metadata: authors: 0: author: Hallegatte,Stephane
1: author: Viguie,Vincent
count: France
docna: 0: docna: Urban infrastructure investment and
            rent-capture potentials
docty: Policy Research Working Paper
subtopic: Urban Housing and Land Settlements,Transport in Urban Areas,Urban Governance and Management,Urban Housing,Urban Transport,Municipal Management and Reform,Transport Services
teratopic: Urban Development,Transport,Public Sector Development,Macroeconomics and Economic Growth,Communities and Human Settlements
abstracts: cdata!: In a context of rapid urbanization and
            energy transition, massive investments will be required to
            develop efficient public transport networks. Capturing the
            increase in land value caused by transport infrastructure
            (for example, through a betterment tax) appears a promising
            way to finance public transport. However, it is no trivial
            task, as it is difficult to anticipate the rent creation.
            This paper uses a simple city model based on urban economic
            theory to compute the rent created by improvements in public
            transport infrastructure in Paris, France. To apply in
            places where models or data are not available, a reduced
            form of the model is shown to provide acceptable
            approximations of the rent creation. Simulations confirm
            that land value capture can finance a significant part of
            transport investments. The simulations also show that value
            capture potentials are influenced by what happens in the
            entire agglomeration. Simultaneous infrastructure
            investments in different parts of the city play a
            significant role, as they change overall accessibility
            patterns. Evolutions taking place in other cities also have
            a comparable influence. Non-local effects can change the
            total potential for land value capture and multiply this
            potential by as much as a factor of two.
geo_reg_key: 517190,517191
repnme: repnme: Urban infrastructure investment and
            rent-capture potentials
display_title: Urban infrastructure investment and
            rent-capture potentials
listing_relative_url: /research/2014/10/20300168/urban-infrastructure-investment-rent-capture-potentials
topicv3: Transport m1326661 135,Mass Transit m1327372 1726,Transport Infrastructure m1327632 1803,Land and Housing m1327648 817,Population, Aging and Health m1327736 640
publishtoextweb_dt: 2014-10-01T00:00:00Z
historic_topic: Urban Development,Transport,Public Sector Development,Macroeconomics and Economic Growth,Communities and Human Settlements
no_of_pages: 41
author_keys: 0: author_key: 1103816
1: author_key: 957932
model_extractions: WPS7067 Policy Research Working Paper 7067 Urban Infrastructure Investment and Rent-Capture Potentials _V. Viguié S. Hallegatte_ Climate Change Group Office of the Chief Economist October 2014
Policy Research Working Paper 7067

# **Abstract**

In a context of rapid urbanization and energy transition, massive investments will be required to develop efficient public transport networks. Capturing the increase in land value caused by transport infrastructure (for example, through a betterment tax) appears a promising way to finance public transport. However, it is no trivial task, as it is difficult to anticipate the rent creation. This paper uses a simple city model based on urban economic theory to compute the rent created by improvements in public transport infrastructure in Paris, France. To apply in places where models or data are not available, a reduced form of the model is shown to provide acceptable approximations of the rent creation. Simulations confirm that land value capture can finance a significant part of transport investments. The simulations also show that value capture potentials are influenced by what happens in the entire agglomeration. Simultaneous infrastructure investments in different parts of the city play a significant role, as they change overall accessibility patterns. Evolutions taking place in other cities also have a comparable influence. Non-local effects can change the total potential for land value capture and multiply this potential by as much as a factor of two.

This paper is a product of the Office of the Chief Economist, Climate Change Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at viguie@centre-cired.fr and shallegatte@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Urban Infrastructure Investment and Rent-Capture Potentials

V. Viguié<sup>a,1,</sup><sup>_∗_</sup> , S. Hallegatte<sup>b</sup> _aCIRED, Ecole des Ponts ParisTech, Paris, France bThe World Bank, Washington D.C._ _Keywords:_ Public transport, urban economics, transport, modeling, public finance _JEL:_ R14, R4, R51, H54, H71

## **1. Introduction**

Efficient public transport systems in cities lie at the core of sustainable development (Wright and Fulton, 2005; Dodman, 2011; Hoornweg et al., 2011; UN-HABITAT, 2011). However, financing both their development and maintenance at a global scale is challenging, because fares do not generally cover full costs. Capturing part of land value increase due to transport infrastructure construction, i.e. land value capture, has often been proposed as a promising alternative method of revenue generation and as a way to address this challenge (see for instance Batt, 2001; Martinez and Viegas, 2007; Peterson, 2009). It might be achieved through different policies such as land value taxes or land betterment taxes (see for instance Plimmer and McGill, 2003; Doherty, 2004; Rybeck, 2004; Medda, 2012).

The concept of land value capture has a history which dates back to Henry George, in the late 19th century (George, 1884; Arnott and Stiglitz, 1979; Arnott, 2004). The basic idea is that successful public transport systems lead to higher land values, as a result of the increase in accessibility, which can be considered as an increase in rent for some landowners. Capturing some or all of this increment in land value (through a betterment tax) can enable the recovery of part of the capital cost of the transport investment. It has been used since then in different cities across the world. Recent examples include Bogotá (Borrero et al., 2011; Peterson, 2009), London (Medda, 2012), Singapore, Hong Kong (Hong Kong SAR, China) (Hui et al., 2004), and various cities in Brazil, Argentina and India (Walters, 2012).

If such a betterment tax can be seen as an equitable, efficient (Mattauch et al., 2013) and easily understood levy, its implementation is however not a trivial task. It can lead to numerous difficulties, among which are political acceptability (Booth, Philip A, 2012), revenue volatility (Peterson,

> _∗_ Corresponding author

> _Email address:_ viguie@centre-cired.fr (V. Viguié)

> 1Tel: (33) 1 43 94 73 03 Fax: (33) 1 43 94 73 70, CIRED, Site du Jardin Tropical, 45bis, Av de la Belle Gabrielle F-94736 Nogent-sur-Marne, France.

_October 16, 2014_
2009), administrative feasibility or inequality creation when for instance some inhabitants are asset-rich but cash-poor (Medda, 2012). Another main issue is revenue predictability, i.e. the fact that it is difficult to anticipate the rent creation. We focus this study on this last point.

Econometric studies have been extensively used to analyze past impact of transport developments on housing and land prices (see reviews in Smith and Gihring, 2006; Stefania Radopoulou et al., 2011; Salon and Shewmake, 2011). This enables to retrospectively quantify the benefits that land value capture could have brought, and serves as a basis for future estimations. However, it is not directly possible to fully rely on econometric-based coefficients to anticipate land value potential. Indeed, transport is not a “standard” public amenity: contrary to, say, the creation of an attractive park or the presence of a good school, accessibility is not an amenity per se (Mohring 1961). Accessibility is only valuable if it enables travel to interesting places. To adequately analyze benefits provided by increased accessibility, and therefore how much can be potentially capitalized in land prices, the whole city should be taken into account and whole-city models should be used.

Using first a theoretical urban economic model, and then an applied model, Nedum-2D, calibrated on Paris urban area, we propose here to show how urban models can capture important mechanisms that cannot be taken into account when using only constant elasticities (of land prices in response to transport investment) from the literature. To do that, we analyze and compare with the model rent creation and land value capture potential corresponding to different public transport investments in the Paris urban area. The results of the applied model that we use here are fully consistent with urban economic theory, but they provide numerical estimations, taking into account geographical specificity of a given city.

The same land price increase will have very different impacts on households when they are landowners or when they are renters. In one case, it is indeed an increase in wealth, whereas in the other case it is often translated into an increase in housing expenditures. The impact of transport improvement on household welfare therefore strongly depends on the hypothesis about who owns the land (Brueckner, 2005; Gusdorf and Hallegatte, 2007). In this study, we do not consider welfare impacts, and focus only on land price variations. We also consider that absentee landowners own the land, i.e. that land price variations do not directly impact household income, and that rents are not redistributed to the city population.

These simulations show that a significant fraction of infrastructure costs (almost all costs, under optimistic hypotheses) can be extracted from land value increase, and that one of the main factors influencing rent creation is population increase in the city due to the infrastructure construction. Simulations also show that land value capture potential is affected by other transport infrastructure developments in the city. Especially, in a sequence of several infrastructure developments, land value capture potential depends on the timing of each investment.

These results are derived on the case of the Paris agglomeration, but the qualitative findings are valid in any large city. Moreover, we are well aware that the data and models needed to replicate this analysis are not available in all locations, especially in developing countries. This is why we also propose a reduced form model that approximates the full model (within a 15% error margin in the case of Paris) and can be applied with limited resources and little data. This reduced form model can be used to assess the potential for rent capture and identify cases where more in-depth studies are necessary.

2
Section 2 recalls main urban economics results on rent creation, section 3 presents simulations on Paris, section 4 discusses our results and section 5 concludes.

# **2. A simple theoretical model**

Let us first recall how the construction of a new public transport line can induce betterment, using a simple model derived from standard urban economic modeling (Alonso, 1964; Mills, 1967; Muth, 1969). Such a framework is particularly suited because it aims at explaining the link between accessibility and land price.

