de…ned.

Note that the supermodularity of U (!) is equivalent to the inequality ! > 0. Indeed, di¤erentiating (A.8) w.r.t. and using (A.9) yield:

! (x; !t(x); U (!))j ! =0 = t H @(t=Q U )=@ U ! = t H @U ! =@ U ! > 0:
(iv) The equilibrium shares. Assume that there are n 2 points x 1 6 = x 2 ::: 6 = x n exist such that (x 1 ) = (x j ) for j = 2; :::; n. Since h (x; ! (x)) = H [ (x); ! (x); U (! (x))], the condition (4) may be rewritten as follows: If ! (x 1 ) = ! (x j ) for j = 2; :::; m, then H [ (x 1 ); ! (x 1 ); U (! (x 1 ))] = H [ (x j ); ! (x j ); U (! (x j ))] ; j = 2; :::; n: s(x 1 ; ! (x 1 )) s(x k ; ! (x k )) = jd (x k ) =dx k j jd (x 1 ) =dx 1 j ; k = 2; :::; n:

s(x; ! (x))f (! (x))H [ (x); ! (
Since P m j=1 s(x j ; !) = 1, the above system of equations has a unique solution given by s (x j ; !) = 1 n P k=1 1 jd (x k )=dx k j 1 jd (x j ) =dx j j ; j = 1; :::; n:

A. [START_REF] Allen | Trade and the topography of the spatial economy[END_REF] The land gradient

Plugging Q b into (A.1) leads to

x (x; !; U (!)) = !t H U Q U !t B(x) T (x) :
Substituting t as given by (A.9), we obtain:

R x (x) = x [x; ! (x); U (! (x))] = ! (x)t H 1 E U;! B(x) T (x)
; a:e:

Using (A.11) and ( 13), we have

E U;! (x) = ! (x)t(x)= ! (x)t(x) hR (x) > 1: Plugging this expression into R x (x) leads to R x (x) = ! (x)t H 1 hR (x) ! (

B.1 Commuting and travel times

To estimate the commuting time for each household, we use the tax register information, which provides information on individual jobs and the number of hours worked in each …rm for each year. From the ABR Regio dataset, we get information on all …rms which provide information on each establishment in the Netherlands, such as its exact location, the industrial sector, and the estimated number of employees in each establishment. Since we do not know the exact establishment, only the …rm people work for, we assume that they work at the nearest establishment of the …rm. This assumption may be problematic for …rms having a large number of establishments (e.g., supermarkets or large banks). Therefore, we keep only …rms with a maximum of 15 establishments throughout the Netherlands. As many such …rms have establishments in di¤erent cities, it is reasonable to assume that people work in the nearest establishment. 19 Overall, we are left with 95% of …rms. To avoid miscoding and to exclude employment agencies (where people do not actually work), we consider establishments with no more than 10 thousand employees, which correspond to 0:1% of the total number of establishments and 1% of total employment. Hence, excluding those establishments is unlikely to a¤ect our results.

We …rst calculate the commuting time from each home location x to each job location i for each year. Then, we determine the commuting time of each household by computing the average commuting time of each adult household member weighted by the number of hours (s)he worked. To calculate the travel time (as well as the time to travel to amenities) we obtain information on the street network from SpinLab, which provides information on average free- ‡ow speeds per short road segment (the median length of a segment is 96m), which are usually lower than the speed limit.

The dataset from SpinLab provides information on free- ‡ow driving speeds for every major street in the Netherlands. The actual speeds are usually well below the free- ‡ow driving speeds, due to tra¢ c lights, roundabouts and intersections. For each neighborhood we calculate the straight-line distance to the nearest access points on the network and then calculate the network distance. The median distance from an observation in the dataset to the nearest access point of the network is 195m (the average is 326m). We assume that the average speed to get to the nearest access points is 10km/h. This is the speed based on the Euclidian distance. In reality, the distance to get to the network will be higher because streets are usually curved. Hence, the assumption of 10km/h seems reasonable as the minimum speed on roads in the network is 20km/h. Furthermore, because of the dominance of the bicycle, this would be close to the average cycling speed. Using these information, we calculate the total driving time, which is the sum of the driving time to access the network, the network driving time and the time it takes from the network to arrive at the destination. Alternatively, we calculate for each location pair the Euclidian distance and assume again an average speed of 10km/h.

We also calculate the travel time using the train, using a similar approach. The median distance of each centroid to the nearest station is 5:25km. We then choose the minimum of the travel time over the road, using the train or taking the Euclidian travel time.

[Figure B.1 about here]

The correlation between travel time and Euclidian distance is modest ( = 0:643). For short distances (< 10km) the correlation is, however, much higher ( = 0:862). We plot the relationship between distance and travel time in Figure B.1a. This relationship is monotonic.

