Congestion pricing and long term urban form: Application to Ile-de-France

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Congestion pricing and long term urban form:

Application to Ile-de-France

Michel de Lara, André de Palma, Moez Kilani, Serge Piperno

To cite this version:

Michel de Lara, André de Palma, Moez Kilani, Serge Piperno. Congestion pricing and long term urban form: Application to Ile-de-France. 2008. �hal-00348439v2�

Congestion pricing and long term urban form Application to Île-de-France

Michel De Lara André de Palma

Moez Kilani Serge Piperno

Décembre 2008

Cahier n°

ECOLE POLYTECHNIQUE

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE

DEPARTEMENT D'ECONOMIE

Route de Saclay 91128 PALAISEAU CEDEX

(33) 1 69333033

http://www.enseignement.polytechnique.fr/economie/

mailto:[email protected]

Congestion pricing and long term urban form:

Application to Île-de-France

Michel De Lara

André de Palma

Moez Kilani

Serge Piperno

Décembre 2008

Cahier n° 2008-14

Résumé: Nous proposons un algorithme de résolution du modèle monocentrique de transport avec congestion. Nous utilisons cet algorithme afin d'explorer l'impact de différents schémas de tarification de la congestion sur la forme urbaine, et par conséquent, sur les véhicules- kilomètres (émissions de CO2) à long terme. L'application empirique concerne la région Île- de-France. Quatre régimes de tarification sont considérés : (i) absence de tarification, où une taxe linéaire reflète le coût d'usage du véhicule ; (ii) péage cordon, où les voitures payent pour passer à l'intérieur d'une zone donnée ; (iii) taxe linéaire optimale, proportionnelle à la distance parcourue (optimale dans la classe des taxes linéaires) ; et (iv) taxe optimale (optimum de premier rang). Par rapport à (i), la taxe optimale aboutit à des réductions de 34%

et 15%, respectivement pour le rayon de la ville et la distance parcourue moyenne.

Abstract: We propose an efficient algorithm that solves the monocentric city model with traffic congestion, and use it to explore the impact of congestion pricing on urban forms and, hence on transport volume, emissions and energy consumption. The application focuses on the region Île-de-France. Four pricing policies are considered: no toll, where transport cost is equal to the vehicle operating cost, cordon toll where users pay the toll when they drive inside cordon region (location and value of the toll are both optimized) linear toll (optimal under the class of linear tolls) and optimal toll (or first-best toll). Our analysis shows that the linear toll is particularly effective in that it yields about 93\% of the welfare gain of the first-best toll. By comparison to the no-toll situation, optimal congestion pricing reduces the size of the city and the average travel distance by 34\% and 15\%, respectively.

Mots clés : Modèle monocentrique ; Calcul d’équilibre ; Tarification de la congestion ; Effets de long terme

Key Words : Monocentric model; Equilibrium computation; Transport pricing; Long term impacts

Classification JEL: R21 ; R41 ; R48

1 Université Paris-Est, CERMICS, Ecole des Ponts ParisTech, 6-8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France. Emails : M. De Lara ([email protected]) ; M. Kilani ([email protected]) ; S.

Piperno ([email protected])

2 ENS Cachan and Ecole Polytechnique, 61 avenue du Président Wilson, Cachan 94230, France. Email : A. de Palma ([email protected])

While the literature on road priing has been abundant in the last deades,

long term impat on housing and business loation have not reeived so

muh attention. Reent implementation of anarea-based harge in London

and a few other experiments have raised onern about the overall impat

on ongestion, business ativities and environmental onditions in the long

run (f. Santos & Fraser 2006). At the same time, the alarming levels of

pollutionreahed in many metropolitanareas and the important inrease of

energy ost ontribute to making the optimization of urban forms and the

regulation of transport animportantissue (f. Mithellet al.2005).

This paper explores the impat of transport priing on the urban form,

and, hene, on transport volume, CO2 ^{emissions and energy} onsumption.