# _2.1. Urban economics framework_

Let us use here a simple monocentric model (Fujita, 1989) and suppose that households choose their accommodation location and size by making a trade-off between the time and money they spend in transport (i.e. to commute to their jobs, which are all supposed to be in the center of the city) and the real estate price level (or, equivalently, between the proximity to the city center and the housing surface they can afford). Let us also suppose that households budget is divided between transport _T_ ( _x_ ), housing _q_ ( _x_ ) _.R_ ( _x_ ) (dwelling size _q_ ( _x_ ) multiplied by unitary rent _R_ ( _x_ )) and composite good _Z_ ( _x_ ) consumption: where _x_ is the location in the city. We can model the household trade-off using the following utility function: which is increasing with respect to _Z_ and _q_ . We suppose that, given rent level _R_ ( _x_ ), and transport costs _T_ ( _x_ ), households choose their housing _q_ ( _x_ ) and composite good _Z_ ( _x_ ) consumption to maximize their utility level. If several transport means are available, transport means choice can be included in the maximization, which will result in the choice of the cheapest transport mode.<sup>2</sup> Finally, we suppose that absentee landowners own the land, and try to maximize the housing rent paid by inhabitants, under the condition that utility _U_ should be constant across the city. This leads to a bid-rent function _R_ <u>(</u> _T_ ( _x_ ) _,U_ ), where _U_ is a constant equal to the value of utility. For a given _U_ , the function _R_ ( _T_ ( _x_ ) _,U_ ) decreases as transport costs _T_ ( _x_ ) increase (Fig. 1a).

The constant _U_ is given through a condition on the total population of the city. If _H_ ( _x_ ) is total dwelling space available at each location _x_ , population density is equal to _n_ ( _x_ ) =<sup>_H_</sup> _q_ (<sup><u>(</u></sup> _x_<sup>_x_</sup> )<sup><u>)</u>andtotal</sup> _H_ <u>(</u> _x_ <u>)</u> population in the city is equal to _P_ = ́ _n_ ( _x_ ) = ́ _q_ ( _x_ )<sup>.</sup><sup>_q_(</sup><sup>_x_)implicitlydependsonrentleveland</sup> transport costs, and for given transport costs _T_ ( _x_ ) and a given total population _P_ 0, utility level _U_ can be obtained by solving the equation:

> 2Transport cost should rather be named a "generalized" transport cost, and includes transport monetary cost as well as costs associated to travel time and travel comfort.

3
Any change in the city structure (through a variation in transport prices, or in income, for instance) will lead to a change in this equation. This can be either reflected by a change in _U_ , keeping _P_ constant (“closed city case”) or by a change in _P_ keeping _U_ constant (“open city case”).

It is often convenient to express all equations in terms of _R_ 0 instead of _U_ , where _R_ 0 is the rent level in the center of the city (i.e. where _T_ ( _x_ ) = 0 ). This can be done because there is a one to one relationship between the two values, and _R_ 0 strictly decreases when _U_ increases. This is what we will do in the rest of this paper. Intuitively, increasing _R_ 0 increases the total population in the city (it makes _q_ ( _x_ ) decrease everywhere), whereas decreasing _R_ 0 decreases total population.

A more detailed description of the model is given in Appendix A, and more information about the model and its resolution can be found in the classic book by Fujita (1989).

# _2.2. Impact of new transport infrastructure_

To see how the construction of a new public transport line can induce betterment, let us compare a city _a_ to a city _b_ exactly similar, except for the existence of better transport infrastructure.

Fig. 1 illustrates this situation, in a simple case of a one-dimensional city where transport costs increase linearly in city _a_ (Fig. 1a) and where one public transport station is constructed in city _b_ (Fig. 1b).<sup>3</sup> Intuitively, in places where the new infrastructure provides increased accessibility, _T_ ( _x_ ) decreases which makes rents _R_ ( _R_ 0 _, T_ ( _x_ )) increase. Therefore, in places where people use the new infrastructure, rents increase. This first phenomenon shows how increased accessibility can be capitalized in land rents, and therefore, land prices. However, another phenomenon can occur, which should not be forgotten : a variation of _R_ 0. This variation is determined by a hypothesis which should be made on the city population evolution.

# _2.2.1. Two different cases_

_Closed city case._ Let us first suppose that the population of the city remains constant, or, equivalently, that city population evolution is exogenously given, and is similar in city _a_ and city _b_ . Because of increased rents, housing consumption _q_ ( _x_ ) will decrease in places with improved accessibility, and population density will increase. To keep total population constant, rent level _R_<sup>_b_</sup> 0 in the center of the city _b_ therefore has to be smaller than in city _a_ . This makes rents decrease in city _b_ in places where accessibility has not improved.<sup>4</sup> This decrease in _R_<sup>_b_</sup> 0<sup>corresponds to a gain</sup> in utility _U_<sup>_~~b~~_</sup> : because of transport improvement households utility has increased.

> 3We speak here about "public transport stations" and about "infrastructure", but of course the transport improvement modeled here can be in practice any possible transport improvement, be it the construction of a new road, the increase of traffic speed, the increase in public transport comfort (leading to a reduction if transport generalized cost) etc.

> 4For instance, if utility is given by a Cobb-Douglas function of _Z_ and _q_ , rents are all divided by the same factor in city _b_ in places where accessibility has not improved. See Appendix C.1.

4
<!-- Start of picture text --> Travel costs<br>:ihoo Bid-rent<br>vt \<br>v .<br>Le oa 7 N SO .<br>Job center<br><!-- End of picture text --> <!-- Start of picture text --> . .<br>Travel costs Location of public<br>a transport station a<br>Overrent decrease bid- Jwih Local increase<br>a close to the<br>“Yt s tT station<br>Y? ¢ \ SS :: url7’ . ::<br>population who : population who<br>commutes directly. i: uses the public.<br>to job center transport station<br><!-- End of picture text -->
initial transport network (Arnott and Stiglitz, 1981).<sup>6</sup> A general land tax in the entire city may thus be inappropriate to finance the transport infrastructure.

_Open city case._ Let us now suppose that the population of the city does not remain constant. Because of the transport investment, it is for instance reasonable to anticipate that the city will become more attractive, and that its population will increase (we suppose here that all new inhabitants can get a job, i.e. the number of jobs in the city increases together with its population). If total population is larger in city _b_ than in city _a_ , it is no longer possible to use the former argument, and there is no reason that _R_<sup>_b_</sup> 0<sup>_< Ra_</sup> 0<sup>.</sup> A useful hypothesis is to suppose that the utility in the world outside of the city is constant, and that people are free to move in and out of the city. In this case, the population of the city will vary so that the utility of the inhabitants remains constant and equal to this utility in the “rest of the world” (the "open city" case).<sup>7</sup> It is easy to prove that, in this case, _popa < popb_ and _R_<sup>_b_</sup> 0<sup>=</sup><sup>_Ra_</sup> 0<sup>(seeagainAppendixB):the</sup> transport investment increases rents where transport accessibility increases, and does not impact them in the rest of the city. In this case, there is therefore a true capitalization of accessibility gains into land and housing rents. However, it is important to note that the change in total population in the city is the key factor, and determines whether capitalization will fully take place or not, and if rents will increase in all places where accessibility has increased or not.

# _2.2.2. Rent variation_

To sum up, rent variation in the city will be the sum of the actual capitalization of accessibility change and of a global decrease of rents due to _R_ 0 change, which will depends on the hypothesis made on city population evolution. It can actually be shown (see Eq. A.3 in Appendix A) that rent variation can be written the following way:<sup>8</sup> (1) Whereas the second term can lead to value capture, the first term prevents full capitalization (and thus capture) of accessibility gains. This first term depends on the scale of the transport investment (it increases with this scale) and decreases if city population is increased by the transport

> 6 _R_ 0 is always lower in city _b_ than in city _a_ , but the average of rents across the city can be higher or lower in city _b_ than in city _a_ .

> 7Another interesting possibility, in a slightly different framework, is considered to derive Henri George Theorem (George, 1884; Arnott and Stiglitz, 1979; Arnott, 2004). It is the case in which the city population remains optimal with respect to households utility, when taking into account production and consumption of a pure local public good, whose cost is divided among all inhabitants, but whose benefit is shared by everybody (i.e. in a non-rivalrous way). In such a framework, aggregated land price increase due to transport infrastructure improvements is strictly equal to transport infrastructure improvements cost for the city (considered as an increase in pure local public good cost). This theorem can be extended to the macroeconomic scale (Edenhofer et al., 2013; Mattauch et al., 2013).

> 8where ε<sup>_q_</sup> _R_ 0<sup>=</sup> ∂<sup>∂</sup> _R_<sup>_<u>q</u>_</sup> 0 _Rq_ <u>0</u><sup>is dwelling size elasticity to rent in the city center.</sup>

6
investment. It is only in the limiting case of the open city that _dR_ ( _x_ ) = _− q_ (<sup><u>1</u></sup> _x_ )<sup>_dT_(</sup><sup>_x_) and that all the</sup> added value can be captured.