B.2 Land prices and lot sizes

Information on land values and lot sizes is not directly available but may be inferred from data on home sales. We use information on home sales from NVM (The Dutch Association of Realtors), which comprises the large majority (about 75%) of owner-occupied house transactions between 2003 and 2017. We know the transaction price, the lot size, inside ‡oor space size (both in m 2 ), the exact address, and a wide range of housing attributes such as house type, number of rooms, construction year, garden, state of maintenance, and whether a house is equipped with central heating. We make some selections to make sure that our results are not driven by outliers. First, we exclude transactions with prices that are above e1 million or below e25; 000 and have a price per square meter which is above e5; 000 or below e500. We also leave out transactions that refer to properties that are larger than 250m where (x; y) is an i.i.d. error term that is assumed to be uncorrelated to land rents and housing characteristics, while 1 are parameters to be estimated. As log R x are given by the very local …xed e¤ects, we do not impose any structure on how land rents R x vary across locations.

For about 80% of the data we do not observe land prices directly, because either there were no multiple sales in our study period or because there is no owner-occupied housing in the respective postcode. We therefore also estimate the above equation with neighborhood …xed e¤ects instead.

[Tables B

B.3 Amenities

Picture density. Here we investigate whether there is a meaningful correlation between picture density and observed proxies for amenities. We gather data on the share of a neighborhood in a historic district, the density of listed buildings, as well as the share of the neighborhood occupied by buildings or water. We use similar speci…cations as reported in Table 2, but take the picture density as dependent variable. Table B.3 reports the results.

[Table B.3 about here]

In column (1) we regress picture density on our proxies for amenities. We …nd that a standard deviation increase in listed building density is associated with an increase in picture density of 8%. When the share of the area in a historic district increases by 10 percentage points, this is associated with an increase in the picture density of 30%. Hence, the impact of historic amenities on the revealed amenity level is large. The share of built-up land and water bodies in the neighborhood are also positively related to the picture density: a 10 percentage point increase in the share of built-up land or water bodies is associated with an increase in the picture density of about 33%. In column (2) we include commuting time, demographic characteristics, housing attributes and travel-to-work-area …xed e¤ects, leading to very similar estimates.

In column (3) we instrument for commuting time. We observe that commuting time does not seem to be statistically signi…cantly correlated to the amenity level in a certain location.

However, once we include job characteristics and workplace …xed e¤ects, we …nd a weak relationship between commuting time and amenity level: a shorter commuting time is generally associated with a higher picture density. We think this makes sense as the most central locations generally provide a high amenity level and a somewhat better accessibility of jobs. However, we reiterate that this relationship is not statistically strong.

A hedonic amenity index. We test whether our results are robust to using an alternative hedonic amenity index, rather than relying on geocoded pictures. Following Lee and Lin (2018),

we construct an aggregate amenity index that describes the amenity level in every neighborhood

x. 20 We will make a distinction between historic amenities and natural amenities.

Let A(x) be a set of variables that describe amenities of property x (so the location is more detailed than the neighborhood x). For example, we calculate the share of historic districts, the number of listed buildings, water bodies and open space within 500m of each property.

Let P(x; y) the house price, while C(x; y) are housing characteristics of property x, and #(y) are year y …xed e¤ects. We also include neighborhood …xed e¤ects &(x), and thus we identify the e¤ects of amenities on prices within neighborhoods. Speci…cally, we estimate the following equation:

log P(x; y) = 0 A(x) + 1 C(x; y) + #(y) + &(x) + (x; y); (B.3.1)
where 0 and 1 are parameters to be estimated and (x; y) is an i.i.d. error term. We then use b 0 and A(x) to predict the amenity level in each location x in the Randstad:

b(x) = 1 N (x) Nx X x=1 b 0 A(x); (B.3.2)
where b(x) is the (alternative) amenity value at x and N (x) are the number of observations in neighborhood x. Hence, we take the mean amenity value of all locations x within neighborhoods

x.

We use data on the universe of housing transactions in the Netherlands between 2010 and 2015 from the NVM. Additional descriptive statistics of the NVM data are reported in Table suggest that the approach using historic instruments addresses omitted variable bias, as the bias-adjusted coe¢ cients lead to very similar outcomes as the 2SLS estimates with historic instruments. We note that Oster's (2019) methodology does not account for measurement error in amenities or commuting time or reverse causality. It may, therefore, still be important to apply our instrumental variables strategy using historic instruments.

C.3 Sensitivity: other checks

Table C.3 reports the results of additional robustness checks. Our dataset contains observations on households. When calculating the commuting elasticity and when including workplace …xed e¤ects, we calculate average commuting times based on working hours of di¤erent jobs of adults in the household. In column (1) we only include households that are associated with one job (location). This does not lead to signi…cant di¤erences in the outcomes.

Recall that we calculate the commuting time to the nearest plant when the …rm has multiple establishments. We test whether this introduces error by only including households that are associated with one job in a single plant …rm in column [START_REF] Ahlfeldt | The economics of density: Evidence from the Berlin Wall[END_REF]. In this way we address any measurement error in commuting time. Again, the estimates are very similar, although the number of observations is considerably lower.

[Table C.3 about here]

Our measure of commuting time relies on the minimum travel time by road and rail. However, in almost all cases travel time by car is shorter. To make sure that households actually consider this travel time, we only keep households having a company car in column (3). This does not change the results.

Column (4) replaces the dependent variable income by the share of adults in the household that have a college degree or more. We …nd very similar e¤ects. For example, when the picture density doubles, this increases the share of highly educated households by 1:1 percentage points.