We onsider a monoentri model with tra ongestion where all the eo-

nomiativityisloatedintheentralbusinessdistrit(CBD).Therearetwo

mainators: households,whoseutilityisinreasingwithhousingarea,anda

governmentthatdeideshowmuhlandisdevotedtoroads. Thegovernment

ollets a population tax, whih is the same for all households, and a loa-

tion tax that depends on where the household lives.

1

Transport ongestion

introduesan externality that requires publi intervention for regulation.

Transport ongestionwasintroduedinthemonoentrimodelbyStrotz

(1965) and Mills(1967). Inthe followingdeade, there wasgrowing interest

in seond-best alloations of land between housing and roads.

2

A synthesis

of this problem may be found in Kanemoto (1980). Reently, Mun et al.

(2003) have shown that seond-best priing shemes are almost as eient

asrst-bestpriing. Theironlusion hasbeenonrmedby Verhoef (2005).

Bothmodels,however, are ratherrestritiveformsofthemonoentrimodel.

Munetal.(2003)donotonsideravariablehousingarea,andVerhoef(2005)

assumes that the amount of land alloated to transportation is xed. The

monoentri modelhas been used mainlyfor theoretial and normative dis-

ussions, and verylittlefor empirialappliations.

3

1

Onpratialgrounds,road priingmay ontributetoraisingfunds forthetransport

setor(f.DePalmaetal.2007,dePalma&Quinet2005).

2

RepresentativepapersareMills &Ferranti(1971),Solow(1972,1973),Riley(1974),

Robson(1976),Kanemoto(1977), Arnott&MaKinnon(1978),Arnott (1979),andSul-

livan(1983).

3

Empirial appliations inlude Baum-Snow (2007), Boarnet (1994), MMillen et al.

(1992)andRouwendal&vanderStraaten(2008).

no toll cordon toll linear toll first-best

0 CBD radius of

the city

distance to the city center (km)

households/km²

Figure1: Impats of ongestion priing.

We adopt the monoentri ity framework using the formulation of Fu-

jita (1989),and ontributeto the literature attwo stages. First,we propose

a exible and eient algorithm to ompute the optimal solution. The so-

lution approah underlying the algorithm replaes the standard optimality

onditions (f. Fujita 1989) by a set of rst-order dierentialequations that

an be solved eiently by standard numerial tehniques.

4

The algorithm

is exible enoughto be used for anumberof priing rules.

Seond, we undertake an empirial appliation on the agglomeration of

Île-de-Frane(IDF).Inpartiular,wefeedthemodelwithdatafromIDFand

ndthatitsueedsinadequatelyapturinganumberofurbanfeatures. On

thebasis ofthealibratedmodel,wequantify theimpatsofdierentpriing

rules: ordon, linear and rst-best tolls. All poliies lead to a smaller ity

and a redued average trip-distane.

Figure1 illustratesthe impats of ongestion priingon the distribution

of households. Eah urve reets the distribution of households under a

givenregime. Roadpriingmotivateshouseholdstomovelosertothe CBD.

Linear tolldepends ononlythe traveldistane, whilerst-best toll,whihis

non-linear, depends on the ongestion or external ost reated by the trip.

A unit oftrip-distaneinaongested area istolled morethanthe same unit

4

The model was solved under a partiular set of parameter values in Riley (1974),

Robson(1976)andKanemoto(1977),butnogeneralsolutionmethodhasbeenproposed.

aroundthe CBD and itis therethat the dierenewithlinear tollsemerges.

In Figure 1, the impats of the (optimal) linear toll and rst-best toll are

rather similar in the outer part of the ity, but they beome quite distint

around the CBD.