This is an important idea when implementing land value capture because this means that, except if the transport investment has a significant impact on the city population evolution, land value capture has a smaller potential as a financing tool for large scale investment than for local investments.

Formula 1 can be greatly simplified when utility function is explicitly specified (Appendix C). A case which is especially interesting for its simplicity is the case in which utility function is given by a Cobb-Douglas function ( _U_ ( _Z, q_ ) = _Z_<sup>α</sup> _q_<sup>β</sup> with α + β = 1). In this case, we have (see Eq. C.2 in Appendix C) : This formula enables to easily compute orders of magnitude of rents variation due to transport accessibility changes in the open city case. For instance, if β = 0 _._ 3 (this is approximately the case in Paris, for instance) and if transport represents, say, 15% of a household’s budget<sup>9</sup> and<sup>_<u>dT</u>_</sup> _T_<sup>= 10%</sup> (i.e. _YdT−T_<sup>=</sup><sup>_−_1</sup><sup>_._76%), then</sup><sup>_<u>dR</u>_</sup> _R_<sup>_≈_6% :a 10% reduction in transport (generalized) cost leads to a</sup> 6% increase in rents. This formula can also be used when variation in _R_ 0 is negligible, i.e. when transport improvement is very localized in the city.

Computing an approximation of the expected variation in _R_ 0 in a closed-city case (or, equivalently, population variation in the open-city case) can be done with a few approximations (see Appendix C.2.1). If we suppose that _Y ≫ T_ (i.e. assume that most of the population lives in areas not too far from city center) and that<sup>_<u>dT</u>_</sup> _Y_<sup>isaconstantτforafraction</sup><sup>_F_oftotalpopulation(we</sup> approximate that transport price variation has a certain negative value for a fraction of the city population, and is equal to zero for the rest of the population), then _R_ 0 variation is approximately given by: where _P_ is city population. Therefore, in the closed city case,<sup><u>δ</u></sup> _R_<sup>_R_</sup> 0<sup><u>0</u></sup><sup>_≈_</sup><sup><u>α</u></sup> β<sup>_._τ</sup><sup>_.F_and, in the open-city</sup> case, population variation is approximately given by<sup><u>δ</u></sup> _P_<sup>_<u>P</u>≈−_</sup><sup><u>α</u></sup> β<sup>_._τ</sup><sup>_.F_.Forinstance,withprevious</sup> values, if transport cost is decreased by 10% for 12% of the population, then<sup><u>δ</u></sup> _R_<sup>_R_</sup> 0<sup><u>0</u></sup><sup>_≈−_</sup> 0<sup><u>0</u></sup> _._<sup>_<u>.</u>_</sup> 3<sup><u>7</u></sup><sup>_×_15%</sup><sup>_×_</sup> 10% _×_ 12% = _−_ 0 _._ 42%. This decrease in _R_ 0 is much smaller (more than 10 times) than the increase in _R_ that we have computed with Eq. 2 for places with improved accessibility.

Combining Eq. 2 and Eq. 3, we get: 9It should be noted that this share is defined here as transport generalized cost (including cost of time, for instance) divided by households budget. It can differ sensibly from actual transport cost share of households budget.

7
<!-- Start of picture text --> Travel cost to go to the city center (€)<br>50 A , 08 = 0 U00 NX<br>40<br>s 50000 50g p<br>£ 30 40000 %. QO<br>SWS SFSj g 30009. ¥%&,<br>8 SPS SF os °<br>= 10 i yr of 3 \<br>% 0 s C $e) 33<br>sO7 -102 3 0Q §s é<br>& 2 % 00 Ss<br>© -20 % cS<br>2 % ?<br>> -30 %, 30000 RS<br>° 40000 Ss Ss<br>-40 S<br>5 \ Nop 50000 S y<br>0 -40 -30 -20 -10hana0 10 20 30 40 50<br>Horizontal distance (km)<br><!-- End of picture text --> <!-- Start of picture text --> Travel cost to go to the city center (€)<br>50WA ; ; RS ; 0, ="Coy NI<br>40 se wy se oF Public transport<br>30 poo stations ?<br>Z20/s£ “Ub, Cp,0% %<br>8 SE S % @<br>= 10 S Se ¥ § of y<br>ao 8 ( 3 8<br>3s7-10 8 3ro 0,a S$m § }<br>2 00 s é<br>© -20 % Ss §<br>Q % 30000 Ss §<br>> -30 XD, Oo s<br>-40\>, eero ogy 4000050000 oo vy,Ss<br>50 -40 -30 -20 00na-10 0 10 eo20 30 40 50<br>Horizontal distance (km)<br><!-- End of picture text -->
<!-- Start of picture text --> Rent level variation (€/m?/year)<br>50 10<br>40 W y,yoyo * 2. ~—_—— Public transpolpol<br>30 LONG a stationsNG a stations a stations stations 8<br>ea=~=~ 20 Z<br>840 6<br>5<br>a 0 ? 4<br>2410 | oS : 6<br>S \ —_ f 2<br>t -20 \ J<br>> 30 x Z<br>YS 0<br>-40 — _~*<br>55 -40 -30 -20 -10 0 10 20 30 40 50 2<br>Horizontal distance (km)<br><!-- End of picture text -->

<!-- Start of picture text --> Rent level variation (€/m?/year)<br>50 10<br>40 Jaa ? ~—__ Publici Publicii transpo!<br>30 ( “ey stations 8<br>é== 20 “oycohcoh Za<br>6<br>855 10<br>a 0 4<br>= 40<br>s 2<br>t -20<br>> 30<br>0<br>-40<br>a -40 -30 -20 -10 0 10 20 30 40 50 2<br>Horizontal distance (km)<br><!-- End of picture text --> <!-- Start of picture text --> Rent level variation (€/m?/year) Rent level variation (€/m?/year)<br>50 10 50 10<br>40 W y,yoyo * 2. ~—_—— Public transpolpol 40 Jaa ? ~—__ Publici Publicii transpo!<br>30 LONG a stationsNG a stations a stations stations 8 30 ( “ey stations 8<br>ea=~=~ 20 Z é== 20 “oycohcoh Za<br>840 6 6<br>5 855 10<br>a 0 ? 4 a 0 4<br>2410 | oS : 6 = 40<br>S \ —_ f 2 s 2<br>t -20 \ J t -20<br>> 30 x Z > 30<br>YS 0 0<br>-40 — _~* -40<br>55 -40 -30 -20 -10 0 10 20 30 40 50 2 a -40 -30 -20 -10 0 10 20 30 40 50 2<br>Horizontal distance (km) Horizontal distance (km)<br>Spatial distribution of housing prices variation values Spatial distribution of housing prices variation values<br>5000 ————— 5000 =<br>MHousing prices decrease Ml Housing prices decrease<br>Mil Housing Housing prices increase Ml Housing Housing prices increase<br>4000 4000<br>EE<br>=30003000 = 3000 3000<br>oS iS]<br>22<br>©@<br>‘se 2000 ‘s 2000<br>3° °<br>FE Ee<br>1000 1000<br>0 0<br>-10 0 10 20 30 40 50 -10 0 10 20 30 40 50<br>Rent variation (€/m?/year) Rent variation (€/m?/year)<br><!-- End of picture text -->

<!-- Start of picture text --> Spatial distribution of housing prices variation values<br>5000 —————<br>MHousing prices decrease<br>Mil Housing Housing prices increase<br>4000<br>=30003000<br>oS<br>©@<br>‘se 2000<br>3°<br>FE<br>1000<br>0<br>-10 0 10 20 30 40 50<br>Rent variation (€/m?/year)<br><!-- End of picture text --> <!-- Start of picture text --> Spatial distribution of housing prices variation values<br>5000 =<br>Ml Housing prices decrease<br>Ml Housing Housing prices increase<br>4000<br>E<br>= 3000 3000<br>iS]<br>@<br>‘s 2000<br>°<br>Ee<br>1000<br>0<br>-10 0 10 20 30 40 50<br>Rent variation (€/m?/year)<br><!-- End of picture text -->
|coeffcients|Value|
|---|---|
|α|0.3|
|β|0.7|
|Population|5 000 000 households|
|_Y_|50 000e/year|
|_T_(_x_)|100e/km/month|
|_H_(_x_)|constant, 1 m2 foor space/m2 ground|
||space|

Table 1: Coefficients of the numerical simulation. A Cobb-Douglas utility function is used: _U_ ( _Z, q_ ) = _Z_<sup>α</sup> _q_<sup>β</sup> togram of rent changes (Fig. 3d) is similar to the histogram in the closed city case (Fig. 3c), except that the negative part has disappeared, and that rents increase are slightly higher in each location. Overall land price variation is therefore positive: total aggregated yearly land rent in the city increases by 1.5% (i.e. almost 700 million euros). In this simulation, to maintain a constant utility level, the population has to increase by 2%.