Doubling commuting times decreases the share of highly educated households by 32:3 percentage points.

Column [START_REF] Allen | Trade and the topography of the spatial economy[END_REF] tests whether the results are robust when using commuting time by rail instead of commuting time by road. The results are comparable.

Finally, in column (6), we only keep locations that are within 15km of a city center with at least 100; 000 inhabitants to make sure that our results are not driven by rural locations.

We …nd that the coe¢ cients are very similar, except that standard errors of the coe¢ cients are considerably higher. Hence, including locations throughout the Netherlands has the bene…t that it leads to more precise estimates. 3) and ( 5)-( 6) is the log of gross yearly income.

In column (4) it is the share of the adults in the household that has a bachelor's degree or higher. Household controls include household size, mean age of adults, mean gender, household type (couple, single, kids), the share of the household that is foreignborn. Job controls are the total hours worked, whether the household has a company car, the share of full-time contracts, the share of permanent contracts. Housing controls include house type, height of the building, construction year dummies and whether a building is listed. Standard errors are clustered at the neighborhood level and in parentheses. *** p < 0.01, ** p < 0.05, * p < 0.10

( 1 )

 1 = ), and ~ (x) log (x). Note that we refrain from interpreting 3 as being equal to 1 + (1) = because the number of working hours may have a direct e¤ect on income, e.g., by working hard in order to get a wage raise (Bell and Freeman, 2001). At the same time, changes in wages may a¤ect labor supply.

Figures

  Figures

Figure 2 -Figure 3 -

 23 Figure 2 -The location quality index

  x); U (! (x))] jd!j = jdxj ; a:e: Using (18), we get s(x; ! (x))H [ (x); ! (x); U (! (x))] g( (x)) d (x) dx = 1 a:e: (A.15)

  x)t(x) B(x) T (x) ; a:e: Given (20) and h (x; ! (x)) = 1= [s(x; ! (x))g( (x)) jd =dxj], we get R x (x) = s(x; ! (x))g( (x)) jd =dxj ! (x)t(x)

Figure

  Figure B.1b shows the share of commuting people who travel at most minutes, which we use to calculate employment accessibility in 1900.

  In this case, each location x is characterized by the share of households who work at i (Ahlfeldt et al., 2015; Monte et al., 2018;

	Thisse et al., 2021). This approach suits well empirical settings with a …nite number of locations
	where a location represents a block or neighborhood occupied by households whose commuting
	behavior may be described by a gravity-like equation. Accounting for this type of heterogeneity
	requires here solving a matching problem between two multi-dimensional continuous spaces
	(income-and-taste on the consumer side, amenities-and-commuting in space), a problem that
	remains unsolved when housing consumption is endogenous. However, we do not see that as
	a major weakness of our approach. As we work with a continuum of locations, households
	separated by a positive but small distance may work in di¤erent employment locations, which
	meets the empirical requirement that people living close to each other need not work at the
	same place. In addition, we will see that households who share the same income need not live at
	the same place. Therefore, such households typically work in di¤erent employment locations. In
	sum, people may have the same income while working and living in di¤erent workplace-residence
	pairs.

  . An increase in leads to less income inequality. The case of G is more delicate. By de…nition, the density g( ) is equal to integer values over a …nite number of intervals of [ ; ]. Each interval is determined by the -images of some subarcs of N . As increases from to , we assume that the number of subarcs in N where (x) takes low (i.e., close to ) or high (i.e., close to ) values is small, while (x) has intermediate values over a large number of subarcs. This amounts to assuming that g( ) is a quasi-concave step function over [ ; ]. In other words, good and bad areas are few

whereas there are many decent and pretty good areas where to live in, which seems a reasonable description of Dutch cities. In what follows, we assume that the values of (x) are drawn from a Fréchet distribution G( ) = exp ( = )

Table 2 -

 2 A hedonic amenity index (Dependent variable: the log of household gross income)

	+ Job controls	Historic instruments

Table 3 -

 3 Effects on land prices

	+ Job controls	Historic instruments

(Dependent variable: the log of land price)

  Bold indicates instrumented. Household controls include household size, mean age of adults, mean gender, household type (couple, single, kids), the share of the household that is foreign-born. Job controls are the total hours worked, whether the household has a company car, the share of full-time contracts, the share of permanent contracts. Housing controls include house type, height of the building, construction year dummies and whether a building is listed. Standard errors are clustered at the neighborhood level and in parentheses.