Optimalpriing redues the radius of the ity, the average trip-distane

and ongestionby 34%,15%and 13%,respetively. Theoptimallineartoll, 5

whih we all linear toll for short, indues a omparable impat and leads

to a relatively dense ity. But inpratie, the linear tollis equivalent to an

importantinrease ingasolineprie. Suh apoliy islikelytofaeroad user

oppositionand has the inonveniene of depending ononlythe lengthof the

trip and not on its loation (origin/destination pair). For example, urban

and inter-urbantrips (whihinduelessongestion)are tolledthe sameway.

So,underamoregeneraltransportnetworkthelineartollwillbelesseient

than in the model we onsider here. Eieny is measured as the unspent

part of the households'revenue, for a given levelof utility.

Cordon priing is less eient than linear toll but still reahes an a-

eptable eieny level of 62% with respet to rst-best. By ontrast to

linear tolls, ordon tolls onern only highly ongested areas and turn out

to be an attrative alternative for poliy makers. Indeed, similar priing

rules toordontollarealready inuse insomeities (Londonand Singapore,

in partiular), and other implementation projets are under study. From

the simulationwehaveonduted,itappears thatanoptimalurbanformre-

quiresbothasmallerradiusandahigheronentrationofhouseholdsaround

the CBD (f. Figure 1). The rst-best rules satisfy these requirements by

setting the toll equal to the external ost. Linear toll is more eient in

reduing the radius of the ity than in onentratinghouseholds around the

CBD.Ingeneral,underthelinearrule,theoptimaltrade-obetween thetwo

objetives requires anexessive harge onroad users.

A ordon toll lose to the CBD does not have a strong impat on the

radius of the ity. At the same time, a ordon toll away from the CBD

has substantial impat on the radius of the ity but does not indue any

signiant variation in the onentration of households inside the ity. In

most ases,and fordata relatedtoÎle-de-Frane,wefound thatit isoptimal

to set the ordon tollat adistane about 21kmfrom the ity enter.

Priingreduesthesizeoftheitybuttheaverageareaoupiedbyhouse-

5

That is,optimalamonglineartolls.

housing and transportationis lost. However, this lostarea is not very large

sine, asthe empirialobservation shows, the availableland for housingand

transportation gets smaller as we move away from the ity enter. On the

other hand, with ongestion priing, the surfae of land alloated to roads

dereases and larger areas are available for housing. Overall, both impats

haveomparable magnitudesand the resultingvariationin the housingarea

remains, ingeneral,small.

Onmoregeneralgrounds,priingongestionontributestodereasingthe

level of pollution sine it leads to smaller and more ompat ities. Indeed,

energy onsumption per household dereases as urban density inreases (f.

Newman &Kenworthy 1989). Sine CO2 ^{emissionsare orrelated with trip-}

distane, ongestion priing has an appreiable environmental benet. The

set of simulations we have onduted shows that ongestion priingredues

the level of emissions by 15%, and has a omparable impat on ongestion.

Thepaperisorganizedasfollows. InSetion2weintroduethenotationand

providethe solution proedure forthe land-use equilibrium. The alibration

of the model to IDF is undertaken in Setion 3. In Setion 4 we disuss the

impat of ongestion priing. Wenally onlude inSetion 5.

2 A general method to ompute a ompensated

equilibrium

2.1 The basi framework

The analysisisarriedout underthelassialmonoentri model. Weadopt

theformulationofFujita(1989)anddenotethemodelbyHS_T^{.6 Thenumber}

of households living in the ity is xed and equal to N ^{(losed ity). The}

variable r ^{denotes the distane from the enter of the ity. Eah household}

makesdailytrips fromits loation,at distane r ^{fromthe enter of the ity,}

to the Central Business Distrit (CBD) that extends to distane rc ^from

the enter of the ity. Inside the CBD, we assume that transportation is

ostless. The radius of the ity is denoted by rf^. N(r) ^{is the number of}

households loated further than distane r ^{from the ity enter.} L(r) ^{is the}

amount of land available for housing or transportation at r^. LT(r) ^{is the}

6

Fujita(1989)referstothemodelastheHerbert-Stevensmodelwithtraongestion.