# **3. Application to the Paris urban area**

With standard urban economic modeling, when transport accessibility in a city is improved, rent variation is given by the competition between two opposing forces: accessibility capitalization and a global, negative, rent change. How the two effects compare, however, is unclear, and it depends on the city characteristics, i.e. on the calibration of this economic model.

To push the analysis further, and see what practical consequences these conclusions can have, we therefore need to focus on a case study, and calibrate a model derived from urban economic framework to compute rent creation and land value capture potential in a realistic case. We use in the following analysis a land-use transport interaction model (LUTI) called NEDUM-2D.

LUTI are applied models which aim at describing how transport investments can change a given city structure. Numerous models exist (for a review, see for instance Hunt et al., 2005; Iacono et al., 2008), and NEDUM-2D has the particularity of being a simple model entirely based on urban economics framework. A few other LUTI models share these characteristics to some extent, and have been proposed and used in similar situations (cf. Anderstig and Mattsson 1992; Anas 1995; Elhorst and Oosterhaven 2002 ).

# _3.1. NEDUM-2D_

NEDUM-2D (Viguié, 2012; Viguié and Hallegatte, 2012) is a land-use transport interaction model which relies on the classical urban economics’ framework. It aims at explaining the spatial distribution – across the city – of the costs of land and of real estate, housing surfaces, population densities and building heights and densities.

The equations of the model are slightly more complex than the equations presented in Sec. 2.1. One major difference is that housing construction _H_ ( _x_ ) is endogenous and driven by changes 10
in rents.<sup>10</sup> Another difference is that households can choose between different transport modes (a discrete choice model is used to compute modal shares). A third difference comes from the introduction of transaction costs and a rent border which have an influence on rent levels and city size.

We use this model calibrated on the Paris urban area, with a grid of 100km2 with cells of 0.25 km2. The model accounts for land-use constraints on the Paris agglomeration. Transportation costs include monetary costs such as the cost of gasoline and the cost of time. We assess them using the spatial structure of the Paris transportation networks (roads and public transport).

As described in Appendix D, a validation of the model over the 1900–2010 period shows that the model reproduces the available data on the city’s evolution fairly faithfully and captures its main determinants. It also reproduces the spatial distribution of dwelling size, population density, and rents in the urban area fairly well. These results suggest that this tool can be used to inform policy decisions. This model, calibrated on Paris, has been used in several other studies (Viguié and Hallegatte, 2012; Viguié, 2012; Avner et al., 2013; Viguié et al., 2014).

# _3.2. Increase in the speed of one local train_

Let us analyze the effects of a theoretical increase by 25 % in the speed of one local train line (“RER B”) of Paris urban area. "RER B" is one of the main transport axis of Paris urban area, crossing Paris city, and linking this city with many of its suburbs. It has 47 stations, and an average 900,000 commuters daily traffic, which makes it the second most used public transport line in the Paris urban area.<sup>11</sup> Due to the planned important development of the places this line crosses, important modernizing works have been launched.<sup>12</sup> The 25 % speed increase we are considering here is a theoretical example, that we will simply use as an illustration of potential improvement. Whereas we call it a “speed improvement”, it is important to note that, in the simulations, this improvement simply corresponds to a decrease in generalized transport costs, which could equivalently be interpreted, for instance, as improvements in comfort (AC, security etc.).

_Closed city case._ Let us first do an analysis in a closed city case, i.e. let us suppose that city population is exogenously determined and is not impacted by the transport investment. Fig. 4 shows a map of rents change due to the improvement in transport. As was observed in Sec. 2.3, rents increase in places with improved accessibility, and this is counterbalanced by a general decrease everywhere else. Rents decrease occurs over a much wider area (5800 km2) than rents increase (1000 km2), but the value of the decrease is extremely small and almost negligible compared to the increase : taking into account population density, averaged housing price decrease and increase

> 10To sum up, two main mechanisms drive the model. First, households choose where they will live and the size of their accommodation by assessing the trade-off between proximity to the city center and housing costs. Living close to the city center reduces transportation costs, but housing costs (per unit of area) are higher there. Second, we assume that landowners combine land with capital to produce housing: they choose to build more or less housing (that is, larger or smaller buildings) at a specific location depending on local real estate prices and construction costs.

> 11Source: RATP (www.stif.org/IMG/pdf/Presentation_de_la_SNCF_et_de_la_RATP_au_comite_de_ ligne_du_RER_B_du_3_novembre_2009.pdf).

> 12 See for instance http://www.modernisation-rerb.fr/.

11
<!-- Start of picture text --> N Pontoise @ an ne<br>Nee C Sarcelles fo ; f<br>vovwNanterre e-trl ad oe, pii<br>ffA<<br>\a Paris ey, J Jorcy<br>Versailles os<br>ow eeJ 4] | + "RER B" stations<br>) Palaiseau yp | "RER B" lines<br>7 wi peovonsend ~~~ Administrative boundaries<br>ao % Rent variation<br>i (_j0<br>a (1+2.5%<br>\ Evry |mm+5%<br>5im [ee0 5 10ee ore15| km = ; ee‘0<br><!-- End of picture text -->
created, for an estimated 7 billion euros<sup>14</sup> . In a rough approximation, this means that the order of magnitude of the estimated cost of planned “RER B” train line improvements is around a billion euros. Rent value capture potential therefore seems comparable with this cost: a 1 billion euros sum could be covered in about 8 years, if, say, half of annual rent increase is captured. What should be done in places where rents, and therefore land prices, decrease? There is in theory no easy answer, as a decrease in rent level is a gain for the inhabitants (they pay less for similar dwelling characteristics, including accessibility), but is a loss for landowners. In practice, however, the simulated rent decrease is so small that it is almost negligible, and this issue might pragmatically be discarded.

_Comparison with the open city case._ Let us now conduct a similar analysis in an open city case. As was explained in Sec. 2.3, in this case, rents increase in places with improved accessibility and do not change in the rest of the city.<sup>15</sup> Rent increases are uniformly larger than in the closed city case. Quantitatively, we simulate that averaged (taking into account population density) and maximum housing price increases are respectively +4.6 ~~C~~ /m2/year and +20 ~~C~~ /m2/year. Housing prices increase over 1000 km2.

To get these results and maintain a constant utility level in the city, the total population has to increase by 3.2 %, i.e. almost 200,000 households. In this case, total rent creation induced by the transport improvement is equal to 290 million euros per year. The maximum potential land value increase which can be captured is therefore almost 16% higher than in the closed city case.

In practice, it is difficult to anticipate whether the transport infrastructure will lead to a significant city total population increase, or not. The link between transport network and a city attractiveness is far from direct and a number of different factors play a role. Important factors, among many others, are the evolutions taking place in other cities. Population increase due to transport improvement might indeed be zero if all other cities undertake similar transport improvements.

There are several ways to tax land value increase (Peterson, 2009). One way is to tax land value changes, in all places within a certain radius from newly built transport stations. Fig. 6 shows simulated total potential annual land value capture as a function of this radius. In the closed-city case, this value ceases to increase after about 1200m due to the inclusion within the radius of places where rent decrease.

In the open-city case, on the contrary, no such phenomenon occurs and the value keeps increasing as all places where rents increase tend to be included. However, the increase after 1200m is much slower than the increase before this threshold. This is for instance coherent with Ovenell (2007) (cited in Senior, 2009), who found a positive price effect on house prices of Metrolink Light Rail System construction in Greater Manchester, for houses located 0.5–1 km away from Metrolink stations.

In practice, of course, a betterment tax could include a legal provision that it does not apply in places where rents or land price decrease because of the transport investment. However, it is

> 14 See (Office of the French prime minister, 2013) or http://www.societedugrandparis.fr/.

> 15In this study, we do not consider congestion change due to increased population. City population increase due to the infrastructure construction could increase global congestion levels in the city, i.e. reduce accessibility in the rest of the city. We neglect this phenomenon here.