		OLS	OLS	2SLS	2SLS
	Pictures per ha (log)	0.1874***	0.0960***	0.2052*** 0.1910***
		(0.0054)	(0.0032)	(0.0212)	(0.0208)
	Commuting time (log)	-0.0357***	-0.0223***	-0.3114	-0.3854*
		(0.0033)	(0.0016)	(0.2518)	(0.2063)
	Household controls	Yes	Yes	Yes	Yes
	Housing controls	Yes	Yes	Yes	Yes
	Job controls	No	Yes	No	Yes
	Year fixed effects	Yes	Yes	Yes	Yes
	Travel-to-work-area fixed effects	Yes	Yes	Yes	Yes
	Workplace fixed effects	No	Yes	No	Yes
	Number of observations	2,196,324	2,196,324	2,196,324	2,196,324
	R-squared	0.2257	0.6007
	Kleibergen-Paap F -statistic			11.56	12.43

Notes: *** p < 0.01, ** p < 0.05, * p < 0.10

Table 4 -

 4 Identification revisited(Dependent variable: the log of household gross income)

	Only	+ Municipality	Control for	1909	Other	Reclaimed
	Randstad	fixed effects	current land use	skills	instruments	land

  Bold indicates instrumented. Household controls include household size, mean age of adults, mean gender, household type (couple, single, kids), the share of the household that is foreign-born. Job controls are the total hours worked, whether the household has a company car, the share of full-time contracts, the share of permanent contracts. Housing controls include house type, height of the building, construction year dummies and whether a building is listed. Standard errors are clustered at the neighborhood level and in parentheses. *** p < 0.01, ** p < 0.05, * p < 0.10

		2SLS	2SLS	2SLS	2SLS	2SLS	2SLS
	Pictures per ha (log)	0.0300***	0.0240***	0.0206**	0.0229***	0.0305***	0.0191
		(0.0102)	(0.0072)	(0.0080)	(0.0069)	(0.0070)	(0.0143)
	Commuting time (log)	-0.2427** -0.2230***	-0.5618***	-0.2962*** -0.1647**	-0.0561
		(0.1146)	(0.0861)	(0.1121)	(0.0865)	(0.0790)	(0.1077)
	Share built-up land			-0.1632***
				(0.0238)
	Population per ha (log)			-0.0234***
				(0.0041)
	Share of medium-skilled workers in 1909				-0.2068***	-0.1747***
					(0.0287)	(0.0265)
	Share of high-skilled workers in 1909				-0.1187	-0.0822
					(0.0990)	(0.0956)
	Share protestants in 1899				-0.0008	-0.0072
					(0.0113)	(0.0102)
	Population accessibility in 1900 (log)				-0.0053	0.0002
					(0.0056)	(0.0051)
	Household controls	Yes	Yes	Yes	Yes	Yes	Yes
	Housing controls	Yes	Yes	Yes	Yes	Yes	Yes
	Job controls	Yes	Yes	Yes	Yes	No	Yes
	Year fixed effects	Yes	Yes	Yes	Yes	Yes	Yes
	Travel-to-work-area fixed effects	Yes	Yes	Yes	Yes	Yes	Yes
	Municipality fixed effects	No	Yes	No	No	No	No
	Workplace fixed effects	Yes	Yes	Yes	Yes	Yes	Yes
	Number of observations	4,340,639	10,213,540	10,213,540	10,213,540	10,213,540	270,106
	Kleibergen-Paap F -statistic	7.500	18.18	10.05	16.08	21.49	0.264

Notes:

Table 5 -

 5 Structural parameters

	(1)

  2 or smaller than 25m 2 , or have lot sizes above 5000m 2 . These selections consist of less than 1% of the data and do not in ‡uence our results. We follow a similar procedure as Rossi-Hansberg et al. (2010), implying that we can only use information on residential properties with land. We are left with

	1; 337; 445 housing transactions.
	Let P(x) denote the house price in year y, H(x) the observed lot size and C(x) the housing characteristics of property x. The log land rent R(x) is equal to the …xed e¤ects at the level of
	the postcode (about 15-20 addresses), which we denote by &(x), while #(y) denote year y …xed
	e¤ects. We estimate:
	log	P(x; y) H(x; y)	=

1 C(x; y) + &(x) + #(y) + (x; y); (C.2.1)

  .1 and B.2 about here] Descriptive statistics for the housing sample are reported in Table B.1. Coe¢ cients 1 related to the housing attributes are reported in Table B.2. It appears that the house priceper square meter of land is generally a bit lower when the property is larger. However, the house price per square meter of land of properties that are (semi-)detached is generally higher. Furthermore, when the maintenance state of a property is good, prices are about 502=1269 = 40% higher. When a property has central heating, the price per square meter is about 5:1% higher. The dummies related to the construction decades show the expected signs. Properties

constructed after World War II until 1970 generally have lower prices because this is a period associated with a lower building quality. Lot sizes are inversely related to pattern of land prices ( = 0:245). In other words, more expensive locations generally have smaller lots, which makes sense.

  Table B.2 -Estimating land prices and lot sizes (Dependent variable: the log of land price per m 2 )

		(1)
	Rooms	-6.1664***
		(0.4506)
	Terraced property	702.4875***
		(6.5087)
	Semi-detached property	510.0447***
		(6.5516)
	Detached property	360.7740***
		(6.7580)
	Private parking space	-56.3558***
		(1.9988)
	Garage	-42.8166***
		(2.0556)
	Garden	47.5907***
		(2.8356)
	Number of bathrooms	17.3274***
		(0.9885)
	Number of kitchens	-7.2575***
		(1.0818)
	Number of balconies	47.8147***
		(1.5204)
	Number of roof terraces	109.0801***
		(1.8878)
	Number of floors	94.9407***
		(1.0148)
	(Internal) office space	-55.3454***
		(6.3595)
	Maintenance score of the outside	29.5137***
		(6.3366)
	Maintenance score of the inside	501.7345***
		(5.8143)
	Number of types of insulation	8.3945***
		(0.3138)
	Central heating	65.8404***
		(1.7719)
	Listed building	27.9334***
		(6.2691)
	Newly built property	-13.3758***
		(4.3108)
	3 th -order polynomial of property size	Yes
	Construction decade dummies	Yes
	Year fixed effects	Yes
	Postcode fixed effects	Yes
	Observations	1,280,031
	R 2	0.8295

Notes: Standard errors are in parentheses. *** p < 0.01, ** p < 0.05, * p < 0.10.