amountof land alloatedfor transportation atr^{. Eah household onsumes}

twogoods,housings^{andaompositegood}z^{,andgetsautility}U(z, s)^where

∂U(z, s)/∂z >0 ^and ∂U(z, s)/∂s >0^{. All households have the same utility}

funtion and the same (pretax)revenue Y^{. The prie of the omposite good}

is normalized to 1 and the unitary prie of land, or land rent, at distane r

fromthe ityenterisR(r)^{. The}opportunityostofland,ortheagriultural rent, isdenoted by RA^{. The amount of omposite goodneessary toahieve}

utility level u ^{when the housing area is equal to} s ^is Z(s, u)^{, whih is the}

solution of U(z, s) = u ⁱⁿ z^{. Let} I ^{denote the revenue net of taxes. The}

household bid rent funtionψ(I, u) ^{isgiven by} ψ(I, u) := max

s≥0

I−Z(s, u)

s , ⁽¹⁾

where the maximum is reahed atthe bid-max lot size S(I, u) S(I, u) := arg max

s≥0

I−Z(s, u)

s . ⁽²⁾

The government is responsible for providing the transportation infrastru-

ture, LT(r)^{, and has the} possibility of levying two kinds of taxes: a popula- tion tax that doesnot depend onr ^{and is denoted by} g^{, and a loation (or}

ongestion) tax that depends onr ^{and is denoted by} l(r)^.

The road oupany at r ^{is dened by the ratio of the number} N(r) ^of

households loated further away than r ^{from the ity enter to the amount} LT(r)^{oflanddevotedtotransportuseat}r^{. Ateahdistane}r^{,thetransport}

ostdependsontheroadoupany atr^: c(N(r)/LT(r))^{,wherethefuntion} c^{is assumed tosatisfy} c(w)>0^, c^′(w)>0^and c^′′(w)>0^forall w≥0^{. The}

transportost fromdistane r ^{to the CBD is} τ(r) =

Z r rc

c

N(x) LT(x)

dx. ⁽³⁾

Dene the bid rent of the transport setor ψT ^{at eah distane} r ^{as the}

marginalbenet of landfor transportationatr^: ψT

N(r) LT(r)

=c^′

N(r) LT(r)

N(r) LT(r)

2

. ⁽⁴⁾

The bid rent ψT(N(r)/LT(r)) ^{represents the umulated gain for the} N(r)

ommuters (away from r^{) froma unit inrease of roads at} r^.

The household's problem is to maximize the utility funtion U(z, s) ^over r^, z ^ands ^{subjet tothe revenue onstraint}z+R(r)s=Y −g−l(r)−τ(r)^{. If}

we replaeI ^in(1)by7 Y −g−l(r)−τ(r)^{, weobtainthe householdbid rent}

at distane r

ψ(Y −g−l(r)−τ(r), u) = max

s

Y −g−l(r)−τ(r)−Z(s, u)

s , ⁽⁵⁾

and the orresponding bid-max lotsize S(Y −g−l(r)−τ(r), u)^{. Appendix}

A provides an interpretation of the HST ^{model and the role played by the}

populationtax g^{. Sine all households are idential, it is onvenient to as-}

sume that they all reah the same utilitylevel atan optimalsolution.

8

The

objetive of the entralplanner is to maximizethe total surplus in the ity.

Let n(r)^{denote the number of households inan annulus of unit width at} r^.

The objetive funtionto be maximizedover (nonnegative)quantities n(r)^, s(r)^,LT(r) ^and rf ^{is the followingtotal surplus} S:

S = Z rf

rc

{[Y −τ(r)−Z(s(r), u)−RAs(r)]n(r)−RALT(r)}dr. ⁽⁶⁾

Any distributionn(r)^{of households should satisfy the following} onstraints.