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<!-- Start of picture text --> Spatial distribution of housing prices variation values<br>400<br>8<br>350<br>“0 Total housing annual price<br>300 E 6 increase € 250 millions<br>4 =<br>= 250 =<br>Ss o4<br>2 2<br>‘S 200 2<br>a=<br>5 =?<br>= 150<br>=<br>100 8% tsa2% svat4% ll6% ahMi8% sttts 10%atts a<br>t<br>By, A% 0% “1% 2% 3%. A%_ 5% 6% 7%_8% 9% 10% 1%<br>Real estate price variation, due to a 25% increase in RER B speed<br><!-- End of picture text --> Spatial distribution of housing prices variation values <!-- Start of picture text --> 70<br>12<br>60<br>*<br>— 10 Totalincrease housing€ 560 annualmillions price<br>,E 50 =Fe 8<br>= 3 6<br>2 40 a<br>° 2&4<br>a —<br>3* = 2<br>8% 2% 4% 6% 8%<br>° tT<br>Be 1% 0% 4%- 2% 3% 4% 5% 6% 7% 8%. 9% --~<br>Real estate price variation, due to a 25% increase in RER B speed<br><!-- End of picture text -->
<!-- Start of picture text --> 400 r 1 1 1 +<br>@ 350<br><<br>2<br>= 300<br>S200 L<br>& 150 /<br>i<j<br>'S<br>>100 /<br>3 — Closed city case<br>F 50 -----Open city case<br>% 500 1000 1500 2000 2500 3000<br>Distance to public transport stations (m)<br><!-- End of picture text -->
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# _3.4. Comparison with reduced-form model_

It is interesting to note that, because this model is based on urban economics equations, an approximation similar to the the one presented in section 2.2.2 enables to approximate rent variations simulated by this model (see Appendix C and especially Appendix C.2.2) : and where α and β are the coefficient of the utility function, and _a_ and _b_ the coefficient of the housing production function (see Appendix D for a description of the meaning of these parameters). This reduced form is useful to get quickly orders of magnitude of rent value capture potential, without requiring complex numerical simulations. Let us compare the results of the reduced form to the simulations of the previous sections. For α, β, _a_ and _b_ , we use the same values as in NEDUM-2D (cf. Tab. D.3), and we assume that τ = _−_ 3 _._ 75% and _R_ 0 = 27 ~~C~~ _/m_<sup>2</sup> _/_ month.

The value to give to _F_ , the fraction of population with significant improved accessibility, is not straightforward. In Sec. 3.2 and 3.3, the fractions of population with reduced transport costs are respectively 16% and 83%. However, only part of these fractions actually experience the maximum decrease by 25% of transport costs. We suppose here that it is only for about 10% of the population with reduced transport costs that accessibility improves significantly.

Using these values, the results of the approximations are close to the results of the simulations (see Tab. 2) : less than 10% difference for variations in population _P_ and rent in the center _R_ 0, and less than 15% difference for rent increase close to stations of improved public transport lines.

These approximations are computed supposing endogenous building construction driven by rents change. The approximations are different when supposing that buildings are exogenously given, and unaffected by transport infrastructure change (this case is examined in Sec. 2.2.2).

# _3.5. Land value capture and timing of investments_

Let us finally have a closer look at the efficiency of land value capture as a financing tool, and assume that, as in Sec. 3.5, it is decided to capture land value changes within a certain radius of transport stations, to finance transport investment for the corresponding line. Fig. 8 shows potential value capture around “RER B” line stations when the speed of all local train lines is increased<sup>16</sup> , and compares it with the curve of Sec. 3.2.

In the closed city case (Fig. 8a), the value which can be captured appears to be 30% lower when all train lines are simultaneously improved: land value increase around one line is strongly dependent on what happens in the rest of the city, and, as noted in the previous section, land value capture is less efficient as a financing tool when the transport improvement increases in scale. The

> 16To prevent double-counting, this graph does not consider places which are within the given distance from the “RER B” stations, but which are closer to a station from another improved transport line. Indeed, it is supposed that land value capture in these places would be used to finance this other line.

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|Coeffcients|Change in one local<br>train line speed|Change in all local<br>train lines speed|
|---|---|---|
|α,β,_a_and_b_|cf. Tab|. D.3|
|fraction of population with<br>signifcant improved accessibility|**2%**|**9%**|
|_R_0<br>τ|27<br>~~C~~/_m_<sup>2</sup><br>-3.7|/month<br>5%|
|δ_R_0as computed by Eq. 5, whenδ_P_=0.|-0.8<br>~~C~~/_m_<sup>2</sup>/year|-3.6<br>~~C~~/_m_<sup>2</sup>/year|
|Simulated values in Sec. 3.2and3.3|-0.8<br>~~C~~/_m_<sup>2</sup>/year|-3.8<br>~~C~~/_m_<sup>2</sup>/year|
|δ_P_<br>_P_ <sup>in the open city case, based on Eq. 5</sup>|3.5%|15.7%|
|<br>Simulated values in Sec. 3.2and3.3|3.2%|14.6%|
|_dR_<br>_R_ <sup>as computed by Eq. 4</sup>|12.26%|11.40%|
|<br>Maximum rent increase in NEDUM-2D simulations|10.76%|9.66%|

Table 2: To compute the value of τ, we assume that transport generalized cost typically represents 15% of income and that transport time, which represents the biggest share of transport generalized cost, decreases by 25%, therefore τ = _−_ 15% _×_ 25% = _−_ 3 _._ 75%. We choose the same coefficients α,β, _a_ and _b_ as in NEDUM-2D simulations (see Tab. D.3). _R_ 0 is taken as the maximum rent in NEDUM-2D simulation (see Fig. D.12a).

explanation is simple: as explained in Sec. 2.2, land value capture potential decreases with the scale of the investment, because of a greater impact on _R_ 0 of large investments.

In the open city case (Fig. 8b), the value which can be captured in the “all lines improvement” scenario becomes almost equal to the value which can be captured in the “RER B improvement” scenario. Indeed, the open city hypothesis cancels out the variation of _R_ 0, and this makes rent variation truly reflect accessibility variation, which is the same in both scenarios. The slight difference (3%) comes from the overlap between different transport lines, which makes that, as noted in previous section, improving all public lines is less efficient in terms of accessibility change than improving one given line.

_Timing of investments._ Let us finally imagine a third, intermediate, case. It is a scenario in which the speed of “RER B” line is increased, such as in Sec. 3.2, but after all other local train line speeds have been similarly increased. This means that after the speed increase of line “RER B”, the situation will be exactly identical to the situation in Sec. 3.3.

This scenario is also represented on Fig. 8: both in open and closed city cases. It leads to a potential land value capture which is smaller than in the first scenario, but bigger than in the second. The reason is that, due to the overlap between transport lines, accessibility gained through the improvement of one line is bigger if the other lines have not been improved yet.

This shows that, when analyzing rent creation, the timing of transport investment is important : when improving/creating several transport lines, land value increase along a given line will be proportionally larger if this line is the first to be improved/created than if it is the last. Land value 18
<!-- Start of picture text --> 400 1 1 1 r<br>@ 350<br>300<br>£<br>$200 [gore<br>5 150 “Increase(Pee in speed:eee<br>2100 4) ——RER line "B"<br>s fr ---- All public transport lines<br>F 50 (iv __..RER line "B" after an increase in<br>0<br>0 500 1000all other public1500 transport2000 lines2500 speed3000<br>Distance to public transport stations (m)<br><!-- End of picture text --> <!-- Start of picture text --> 400 1 1 1 '<br>@ 350<br>= 300<br>£<br>$200<br>5 150 Increase in speed:<br>2100 —RER line "B"<br>s ---- All public transport lines<br>F 50 ___.RER line "B" after an increase in<br>00 500 1000all other1500 public transport2000 lines2500 speed3000<br>Distance to public transport stations (m)<br><!-- End of picture text -->
increases or decreases in rents and land prices, possibly changing our conclusions. Taking into account this phenomenon, however, requires a much finer analysis than ours.

Similarly, our analysis did not consider amenities, i.e. places that households value per se. We therefore do not consider here how transport improvement could increase or decrease local quality of life (through increased or decreased sound or pollution nuisances, for instance). This effect is entangled with the former one, as amenities can be valued differently by different income classes (Brueckner et al., 1999).

Finally, as mentioned in Sec. 3.3, the fact that we only consider here one employment center has some implications on our assessment of potential land value capture, as it overestimates the overlaps between public transport lines accessibility. This will be addressed in an on-going study.

_Impact on jobs._ Our analysis only models household location choices : we do not consider firms choices and only investigate added value from housing. Transport improvement could have a strong impact on jobs and activities locations. This is roughly analyzed here by the comparison between "closed city case" and "open city case", but we did not do not explicitly model new economic activity generated by better infrastructure. In the open city case, we suppose that economic activity is a by-product of the increase in population, not the driving factor of population changes.

Economic activity change would impact rents and land prices either directly, through commercial demand for land, or indirectly, through an increased attractiveness for households. We therefore a priori underestimate land value changes.

_Perfect land market._ In our analysis, only economic processes drive land prices, and population density evolutions (through buildings construction or dwelling size changes). In practice, these evolutions are constrained by numerous regulations and market imperfections. NEDUM-2D only takes into account a few of these constraint (a limitation of maximum buildings density in Paris city center), but not all of them. This could have an impact on our conclusions, because this could limit land price increases in places where increased demand would exist, but where construction prospects or households movements are limited by such restrictions.

_Inertia and dynamics._ We use here a static equilibrium model, with no inertia on price evolution or population density change. In practice, this inertia could play an important role. For instance, land prices downward rigidity (Gao et al., 2009) could reduce the difference between the “closed” and the “open” city case, by preventing any significant price decrease. Similarly, any rigidity on households dwelling size evolution or building construction can cancel at least part of the land price decrease. Also, dynamic effects can create significant distributional effects, as explored in Gusdorf et al. (2008).

Exploring such a phenomenon would be an interesting complement to this work. Dynamic urban models such as the model defined in Viguié et al. (2014) could be used to do such an analysis.