  TableB.5 -Determining the hedonic amenity index (Dependent variable: the log of house price per m 2 ) Housing controls include house type, house size, whether the property has a garage, garden and/or central heating, the number of layers of insulation, the maintenance quality, the number of rooms, construction year dummies and whether a building is listed. Standard errors are clustered at the neighborhood level and in parentheses. *** p < 0.01, ** p < 0.05, * p < 0.10Notes:The number of observations is 10,213,540. For land price and lot size the number of observations is 2,196,280. Because of confidentiality restrictions the minimum and maximum values refer to the 0.01% and 99.99% percentile. This implies that we exclude the bottom and top 1,024 observations.

		(1)		(2)	(3)
		OLS	OLS	OLS
	Share land in historic district <500m 0.1796***		0.1710***
		(0.0210)		(0.0204)
	Listed buildings <500m	0.0047**		0.0052**
		(0.0024)		(0.0024)
	Share water bodies <500m		0.3014***	0.2824***
				(0.0255)	(0.0253)
	Share open space <500m		0.0604***	0.0636***
				(0.0084)	(0.0084)
	Housing controls	Yes		Yes	Yes
	Neighborhood fixed effects	Yes		Yes	Yes
	Year fixed effects	Yes		Yes	Yes
	Number of observations	695,709	695,709	695,709
	R 2	0.8206	0.8207	0.8217
	Table B.6 -Descriptive statistics
		(1)	(2)	(3)	(4)
		mean	sd	min	max
	Gross income (in e)	91,535 53,683	3,589	999,897
	Land price (eper m 2 )	1,312	752.2	0.00753	22,418
	Lot size (m 2 )	364.3	923.8	25	24,824
	Pictures per ha	2.189	8.840	0	231.9
	Commuting time in minutes	26.39	17.18	0	120.0
	Hedonic amenity index	2.821	0.0915	2.723	3.885
	Share historic district	0.0347	0.139	0	1
	Listed building	0.0941	0.699	0	17.06
	Share built-up land	0.449	0.298	0.000856	1
	Share water	0.0496 0.0738	0	0.813
	Total hours worked in household	2,159	913.1	416.1	6,239
	Household has company car	0.149	0.356	0	1
	Works at single-establishment firm	0.443	0.497	0	1
	Number of jobs in household	1.511	0.968	1	18
	Person is male	0.521	0.215	0	1
	Person is foreigner	0.0718	0.217	0	1
	Age of person	41.99	9.008	18	64
	Apartment	0.153	0.360	0	1
	House built <1945	0.192	0.394	0	1

Notes:

  Table B.7 -Descriptive statistics for historic data The number of observations is 10,213,540. Because of confidentiality restrictions the minimum and maximum values refer to the 0.01% and 99.99% percentile. This implies that we exclude the bottom and top 1,024 observations Table C.1 -Pictures and commuting time: first-stage regression results Household controls include household size, mean age of adults, mean gender, household type (couple, single, kids), the share of the household that is foreign-born. Job controls are the total hours worked, whether the household has a company car, the share of full-time contracts, the share of permanent contracts. Housing controls include house type, height of the building, construction year dummies and whether a building is listed. Standard errors are clustered at the neighborhood level and in parentheses. *** p < 0.01, ** p < 0.05, * p < 0.10 Table C.3 -Other robustness checks