First, the total amount of land devoted tohousing and transportation must

belowerthan orequal tothe amountof land available:

n(r)s(r) +LT(r)≤L(r) ^for rc ≤r≤rf. ⁽⁷⁾

Seond, the distribution of householdssatises:

N(r) = Z rf

r

n(r)dr ^for rc ≤r≤rf. ⁽⁸⁾

Finally,all households are loated inside the ity:

N =N(rc) = Z rf

rc

n(r)dr. ⁽⁹⁾

7

Indeed,Y−g−l(r)−τ(r)^{isthepartoftheinomethatremainsfortheonsumption}

ofhousing(s)^andthehomogeneousgood(z)^.

8

Without this assumption, an optimal solution may imply an inreasing utility as

we move away from the CBD (f. Riley 1974, Papageorgiou & Pines 1999). When all

householdsareassumedidentialsuhasituationmayseeminonsistentandtheMirrlees

paradigmoftheunequaltreatmentofequalsappears(f.Mirrlees1972). Weavoidthis

disussionandonsideronlysolutionswithequalutilitiesamonghouseholds.

Sine the bid rent funtion ψ(I, u) ^is ontinuously inreasing in I^{, we an}

dene φ(R, u)^by

φ(R, u) :=I ⇔ψ(I, u) =R. ⁽¹⁰⁾

The quantity φ(R, u)^{isthe aftertax revenue requiredbya householdhaving}

utility level u ^{and willing to pay a land rent} R^{. The optimality onditions}

of this problem (maximize (6) subjet to onstraints (7), (8) and (9)) are

realled in their standard form in Appendix A. They represent onditions

for the ompensated equilibriumin whih the ommon utility u ^{is ahieved}

by a ompetitive land market with ommon loation tax g ^{and an optimal}

loationtaxl(r)^{. Theidea of the approahwepropose isto transformstan-}

dard optimalityonditions (Equations(22a)-(22f)inAppendixA) intoa set

of rst-order dierentialequations. Bruekner (2005) proposed asimilar ap-

proahbutunderaframeworkwheretheproportionoflanddevoted toroads

is xed. Wehave the followingresult.

Proposition1. Letu >0^{beaxedutilitylevel. Thesolutionof theproblem}

whihonsistsinmaximizing(6) subjettoonstraints(7),(8)and(9)anbe

omputed in thefollowingway. Solve, forall positiverf ^{and for}rc ≤r ≤rf^,

the system of bakward dierentialequations :



















R^′(r) =−c^′ Ψ⁻¹_T R(r)

Ψ⁻¹_T (R(r)) +c(Ψ⁻¹_T (R(r)))

∂φ

∂R(R(r), u) N^′(r) =

N(r)

Ψ⁻¹_T (R(r)) −L(r) S(φ(R(r), u), u),

(11)

with terminal onditions R(r_f) = R_A ^and N(r_f) = 0^{. Then, nd} r_f ^suh

that N(rc) = N^{. From these, ompute} LT(r) = N(r)/Ψ⁻¹_T (R(r))^, s(r) = S(φ(R(r), u)) ^and l(r) = Rr

rcc^′(Ψ⁻¹_T (R(r^′)))Ψ⁻¹_T (R(r^′))dr^{′ for} rc ≤r ≤rf^.

Proof. See Appendix B.

This proedure assumes that n(r) > 0 ^and LT(r) > 0 ^{for all} rc ≤ r ≤ rf^{. While the seond ondition is guaranteed at any optimal solution,}

9

it

is possible that households density be equal to zero at some distane r^{. In}

Appendix Cweprovidedetailsonhowtoimplementthisalgorithmandshow

how tohandle the ase where n(r) = 0 ^for r > rc^.