# **5. Conclusion**

In conclusion, using urban economics based models similar to NEDUM-2D provides estimations of land value capture potential, taking into account city systemic effect. The model that we 20
use is very simple, and disregards important phenomena that could strongly impact value capture potential, so caution is needed before using the exact figures we compute. However, a few important ideas emerge from our analysis.

First, the city systemic effect is particularly important to consider for large scale transport investments, and can dramatically change the magnitude of rent creation. It makes potential land value capture dependent on city population change, i.e. city attractiveness compared to other cities. This result warns against analysis realized at a lower scale than the urban area.

Another conclusion is the fact that value capture potential depends on the timing of investments. The value that is created is different when considering first an improvement in one given transport line, and, then, improvements in another transport line, than when considering improvement of both lines at the same time. And when considering land value capture potential for a given line, it can be different if this line is the first to be improved, or if the other line is improved first (Sec. 3.5).

In the theoretical case of the “open city”, i.e. with an adequate total population increase, transport accessibility is truly capitalized into land prices. However, if total population does not change (“closed city” case), capitalization does not fully occur, and the larger the scale of the transport project, the less rent is generated. In practice, if one considers that a reasonable bet is an intermediate evolution between these two limiting cases, it still means that land value capture has a smaller potential as a financing tool for large scale investments than for local investments.

Our results also indicate that, even without considering transport-related negative amenities such as air pollution or noise, there are by nature winners and losers when transport is improved. Except if the city population increases sufficiently (in which case, land price increases in all places), there are some places where land price increases, and some places where it decreases. This idea is actually not specific to transport, and valid for any amenity, such as the creation of a park, for instance. Indeed, in a not fully open city, an increase in land prices somewhere means that there has to be a decrease in land prices somewhere else.<sup>18</sup> However, it should be noted that transport improvement is actually exactly capitalized into land prices, _when land prices are compared across the city_ . Systemic effects can induce a general decrease of land prices in the city, but they do not change the fact that land prices close to the improved transport line will decrease less than land prices far from this line. Transport accessibility is capitalized in the difference in land prices across the city, not in land price evolution across time. The true question is whether it is politically or administratively feasible to capture this relative variation across the city, i.e. to tax a place because prices have decreased less than in other places in the city.

In practice, if a city follows a trajectory of sufficiently increasing land prices, due to external and exogenous factors (economic or demographic growth, interest rate changes etc.), then systemic effects could remain hidden and unnoticed. In such a case, the difference between small-scale and large-scale investment would disappear. This is part of a broader issue: one difficulty, when implementing land value capture, is the fact that land prices are not impacted only by the transport 18Of course, this is true only if dwelling size variation or housing construction are responsive enough to land price increase.

21
improvement, but also but many other evolutions which may take place at the same time. Some of these evolutions are local: increase in quality of life through city infrastructure improvement, better schools or socio-economic changes (such as gentrification). Some are city-wide, such as local economic development and population change. Others are national or even global, such as macroeconomic cycles, housing price bubbles and interest rate change. As indicated in Sec. 3.2, it could be difficult to disentangle land price variation due to the transport investment from land price variation due to other factors. The systemic effect we have analyzed in this study (the variation in _R_ 0 ) acts exactly as these external factors, and prevents an exact capitalization of accessibility gains into land prices. Factors which lead to land price increase a priori favor the use of land value capture, whereas factors which lead to land price decrease prevent its full efficiency.

Our conclusions, here reached on the specific case of Paris, would remain valid in other urban areas, as long as an urban economics standard framework is supposed to be an adequate representation of reality. As indicated in Sec. 4, using a more complex model would enable getting finer conclusions, especially related to dynamic issues and firm choices. Both these questions are the subject of on going work. On the other hand, a full modeling analysis is not always possible, because of resource, time, or data availability issues. In this case, we propose a reduced form model that can help decision makers assess the orders of magnitude at stake, and determine whether more in-depth analyses are necessary.

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# **Acknowledgment**

This study benefited from funding by ANR through MUSCADE (ANR-09-VILL-0003-01) project. The authors are grateful to Didier Rouchaud and Alain Sauvant for their data on transport time in Ile de France, and to Nicolas Coulombel, Joël Hamann and participants to North American Regional Science Council (NARSC) 2013 conference, especially Harris Selod, for interesting discussions. Mook Bangalore, and Paolo Avner provided helpful comments.

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# **Appendix A. Urban economic framework**

_Notations._ Let us recall briefly the main characteristics of classical monocentric city modelling (Fujita, 1989): we want to maximize Under the constraints: Landowners can choose _R_ ( _x_ ), and households _q_ ( _x_ ), _Z_ ( _x_ ) and their location in the city. Variables _q_ ( _x_ ), _Z_ ( _x_ ) and _R_ ( _x_ ) are always positive.

_Mathematical hypotheses._ We suppose that the utility function is twice differentiable, increasing at all _Z >_ 0 and _q >_ 0, and that all indifference curves are strictly convex. For simplicity, we also suppose that there is at least one location _x_ in the city where _T_ ( _x_ ) = 0, i.e. one location where transport costs are equal to zero.

_Equilibrium._ Equilibrium leads to a uniform utility level across the city (see the proof in Fujita, 1989) : The limit of the city is given by the condition: _Y > T_ ( _x_ ). In the city, rent levels are given by the bid-rent function: Where Ψ( _R_ 0( _U_ 0) _, T_ ( _x_ )) is strictly decreasing with _T_ and increasing with _R_ 0( _U_ 0), and where Ψ( _R_ 0( _U_ 0) _,_ 0) = _R_ 0( _U_ 0). _R_ 0( _U_ 0) is a bijective function, strictly decreasing with _U_ 0. It represents rent level in the middle of the city, where transport price is zero.

Dwelling size _q_ ( _x_ ) is given by: Using eq. A.1, we can write _q_ ( _x_ ) as a function of _R_ 0( _U_ 0) and _T_ ( _x_ ), or as a function of _R_ 0( _U_ 0) and _R_ ( _x_ ), due to the fact that there is a one-to-one correspondence between _T_ ( _x_ ) and _R_ ( _x_ ), if _U_ 0 is given.

An important property of the function _S_ , which will be useful in the following demonstrations, is the fact that it is strictly increasing with _T_ , when _U_ 0 is given. Indeed, keeping _U_ 0 constant, an 26
increase in _T_ will induce a non-zero decrease in _R_ . As _U_ 0 is constant, _q_ and _Z_ will move along an indifference curve, and as these curves are strictly convex, a decrease in _R_ will result in an increase in _q_ and a decrease in _Z_ .

_Remark._ Using envelope theorem (here again, see Fujita, 1989) it is possible to show that ∂<sup><u>∂</u></sup> _T_<sup>_<u>R</u>_=</sup> _−_<sup><u>1</u></sup> _q_<sup>_._</sup><sup>_<u>dT</u>_</sup> _dx_<sup>, i.e.</sup> and therefore: _R_ <u>0</u> where ε<sup>_q_∂</sup><sup>_<u>q</u>_</sup> _R_ 0<sup>=</sup> ∂ _R_ 0 _q_<sup>is dwelling size elasticity to rent in the city center.</sup>

# **Appendix B. Propositions and proofs**

_Appendix B.1. Main hypothesis_ We consider here two cities _a_ and _b_ which are strictly identical, except for transport prices _Ta_ and _Tb_ : In all these demonstrations, we will use the same set of joined hypotheses:

**Definition 1.** Hypothesis _H_ : we suppose that there is a non measure-zero set _A_ of _x_ where we have simultaneously _Ta_ ( _x_ ) _> Tb_ ( _x_ ) , _H_ ( _x_ ) _>_ 0, and _Y > Tb_ ( _x_ ).

_H_ simply means that there is at least one location _x_ in the city with some housing space and where transport is strictly cheaper in city _b_ than in city _a_ .

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_Appendix B.2. Close city case_ In the close city case, the sum of inhabitants in the city has to be equal to the total population _pop_ , given exogenously:

**Definition 2.** Hypothesis _P_ : We suppose that total population in cities _a_ and _b_ is identical : _popa_ = _popb_ .

**Lemma 3.** _If hypothesis H and P are true, then R_<sup>0</sup> _a_<sup>_> R_0</sup> _b_<sup>_(i.e.ifthereisaplaceincityawith_</sup> _some housing space and where transport is strictly more expensive than in city b, then the rent in the city center is higher in city a than in city b)._

_Proof._ Let us do a proof by contradiction and suppose that _R_<sup>0</sup> _a_<sup>_≤R_0</sup> _b_<sup>.In this case, in each location</sup> _x_ , we have: Therefore: However, for any _x_ in _A_ , this inequality is strict, as _Ra_ ( _x_ ) _≥ Rb_ ( _x_ ) and _Ta_ ( _x_ ) _> Tb_ ( _x_ ) (cf. Appendix A). Indeed: and therefore: Since, we also have, from Eq. B.1: This means that: whereas, by hypothesis, _POPa_ and _POPb_ should be equal.