				(1)	(2)	(3)	(4)
		One job	+ Single	mean Company sd	min Education	max Commuting	City center
		households	plant firm		car	level	by rail	<15km
	Employment accessibility in 1909 Share of high-skilled workers in 1909 (1) (2) Share of medium-skilled workers in 1909 2SLS 2SLS	38,029 23,884 0.0298 0.0285 (3) 0.216 0.128 2SLS	1,494 0 (4) 0.00386 2SLS	163,349 (5) 0.197 0.688 2SLS	(6) 2SLS
	Population accessibility in 1900		89,184 62,641	3,008	422,544
	Share built-up land in 1900 Pictures per ha (log) 0.0239***	0.0432 0.0230*** 0.0207** 0.103	0 0.0165*	0.930 0.0294***	0.0229*
	Share water in 1900 (0.0065)	(0.0068)	0.0591 (0.0085) 0.175	0 (0.0090)	1 (0.0061)	(0.0119)
	Share locals in 1899 Commuting time (log) -0.2127*** -0.1869*** -0.2285** -0.4740*** 0.643 0.102 0.217	0.950	-0.3173**
	Share protestants in 1899 (0.0643)	(0.0691)	0.518 (0.1009) 0.337	0 (0.0927)	0.998	(0.1440)
	Commuting time by rail (log)						-0.1045**
							(0.0454)
	Household controls	Yes	Yes		Yes	Yes	Yes	Yes
	Housing controls	Yes	Yes		Yes	Yes	Yes	Yes
	Job controls	Yes	Yes		Yes	Yes	Yes	Yes
	Year fixed effects	Yes	Yes		Yes	Yes	Yes	Yes
	Travel-to-work-area fixed effects	Yes	Yes		Yes	Yes	Yes	Yes
	Workplace fixed effects	Yes	Yes		Yes	Yes	Yes	Yes
	Number of observations	6,706,524	3,532,906	1,523,567	7,626,355	10,213,540	6,023,886
	Kleibergen-Paap F -statistic	17.63	14.56		7.026	17.02	10.15	5.657
			Dependent variable:	Dependent variable:
			the log of pictures per ha	the log of commuting time
			(1)		(2)	(3)	(4)
			OLS	OLS	OLS	OLS
	Share built-up land in 1900	4.2960***	4.2413*** -0.3319*** -0.3119***
			(0.3192)	(0.3075)	(0.0364)	(0.0348)
	Share built-up land in 1900, 0-500m	0.6415	0.7537	0.3573**	0.2261
			(1.2821)	(1.2279)	(0.1483)	(0.1482)
	Share built-up land in 1900, 500-1000m 7.2817***	6.5271*** -1.0300*** -1.1923***
			(1.4426)	(1.3754)	(0.1837)	(0.1869)
	Share water in 1900		0.8896***	0.9869***	0.1885***	0.1601***
			(0.3413)	(0.3222)	(0.0452)	(0.0402)
	Employment accessibility in 1909 (log)	0.1472**	0.1624*** -0.0353*** -0.0499***
			(0.0593)	(0.0569)	(0.0084)	(0.0096)
	Household controls		Yes	Yes	Yes	Yes
	Housing controls		Yes	Yes	Yes	Yes
	Job controls		No		Yes	No	Yes
	Year fixed effects		Yes	Yes	Yes	Yes
	Travel-to-work-area fixed effects	Yes	Yes	Yes	Yes
	Workplace fixed effects		No		Yes	No	Yes
	Number of observations		10,213,540 10,213,540 10,213,540 10,213,540
	R 2		0.4907	0.5291	0.0357	0.1496

Notes:

Notes: Notes: Bold indicates instrumented. The dependent variable in columns (

1

)-(

Table C .

 C 4 -Preferences for land(Dependent variable: the log of land price)

		(1)	(2)	(3)	(4)
		NLS	NLS	NLS	NLS
	μ	0.1036*** 0.0955*** 0.1485*** 0.1375***
		(0.0003)	(0.0003)	(0.0004)	(0.0005)
	Discount rate	0.035	0.035	0.050	0.050
	Minimum lot size, h	10	25	10	25
	Number of observations 2,201,524	2,201,524	2,201,524	2,201,524

Notes: NLS stands for Nonlinear Least Squares. Standard errors are clustered at the neighborhood level and in parentheses. *** p < 0.01, ** p < 0.05, * p < 0.10

In Gaigné et al. (2020), we discuss how incomes can be determined from the interaction between individual skills and local productivity factors.

Note that b(x) may also include local disamenities as long as b(x) > 0.

For any function f (y; z), let f y (resp., f yz ) be the partial (cross-) derivative of f with respect to y (resp., y and z).

Note that the function t is not di¤erentiable at the intersection points between any two functions t i and t j .Since we assume that t is never ‡at over a positive measure set, t is a.e. di¤erentiable.

When no ambiguity may arise, we do not specify the independent variables.

We furthermore obtain information on the educational level of adults in the household. This is available for only 75% of the population, but our main speci…cations will not use these data so this does not appear to be an issue.

We exclude transactions with prices that are above e1 million or below e25; 000 and have a price per square meter which is above e5; 000 or below e500. We furthermore leave out transactions that refer to properties that are larger than 250m 2 of inside ‡oor space, are smaller than 25m 2 , or have lot sizes above 5000m 2 . These selections consist of less than one percent of the data and do not in ‡uence our results.

[START_REF] Ahlfeldt | Urbanity[END_REF] shows that in Berlin and London the picture density is strongly correlated to the number of restaurants, music nodes, historic amenities and architectural sites, as well as parks and water bodies.

The location of most pictures is determined through the GPS of cameras. Hence, in a continuous space, the probability that several pictures are taken at exactly the same location is zero. Observing multiple pictures at the same location is therefore likely caused by inaccurate geocoding.

We introduce here the number of working hours w i (x). Hence, ! captures the gross income per hour. This implies that in our theoretical set-up t (0) is normalized to 1. We are agnostic about the reasons that explain why w i (x) varies with x. According toRosenthal and Strange (2008), professionals work is longer in denser areas, while Black et al. (2014) observe that married women work more when commutes are shorter.