9

Ifnot,N(r)/L_T(r)^{willbeunbounded induingaveryhigh}transportationost.

relaxtheanalytialformintherstequationof(11)byintroduingthemore

exible rule:

H(r) =







c^′(Ψ⁻¹_T (R(r)))Ψ⁻¹_T (R(r)) ^(rst-best)

κ ^{(linear toll)}

ξdI{rd}(r) ^{(ordon toll)},

(12)

where κ ^and ξd ^{are positive onstants and} I_{rd}(r) ^{the funtion that takes}

valueone atrd ^{and zero elsewhere along with replaing the rst equationin}

(11) by R^′(r) = −(H(r) +c(Ψ⁻¹_T (R(r))))/(∂φ(R(r), u)/∂R)^{. Then, instead}

of (11),wesolvethe system of dierentialequations given by



















R^′(r) =−H(r) +c(Ψ⁻¹_T (R(r)))

∂φ

∂R(R(r), u) N^′(r) =

N(r)

Ψ⁻_T¹(R(r)) −L(r) S(φ(R(r), u), u).

(13)

The seond priing rule in(12) orresponds toa harge that is proportional

tothelengthofthetrip,whereκ^{isthehargeperunitofdistane. Suhmay}

reet a harge implemented as a gasoline tax. Notie that the linear toll

does not depend onthe originand destination of the trip. The thirdpriing

rule in (12) reets ordonpriing. Eahdriverpays ξd ^{forrossing the ring}

of radius r_d^{. Households living insidethis ring donot pay the harge.}

3 Calibration on Île-de-Frane

In this setion, we alibrate the modelparameters to math seleted target

variablesrelated to the IDF (Île-de-Frane) region. The monoentri model

may be ritiized as being based on unrealisti assumptions. Indeed, many

metropolitan regions have a polyentri struture, and many authors on-

sider that the main eort should therefore fous on polyentri models (f.

Mieszkowski & Mills 1993, forexample). The monoentri framework, how-

ever, remains very useful for at least three reasons. First, for the ase of

IDF, as we disuss below, there is a high onentration of (non-industrial)

ativities in the CBD loated inside Paris. Seond, the monoentri model

transportation. In partiular, in IDF, most eonomi ativities with highly

skilledemployees areonentratedinthe CBD.This issueispartiularlyrel-

evant sine polyentri models have not yet been used suessfully. Third,

giventhatthetheory underlyingthemonoentri modelismuhmoreoher-

ent and omplete (many theoretial insights have already been gained), the

empirialexerisean beevaluatedmuhmoreauratelythan ifpolyentri

models are used. We do not intend to say that the monoentri model is

superior topolyentri models,but we arguethat thereare many lessonswe

an draw from it if we remain aware of its limitations. Moreover, empiri-

al observations still onrm the high onentration of eonomi ativities

in smallareas. For the ase of IDF, a reent reportby Pottier et al.(2007)

statesthatmorethanthreemillionhouseholds(amongatotalofvemillion)

are working inthe twenty distritsinside Paris. The ratio iseven higher for

highlyskilledemployees,whogenerallyuse privatears relativelyfrequently.

Moreover, maps fromAIRPARIF show a high onentration of emissionsin

theCBD andtheregionaround. Onthebasisoftheseobservations,wethink

that many urban attributes of IDF an be explored within the monoentri

framework.

3.1 A spei model

The related literature has extensively onsidered the Cobb-Douglas utility

funtion:

10

U(z, s) =z^αs^{β with} α >0, β >0. ⁽¹⁴⁾

From U(s, z) =u^{, we havethe quantity of omposite good}

Z(s, u) = u^1/α s^−β/α, ^(15a)

and the solutionof (2) yields

S(I, u) = (α+β

α )^αβu^1βI^−αβ. ^(15b)

Substituting itin (1) yieldsthe bid rent funtion

ψ(I, u) = β

α+β( α

α+β)^αβu^−1βI^α+ββ . ⁽¹⁵⁾

10

SeeRobson(1976),Verhoef(2005)andKanemoto(1977).