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**Proposition 4.** _If hypothesis H and P are true, then in places where Ta_ ( _x_ ) = _Tb_ ( _x_ ) _, Ra_ ( _x_ ) _≥ Rb_ ( _x_ ) _. More specifically, in theses places, either Ra_ ( _x_ ) _> Rb_ ( _x_ ) _or Ra_ ( _x_ ) = _Rb_ ( _x_ ) = 0 _(i.e., x is outside of the city)._

_Proof._ Let us call Ψ the bid-rent function expressed as a function of _R_<sup>0</sup> and _T_ . It is a classical result that Ψ is an increasing function of _T_ and a strictly increasing function of _R_<sup>0</sup> (cf. Fujita (1989)) inside the city, i.e. where _Y > T_ ( _x_ ). When _Y ≤ T_ , Ψ( _R_<sup>0</sup> _, T_ ) = 0 for any _R_<sup>0</sup> . Following lemma 3, we have:

**Proposition 5.** _If hypothesis H and P are true, then the larger the area where Ta_ ( _x_ ) _> Tb_ ( _x_ ) _, or/and the larger the decrease in transport prices, the bigger the decrease in rents in areas where transport prices do not change._

_Proof._ Let us imagine a third city _c_ where _Ta_ ( _x_ ) _≥ Tb_ ( _x_ ) _≥ Tc_ ( _x_ ), with a non measure-zero set of _x_ where we have simultaneously _Tb_ ( _x_ ) _> Tc_ ( _x_ ) , _H_ ( _x_ ) _>_ 0, and _Y > Ta_ ( _x_ ) (and therefore _Y > Ta_ ( _x_ ) _≥ Tb_ ( _x_ )). Applying proposition 4 to cities _b_ and _c_ leads to: in places where _Ta_ ( _x_ ) = _Tb_ ( _x_ ) = _Tc_ ( _x_ ).

# _Appendix B.3. Open city case_

In the open city case, utility level _U_ 0 in the city is given exogenously. The sum of inhabitants in the city varies to maintain this constant utility level.

**Definition 6.** Hypothesis _O_ : We suppose that utilities in cities _a_ and _b_ are identical, therefore : _Ua_ = _Ub_ .

In an open city, rents are unaffected by the new transport infrastructure in places where people do not use it, and rents are increased in places where people use it:

**Proposition 7.** _If hypothesis H and O are true, then in places where Ta_ ( _x_ ) = _Tb_ ( _x_ ) _, Ra_ ( _x_ ) = _Rb_ ( _x_ ) _. In places where Ta_ ( _x_ ) _> Tb_ ( _x_ ) _and Y > Ta_ ( _x_ ) _, Ra_ ( _x_ ) _< Rb_ ( _x_ ) _._

_Proof._ Let us consider a location _x_ . If _x_ is outside the city (i.e. _Y < Ta_ ( _x_ ) ), we have _Ra_ ( _x_ ) = _Rb_ ( _x_ ) = 0.

If _x_ is inside the city, let us suppose that _Ta_ ( _x_ ) = _Tb_ ( _x_ ). In this case, we have: 29
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# **Appendix C. Approximations and simplifications**

Without any complex city modelling, it is possible to compute easily an approximation of the change in _R_ 0 and population in the closed and open city cases with a few hypotheses. If households utility function is a Cobb-Douglas function, it is possible to simplify the formulas even further.

# _Appendix C.1. Rent variation_

Let us suppose that ε<sup>_q_</sup> _R_ 0<sup>is constant over the urban area (this is for instance what we get if house-</sup> holds utility function is a Cobb-Douglas function, see below). Eq. A.3 can be greatly simplified: Let us now suppose further that households utility function is a Cobb-Douglas function with constant returns ( _U_ ( _Z, q_ ) = _Z_<sup>α</sup> _.q_<sup>β</sup> with α + β = 1). In this case, we have: this enables to compute that, as noted above, ε<sup>_q_</sup> _R_ 0<sup>is constant over the urban area, and equal to</sup> _−_ 1. Eq. C.1 then becomes: _Open city case._ If _R_ 0 does not change, but _T_ changes, then we have<sup>_<u>dR</u>_</sup> _R_<sup>=</sup><sup>_−_</sup> β _._ ( _YdT−T_ )<sup>.This formula</sup> enables to easily compute orders of magnitude of rents variation due to transport accessibility changes in the open city case (see section 2.2.2).

_Closed city case._ In the closed-city case, we have to add to the last formula the effect of the variation in _R_ 0. We can note that, in places where _T_ remains the same, Eq. C.2 shows that rents are all multiplied by the same factor (i.e.<sup>_<u>dR</u>_</sup> _R_<sup>=</sup><sup>_<u>d</u>_</sup> _R_<sup>_R_</sup> 0<sup><u>0</u>).Next section shows how to compute an order</sup> of magnitude of the variation in _R_ 0.

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_Appendix C.2. Variation in R_ 0 _Appendix C.2.1. Exogenous building construction_ Let us first consider, as in Sec. 2, that buildings are exogenously given, and unaffected by transport infrastructure change (it can also be regarded as a description of short-term variations in rents). Population is given by: _P_ = ́ _<u>Hq</u>_<sup>, therefore</sup> where _n_ =<sup>_<u>H</u>_</sup> _q_<sup>ispopulationdensity,andε</sup><sup>_q_</sup> _R_ 0<sup>=</sup> ∂∂ _Rq_ 0<sup>_.R_</sup> _q_<sup><u>0</u>andε</sup> _Y_<sup>_q_</sup> ̄<sup>=</sup> ∂( _Y_ ∂ _−q T_ )<sup>_.Y−_</sup> _q_<sup>_<u>T</u>_</sup> dwelling size elasticity to respectively rent in the center of the urban area and to available income _Y_<sup> ̄</sup> (income minus transport cost: _Y_<sup> ̄</sup> = _Y − T_ ).<sup>19</sup> If ε<sup>_q_</sup> _R_ 0<sup>and ε</sup> _Y_<sup>_q_</sup> ̄<sup>are constant over the urban area (this is also what</sup> we get if households utility function is a Cobb-Douglas function), then, noting that _P_ = ́ _n_ , we have: To approximate the variation in _R_ 0, we have to do two other approximations. First, we suppose that _Y ≫ T_ , i.e. we assume that most of the population lives in areas not too far from city center. Second, we assume that<sup>_<u>dT</u>_</sup> _Y_<sup>isaconstantτforafraction</sup><sup>_F_oftotalpopulation(weapproximate</sup> that transport price variation has a certain negative value for a fraction of the city population, and is equal to zero for the rest of the population). In this case, we have: This equation can also be used to compute<sup><u>δ</u></sup> _P_<sup>_<u>P</u>_intheopencitycase:</sup><sup><u>δ</u></sup> _P_<sup>_<u>P</u>_= ε</sup> _Y_<sup>_q_</sup> ̄<sup>_._τ</sup><sup>_.F_.Thereis</sup> another way to write this equation: if δ _R_ 0 is _R_ 0 variation in the closed-city case, then population variation in the open-city case is given by

> 19In this computation, we assume that _Y_ does not change, so _d_  ̄ _Y_ = _d_ ( _Y − T_ ) = _−dT_ . If _Y_ can vary, too, then other terms have to be added to the equation.