In Appendix C.4 we consider some robustness checks with respect to di¤erent values of the minimum lot size and the discount rate. This hardly a¤ects the estimate for .

s(x; ! (x))g( (x)) jd =dxj h 1 (1)s(x; ! (x))g( (x)) jd =dxj h B(x) T (x) ; a:e:

Alternatively, we could consider a distance-decay average of distances to the …rm's establishments. Instead, we test robustness by keeping households that have only one working-member who works during the whole year in a single-establishment …rm. This leads to nearly identical results.

[START_REF] Albouy | What are cities worth? Land rents, local productivity, and the total value of amenities[END_REF] uses information on wages and housing costs to infer the level of amenities for U.S. cities. However, his approach is not applicable here because we are interested in intra-city variation in amenities and incomes.

The authors acknowledge, respectively, …nancial supports from the Agence Nationale de la Recherche (ANR-12-INEG-0002

Appendix B. Data

In this appendix, we detail the construction of the various datasets. In Appendix B.1 we elaborate on how we calculate network distances and show the relationship with Euclidian distance. Appendix B.2 continues by explaining how we measure land prices and lot sizes for all locations. This is followed in Appendix B.3 by more information on our proxies for amenities: the picture density and the construction of the hedonic amenity index. In Appendix B.4 we introduce the historical data based on 1900 land use maps. Appendix B.5 reports distributions of the variables of interest. B.4. We have 695; 709 observations and the average house price is e229 thousand.

[Tables B. [START_REF] Albouy | Housing demand, costs-of-living inequality, and the a¤ordability crisis[END_REF] and B.5 about here] In Table B. [START_REF] Allen | Trade and the topography of the spatial economy[END_REF] we report the results of the regression of equation (B.3.1). We …rst investigate the impact of listed buildings. It can be seen that the share of historic districts leads to higher price. A 10 percentage point increase in the share of land designated as historic district increases prices by 1:8%. Listed buildings do have a small additional e¤ect of 0:5% per listed building.

In column [START_REF] Ahlfeldt | The economics of density: Evidence from the Berlin Wall[END_REF] we investigate the impact of water bodies and open space. For a 10 percentage point increase in water bodies, prices rise by 3%. Moreover, a 10 percentage point increase in open space implies a price increases of 0:6%, so this e¤ect is considerably smaller. When we put historic amenities and natural amenities together, the coe¢ cients are essentially una¤ected.

We consider this as our preferred speci…cation.

B.4 Historic data

To instrument current amenity levels and commuting time we use information on land use, the 10%. We also use information on municipal population in 1900 from NLGIS. Municipalities were much smaller at that time and about the size of a large neighborhood nowadays. We impute the local population distribution using the location of buildings and assuming that the population per building is the same within each municipality. We further use information on railway stations from [START_REF] Koopmans | An accessibility approach to railways and municipal population growth, 1840-1930[END_REF]. We enrich these data by adding missing stations from various sources on the internet and create a network with travel times. To approximate the speed, we …t a regression of the length of (current) railway segments between stations on current travel time on the railway network. Based on historic sources, it appears that the average speed is about 50% of what it is currently (about 70km/h).

We show a map of land use and the railway network for the Netherlands in 1900 in Figure

B.5 Descriptive statistics

We report descriptive statistics of the 10; 213; 524 households of our sample in Table B.6.

[Table B.6 about here] The average (median) yearly income is e91; 535 (e86; 732). The average land price in the sample is e1; 312 , but there are stark spatial di¤erences. For example, in the capital Amsterdam, it is e3; 046, while in the rural province of Friesland it is only e716. As expected, land prices are generally higher in cities. The pattern for incomes is less clear, but generally speaking we …nd that wealthier households locate close to or in cities.

[Figure B.4 about here]

As expected, the correlation between the estimated land price and lot size is negative ( = 0:245). The average lot size is 364m 2 . However, in Amsterdam it is only 253m 2 , which corresponds to the higher land values in this city. About 15% of households occupy apartments and the correlation between occupying an apartment and the land price is positive ( = 0:153).

The picture density, i.e., the proxy for amenity endowments, range from 0 to 231 pictures per hectare. Only 0:2% of the households live in neighborhoods that do not have any pictures.

We will disregard those households. The average picture density in Amsterdam (22:7) is much higher than in Rotterdam (9:63), The Hague (6:17), and Utrecht (7:66). Recall that we only use pictures outside a building taken by residents in determining the amenity index. It appears that 80% of the pictures are taken outside a building while about 60% of the pictures are taken by local residents. Going back to Table B.6, we see that the average commuting time is 26 minutes, which is very close to statistics provided by other sources (Department of Transport, Communications and Public Works, 2010). The unconditional correlation of picture density with income is close to zero ( = 0:0533), but this is not very informative as we do not control for household characteristics. The correlation of picture density with land prices is substantially higher ( = 0:431). Finally, households that have a short commute do not seem to live in high amenity locations as the correlation between picture density and commuting time is low ( = 0:0454).

In Table B.7 we provide descriptives for all instruments. The average share of built-up area in 1900 was 4:3%.

[Table B

C.1 Pictures and amenities: the …rst-stage results

We report …rst-stage estimates in Table C.1 related to the baseline results reported in Table 2.

In columns ( 1) and ( 2) we focus on the impact on picture density. We …nd strong positive impact of share of built-up land in 1900 and share water in 1900 on current picture density.

For example, when we include workplace and travel-to-work-area …xed e¤ects in column (2), increasing the share of built-up land in 1900 by 10 percentage points in the neighborhood is associated with an increase in the picture density of 42%. This makes sense: locations with many historic buildings are often locations which were already developed around 1900. The impact of water bodies is also positively correlated to the picture density, although the impact is somewhat smaller in magnitude: a 10 percentage point increase in the share of water in 1900 leads to an increase in the picture density of 9:9%.

[Table C.1 about here]

In columns (3)-( 4), Table C.1 we take commuting time as a dependent variable. The instruments for commuting time are relevant. We …nd a strong positive e¤ect of the share of built-up land in 1900 between 500 and 1000m on accessibility. Likewise, employment accessibility in 1909 has a strong negative e¤ect on current commuting time. More speci…cally, in column (4), doubling employment accessibility in 1909 is associated with an decrease in commuting times of 3:5%.

C.2 Bias-corrected estimates

Many non-experimental papers use coe¢ cient movements after the inclusion of control variables to investigate whether omitted variable bias is important. Oster (2019) argues that coe¢ cient movements alone are not a su¢ cient statistic to calculate the bias. Instead, she argues that whether omitted variable bias is a concern depends on the variance of the added control variables, as well as coe¢ cient movements. In other words, changes in the coe¢ cient(s) of interest after adding controls should be scaled by the change in the R 2 . Oster (2019) then derives an estimator to correct estimates for omitted variable bias under the assumption that the relationship between the variables of interest and unobservables can be recovered from the relationship between the variables of interest and observable control variables. In our context, this assumption makes sense as control variables that are added bear some potential relationship to unobservables. To be precise, we add many housing, demographic and job controls, as well as workplace and travel-to-work-area …xed e¤ects, which are likely to be correlated to unobservables.

Oster (2019) then derives a GMM estimator to derive bias-corrected estimates of the impact of amenities and commuting time on incomes. There are two key input parameters that have to be determined. First, a parameter must be chosen that determines the relative degree of selection on observed and unobserved variables, which we denote by . Although this parameter is fundamentally unknown, Altonji et al. (2005) and Oster (2019) show that = 1 is a reasonable (upper-bound) value. Second, there is the maximum R 2 from a hypothetical regression of income on amenities, commuting time and controls, which we denote as R 2 max . Because of measurement error and random variation in incomes, R 2 max is usually well below 1. Oster (2019) considers to set R 2 max = R2 , where R2 is the R 2 obtained from the regression of log income on the variables of interest and all controls and …xed e¤ects. She provides some evidence based on experimental studies that 1:3. To be on the safe side, we consider = 1:5 and = 2.

We report results in Table C.2

[Table C.2 about here]

In column (1) we consider = 1:5. We …nd that the bias-corrected 1 = 0:026 , which is virtually identical to the preferred estimate with historic instruments. The bias-adjusted impact of commuting time is now negative ( 1 = 0:058), in line with instrumented estimates reported in Table 2. However, the magnitude is somewhat smaller: doubling commuting time attracts households whose incomes are 4% lower. However, when the hypothetical R 2 max is higher, so that it is twice the R 2 obtained from the regression of log income on the variables of interest and all controls and …xed e¤ects ( = 2), we see that the impact of commuting times becomes considerably stronger. The coe¢ cient in column [START_REF] Ahlfeldt | The economics of density: Evidence from the Berlin Wall[END_REF] indicates that a 100% increase in commuting time attracts households whose incomes are 31% lower, which is very much comparable to the baseline IV results. The impact of amenities with = 2 is somewhat stronger: doubling the picture density attracts households whose incomes are 5:9% higher.

In column (3) of Table C.2 we further investigate whether these bias-adjusted estimates are robust to the inclusion of more detailed …xed e¤ects. More speci…cally, we further include municipality …xed e¤ects and assume = 1:5. The impact of amenities is very similar to the result reported in column (1) (as well as the baseline result). The coe¢ cient regarding commuting time is about twice as strong, but somewhat in between the results reported in columns ( 1) and ( 2). Overall, the results are robust.

In other words, although we do not know the 'true' value for , these results strongly

C.4 Preferences for land

Table C.4 below reports the estimated values for given di¤erent assumptions on the discount rate and the minimum lot size.

[Table C.4 about here]

We show that with h = 10m 2 , and a discount rate of 3:5% (see [START_REF] Koster | How do households value the future? Evidence from property taxes[END_REF], ^ = 0:1036, which con…rms that richer households spend less of their income on land [START_REF] Albouy | Housing demand, costs-of-living inequality, and the a¤ordability crisis[END_REF]. Note that ^ may seem low, but we only include payments to land, not to housing itself. If we set h = 25m 2 , ^ = 0:0955, so it is hardly a¤ected. We consider this to be the preferred estimate. When we increase the discount rate to 5%, the estimates of ^ slightly increase to about 0:14.

Appendix tables