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_Case of a Cobb-Douglas utility function.._ If households utility function is a Cobb-Douglas function with constant returns, then we have ε<sup>_q_</sup> _R_ 0<sup>=</sup><sup>_−_1 and ε</sup> _Y_<sup>_q_</sup> ̄<sup>=</sup><sup>_−_</sup><sup><u>α</u></sup> β<sup>, and Eq.C.3 becomes:</sup> so, in the closed city case δ _R_ 0 _≈_<sup><u>α</u></sup> β<sup>_.R_0</sup><sup>_._τ</sup><sup>_.F_and, in the open city case,</sup><sup><u>δ</u></sup> _P_<sup>_<u>P</u>_=</sup><sup>_−_</sup><sup><u>α</u></sup> β<sup>_._τ</sup><sup>_.F_.</sup> _Appendix C.2.2. NEDUM-2D model approximation: Endogenous building construction_ Equations C.3 or C.4 cannot be directly used to approximate results of NEDUM-2D model because of endogenous building construction driven by rents change (Muth, 1969; Fujita, 1989). However, a similar result can be derived, noting that in this case: And where ε<sup>_H_</sup> _R_ 0<sup>=</sup> ∂<sup><u>∂</u></sup> _R_<sup>_<u>H</u>_</sup> 0<sup>_.R_</sup> _H_<sup><u>0</u>andε</sup> _Y_<sup>_H_</sup> ̄<sup>=</sup> ∂( _Y_ <u>∂</u> _−HT_ )<sup>_.Y−_</sup> _H_<sup>_<u>T</u>_</sup> . If, here again, we suppose that ε<sup>_H_</sup> _R_ 0<sup>andε</sup> _Y_<sup>_H_</sup> ̄<sup>are</sup> constant over the urban area, which is for instance what we get if housing production function and utility function are both Cobb-Douglas functions, we have: As with Eq. C.3, this equation can be used to compute<sup><u>δ</u></sup> _P_<sup>_<u>P</u>_in the open city case:if δ</sup><sup>_R_0is</sup><sup>_R_0</sup> variation in the closed-city case, then population variation in the open-city case is given by: 33
_Case of Cobb-Douglas functions._ If households utility and housing production functions are both Cobb-Douglas functions with constant returns (use the same notations as in Appendix D: utility function _U_ = _Z_<sup>α</sup> _q_<sup>β</sup> and housing production function _H_<sup>�</sup> = _AL_<sup>_a_�</sup> _K_<sup>_b_</sup> and _H_ =<sup>_<u>H</u>_</sup> _L_<sup>�and</sup><sup>_K_=</sup><sup>_<u>K</u>_</sup> _L_<sup>�), then we</sup> have ε<sup>_q_</sup> ̄<sup>_<u>b</u>_</sup><sup>~~ε~~</sup><sup>_H_</sup> _<u>b</u> R_ 0<sup>=</sup><sup>_−_1 and ε</sup> _Y_<sup>_q_=</sup><sup>_−_</sup><sup><u>α</u></sup> β<sup>, as above, and ε</sup> _R_<sup>_H_</sup> 0<sup>=</sup> _a_<sup>and</sup> _Y_<sup>=</sup> _a._ β<sup>, because</sup> Eq. C.5 then becomes: _<u>b</u>_<sup><u>α</u></sup> so, in the closed city case δ _R_ 0 _≈ a.ba_ <u>β</u><sup>++1</sup> <u>β</u><sup>_.R_0</sup><sup>_._τ</sup><sup>_.F_and, in the open city case,</sup><sup><u>δ</u></sup> _P_<sup>_<u>P</u>_=</sup><sup>_−_</sup> � _ab._ β<sup>+</sup><sup><u>α</u></sup> β � _._ τ _.F_ .

# **Appendix D. Description of NEDUM-2D model**

For more information on NEDUM-2D model, see Viguié (2012); Viguié and Hallegatte (2012).

_Appendix D.1. Equations_ We model the household trade-off using the following utility function: where α and β are coefficients (α + β = 1), _q_ the surface of the households’ dwelling and _Z_ the money remaining after the household has paid its rent and a commuting round-trip per day to the center of Paris. The cost of transportation includes the monetary cost of transportation and the cost associated with the trip duration, which we consider as an actual loss of income. Such a functional form is consistent with the fact that the share of household income devoted to housing expenditures is relatively constant over time and space (Muth, 1969; Thorsnes, 1997). Household income constraint reads: where _Y_ is the average household income, _R_ is the rent per square meter, and _tr_ the transportation costs (monetary cost added with time cost).

We assume that absentee landowners own the land, and that they combine land with capital to produce housing. The housing production function reads, in a classical way: where _A_ , _a_ and _b_ are coefficients ( _a_ + _b_ = 1), _H_<sup>�</sup> the housing surface built, _L_ the land surface occupied by the buildings and _K_<sup>�</sup> the financial capital used for construction. If we call _H_ =<sup>_<u>H</u>_</sup> _L_<sup>�and</sup> _K_ =<sup>_<u>K</u>_�</sup> _L_<sup>housing surface and capital per land unit, the benefit of land owners reads therefore :</sup>
|Main Data||
|---|---|
|Urban area population|5,101,300 households|
|Fraction of ground surface devoted to housing|0.45|
|Households average income|56,098<br>~~C~~|
|Transport times and costs in Paris urban area|cf. Appendix D.2|
|Interest rate|δ=5%|
|Built capital depreciation time|ρ=0_._5%|
|Calibrated paramet|ers|
|Households utility function parameter|β=0_._3<br>|
|Coeffcients of construction cost function|_A_=1_._57_∗_10<sup>_−_3</sup>and_a_=0_._07|
|Maximum foor-area ratio|cf. Appendix D.2|
|Cost associated with travel time|cf. Appendix D.2|
|Rent determining city border|_R_0=12<br>~~C~~_/m_<sup>2</sup>|

Table D.3: Summary of main data and calibration parameters Π is the profit per land unit, ρ represents the joined effect of real estate capital depreciation and annual taxes payed by land owners on the real estate capital, and δ the interest rate. The metropolitan area boundary is defined by a rent _R_ 0, below which it is not profitable to build housing building (this value corresponds both to other use of the land like agriculture and to transaction costs in the building and renting process). Developers build to maximize their profit: at each point of the metropolitan area they construct, i.e. choose _K_ , to maximize Π under the constraint that<sup>_<u>H</u>_</sup> _L_<sup>�ratio is</sup> limited by a land-use constraint (see detail below).

# _Appendix D.2. Calibration_

Tab. D.3 presents the numerical data we used in our simulations. In absence of adequate data for some parameters, for instance the cost of time and construction costs, these parameters have been calibrated on the Paris structure in 2006. A detailed comparison of model results with available data is provided below, and shows a good agreement on the model with observed urban evolutions.

_Maximum floor-area ratio in the center of Paris._ We include in the model a constraint on the maximum density allowed. This constraint reflects actual land-planning constraints which prevents building which are too high. Calibration process leads to 1.5 as the value of this maximum floor-area ratio. In the simulations, this constraint is only binding in the center of Paris.

This value may seem low as most buildings in Paris have approximately 6 floors, which would induce a ratio of about 6 at the center of Paris. However, our ratio is only taking into account housing surface, and not the total built surface, and the discrepancy is simply caused by built surface intended for purposes other than housing (it includes, on the one hand, corridors and lobbies in buildings dedicated to housing and, on the other hand, all buildings not dedicated to housing: offices, shops, museums, train stations, office buildings, schools, universities, etc.).

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<!-- Start of picture text --> 4000<br>3<br>£ 3500<br>S<br>2 3000 +——<br>3<br>2 2500<br>82<br>+ £2000<br>ow<br>i)<br>c 1500<br>2<br>S 1000<br>7 —CSTB data<br>5 500 . .<br>6 — Construction costs in the model<br>0<br>12 3 4 5 6 7 8 9 10 14 12<br>Number of floors<br><!-- End of picture text -->
_Cost of time._ In the model, rents (per surface unit) decrease when moving away from the center of Paris because households have to pay a generalized transportation cost, which is the sum of a perceived monetary cost (interpreted here as the cost of fuel) and of the cost associated with transport time, assuming that households do a round-trip per day towards the center of Paris. In the simulation, cost associated with transport time represents generally the bigger part of generalized cost, and the way we assess this cost has an important role in our results.

Numerous studies have dealt with this issue, but no conclusive result exists on this complex subject. In Ile-de-France, French Government’s Strategic Analysis Center proposed to use net hourly wage as an estimate for commuting time cost, but explained that the value of actual commuting time cost depends greatly on several factors such as households characteristics or modal choice. (Boiteux and Baumstark, 2001) Due to the importance of time cost choice in the simulation, we calibrated time cost instead of using an a priori fixed value. We computed this cost using our data on rent spatial distribution: out of these data, assuming our model perfectly exact, it is indeed possible to estimate a theoretical generalized transportation cost. Assuming that this generalized cost reflects the sum of the direct cost of transport and of the cost associated with transport time, and assuming that households do a round-trip per day towards the center of Paris, the transport time cost was estimated as a function of journey time.

Marginal time cost seems to decrease with travel time, and we chose to model simply this decrease using a piecewise affine function. This representation leads us to use a cost time worth 105% of the net hourly wage when the travel time is less than 25 min (or, equivalently, when the distance to the center of Paris is less than 15 km), then a very low cost (6.6% of the net hourly wage) for portions of journey in excess of this limit. The value of time for journeys during less than 25 min is therefore very close to commuting time cost in Ile de France according to French Government’s Strategic Analysis Center.

This observed decrease in marginal time cost can be attributed to the limits of our approach, in particular to the monocentric framework and to the hypothesis that households do a round-trip per day towards the center of Paris. In the real world, in places where travel time exceeds 25 minutes, a large fraction of households do not commute to the center of Paris. This leads to a shorter average trip length than in the mono-centric case, and using actual average trip length would enable to use more realistic time cost values and smaller total fuel costs for locations far from Paris city center. In absence of needed data, we did not take into account explicitly this variation in trip length, and modeled it with a non-linear time cost.

# _Appendix D.3. Validation: urbanized surface evolution_

As can be seen on Fig. D.11a, the model reproduces well the current Paris urban area. The main mismatch is in the west (Mantes la Jolie) and in the south of the urban area (Melun), where the model does not capture observed urbanized areas. These two zones correspond to cities which were built long before being included in Paris urban area, and are important employment centers on their own, whereas the model only represents built areas due to Paris urban area sprawl.

It is possible to use this model to simulate city evolution from 1900 to 2006. For instance, Fig. D.11d, Fig. D.11c and Fig. D.11b compare simulated urban area with actual urbanized area, in 1900, 1960 and 1982, respectively. Because of the lack of data, we used the same transport 37
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